DAC

Direct Acoustic Digital-To-Analogue Conversion from a Digital Transducer Array Loudspeaker

JORGE MENDOZA LÓPEZ

A thesis submitted in partial fulfilment of the requirements of the University of Brighton for the degree of Doctor of Philosophy

January 2007

THE UNIVERSITY OF BRIGHTON IN COLLABORATION WITH B&W GROUP LTD.

Abstract

Direct-converting digitally driven loudspeaker array prototypes made of moving-coil transducers have been studied by a variety of experimental, simulation and theoretical techniques. Simulation techniques were developed to allow system parameters to be extended beyond those possible purely by experiment. Distortion in the three-dimensional radiated sound pressure field has been characterised as functions of transducer properties, array geometry and digital driving parameters.

The dependence of the Total Harmonic Distortion (THD), Frequency Response Functions (FRFs) and sound pressure directivity responses on specific system parameters including the transducer impulse and frequency responses, the transducer and array geometries, sampling rate, number of bits and diffraction effects have been quantified. Results show that the most significant parameter affecting THD is the array size, determined by the transducer size, in relation to the listening position.

For the multi-bit converting topology and moving-coil transducer type, acceptable direct-converting array performance can only be expected with currently unrealisable transducer performance and geometry. Values of THD significantly in excess of those expected solely from quantisation distortion were found in all practical implementations of the system. The relationship between THD and transducer frequency response showed that a gentle low-pass characteristic is beneficial in reducing distortion at the expense of reconstructed bandwidth. On the other hand, the effect of cone break-up was found to be detrimental for reconstruction when the Nyquist frequency was above the break-up frequency.

Theoretical simulation and practical implementations showed that the low-pass filter action of the transducer FRF acts as a desampling filter for the reconstructed signal, so that no acoustic desampling filter is required. On the basis of the obtained results, the criteria of an array size of the order of ten times smaller than the closest listening distance and a sampling rate as high as possible below the cone break-up frequency are established for an acceptable performance which leads to on-axis distortion figures of the order of the quantisation distortion.

Future research directions for direct acoustic digital-to-analogue conversion loudspeaker arrays are envisaged by integrating the transducer and array manufacturing processes with MEMS and audio spatialisation techniques.

COPYRIGHT © Jorge Mendoza López 2007

Contents

ABSTRACT ........................................................................................................................................... 2 CONTENTS........................................................................................................................................... 4 LIST OF ABBREVIATIONS............................................................................................................... 7 ACKNOWLEDGEMENTS.................................................................................................................. 8 1 INTRODUCTION.......................................................................................................................... 9

1.1 HYPOTHESIS.............................................................................................................................. 9 1.2 RESEARCH QUESTIONS ........................................................................................................... 10 1.3 EXPERIMENTAL CONTEXT ...................................................................................................... 11 1.4 SIMULATIONS CONTEXT ......................................................................................................... 11 1.5 CONTRIBUTION TO KNOWLEDGE............................................................................................ 12

2 LITERATURE REVIEW ........................................................................................................... 13 2.1 RELEVANT DIRECT-CONVERTING ELECTROACOUSTIC SYSTEMS .......................................... 13 2.2 TRANSDUCER DEVELOPMENTS............................................................................................... 15 2.3 RELEVANT TRANSDUCER ARRAY LITERATURE ..................................................................... 16 2.4 SOUND FIELD RADIATION BY TRANSDUCER ARRAYS............................................................ 18 2.5 DITHERING AND NOISE SHAPING............................................................................................ 19 2.6 AUDIO SPATIALIZATION ......................................................................................................... 20

3 THEORETICAL BACKGROUND............................................................................................ 21 3.1 THE MULTI-BIT DAC.............................................................................................................. 21 3.2 QUANTIFICATION OF DAC PERFORMANCE ............................................................................ 23

3.2.1 Total Harmonic Distortion (THD) .................................................................................. 23 3.2.2 Signal-to-Noise Ratio (SNR)............................................................................................ 24 3.2.3 Spurious-Free Dynamic Range (SFDR) .......................................................................... 24 3.2.4 Signal-to-Noise and Distortion (SINAD)......................................................................... 24 3.2.5 Effective Number of Bits (ENOB).................................................................................... 24

3.3 ARRAY THEORY...................................................................................................................... 25 3.4 TRANSIENT RADIATION BY PLANAR PISTONS ........................................................................ 27 3.5 QUANTISED PULSES ................................................................................................................ 29 3.6 EXCITATION IN AN IDEAL DTA .............................................................................................. 31

3.6.1 Harmonic Excitation ....................................................................................................... 31 3.6.2 Broadband Excitation...................................................................................................... 34

3.7 REAL TRANSDUCER EFFECTS ................................................................................................. 34 3.7.1 Acoustic Path Length Differences ................................................................................... 34 3.7.2 Transducer FRF Non-Uniformities ................................................................................. 35 3.7.3 Transducer Mismatches................................................................................................... 37 3.7.4 Baffle Effect ..................................................................................................................... 37

3.8 BANDWIDTH AND RESOLUTION.............................................................................................. 37 3.8.1 Sampling Rate.................................................................................................................. 37 3.8.2 Number of Bits................................................................................................................. 38

3.9 NOISE SHAPING....................................................................................................................... 38 3.10 SUMMARY ........................................................................................................................... 40

4 IMPLEMENTATION ................................................................................................................. 42 4.1 MEASUREMENT SYSTEMS....................................................................................................... 42 4.2 DIGITAL SIGNAL PROCESSING PLATFORM.............................................................................. 42 4.3 AMPLIFIER SET ....................................................................................................................... 43 4.4 TRANSDUCERS ........................................................................................................................ 47 4.5 FULL-RANGE LOUDSPEAKERS ................................................................................................ 52 4.6 TRANSDUCER ARRAYS ........................................................................................................... 55 4.7 SUMMARY............................................................................................................................... 57

5 EXPERIMENTAL RESULTS.................................................................................................... 58 5.1 RECONSTRUCTION WITH ONE DRIVER.................................................................................... 59

5.1.1 Individual Bitstream Responses ...................................................................................... 60 5.1.2 Digital Reconstruction with Moving-Coil Tweeters ........................................................ 62 5.1.3 Digital Reconstruction with Full-Range Systems............................................................ 65 5.1.4 Transducer Interspacing ................................................................................................. 67 5.1.5 Transducer Mismatches................................................................................................... 69 5.1.6 DTA Far-Field Directivity............................................................................................... 72 5.1.7 Multiple Frequency Excitation ........................................................................................ 73

5.1.7.1 White Noise................................................................................................................................................73 5.1.7.2 Two-Tone...................................................................................................................................................73

5.2 RECONSTRUCTION WITH REAL-TIME PROTOTYPES................................................................ 76 5.2.1 Electronic Binary-Weighting: the One Bit-Per-Transducer DTA ................................... 76

5.2.1.1 Frequency Responses and Total Harmonic Distortion ...............................................................................77 5.2.1.2 Directivities ................................................................................................................................................77 5.2.1.3 Swept-sine Responses ................................................................................................................................80

5.2.2 Acoustic Binary-Weighting: the Bit-Grouped DTA......................................................... 82 5.2.2.1 Frequency Responses and Total Harmonic Distortion ...............................................................................83 5.2.2.2 Directivities ................................................................................................................................................85 5.2.2.3 Swept-Sine Responses................................................................................................................................86 5.2.2.4 Effect of the Baffle .....................................................................................................................................87

5.3 SNR ENHANCEMENT WITH NOISE-SHAPING .......................................................................... 88 5.4 SUMMARY............................................................................................................................... 90

6 EXPERIMENTAL RESULTS DISCUSSION .......................................................................... 92 6.1 RECONSTRUCTION WITH ONE DRIVER.................................................................................... 92 6.2 ELECTRONIC BINARY WEIGHTING: ONE TRANSDUCER-PER-BIT DTA.................................. 94 6.3 ACOUSTIC BINARY-WEIGHTING: THE BIT-GROUPED DTA.................................................... 95 6.4 SOURCES OF EXPERIMENTAL ERROR...................................................................................... 96

6.4.1 Measurement Distance Compared to Main Array Dimension ........................................ 96 6.4.2 Transducer Mismatches................................................................................................... 96 6.4.3 Hypersensitivity to Sweet-Spot Misalignment ................................................................. 97 6.4.4 Binary-Weighting Error .................................................................................................. 97 6.4.5 Bitstream Noise ............................................................................................................... 97 6.4.6 Transducer Non-Linearities ............................................................................................ 97

7 SIMULATION RESULTS .......................................................................................................... 98 7.1 SIMULATION APPROACH......................................................................................................... 98

7.1.1 Idealised Transducer FRF............................................................................................... 99 7.1.2 Semi-Experimental Transducer FRF............................................................................. 100 7.1.3 Time-Domain DTA Response ........................................................................................ 101 7.1.4 Limitations..................................................................................................................... 102

7.2 DTA SOUND FIELD CHARACTERISATION ............................................................................. 104 7.2.1 Coordinate System and Mapping Parameters............................................................... 104 7.2.2 Simulated DTA Types .................................................................................................... 105 7.2.3 Sound pressure level and THD Maps ............................................................................ 107

7.3 TRANSDUCER FRF NON-UNIFORMITIES............................................................................... 113 7.4 TRANSDUCER MISMATCHES ................................................................................................. 117

7.5 SAMPLING RATE ................................................................................................................... 117 7.6 NUMBER OF BITS .................................................................................................................. 121 7.7 ARRAY SIZE REDUCTION ...................................................................................................... 121 7.8 BINARY-WEIGHTING ERROR ................................................................................................ 122 7.9 NOISE SHAPING..................................................................................................................... 125 7.10 SUMMARY ......................................................................................................................... 127

8 CONCLUSIONS AND FUTURE WORK ............................................................................... 129 8.1 FUTURE WORK...................................................................................................................... 131

8.1.1 Transducer Development............................................................................................... 131 8.1.2 Integration with Spatial Audio ...................................................................................... 131 8.1.3 Room Acoustics and Influence on Reconstruction ........................................................ 131 8.1.4 DSP Algorithm Development ........................................................................................ 132

9 REFERENCES........................................................................................................................... 133 APPENDIX I – DSK6713 GPIO USE ............................................................................................. 140 APPENDIX II – THE SWEPT-SINE TECHNIQUE..................................................................... 142 APPENDIX III – PUBLICATIONS ................................................................................................ 146

List of Abbreviations

ADC Analogue to Digital Conversion / Analogue to Digital Converter

ASIC Application-Specific Integrated Circuit B&W Bowers and Wilkins Ltd. BEM Boundary Element Method CBT Constant Beamwidth Transducer CMOS Complementary Metal Oxide Semiconductor

DAC Digital to Analogue Conversion / Digital to Analogue Converter

DSP Digital Signal Processing / Digital Signal Processor dSP Digital Sound Projector DSR Direct Sound Reconstruction DTA Digital Transducer Array DUT Device Under Test ENOB Effective Number of Bits FFT Fast Fourier Transform FIR Finite Impulse Response FPGA Field-Programmable Gate Array FRF Frequency Response Function GPIO General Purpose Input and Output HD Harmonic Distortion HIR Harmonic Impulse Response HOA High Order Ambisonics IR Impulse Response LSB Least Significant Bit MDF Medium Density Fiberboard MEMS Micro-Electro Mechanical Systems MSB Most Significant Bit PCM Pulse Code Modulation PVC Polyvinyl Chloride PWM Pulse Width Modulation SDLA Smart Digital Loudspeaker Array SDM Sigma Delta Modulation SFDR Spurious-Free Dynamic Range SINAD Signal-to-Noise-and-Distortion SNR Signal-to-noise Ratio SONAR Sound Navigation and Ranging SQNR Signal-to-Quantisation Noise SSM Simple Source Method THD Total Harmonic Distortion WFS Wave Field Synthesis

Acknowledgements

The author of this work wishes to acknowledge the following people for having offered support of any kind during the course of this project.

All the staff at B&W Group Ltd. Research Centre in Steyning: Jon Moore, Gary Geaves, Tom O’Brien, Martial Rousseau, John Dibb, Krestian Pedersen, Steve Marks, Albert Yong, Stuart Nevill, Peter Fryer, John Vanderkooy, David Henwood, Steve Pierce, Penny Davies, Steve Roe, Mike Geoff, Thaiman Beard and Kathrin Ireland.

Antonio Mejías “Anter” for effective computing support.

His supervisory team and other research staff at the University of Brighton: Simon Busbridge, David Lawrence, Chris Garrett, Sharon Gunde, Alison Bruce and David Stansbury.

His parents, Fernando and Luisa, his sister Elena, and the rest of his family.

His colleges from the University of Brighton Luis García-Gancedo, Francisco Prados, Haihua Zhang, Romain Demory, Stefania Rosso, Antonio Emmanoulis, Koray Ozcan, Alexis del Rio, Reza Mortezaei, Daniel Judson, Mark Bailey, Michael Maelzer, Mike Taylor, Dionisis Lefkaditis, Khayzuran Iqbal and Romain Elsair.

The Brighton Capoeira Angola Group: M. Laercio, Ed Wade-Martins, Tim Mayhew, Rodrigo Amorim, Penny Zikic, Susi Gray, Alistair Wallis, Suzanne Mehdi, Adele Renault, Patrick Hoad, Oliver Marlow, Sera Marlow, David Parker, Eamonn “Alma” Canning, Lief Miller, Becky Rastall, Pablo Sotelo, Cicely Taylor, Pippa Martin and Rob Davies.

His friends Alexandre Roland, Miranda Fairbairn, Miguel Pérez, María Yusty, Fernando Sancho, Elena Sánchez, Victor Morales, Agustín Agüera, Laura Donaire, Alba García, Fernando Vega, Esther Mesas, Eva Marco, Olga Faura, Iñigo Lobera, Marta Reina, Aitana Rodríguez, Soledad Pérez, Santi Rojas, Violeta Tirado, Sergio Tirado, Juan Garrido, Felipe Iglesias, Juan Luis Celis, Gonzalo Rodríguez, Pablo Bouzada, Moisés Barrero, Israel Cano, Javier Alberola, Elisa Grimaldi and Angelique Facondini.

I declare that the research contained in this thesis, unless otherwise formally indicated within the text, is the original work of the author. The thesis has not been previously submitted to this or any other university for a degree, and does not incorporate any material already submitted for a degree.

Jorge Mendoza López

Jorgemendozalopez AT hotmail DOT com

Brighton, 30 - 9 - 2006

Chapter 1 Introduction

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1 Introduction

Digital loudspeakers constitute the missing link in the implementation of the all-digital audio reproduction chain. Recent developments in digital audio and advances in semiconductor and network technology are determining the evolutionary trend of the audio chain. However, two ends of the current recording and reproduction chains still remain in analogue form: the microphone and the loudspeaker. Analogue-to-digital converters (ADC) and digital-to-analogue converters (DAC) as well as microphone and loudspeaker manufacturing technologies remain the prices to pay in terms of quality in digital audio recording and reproduction. Further advantages in reproduction such as reduced conversion errors, greater channel transparency and greater system flexibility may result as a consequence of implementing truly digital loudspeakers.

As a truly digital loudspeaker, the digital transducer array (DTA) differs from a normal loudspeaker in that no standard DAC is employed throughout the audio chain and the transducers are directly driven by digital signals. Direct conversion is thus achieved by superposition of the acoustic waves which represent the different bits of the input signal. The quality of the radiated sound field is spatially and frequency dependent. The extra high-frequency harmonic content of the driving bitstreams must be cancelled out to aid reconstruction.

A different strategy for direct converting loudspeaker arrays consists of filtering and processing signal bitstreams in such a way that acoustic beams are synthesised and radiated [1,2]. Bitstream high-frequency content is therefore removed by filtering or further digital processing.

Converting a digital signal directly to the acoustic domain by driving a loudspeaker array directly with bitstreams implies broadband signal radiation from each array element. Owing to the principle of superposition, the resulting acoustic disturbance consists of the sum of individual broadband signals. Because of the transducer finite size, each radiated bitstream travels a different path to a common listening point: acoustic path length differences introduce distortion which adds to the inherent quantisation distortion given by the number of bits and improved by any dithering or noise shaping in place. The resulting distortion manifests itself in the form of odd and even order harmonic distortions. Furthermore, real transducer properties such as FRF non-uniformities or transducer mismatches contribute to the degradation of the reconstructed sound field. In fact, perfect reconstruction over the free-field space can be obtained under the ideal assumption of perfectly uniform FRFs and exactly equal acoustic path lengths.

1.1 Hypothesis

Practical transducer properties such as transducer bandwidth, breakup frequency, transducer mismatches and departures from ideal and linear behaviour limit and introduce distortion in direct acoustic DAC process carried out with digital transducer arrays. Direct DAC systems are believed to be highly influenced by the real properties of the array transducers, though no previous scientific research has been found dealing with the matter.


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It is a known fact that by reproducing an electric signal with an acoustic transducer the result is a band-pass filtered acoustic signal whose spectral content is determined by the transducer frequency response characteristics as well as any existing non-linear or non-ideal transducer behaviour. In other words, transducers will introduce their own signature in any reproduced signal, and this will be the case even for bitstreams. However it is not yet known precisely how and up to what extent a particular transducer will condition and / or limit the reconstructed sound field quality in a direct acoustic DAC. Relationships between measured transducer properties and the obtained sound field quality in a direct DAC will be presented in this thesis, contributing to the existing knowledge in audio engineering.

It is also a known fact that any signal can be decomposed into an infinite sum of harmonics. In a DAC, all bitstream harmonics should be accounted for to be left with perfectly square bitstreams whose magnitudes and phases add up in phase to cancel extra bitstream harmonics. With perfectly linear transducers, the best only remaining distortion would then be the inherent quantisation distortion given by the number of bits if no dithering or noise shaping was applied. The DTA acoustic output with real transducers must be the sum of each bitstream as filtered by each transducer together with the contributions of any other real effects such as acoustic path length differences, transducer mismatches, misalignments, baffle diffraction or room acoustics. If the transducer pass-band allowed enough bitstream harmonics for perfect reconstruction on the sweet-spot to a given number of bits, this would be reflected in relatively low harmonic distortion figures. On the other hand if there were not enough harmonics in the transducer pass band then harmonic distortion figures would rise from the minimum level set by quantisation, bitstream harmonics would not cancel out and the reconstruction would be compromised.

The practical distortion sources affecting a direct acoustic DAC with moving-coil transducers will be investigated by implementing, measuring and simulating DTAs made of moving-coil transducers.

1.2 Research Questions

The fundamental questions addressed in this research are the following:

How and to what extent is the DTA sound field modified by real transducer characteristics such as bandwidth, FRF non-uniformities, transducer mismatches, breakup, transducer size and interspacing?

How does transducer bandwidth limit the reconstructed signal?

How does the DTA geometry and transducer-to-bit assignment affect the radiated sound field?

What spatial characteristic has the resultant DTA sound field?

Until what extent could harmonics be reduced by means of noise shaping technology?

Answers to these questions provide information on the characteristics of transducers, bandwidth, sampling rate, number of bits and noise shaping parameters to be used in direct-converting digital transducer arrays, as well as the spatial regions where direct acoustic conversion could be usefully applied. Proposal ideas for future work are established on the basis of these answers.


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1.3 Experimental Context

The Digital Transducer Array loudspeaker is one of the possible digital loudspeaker implementations which currently represent the next significant development towards realizing an all-digital audio chain. The other different implementation approach is that of the multiple voice-coil system described in [3-9].

Previous work on the subject of the digital transducer array considered its relationship to existing digital formats [10,11], digital signal processing (DSP) strategies for realising simultaneous coherent beams [1,2,12] or directivity and distortion simulations [13]. All of these works assumed idealised transducer behaviour.

Experimental results concerning real DTA implementations have so far only been reported using micro-electro-mechanical system (MEMS) techniques [14,15] in the context of headphone audio systems, although not in the contexts of home audio or consumer multimedia.

Our experimental work on the other hand addresses the questions of what transducer properties are required for DTA implementations with moving-coil transducers, and how real transducer properties such as bandwidth, transducer departures from ideal and linear, transducer mismatches and breakup peak affect the quality of the reconstructed DTA sound field. To this end, several DTA prototypes were built with different moving-coil transducer types. Experimental results proved the direct digital-to-acoustic conversion concept and showed that conversion was affected by real properties of moving-coil transducers which had not previously been taken into account by previous simulations, namely: transducer frequency response non-uniformities seen in both magnitude and phase, transducer non-linearity, transducer mismatches, array geometry, measurement point misalignments, baffle diffraction effects, and the relationship between transducer breakup and system bandwidth.

This experimental work extends all previous knowledge by providing:

Proof-of-concept prototype DTA implementations with moving-coil transducers.

The results of a noise shaper DSP implementation used in conjunction with the DTA prototypes.

Quantification of DTA radiated sound field quality in terms of frequency and swept-sine responses, directivity and THD figures.

Quantification of different sources of harmonic distortion in isolation.

1.4 Simulations Context

In order to complement the results of direct digital-to-acoustic conversions obtained with real DTA prototypes and compare them to what would be obtained with ideal transducers, a simulation was developed that allowed inputting a variety of transducer FRFs: on one hand, a purely experimental frequency response (real); on the other hand, a perfectly uniform and linear phase frequency response (ideal). Performance bounds on the ideal transducer size needed for the DTA were thus derived.

Experimental DTA responses at different points in space were compared to simulated DTA responses obtained under different simulation assumptions. Observed differences in the radiated sound fields were used to assess the effect of the assumptions considered. With the knowledge of how simulations differ from reality under the different assumptions, sound pressure level and THD maps were


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predicted over the free-field space. These maps helped quantifying the differences between the idealised and the more realistic assumptions. Simulations complement the experimental work throwing light on the effect of different DTA geometries and sizes as well as relationships to important digital system variables such as the sampling rate or the number of bits, and assessing the errors incurred in binary weighting gains or transducer mismatches.

1.5 Contribution to Knowledge

This thesis is the first document dedicated to multi-bit direct-converting digital loudspeaker arrays made with moving-coil micro transducers. Existing knowledge in audio engineering is hereby extended by providing experimental and simulation results of DTAs made of moving-coil transducers and relating the results to the quality and spatial characteristics of the radiated sound field.

Several DTA prototypes were developed and their performance was measured with THD, frequency responses and directivity figures. Experimental results proved the direct acoustic conversion concept, quantified the limits for transducer size and performance and the relationship between array size and geometry and sweet-spot location. The relationship of the DTA sound field quality to its constituent transducer characteristics was established and the main sources of error underlying the direct conversion process were analysed.

An interactive DTA simulator in the form of computer programs which can be extrapolated to different geometrical scenarios was developed and validated with experimental results. Several DTA topologies had been simulated in the past under different idealised simulation approaches which did not account for the practical effects introduced by real transducers. On the other hand, our simulation considered transducer FRFs which derived directly from experimental measurements, which constitutes a more realistic simulation approach. The effect of transducer FRF non-uniformities on the reconstructed signal was thus isolated and it was seen that THD was increased over the free-field space under non-uniform transducer FRFs. The effect of independent system parameters such as array size and geometry, sampling rate and number of bits was also established.

In this thesis previous knowledge is extended by:

Using moving-coil transducers to report on the first experimental implementation of multi-bit DTA prototypes.

Comparing real DTA responses with those obtained with idealised transducers, deriving performance bounds on how a DTA recreates a sound field and on the ideal transducer characteristics best suited to the DTA needs.

Judging the quality and value of a direct acoustic DAC by the value of individual harmonic distortion components and THD.

An improved, realistic characterisation of the DTA sound field over the free-field space for various geometries is provided via experimental and simulation results, by accounting for real transducer properties and relating these properties to the radiated sound field. Critical system parameters and errors are explored, quantified and interrelated. Results should prove of valuable interest for future implementations and developments of the multibit direct-converting technology.

Chapter 2 Literature Review

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2 Literature Review

A critical analysis of the most relevant literature found to date on the field of direct digital-to-acoustic conversion is now presented. Several direct-converting systems are commented, linking whenever possible with relationships between practical transducer properties and output data, and those particular properties that impinge an acoustic signature on the output. The transducers constituting the only relevant direct-converting systems which were found implemented to date [14-19] were developed specifically for each particular application. Therefore relationships and comparisons with the properties affecting real transducers cannot be drawn. Moreover, existing literature on the DTA subject [1,10-13,20,21] has so far only considered idealised transducer types for simulation results, after which no practical considerations of transducer properties affecting radiated sound fields could be obtained.

In section 2.3 the main works to date dealing with the theory of transducer arrays are presented. In section 2.4, concepts of interest in loudspeaker array modelling are reviewed. To conclude, a review of DSP techniques relevant to the DTA field namely noise shaping and spatial audio is presented.

2.1 Relevant Direct-Converting Electroacoustic Systems

The idea of direct digital to analogue conversion (DAC) was proposed by Flanagan in 1980 [22], with the aim of simplifying the operation of telephone transducer systems. Other examples of direct DAC and direct ADC have been reported such as those in [17,19,23]. This section will critically comment on the most important ones found to date.

Morgan et. al [17] described a 5-bit capacitive transducer / filter system made of concentric circular rings with binary-weighted areas. The acoustic desampling filter associated to the transducer consisted of consecutive cylindrical chambers of increasing thicknesses and diameters. The whole system occupied a maximum of 10 cm diameter and presented a maximum bandwidth of 8 kHz although the presence of resonances in reported FRFs, possibly due to the different responses of each ring, indicated low overall system quality.

In a different approach [18], simulations and experiments were presented for line and circular arrays in which the technique of multidimensional delta-sigma modulation was used to generate bit streams that drove the arrays. However, no information was reported regarding the effect of transducer performance on the reconstructed signals.

Sakai [16] described an 8-bit all-digital piezoelectric headphone with acoustic desampling filter, a 10 kHz bandwidth and an SNR of near 48 dB. The obtained third harmonic distortion was less than 30 dB in the band from 200 Hz to 20 kHz. Second harmonic distortion component was higher, with a peak value near 48 dB. The fundamental FRF presented a resonance peak of 100 dB at a frequency of 200 Hz. Overall system performance was acceptable with regards to the maximum theoretical dynamic range of 49.8 dB that can be achieved with an ideal 8-bit system.

CMOS–MEMS techniques for large-scale micro transducer fabrication and compensation were described by Neumann et al. [14]. Based on a multi-bit addressing and reconstruction scheme, the


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results showed acceptable repeatability, uniformity and linearity across the micro transducers in the arrays described, which could be further optimised with integrated on-chip electronics. The overall size of an 8-bit 255 transducer array was only around a few mm. An application for a patent giving details on micro transducer fabrication was made shortly after by the same authors [15]. Further work on the subject is given in [23,24].

A different electroacoustic system is the digital microphone described in [19]. The system was a 4-bit direct ADC, as opposed to previously described direct DAC systems, with integrated sigma-delta modulation. The system represented a proof of concept for the direct acoustic ADC principle and a future proposal for adaptation with MEMS electroacoustic devices.

The main conclusion to be drawn from references [14,16,19] is that direct conversion improves in small sized electroacoustic systems in which path lengths from each transducer to a common listening point are similar to each other. This implies far-field listening or time delay compensation for each array element.

Hawksford [12] addressed the theoretical problem of filter bank design to produce diffuse and directional radiation from a digital loudspeaker array. The transducers needed to produce these results would have to be of extremely small size whilst powerful enough to be heard properly at a listening distance much larger than the array size. No details were given on transducer performance, bandwidth and its effect on the reconstructed sound field. The term “Smart Digital Loudspeaker Array” (SDLA) was used to describe a direct DAC system with extended functionality, principally by the design of a digital filter bank able to achieve simultaneous directional acoustic beams or diffuse radiation. The array size determines the low-frequency bandwidth while the interspacing determines the upper bandwidth, above which spatial aliasing distortion occurs. The observed spatial frequency is a function of the listening position. Therefore the upper audio frequency is limited by the relative angle between observer and array plane. The upper interspacing limit to avoid spatial aliasing is 2/λ [1,25].

Previous work on DTAs has not yet considered transducer interspacings other than constant, such as logarithmic spacing, known to improve directivity [26,27]. Array geometries such as the circular array [28] or the circular-arc line array [29], as well as practical driving electronics or implementation details have not been reported in detail.

A similar system, able to produce separate simultaneous steerable beams of sound from digital inputs was the one described by Hooley et al. [2,30]. The system described the necessary DSP stages to convert surround encoded signals to at least five independent steerable beams of sound. Conversion was obtained through transducer FRF filtering compensation, noise shaping and digital delays implementations with three DSPs and one FPGA module, although no reference was made to the practical effects of transducer FRF compensation nor to performance results without the FRF compensation in place. A subwoofer driving signal was obtained by low-pass filtering each serial bitstream and adding the resulting signals. Each surround channel was oversampled and noise shaped, delayed and converted from serial PCM to PWM format. The resulting serial PWM bitstream drove a class-D type amplifier whose output was fed directly to a transducer or transducer group in the array without electronic low-pass filtering. According to the authors, optimum transducer size and interspacing are both contained in the range of 25 mm to 45 mm. Transducer arrangements as a honeycomb were encouraged on the basis of providing a more densely packed structure.

Distortion and directivity in the bit-grouped DTA were studied in simulations by Huang et al. [13] under the assumption of omnidirectional point sources. The bit-grouped DTA is based on assigning binary weighted numbers of transducers to each bit. It requires 2N-1 transducers, where N is the number of bits. Its operation is closer to the all-digital transduction concept since none of the traditional DAC stages is performed in the electronic domain. Very high distortion levels above 15%


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on-axis were numerically predicted even treating transducers mathematically as point sources. The distortion was found to increase with increasing frequency and transducer selection details. This indicates that if a digital filter bank and/or noise shaping or any other DSP tools to allow beamforming were not present in a real DTA system implementation, obtained results will be poor. Different transducer locations within the array and different driving strategies were considered. Driving strategy was found to play an important role in final acoustic predictions, especially in the near-field.

Tatlas et al. [11] envisaged an all-digital audio chain including other elements such as wireless receivers and all-digital amplifiers, which are currently in a more advanced state than the digital loudspeaker. The main advantage of the all-digital chain is that it leads to increasingly flexible systems controlled totally by software, which makes things simpler and cost effective for application developers. Simulation results were reproduced describing the same arrays as in [13], particularly the relationship between different digital formats and possible direct acoustic converter topologies, including the number of transducers and amplifiers required. The option of one bit sigma-delta modulation seemed to require the least number of transducers, although transducer performance should be of critical importance here since they would be fed with extremely fast pulses at frequencies of the order of GHz. Further work on the subject is contained in [10].

2.2 Transducer Developments

The ideal transducer intended for a DTA application must be fast enough to switch at the specified oversampled rate. In a direct-converting system, the maximum switching frequency is limited by transducer damping and driving electronics. Transducer responses should be repeatable over time and across all different transducers in the array. It is thought that the broader the transducer bandwidth, the broader the direct acoustic DAC operating band, though no relevant literature was found to assert this fact.

In order to increase the efficiency of moving-coil transducers, which are still the weakest link in the audio chain, and possibly to correct other problems such as transducer matching and step response departures from ideal, research trends have addressed new transducer types and devised a piezoelectric “Superhelix Actuator” [31-33] for loudspeaker array and phased array antennae applications, and a simple unary form of digital signal coding, consisting of firing, for each digital sample, an integer number of speaklets proportional to voltage amplitude. The main problem of the unary approach is resolution because it is limited by the number of transducers present in the system. Although the “Superhelix Actuator” [34] is significantly smaller and more efficient than traditional moving-coil loudspeakers (approximately the size of a small coin), its size vs. efficiency ratio is not near that required for a multi-bit direct-converting DAC application. Experimental results giving the effect of the new transducer compared to the effect of traditional moving-coil transducers on reconstructed signals were not reported.

The use of wideband miniature moving-coil transducers was reported by Keele [29] regarding constant beamwidth transducer (CBT) arrays. These transducers, originally intended for multimedia applications in laptop PCs, are believed to have a reasonable performance in terms of bandwidth and size for a DTA, amongst moving-coil type transducers. Their main properties will be seen in section 4.4 and the minimum THD produced after direct reconstruction with one such transducer will be shown in section 5.1.

From the previous work in which a direct DAC system was implemented, it seems that the far-field is reached at significant distances compared to the array size, suggesting that MEMS technology [14,15,23,24] may offer the best practical solution. However, no key evidence was found on the


- 16 -

practical effect of moving-coil transducers on reconstructed sound fields, nor on the characterisation of these sound fields.

2.3 Relevant Transducer Array Literature

Analogue line arrays have traditionally been researched in the audio field to provide increased directivity in the contexts of sound reinforcement or sound recording. The main property of analogue arrays which makes them different to digital transducer arrays is that the same frequencies are radiated by all array elements at each time instant, therefore the same radiation impedance applies to all array elements. In the case of digital transducer arrays, different radiation impedances are seen at each instant depending on the active number of transducers [21].

It is well known that as soon as two sources radiate next to each other, interference phenomena occur, evidenced by the lobes and nulls present in their combined FRFs, as the measurement point is taken around the sources plane [35]. Constructive or destructive interference occurs depending on the listening point location with respect to the array. The problem is described by the well-known interference theory of light [36]. However, the direct-conversion problem is different since radiated waves are broadband. Their extra high-frequency content must be suppressed to allow reconstruction. If the right conditions are met in terms of array geometry, noise shaping and transducer FRF compensation, the high-frequency content of radiated bitstreams is suppressed in a similar manner as in destructive interference.

Ureda [37] used the simplification of treating line arrays as continuous line sources. Mathematical expressions for null and lobe positions were derived. The following approximation for the far field distances was reported:

82

2 λλ−=

lr where 2

λ≥l

(2.1)

where l is the array length, λ is the wavelength, and r is the distance to the far-field on-axis.

Smith [38] reported results of experiments for both near and far-field pressures of a line array and compared them to those predicted by a point-source model. It was concluded that for arrays used in a home setting, the performance is dominated by near-field considerations, and attention was given to the fact of concentrating on the array’s main beam as opposed to the complete polar curve.

Urban et al. [25] studied the array near-field in terms of Fresnel optics. The context of the work was the use of loudspeaker arrays for outdoor concert sound reinforcement, although the criteria it established were also valid for other types of loudspeaker arrays. First of all a distinction between near-field, chaotic zone, and far-field produced at different listening distances depending on sound source positioning and array size was provided. Criteria as to where and how close to place transducers were arrived at. A line-array that respects these criteria will successfully approximate a continuous line source and will lack the chaotic transition region between near and far-field. The main condition for “arrayability” given in this paper can be stated as follows:


- 17 -

22min

max

λ=≤

fcd

(2.2)

where d is transducer interspacing, c is the speed of sound, and maxf and minλ are the maximum signal frequency and minimum signal wavelength before spatial aliasing occurs. Alternatively, according to the authors, the condition of the sum of individual radiating areas adding up to at least 80% of the array’s surface can be fulfilled instead of the previous one. Therefore for an array of 26 mm diameter tweeters which are spaced by a constant distance of 50 mm, the maximum signal frequency before spatial aliasing is 3.44 kHz, a very low limit for audio purposes. If the mass-controlled region of these tweeters starts just after 1 kHz, test signals in the band of 1 – 3 kHz should be employed.

The problem of coherently summing the outputs from a number of transducers is known as beamforming. This problem is of great importance to arrays and has been studied for underwater line arrays in [39]. From general beam and optical theory, the far-field radiation pattern of a radiating aperture is given by the Fourier transform of the aperture function:

∫+∞

∞−

−⋅= dxexfsF jkxs)()(

(2.3)

where f(x) is the aperture function, F(s) is its correspondent far-field radiation pattern, k is the wavenumber and x is the spatial dimension along the aperture.

An array of omnidirectional radiating elements is just a sampled version of a continuous aperture, allowing replacement of the previous integral by a sum:

∑=

⋅=N

n

sjkxn

nefsF1

)(

(2.4)

A given beam can be shaped by two methods. Varying transducer amplitudes results in periodic arrays for which the aperture function can easily be solved by FFTs, making beamforming computationally inexpensive for large arrays. On the other hand, changing transducer locations results in non-uniform arrays for which the radiation pattern can no longer be cheaply solved using FFTs, although an equal beam can be produced using fewer transducers with interspacings greater than 2/λ . This approach is called space tapering, density tapering or spatial shading. Its application seems therefore ideal for analogue loudspeaker arrays made of moving-coil tweeters, due to their big physical size. However, as the number of elements is increased, simulations become more expensive. McGehee et al. [39] provided an optimisation for finding the beam pattern that generated the best beam, in the context of underwater-acoustics. A similar optimisation approach could be taken to the digital audio field yielding originality.

An example of transducer amplitude variation in an analogue loudspeaker array was the application of the so-called Constant Beamwidth Transducer (CBT) theory to the production of coherent beams in [29]. CBT array theory prescribes that each transducer in the array be driven with different levels following a Legendre shading function. The results of measurements are presented showing a


- 18 -

passively shaded array providing uniform wideband coverage control with constant beamwidth in both x and y planes as well as constant directivity characteristics over significant frequency and distance ranges. These results are significant since they approximate the ideal performance of an analogue transducer array.

A very interesting field concerned with the design and theory of arrays is that of antennas. A good mathematical theory for unequally spaced arrays was developed by Ishimaru [26]. It was based on the use of Poisson’s sum formula and the introduction of a function called the “source position function”. Resulting electromagnetic field patterns were expressed as a function of the individual currents input into the radiating elements. The theory was found effective in treating unequally-spaced arrays with a large number of elements, whether on flat or on curved surfaces. The issues of radiation pattern design with specified sidelobe levels as well as secondary beam suppression were effectively covered by this theory. Other optimizations for array element positioning, conceived to minimise radiated sidelobe level or array element number were described by Dooley [40] and Leahy et al. [41], respectively.

2.4 Sound Field Radiation by Transducer Arrays

Prediction of radiated sound-fields by loudspeaker arrays is an interesting subject with many potential applications in the DTA field. Analytical models provide computational speed at the cost of a decreased accuracy. As more accuracy is needed, numerical methods based on simplifications of the wave equation such as the Simple Source Method (SSM) or the Boundary Element Method (BEM) might be used, at the cost of computational efficiency. There is a trade-off between accuracy and computational expense: each level of sophistication increases computation time proportionally more than its correspondent accuracy improvement.

A good analytical model for the near-field pressure and the radiation impedances for an infinite phased array of planar pistons on an infinite baffle was produced by Mangulis [42] in the context of SONAR arrays. Results indicated that the resulting pressure became more ondulatory as piston interspacing was increased, for constant piston radii. The radiation impedance of a piston in an infinite array was found to agree well with the average radiation impedance of a piston in a large finite array.

A deep thorough mathematical treatment on analytical solutions such as the piston on infinite baffle or the piston in a sphere was given by Skudrzyk [43]. Other analytical solutions exist in the context of audio for calculating the on-axis sound pressure produced by convex and concave circular pistons on infinite baffles [44], assuming an infinite number of point sources equally distributed along the radiator.

Some of the existing numerical methods are based on solving the wave equation. When a sinusoidal excitation is considered, the wave equation becomes Helmholtz’s equation, whose basic variable is the velocity potential. Solving the equation for velocity potential leads to the calculation of volume velocity and air pressure, which constitute the two acoustic ‘through’ and ‘across’ variables. Helmholtz’s equation can be solved numerically by the SSM or the BEM (a thorough mathematical review of the above-mentioned methods was given by Henwood [45]). For a detailed theoretical review of transient field theory for baffled planar pistons, see [46].

The point source model was shown to be sufficiently accurate and useful for loudspeaker system design, especially at frequencies above several hundred Hertz [47]. However, with this approach, the directivity and frequency response characteristics of the transducers considered are not taken into account.


- 19 -

Morse [48] described a model that accounted for any number of circular pistons on an infinite baffle. The assumption of infinite baffle is however a best-case scenario for back-radiation isolation. More realistic assumptions concerning back radiation isolation include the baffle diffraction step and further dips and peaks introduced in the frequency response [49].

The work presented by Meyer [50] consisted of a three dimensional computer simulation of loudspeaker directivity as a function of frequency, obtained with a mathematical model based on realistic assumptions. Results were validated against relevant case studies. The derived loudspeaker pressure response presented a 1/r dependence which accounted for the intensity inverse square law, a directivity factor given by a first order Bessel function and an exponential which accounted for the sinusoidal excitation and acoustic wave propagation. With the only knowledge of vertical and horizontal loudspeaker polar responses, the response at any point in space was simulated. As shown in the subsequent case studies, this approach represents a good compromise between computational expense and prediction accuracy. As a further application, results of the previous theory applied to a loudspeaker array implementation allowing control of the array directional characteristics in the digital domain was presented in [51].

2.5 Dithering and Noise Shaping

In a digital system, at the quantisation stage, dither refers to a small amount of random noise that is added to the input signal which linearises the quantiser. The amplitude and statistical properties of the dither signal play a crucial role in the presence of quantisation distortion as well as the audibility of quantisation noise in the final output signal. A thorough theoretical treatment on dithering was given by Lipshitz et al. [52]. Further work on dither as applied to digital systems may be found in [52-59].

Noise shaping is a feedback technique which results in the quantisation error shifted into frequency bands higher than the audible range [53,55,56,60-64]. Both dithering and noise shaping are commonly used in audio to reduce the errors and distortion caused by quantisation or requantisation. Quantisation error produces background noise as well as statistical correlation of the error with the signal, causing distortion which may produce alterations in the perceived quality of the signal [53].

With noise shaping the quantizing process of the current sample is modified by the quantising error of the previous sample [53,63,65]. The objective of noise shaping systems is twofold: to apply dither to the input audio signal and to filter quantisation noise in order to obtain a less audible noise level [66], increasing signal-to-noise ratio.

Dithered noise shaping systems with oversampling are capable of reducing noise levels inside the audio band whilst increasing it out of band, where noise is totally inaudible. If oversampling is not applied, noise can only be redistributed inside the audio band, although it cannot be shifted out of band [66].

An oversampling dithered noise shaper shifts audible noise into inaudible frequency bands, preserving audio quality whilst reducing the number of bits, something which should certainly proof critical in direct-converting DAC applications, especially in the bit-grouped DTA, where every time a bit is reduced the number of transducers required halves. The application of noise shaping technology to an electrodynamic digital loudspeaker was reported in [8,9], and it was shown that in this kind of loudspeakers the acoustic characteristics resulting from a 12-bit digital signal with noise shaping were equivalent to those from a 16-bit signal without noise shaping. As it will be seen in section 5.3 the application of noise shaping technology to the digital transducer array is one of the novel experimental results of this thesis [67].


- 20 -

In theory, if enough feedback filter coefficients are taken into account, the design and implementation of an arbitrarily-shaped noise shaper frequency response function is possible [53]. An example of this was presented in reference [64] in order to compensate for the ear canal resonances of the human auditory system, therefore optimizing perception. Thanks to the increasing power of DSP, a DTA system could be made flexible enough to equalize for non-flat system responses and match transducer responses whilst shifting noise out of band as a result of the feedback process.

2.6 Audio Spatialization

Because proposed DTA systems to date are focussing array systems which have the restriction of a sweet spot, audio spatialization techniques such as Wave Field Synthesis (WFS) [68-71] and High Order Ambisonics (HOA) [70,72-75] constitute possible solution gateways to increase the spatial region of minimum distortion in the DTA sound field. It seems reasonable to take advantage of the fact that three-dimensional sound fields are generally recorded and reproduced with microphone and loudspeaker arrays, respectively. The idea of bitstream radiation by virtual source(s) positioned well behind the listener is of particular interest [76-78].

Possible applications of a technique integrating the DTA and the spatial audio field would increase the direct DAC concept value. The integration of these techniques into the DTA field will follow in the form of DSP algorithm developments and has currently got tremendous potential for future work.

A good review on the mathematics of spatial sound based on spherical harmonics was given by Poletti [79]. A less mathematical review on the techniques and in-depth practicalities of spatial audio was provided by Rumsey [80].

Wave Field Synthesis (WFS) or holophonic theory [68,71] derived from Huygens principle and aims to reconstruct a sound field by means of microphone array recordings and loudspeaker array playback. Acceptable sound field reproductions on a plane were obtained with finite loudspeaker arrays of the order of 100 loudspeakers with this approach.

Periphony [73] is based on a spherical harmonic decomposition of the sound field and in practice it consists of recording the directional components of the sound field (i.e. the pressure field and its derivatives). Periphony is theoretically perfect when infinite harmonics are taken into account. The practical outcome of this theory was the Ambisonics system, which recreated a first-order spherical harmonics representation of a sound field [81]. Higher order ambisonics representations have been proposed although in practice the inherent difficulty of recording high order pressure field derivatives has to be overcome.

Periphony and holophony were shown to be equivalent by Nicol et al. [82]. Their integration with DTA theory remains open for future work.

Chapter 3 Theoretical Background

- 21 -

3 Theoretical Background

In this chapter, mathematical tools describing the direct conversion problem are provided. The operation of a standard multi-bit DAC is reviewed and extended to that of a multi-bit direct-acoustic DAC. General quantities used to quantify the performance of DACs are reviewed. Well known theoretical concepts with a direct use in digital transducer arrays are commented and extended, from array theory to the modelling of transients by planar pistons. Two well known analytical models for the transient response of planar circular pistons are presented with applications to the transient response of a DTA. Viewing DTA radiation as a sum of discrete radiation produced by quantised pulses of air also helps understanding the direct digital-to-acoustic operating principle from a theoretical point of view.

The theoretical output of a DTA is studied under harmonic and broadband inputs. It is shown that as the number of bits and the number of harmonics considered increase, the reconstructed sound field improves with perfectly flat transducer FRFs. A THD map of the on-axis reconstructed output considering different numbers of bits and harmonics is provided.

Theoretically, THD increases as the acoustic path length differences increase, owing to out-of-phase pulse summation, and as the transducer frequency response degrades from purely uniform and linear phase. Other factors occurring in practice such as transducer mismatches or the presence of a baffle are modelled from a theoretical point of view. The relationship to system parameters such as the number of bits and the sampling rate is discussed. Finally, the effect of introducing a noise shaping algorithm in the DTA chain is discussed.

3.1 The Multi-bit DAC

A DTA is a direct signal to acoustic converter, with radiated bitstream signal mixing directly in the air. The theory behind electronic binary-weighted DACs can be extended to acoustic DACs with small modifications. In this section the standard DAC principle is reviewed and expressed mathematically with extended views towards the direct digital-to-acoustic conversion.

Many DAC electronic topologies exist, their common goal being to convert a binary number into a voltage or current which is proportional to the value of the digital input. Although some work has been done proposing the sigma-delta modulator as a possible direct-converting topology [10,18], and other topologies have been suggested [11], the most relevant DAC topology to direct-converting systems and the one this work focuses on is that of the multi-bit DAC.

A standard electronic N-bit binary weighted DAC structure consists of N current sources and N bit switches. The current sources are binary-weighted and the bit switches select the corresponding sources depending on the state of their incoming bitstream. The digital input is a binary word b1b2…bN with b1 the least significant bit (LSB) and bN the most significant bit (MSB). The output current is then given by:


- 22 -

∑=

−

−

⋅⋅=

⋅⋅++⋅⋅+⋅⋅=N

ii

iLSB

LSBN

NLSBLSBOUT

bI

IbIbIbI

1

1

112

01

2

2...22 (3.1)

where ILSB is the current produced for the LSB. For each digital word, the analogue output current results from the summation given above. In practice, because no ideal current source exists and practical current sources must always have finite output impedance, the current delivered into the load is less than the ideal DAC output current. The binary weights are traditionally implemented as a resistor ladder and in a direct acoustic DAC they can be implemented via different transducer areas, transducer numbers or signal amplitudes.

As an example, figure 3.1 shows bitstreams obtained when one cycle of a 1 kHz sine wave is binary coded with 6-bits, expressing negative samples with signed magnitude (top left) and offset binary (bottom left) representations. The result of adding binary-weighted bitstreams in each case (right-column figures), followed by low-pass filtering, is representative of the DAC output. D/A conversion is therefore accomplished by adding binary weighted bitstreams and subsequent desampling (low-pass) filtering to eliminate spectral images and higher frequency components arising from the sampling process.

The standard multi-bit DAC processes of binary-weighting, adding and desampling also apply to a direct-converting DAC. Binary-weights are necessary in an acoustic DAC to resemble the operating principle of an electronic DAC. When a given waveform is sampled, quantised and expressed in binary code, digital words are obtained. A binary weight is assigned to each parallel bitstream according to its bit significance in order to reconstruct the original analogue signal. The quality of the final reconstruction is now affected by the different acoustic path lengths from transducer to receiver. Other factors affecting reconstruction are identified as bitstream filtering carried out by the transducers, transducer mismatches across the array and any other non-uniformities or non-linearities present in the system such as DSP, sound card or amplifier set transfer functions and most importantly, non-linearities arising from transducer operation.

The adding stage of the DAC, where binary-weighted bitstream contributions are added, is performed in direct-conversions in the acoustic domain, where radiated bitstreams are added. Linearity and time-invariability in the air is therefore assumed. At this stage radiated signals have already been filtered by the transducers. Ultrasonic components are therefore present to the extent of the maximum frequency component in the transducer FRF. In fact, the desampling (low-pass) filtering stage of a normal DAC is carried out by the transducer FRF.


- 23 -

Figure 3.1. Binary-weighted bitstreams corresponding to one cycle of a 1 kHz sine wave coded with 6-bits and signed magnitude representation (top left) and offset binary representation (bottom left). On the right column, the output obtained by the addition of the corresponding bitstreams in each case. Subsequent low-pass (desampling)

filtering for signal smoothing is required in standard DAC.

3.2 Quantification of DAC Performance

There are a number of parameters commonly used to quantify the performance of a DAC which will be useful to know throughout this work in order to characterise our direct-converting systems. Their definitions [83,84] are included here as a reference against which further work might be checked.

3.2.1 Total Harmonic Distortion (THD)

Quantity for periodic waveforms which measures how much the waveform departs from a perfect sine wave. For a total harmonic power denoted by Pt with a fundamental power denoted by Pf, the distortion power follows as Pd = Pt - Pf. THD was then defined by Morfey [83] as:

100(%) ⋅=f

d

PP

THD

(3.2)

THD is a useful quantity to specify DAC harmonic output in relation to its harmonic components; however it does not say anything about the relationship between the output and the noise level.


- 24 -

3.2.2 Signal-to-Noise Ratio (SNR)

In a signal consisting of a desired noise component and an uncorrelated noise component, SNR was defined by Morfey [83] as the ratio of the desired component power to the noise power:

⎟⎟⎠

⎞⎜⎜⎝

⎛

⟩⟨⟩⟨

⋅= 2

2

10log10)(nx

dBSNR s

(3.3)

where xs represents the desired signal as a function of time, n represents the noise signal as a function of time, and the brackets indicate time averaging.

3.2.3 Spurious-Free Dynamic Range (SFDR)

Difference between the RMS value of the desired output signal and the highest amplitude output frequency that is not present in the input, expressed in dB.

3.2.4 Signal-to-Noise and Distortion (SINAD)

Combination of the SNR and THD specifications defined as the RMS value of the output signal to the RMS value of all of other spectral components below Nyquist frequency, including harmonics and excluding DC. It can be calculated from the SNR and THD figures according to:

⎟⎠⎞

⎜⎝⎛

+⋅=

− 10/10/10 10101log10 THDSNRSINAD

(3.4)

where SNR and THD are in dB. Because SINAD compares all undesired frequency components with the input frequency, it is an overall measure of DAC dynamic performance. It is sometimes also referred to as total harmonic distortion plus noise (THD+N).

3.2.5 Effective Number of Bits (ENOB)

A converter with a given ENOB performs as if it was an ideal converter with a resolution of ENOB. An ideal DAC would have no distortion and its only noise would be due to quantisation only, so SNR would be equal to SINAD. The theoretical SINAD for an ideal DAC is (6.02n + 1.76) dB, where n is the number of bits. Therefore the ENOB might be calculated from the SINAD figure as:

02.676.1−

=SINADENOB

(3.5)


- 25 -

After having reviewed the definitions of these quantities commonly employed in ADC and DAC specifications, the theoretical basis to assess the performance of direct-converting systems is established.

3.3 Array Theory

The most relevant theory concerning loudspeaker arrays which is at the same time applicable to direct-converting loudspeaker array systems is now presented and discussed.

Loudspeaker or microphone arrays are generally employed in audio applications to achieve increased directivity, which can be controlled digitally [50,51]. Vertical loudspeaker line arrays control dispersion to decrease the amount of room modes which are excited, resulting in an improved direct to reverberant sound field ratio. In addition, greater power sound pressure levels with lower distortion are obtained due to the splitting of the signal through different drivers.

The near-field acoustic behaviour or Fresnel region in a line array can be modelled with cylindrical waves, which present a 1/r pressure intensity dependence and a 3 dB roll-off per doubling of distance [25,85].

The Rayleigh distance in a line array is defined as:

λ

22LdR =

(3.6)

where L is the array length and λ is the signal wavelength. This concept should not be confused with the reverberation distance, reverberation radius or critical distance of a room, which is defined as the distance from the source where the direct sound field level equals the reverberant sound field level [86]. The reverberation distance is an intrinsic property of the reproduction room, whereas the Rayleigh distance relates to the array.

The Rayleigh distance is just a function of the array size and the wavelength and sets the limit between the near and the far-field. In the far-field or Fraunhofer region the acoustic behaviour can be modelled with spherical waves, which present a 1/r2 pressure intensity dependence and a 6 dB roll-off per doubling of distance. In a listening room, excitation of room modes is greater in the far-field, when the reverberation distance has been exceeded and the sound field is mainly diffuse. In order to compensate for this effect and have a greater ratio of direct-to-reverberant sound field for a larger listening distance, the near-field coverage of an array is extended as much as possible.

From optical aperture theory, the far-field radiation pattern of a set of apertures can be obtained as the spatial Fourier transform of the aperture function [36], a result which also applies to acoustic radiating arrays. Since propagating acoustic waves carry information in both spatial and temporal domains, sampling in both domains must be carried out at enough sampling rate to avoid aliasing. In a line array, to avoid spatial aliasing the following condition must be satisfied [25]:


- 26 -

2minλ

≤d

(3.7)

where d is the transducer interspacing and λmin is the minimum wavelength that can be reproduced without spatial aliasing.

While the output of an acoustic monopole point source is omnidirectional, the output of an acoustic pistonic radiator reduces with increasing off-axis angles and increasing frequency. At a fixed elevation ψ0, the ratio D(θ) of the acoustic output of a radiator at an angle θ off-axis to its acoustic output at the on-axis position at θ = 0 is called the directivity function, also known as directivity pattern, directional response pattern or beam pattern [43].

The vertical array directivity function for an N point-source line array with constant interspacing d can be expressed with the following analytical model derived by adding the acoustic pressure contribution of each radiator in the array [43]:

( )( )

⎟⎠⎞

⎜⎝⎛

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛

⋅=⎟⎠⎞

⎜⎝⎛⋅−

2sinsin

2sinsin

2sin1

θ

θ

θθ

kdN

kdNeD

kdNj

(3.8)

where ( )θD is the vertical directivity function, N is the number of sources in the array, d is the transducer interspacing, k is the wave number, and θ is the angle between the acoustic axis and the listening position.

The directivity function given in (3.8) presents nulls at πmn =Δ and peaks at ( ) 2/12 π+=Δ mn , where m is a positive integer greater than zero. The horizontal directivity function is constant for all angles.

In the case of a series of point-sources equally spaced along a circle, the following directivity function at any point P with spherical angular coordinates ( )ϕθ , is obtained by adding the contribution of each source:

( ) ( ) ( ) ( )( ) ( )ϕθ

ϕθθϕθ

nkdJj

nkdJjkdJD

nn

nn

2cossin2

cossin2sin,

22

0

+

++=

(3.9)

where Jn is the Bessel function of nth order and j is the unit imaginary number.


- 27 -

3.4 Transient Radiation by Planar Pistons

The calculation and prediction of transient responses generated by pistons give a tool with which the DTA acoustic field can be characterised.

In this section, two analytical models which result in time-domain expressions for the transient radiation produced by planar pistons are provided. The first model was derived starting from the Green’s function between the source and the receiver and therefore from its acoustic transfer characteristics [87]. Damping is not taken into account by this model. The second model was derived by considering a second order damped mass-spring system and it results in a closer approximation to what the step response of a real transducer looks like [35]. References [42,46,88-92] provide more detailed mathematical models and tools with which transient piston radiation can be calculated, outside of the scope of this thesis.

By integrating elementary pressure contributions in a circular geometry over time, the transient pressure pulse radiated from a piston undergoing a displacement ∆ can be shown [87] to be described by the expression:

( )( )

⎪⎪

⎩

⎪⎪

⎨

⎧

+>

+<<−−−

−Δ

−<

=

θ

θθθθπ

ρ

θ

sin0

sinsinsinsin2

sin0

2222

2

arct

arctarctra

ctrrc

arct

tp

(3.10)

where ρ is the air density, c is the speed of sound, r is the scalar distance from the centre of the piston to the microphone, and θ is the off-axis angle.

This expression results in both a positive and a negative infinite pressure pulse which stretches out in time as the angle theta is increased.


- 28 -

1.5 2 2.5 3

x 10-3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25Analytical Step Response, Morse

t (s)

p (P

a)

θ = 0.87266

δ = 1e-005

Figure 3.2. Analytical piston step response derived from Green’s function integration (Morse).

Radiated pulses are a stretched out version of the Dirac delta function and the greater the off-axis angle, the greater the stretch. On-axis, the radiated pulse presents exactly the same shape as the Dirac delta function. This expression justifies the existence of a sweet-spot in a DTA, with increasing amounts of THD as the off-axis angle is increased, due to the time stretching of radiated pressure pulses. In practice, the shape of an experimental transducer step response depends on the transducer operating principle and its deviation from non-ideal behaviour.

The second model considered applicable to the DTA transient characteristics is a simple second order mass-spring system step response model [35]. The transient solution for the differential equation governing the system can be summarised as:

( ) ( )teM

tp t

MS

Ω⋅⋅Ω

= − sin1 α

(3.11)

The parameters parameters α and Ω are given by:

21 α−=ΩMSMS MC

(3.12)


- 29 -

MS

MS

MR

2=α

(3.13)

where RMS represents the mechanical losses of the driver, CMS represents the mechanical compliance of the driver and MMS represents the effective mechanical mass of the driver including the air load.

The parameter α accounts for the width of the spike. The higher the value of α, the thinner the spike. In order to radiate an acoustic pressure pulse, the value of α should be as high as possible in order to produce a very thin spike. The parameter Ω accounts for the damping in the system and should be as small possible to avoid extra unwanted ripple that would otherwise introduce oscillation if the driving signals were bitstreams. In figure 3.3, the analytical step response is presented with values for α and Ω derived from the measured electromechanical parameters of a moving-coil tweeter. Both parameters are interrelated and by choosing one of them the value of the other one is constricted; best choices are usually compromise solutions.

0 0.5 1 1.5 2 2.5 3 3.5 4

x 10-3

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3Analytical Step Response, Olson

t (s)

p (P

a)

Ω = 3879.2

α = 2594.4

Figure 3.3. Analytical piston step response derived from a second order damped mass-spring system (Olson).

3.5 Quantised Pulses

In a DTA, transducers produce positive and negative pressure pulses, behaving as quantised pistons. It can be shown mathematically that these quantised pistons should behave in the same way as normal radiating pistons, in terms of pressure-versus-frequency dependency.


- 30 -

The sound pressure level produced by a radiating piston of radius a on-axis and in the far-field has the form:

)(0),( krtjeptrp −⋅= ω

(3.14)

where ω is the angular frequency, t is the time, k is the wave number and r is the scalar distance from source to receiver. The term p0 has the form:

⎟⎟⎠

⎞⎜⎜⎝

⎛⋅=⎟

⎠⎞

⎜⎝⎛=

raUka

racUp

2

000 21

21 ωρρ

(3.15)

where 0ρ is the air density, c is the speed of sound, and U is the source strength at r = a.

The source strength U is related to the volume velocity Q through:

arUaQ

=⋅= 24π (3.16)

Substituting eqn. (3.16) into eqn. (3.15), the following expression is obtained:

rcQcp ω

πρ

⎟⎠⎞

⎜⎝⎛=

80

0

(3.17)

Let us now consider the case of a quantised piston producing elementary displacements of air volumes V, with cross-sectional area A, length u·δt, where u is the air velocity (Figure 3.4). The elemental volume velocity associated to this mass of air is therefore q = Au. The volume of this air quantum is given by:

tqtuAV δδ ⋅=⋅⋅= (3.18)

Since the total volume velocity Q is the product of elemental volume velocities q times the number n of air masses needed to produce the total volume velocity, it follows that:

tVnnqQδ

==

(3.19)

Substituting this value of Q into eqn. (3.17), p0 can be related to the elemental air volume properties as:


- 31 -

rtnVp ωδπ

ρ8

00 =

(3.20)

The pressure produced by a quantised piston is thus proportional to frequency. Therefore the pressure radiated by a quantised piston shows the same qualitative frequency dependency as a radiating piston, by comparing this last expression with eqn. (3.17).

dtu

nQu=

Area A

A q

Quantised pistonElementary air volume

·

Volume V

q =

Figure 3.4. Elementary air volume produced by a quantised piston. The air moves with velocity u; q is the volume velocity for the elementary air volume; Q is the total volume velocity of the DTA and n is the number of elementary

air movements necessary to achieve a volume velocity of Q.

3.6 Excitation in an Ideal DTA

3.6.1 Harmonic Excitation

When a harmonic excitation is digitised and its binary words are used as the input to a DTA, it is possible to reconstruct the original harmonic excitation by considering a sufficiently high number of bits and of bitstream harmonics.

The original waveform at an on-axis point at a scalar distance r from the array centre can be expressed as:

)(),( fff rktjeAtry φω +−⋅= (3.21)

It is possible to express any bitstream bn(r,t) as a fundamental component with frequency fω and

phase fφ followed by a series i of bitstream harmonics at frequencies iω and phases iφ as:


- 32 -

∑∞

=

+−+− ⋅+⋅=1

)(,

)( ,),(i

rktjni

rktjfn

niiifff eAeAtrb φωφω

(3.22)

It will be shown that the binary-weighted sum of bitstreams will tend to expression (3.21) as the number of bits and harmonics tends to infinity. The infinite binary-weighted bitstream sum can be expressed as the sum of the fundamental plus infinite harmonics:

∑ ∑∑∞

=

∞

=

+−+−∞

=⎥⎦

⎤⎢⎣

⎡ ⋅+⋅⋅=⋅1 1

)(,

)(

1

,),(n i

rktjni

rktjfn

nnn

niiifff eAeAWtrBitW φωφω

(3.23)

The binary-weighting coefficients have the following analytical form:

nnW −= 2 (3.24)

where n is the bit number with n = 1 corresponding to the MSB. The MSB accounts for half of the signal amplitude and the rest of the bits account for the other half. Owing to the fact that the infinite sum of binary-weights is one:

121

=∑∞

=

−

n

n

(3.25)

then our problem is equivalent to showing that for a finite number of bits N and a finite number of harmonics H the following sum tends to zero:

0,1 1

)(,

, ⎯⎯⎯ →⎯⋅ ∞→= =

+−∑∑ HN

N

n

H

i

rktjnin

niiieAW φω

(3.26)

For a fixed number of harmonics H the last expression reduces to:

0)cos(21

,, ⎯⎯ →⎯+−⋅⋅ ∞→=

−∑ N

N

nniiini

n rktA φω

(3.27)

At any given listening position, the THD can be numerically computed on the DTA output for a fixed number of harmonics and bits. This will measure the harmonic deviation from a sine wave on the output signal. As the THD tends to zero, fewer harmonic components will be present and therefore expression (3.27) will hold.


- 33 -

Owing to the principle of superposition, any waveform as a function of time, in our case a digital bitstream, can be regarded as an addition of infinite harmonic components with different amplitudes and phases. In a real case scenario only a finite amount of harmonics can be taken into account and therefore the original waveform can never be completely reconstructed.

For periodic waveforms, Fourier series harmonic analysis yields the coefficients an and bn from which the original waveform can be synthesised.

Starting from an analogue sine wave at frequency fω , the MSB is a square wave of fundamental

frequency fω whose Fourier coefficients are given by:

oddnn

b

na

n

n

==

∀=

π2

0

(3.28)

which shows that only odd order harmonics with exponentially decreasing amplitudes are present in a square signal.

For the next least significant bits representing the original sine wave there is not such a simple analytical expression since the bitstream waveform gets increasingly complex, with higher number of discontinuities. An FFT was used to numerically compute the Fourier coefficients of these bitstreams and a reconstruction with a finite number of harmonics was calculated.

Figure 3.5 shows the obtained THD map for different number of harmonics and different number of bits.

Figure 3.5. Output THD map for a harmonic excitation in the DTA. As the THD tends to zero for a given number of bits and harmonics, the DTA output is also harmonic.


- 34 -

3.6.2 Broadband Excitation

The individual output yi(ri, t) from a transducer i at position vector ri excited by a weighted bitstream input signal is to a first approximation (assuming linearity) equal to the convolution of the transducer impulse response at position vector ri with the input driving signal:

( ) )(,),( tbthwty iiiiii ⊗⋅= rr (3.29)

where hi(ri,t) is the ith transducer impulse response at position vector ri, wi = 2-i are the binary weights, and bi(t) is the ith bitstream signal at time t. The DTA response y(r, t) at position vector r from the array centre is then obtained as a sum of individual responses:

( )∑∑==

⊗⋅==N

iiiii

N

iii tbthwtyty

11

)(,),(),( rrr

(3.30)

In the z-domain:

( )∑∑==

⋅⋅==N

iiiii

N

iii zBzHwzYzY

11

)(,),(),( rrr

(3.31)

where the z-transforms have been denoted with capitals.

The impulse response for transducer i at a position given by vector ri, hi(ri,t), accounts for the spectral characteristics of this transducer including any possible irregularities and its band-pass frequency response characteristic which in practice acts as an analogue filter. Therefore the DTA acoustic output results from filtering the driving bitstreams (with a Nyquist frequency of half the DSP sampling rate) with the transducer spectral characteristics. Any bitstream components higher than the transducer cut-off frequency will not be radiated.

3.7 Real Transducer Effects

In practice there exist a number of transducer effects relevant to the operation of direct-converting systems worthwhile modelling and discussing from a theoretical point of view. These are: acoustic path length differences, transducer FRF non-uniformities and non-idealities, transducer mismatches and baffle effects.

3.7.1 Acoustic Path Length Differences

In the DTA, the radiated bitstreams have to undergo different acoustic path lengths to arrive to a common listening point, resulting in increased THD due to the delayed radiated pressure pulses.


- 35 -

As the distance from the listening point to the array centre is decreased and it becomes comparable to the array size, the differences in acoustic path lengths increase. On the other hand, as the distance to the listening point increases, path length differences become smaller with respect to the whole acoustic path. In the limit, if our DTA was a point source, all path lengths would be exactly identical and all pressure pulses radiated from the different bitstreams would arrive exactly in phase, allowing perfect reconstruction at all points in space.

Suppose a one transducer-per-bit DTA line array of length L with a total of N drivers. Call the scalar distance from the centre transducer to the listening point r, and the individual scalar distances from each driver to the listening point rn.

In the near-field, r is comparable to L. Therefore, the generic path difference ∆rn = rn+1 – rn is big. The resulting DTA pressure y(r,t) at a point given by vector r and at time t can be expressed as a finite sum over all bits and an infinite sum over all harmonics, considering ideal transducers with perfectly linear phase response and an all-pass magnitude response:

( ) ( ) ∑ ∑∑=

∞

=

+−

=

⋅==N

n i

rktj

n

nin

N

nnn

niniier

AWtyty

1 1

)(,

1

,,, φωrr

(3.32)

Analysing expression (3.32), the modulus of rn is present in two occasions: the factor 1/rn represents the amplitude dependence of the different bitstreams change according to the position the transducers occupy within the array; the factor nirjke− represents different phase delays distorting the resulting wavefront. Both of the rn contributions result in increased THD.

In the far-field, r >> L. Therefore the generic path difference ∆rn can be considered to be negligible and rn can be approximated by r. The resulting DTA pressure can be approximated by:

( ) ( ) ∑ ∑∑=

∞

=

+−

=

⋅≈=N

n i

rktjnin

N

nn

niniieAWr

tyty1 1

)(,

1

,1,, φωnrr

(3.33)

where the phase differences still remain due to the sensitivity of the exponential to small changes although the amplitudes of the different bitstreams are now all weighted by the identical distance factor 1/r.

3.7.2 Transducer FRF Non-Uniformities

Direct digital-to-acoustic conversions with loudspeaker arrays will be affected by any non-uniformity present in the frequency response of the transducers forming the array. In the ideal limit, perfect DTA reconstruction over the whole listening space is possible if the driving transducer frequency response is perfectly uniform and has a perfectly linear phase over the whole spectrum. Any variations from this theoretical limit will degrade the reconstructed DTA response.

Suppose an ideal transducer magnitude response ( )ω,rH and an ideal transducer phase response

( )ωφ ,r . Denoting the real transducer magnitude response (including any irregularity, roll-off and


- 36 -

break-up) by ( )ω,~ rH , and its associated phase response by ( )ωφ ,~ r , then following equation (3.31),

the real frequency domain DTA response which accounts for all transducer effects can be written as:

( ) ( )∑∑==

⋅⋅==N

ii

jiii

N

iii BeHwYY

1

,~

1

)(,~),(~),(~ ωωωω ωφ rrrr

(3.34)

The ideal transducer frequency response for transducer i can be modelled as a first approximation as:

( )( )

i

krtj

ii reC,H

i−

⋅=ω

ωr

(3.35)

where C is a constant and the 1/r factor accounts for the inverse square law. The magnitude of ( )ω,rH described by equation (3.35) is uniform and its phase is linear over the whole spectrum. A

more realistic approximation to transducer response includes the characteristic 12 dB / oct roll-off G(ω) and a directivity function D(θ). For transducer i the following expression holds:

( ) ( ) ( )( )

i

krtj

ii reGDC,H

i−

⋅⋅=ω

ωθωr

(3.36)

The further fluctuations introduced by the real transducer frequency response can be included in an extra frequency dependent magnitude term Mi(ω) and phase term φi(ω). Therefore the real transducer frequency response can be modelled as:

( ) ( ) ( ) ( )( )[ ]

reGDMC,H

ikrtj

iii

ωϕω

ωθωω+−

⋅⋅⋅=r~

(3.37)

The relationship between the real and the ideal transducer frequency response can then be expressed as:

( ) ( ) ( )ωωω ,,~iiiiii RH,H rrr ⋅= (3.38)

where ( )ω,iiR r is the complex fluctuation associated to transducer i.

The DTA frequency-domain output accounting for transducer irregularities can then be expressed as:

( ) ( )∑∑==

⋅⋅⋅==N

iiiiiii

N

iii zBzRzHWzYzY

11

)(,,),(~),(~ rrrr

(3.39)


- 37 -

The transducer irregularities will have an impact on the DTA acoustic output, and this can be seen by taking the inverse FFT of expression (3.39). The result is here expressed as the sum of the convolutions between each individual transducer response and each transducer complex fluctuation.

3.7.3 Transducer Mismatches

Suppose a one bit-per-transducer DTA where the levels and phases of all bits are perfectly matched, where binary-weighting is perfect and path length differences are perfectly compensated for, at least for one point in space. The harmonic content of all bitstreams will ensure that extra bit harmonics are cancelled out and reconstruction is perfect at that point.

Suppose now that in the same system bit responses are such that the level of bit n is off from its theoretically perfect level by a factor of δ. The sum of binary-weighted bitstreams will result in an extra waveform whose harmonics are at a level of δ times the binary-weight associated to bitstream n (2-n), where n = 1 denotes the MSB. So the higher the bit significance, the more important it is to match bit responses correctly, in a binary-weighted manner.

Practically, reaching perfection in bit weighting, be it by electronic binary-weighting or by matching transducer responses is a difficult task. This is perhaps the most important inconvenience in direct-converting DAC systems.

3.7.4 Baffle Effect

The arrangement of a transducer on a baffle produces diffraction effects around the baffle which modifies the system impulse response, introducing a 6 dB baffle diffraction step, where radiation changes from a 4π space to a 2π space, followed by further dips and peaks in the resulting FRF. The location and width of the peaks and the centre of the diffraction step rise depend on the relative position of the transducer in the baffle and the baffle shape and dimension [93].

Several theories have been developed over the years to account for loudspeaker baffle diffraction [49,94,95]. Diffraction is a complex subject in itself and for the purpose of this work it will suffice to say that the frequency response of each transducer in the array will be affected differently by the baffle, and that in practice this will need individual compensation for each element. The theoretical limit for element transfer functions is represented by an infinite bandwidth with a perfectly flat magnitude and a perfectly linear phase.

3.8 Bandwidth and Resolution

3.8.1 Sampling Rate

Theoretically, assuming perfect channel transparency with regards to transducer operation, by increasing the sampling rate the bandwidth is increased, the sampling time is decreased and radiated bitstreams tend to the driving bitstreams. The extra bandwidth in the ultrasonic region could be used to shift unwanted noise components. The transducer will then act as a desampling filter.


- 38 -

By band-limiting the driving bitstreams transducers constitute the main limit in direct-converting systems if the adding stage is performed in the air. Reconstruction is possible up to the highest frequency in the band and improves for lower frequencies. At higher frequencies, the amount of bitstream harmonics available with respect to the available bandwidth is small and reconstruction quality decreases.

A direct-converting loudspeaker system should employ high oversampling to increase the bandwidth and implement noise shaping at the DSP stage, keep the bandwidth as high as possible in the amplifier stage and use the transducers as desampling and band-limiting filters. A smooth transducer roll-off is recommended rather than a prominent break-up peak.

3.8.2 Number of Bits

A digital system with theoretically perfect ADC and DAC has its quality determined at the ADC. Key parameters such as sampling rate and number of bits determine the quality of the conversion at the ADC and the DAC does not make things any worse [65].

In digital systems the higher the number of bits the higher the number of quantisation steps and the lower the quantisation error. In fact, every time the number of bits is increased by one the dynamic range increases by approximately 6 dB. The signal to quantisation noise and distortion theoretical limit for an n-bit system is given by:

ndBSQNR 02.6761.1)( += (3.40)

The maximum theoretical quality of a direct acoustic DAC is set by this limit, although in practice there are several practical factors that might prevent it to be reached, as it will be seen in future chapters.

3.9 Noise Shaping

Dithering and noise shaping are commonly used in audio to reduce the errors and distortion caused by quantisation [53]. It is thought that in the operation of direct-converting systems, noise shaping technology will be required as a means of bit reduction and SNR enhancement.

Quantisation error produces background noise as well as statistical correlation of the error with the signal, causing distortion which may produce alterations in the perceived quality of the signal. Dither consists of adding a small amount of random noise to the signal before quantisation, which linearises the action of the quantiser. Noise shaping is a feedback technique which results in the quantisation error being shifted into frequency bands higher than the audible range. Both techniques can be combined and result in the dithered noise shaper, the operation of which is depicted in figure 3.6. Practical implementations of noise shaping technology to the multiple voice-coil digital loudspeaker have been described in [8,9], and the first known practical implementation of a noise shaper with digital transducer arrays was published in [67], after which a theoretical extract is hereby presented.


- 39 -

A relationship between the noise shaper input and output which is dependent only on the independent parameters H(z), D and E is sought. The quantisation error E can be inferred from the numerical rounding action of the quantiser.

By inspecting figure 3.6, equations (3.41) and (3.42) are derived. Eliminating U, the relationship given by (3.43) is obtained between the noise shaper input X and output Y, the quantisation error E and the filter transfer function H(z), from which it is seen that the signal transfer function is unity and the noise transfer function is given by [1 – H(z)] regardless of the quantiser operating principle.

Figure 3.6. The dithered noise shaper

EzHXU ⋅−= )( (3.41)

UYE −= (3.42)

( )[ ] EzHXY ⋅−+= 1 (3.43)

By controlling the statistical properties and amplitude of the dither signal, the quantisation error can be made statistically independent of the input signal. In particular, by adding dither with triangular probability density function and a peak-to-peak amplitude of 2 LSB, it can be shown [54] that both the quantisation error and quantisation error power are uncorrelated to the input signal.

Consider an N-bit DTA without noise shaping. Denote the z-transform of the transducer impulse response sequence for transducer n by Hn(rn, z), where rn is the vector from the centre of transducer n


- 40 -

to the listening point P. Denote the transducer driving bitstream z-transform by Bn(z). The frequency response of the transducer and associated bitstream spectrum can be found by substituting z = ejωT, where T is the sampling time. The resulting pressure at an arbitrary listening point P is obtained as the convolution of each transducer response with its correspondent bitstream as given by equation (3.44).

( )∑=

⋅=N

nnnn zBzHzP

1

)(,),( rr

(3.44)

where r is the vector from the array centre to the listening point.

The contribution to the sound field from each bitstream may be expressed as a signal term Xq(z), which stands for the quantized input analogue signal, plus another term N(r,z) which represents all the noise and distortion contributions produced by real factors such as transducer mismatches, non-uniformities in the frequency response and phase differences arising from the different path lengths from the array elements to the common listening point. Therefore the sound field at P will be:

),()(),()(),(1

zNzBzNzXzPN

nnq rrr +=+= ∑

= (3.45)

In practice, the transducer arrangement and FRFs must be made such that the extra term N(r,z) is minimized over all listening positions and for all frequencies in the audio band. Practical limitations exist in fabricating transducers with uniform FRFs and the impact of non-uniform FRFs on reconstruction will be quantified in later chapters.

Now consider a DTA with a noise shaping algorithm implemented in a DSP. As a result, the signal term approximates the input analogue signal and the noise and distortion term is altered by the feedback process. The noise shaper noise transfer function is [1 – H(z)]. The DTA input is given by equation (3.43). The resulting pressure at an arbitrary listening point P can now be expressed as:

),('),()(),( zNXzNzXzP NSNSNSNS rrr +=+= (3.46)

where the NS subscript denotes noise shaping. The error and the noise transfer function introduced by the noise shaper are now contained in the term N’NS(r,z). The overall DTA error transfer function is therefore modified by the noise shaper transfer function, which can be designed of arbitrary shape as suggested in [53].

3.10 Summary

This chapter has provided a theoretical base upon which to build existing knowledge on direct acoustic converters. The already known theory presented which is also relevant to the DTA ranges from line array theory to piston transient response modelling, including the DAC operating principles and noise shaping. Novel additions introduced were the sections on DTA broadband and harmonic excitation,


- 41 -

path length difference induced distortion, quantised pulses, transducer mismatches and transducer frequency response non-uniformities, as well as the theoretical relationships to important system parameters such as sampling rate and number of bits.

Chapter 4 Implementation

- 42 -

4 Implementation

This chapter provides details of the most important DTA systems implemented together with technical characteristics of the experimental equipment employed for experiments. A detailed description of the main transducers and transducer arrays employed is given and transducer specifications are presented in terms of transducer frequency, step responses and directivity polar plots. Based on this data, the effect of transducers on reconstruction in DTA systems was experimentally investigated and reported in chapter 5.

All measurements and prototype developments were undertaken in the B&W Loudspeakers Ltd. Research Establishment, Steyning, UK.

4.1 Measurement Systems

The main measurement system used for acoustic measurements was the WinMLS 2004 PC-based audio–acoustic-vibrational measurement system, with a 192 kHz 24 bit LynxTWO sound card, which generated input signals and acquired microphone data, computing FRFs and THD measurements. Directivity measurements were obtained by taking FRFs every 5º from 0º to 180º and subsequently selecting the level at each frequency for every angle.

The commercial software Adobe Audition 1.5 was also used for the reproduction and recording of signals. For transducer electromechanical parameter estimation, the Klippel Analyzer System was employed. For electronic circuit measurements, the Audio Precision APWIN, 24 Bit, 108 kHz was used.

All near-field acoustic measurements were performed in anechoic chamber and recorded with a B&K free-field ¼'' microphone type 4939 and B&K type 2690 microphone preamp, allowing an upper frequency limit of 100 kHz.

4.2 Digital Signal Processing Platform

The Texas Instruments TMS320C6713 DSK DSP platform was used (figure 4.1). At the time of selection, the TMS320C67xTM family offered a high audio performance with floating point operation. The TMS320C6713 DSK operates at 300 MHz and uses a 1.4-volt core supply. It has an on-board audio codec with integrated ADC and DAC with mini-jack sockets for audio connections. In addition, the TMS320C6713 DSK offers a general purpose input-output port (GPIO) which was used in our application in order to output processed digital parallel samples. Sampling rates allowed by the codec were 8, 16, 22, 44, 48 and 96 kHz. The maximum sampling rate of 96 kHz was generally used throughout the experiments, in occasions reducing to smaller values to investigate the effect of changing the DSP sampling rate. However, for the purpose of noise shaping, a sampling rate of 96 kHz might not provide sufficient spare bandwidth into which unwanted noise could be shifted, so higher sampling rate DSP platforms are recommended for future direct-converting system implementations.


- 43 -

Figure 4.1. TMS320C6713 DSP development board

4.3 Amplifier Set

Fast switching class D amplifiers currently offer very efficient operation with regards to power consumption in audio systems, and as such they could bring efficiency savings to the DTA. However, their quality is not as good as that of a class AB amplifier. If adopted for the DTA, a modified class D topology would be needed with digital in and digital out, without the bitstream addition and last low-pass filtering stages. Class AB amps on the other hand offer better fidelity and lower efficiency. For simplicity and convenience, a set of class AB amplifiers were used to develop our prototypes. By using modified class D topologies instead, obtained results could not have been any better in terms of quality although perhaps more efficient in terms of heat dissipation.

The amplifiers consisted of an input preamplifier stage followed by a power stage implemented with the TDA7494 chip. The preamp stage set the gains required, either binary-weighted for the one bit-per-transducer DTA or impedance adjusted gains for the bit-grouped DTA. Further details on preamp gain settings can be seen in table 4.1.

The power amps were made with the TDA7494 power opamp chip, which is a 10 W class AB amplifier intended for high quality sound applications, and produces a uniform output versus frequency up to 20 kHz, rolling off smoothly thereafter at 2 dB / Oct, with a THD of less than 1% throughout the audio band.


- 44 -

Figure 4.2. Aerial view of amplifier set

A direct-acoustic DAC aims to postpone the conventional stages of an electronic DAC (i.e. binary-weighting, adding and desampling) until the acoustic domain.

In the one bit-per-transducer DTA, the amplifier gains were binary-weighted to allow reconstruction. Therefore the first one of the three conventional DAC stages was still kept in the electronic domain, bringing the remaining two to the acoustic domain.

On the other hand in the bit-grouped DTA the binary-weighting stage is taken to the acoustic domain by having a binary-weighted transducer number, which makes the concept closer to the direct-acoustic DAC idea. In its simple form, one amplifier can be used for each bit, keeping the binary-weighted gains, and the transducers in each bit need to be wired up in series-parallel combinations which make the overall bit impedances change with respect to that of a single transducer.


- 45 -

Figure 4.3. Prototype DTA schematics for one amplifier with digital in and digital out. Six of these were implemented and used for the experimental prototype.

Therefore amplifier gains need to compensate for different bit impedances depending on the overall impedance in the bit group. In the extreme, one amplifier could be used for each transducer, all having the same gains, although this would be extremely inefficient.

The resistor values used to set the gains of the preamp modules which performed the binary-weighting in the case of one transducer-per-bit prototypes and the weighting for the bit-grouped DTA are given in table 4.1. The measured frequency response magnitudes for the one bit-per-transducer topology are shown in figure 4.4, whereas measured frequency response magnitudes for the binary-weighted topology are shown in figure 4.5. Amplifier measurements were performed with the APWin measurement system.


- 46 -

One transducer-per-bit arrays preamp module

Bit Gain Level (dBr) R1 (kΩ) R2 (kΩ) R3 (kΩ)

1 1 0 10 10 5.1

2 0.5 -6 10 20 6.8

3 0.25 -12 10 39 8.2

4 0.125 -18 10 82 9.1

5 0.0625 -24 10 160 9.1

6 0.03125 -30 10 330 10

Bit-grouped array preamp module

Bit Gain Level (dBr) R1

(kΩ) R2 (kΩ) R3 (kΩ)

1 1 0 10 10 5.1

2 0.25 -12 10 39 8.2

3 0.25 -12 10 39 8.2

4 0.0625 -24 10 160 9.1

5 0.0625 -24 10 160 9.1

6 0.03125 -30 10 330 10

Table 4.1. Resistor values and gains used for the different DTA topologies preamp modules. R1 is the feedback resistor in an inverting amplifier, R2 is the resistor to the non-inverting input and R3 is the resistor

to the inverting input.

Figure 4.4. Binary-weighted gains set by preamplifier module for the 1 transducer / bit DTA topology.


- 47 -

Figure 4.5. Gains set by preamplifier module for the bit-grouped DTA topology.

4.4 Transducers

Three main transducer types were used for the experiments. Their main characteristics are described here and their detailed electromechanical parameters are contained in table 4.2.

The B&W 800 Series tweeter with aluminium-dome (figure 4.15, middle), is intended for the high end audio market. It is a moving-coil type transducer and has a strong magnet which results in a Bl factor of approximately 2.5 T·m. The diaphragm diameter is 26 mm and the outer tweeter diameter including the magnet is 58 mm. Its break-up frequency is near 30 kHz. It has a specifically designed hard plastic housing which allows optimised back radiation isolation. The housing proved quite useful to avoid baffle effects in the array prototypes made with these tweeters.

The B&W 800 Series tweeter with diamond-dome (figure 4.15, right), also intended for the high end audio market, is also of the moving-coil type but has a stiffer diaphragm coated with a diamond layer. The stiffer diaphragm extends the bandwidth and sets the break-up frequency close to 70 kHz. Its dimensions are the same as the aluminium-dome tweeter.

The Harman Odyssey II transducer (figure 4.15, left), used in multimedia laptops, is a smaller transducer (12 mm diameter), also of the moving-coil type, with a weaker magnet and a bigger magnet gap which makes it less sensitive and inefficient. Its main advantage is its size-versus-bandwidth ratio. Due to its small size and reasonably flat frequency response, this transducer was chosen for the development of bit-grouped array prototype topologies.

The on-axis FRFs, for the three transducer types are shown in figure 4.6, figure 4.7 and figure 4.8. The on-axis and off-axis step responses from 0º to 90º in 15º steps for the three transducer types are shown in figure 4.9, figure 4.10 and figure 4.11. By comparing the step responses of the various transducer types, the most similar to the ideal pressure pulse corresponds to the diamond-dome tweeter. This was


- 48 -

also the transducer type which presented a most uniform frequency response and the one with which less THD was obtained in DTA acoustic measurements (section 5.1). The other extreme was given by the Harman Odyssey 2 transducer whose step response presented significant ringing for several ms after the main pulse. This deficient transient response will indeed introduce THD in the DTA acoustic output as it will be seen in section 5.1.

Directivity polars for each transducer type at 1, 2, 4, 8 and 16 kHz are shown in figure 4.12, figure 4.13 and figure 4.14. For the Odyssey 2 transducer, due to its frequency response rise until 3 kHz the polars at 1 kHz and 2 kHz are at a lower level than the rest.

The presented results are typical for average transducers of each kind. Due to present transducer mismatches, transducer characteristics may vary depending on the manufacturers allowed tolerance and present a statistical distribution, altering the obtained results. Further results on transducer mismatches and how they affect results in a direct conversion are given in section 5.1.5.

Bl (T·m) 2.63 Bl (T·m) 2.42 Bl (T·m) 0.9

Cms (µm/N) 70 Cms (µm/N) 60 Cms (µm/N) 1200

Mms (g) 0.33 Mms (g) 0.52 Mms (g) 0.13

Qms 1.59 Qms 1.57 Qms 2.5

Qes 1.18 Qes 1.43 Qes 1.3

Qts 0.68 Qts 0.75 Qts 0.86 B&

W A

lum

iniu

m

Sd (cm2) 5.31

B&

W D

iam

ond

Sd (cm2) 5.31 H

arm

an O

dyss

ey 2

Sd (cm2) 1.76

Table 4.2. Transducer main electromechanical parameters. Foster values are quoted from the manufacturer whereas B&W values were experimentally determined with the Klippel distortion analyser system.

Figure 4.6. B&W aluminium-dome tweeter on-axis frequency response magnitude


- 49 -

Figure 4.7. B&W diamond-dome tweeter frequency response magnitude

Figure 4.8. Odyssey 2 tweeter frequency response magnitude.


- 50 -

Figure 4.9. B&W aluminium-dome tweeter step responses on-axis and at different angles off-axis.

Figure 4.10. B&W diamond-dome tweeter step responses at different angles


- 51 -

Figure 4.11. Odyssey 2 tweeter step responses at different angles.

-10 0

30

60

90-90

-60

-300

1 kHz2 kHz4 kHz8 kHz16 kHz

Figure 4.12. B&W aluminium-dome tweeter polars

-10 0

30

60

90-90

-60

-300


Figure 4.13. B&W diamond-dome tweeter polars


- 52 -

-10 0

30

60

90-90

-60

-300


Figure 4.14. Harman Odyssey 2 tweeter polars

Figure 4.15. Dimensions and appearance of the three tweeter types. Left: Odyssey 2; middle: B&W aluminium-dome; right: B&W diamond-dome

4.5 Full-range Loudspeakers

In addition to the transducers described, two different full-range loudspeakers were used for experiments. Results on reconstruction with these loudspeakers will be given in section 5.1.3. The specifications as quoted from the manufacturer of these loudspeakers are given in table 4.3. Their measured impulse responses are shown in figure 4.16 and figure 4.18, and frequency responses together with second and third harmonic components are shown in figure 4.17 and figure 4.19.


- 53 -

Figure 4.16. B&W XT4 Impulse response

Figure 4.17. B&W XT4 Measured frequency response, second and third harmonics


- 54 -

Figure 4.18. B&W 800D measured impulse response

Figure 4.19. B&W 800D measured frequency response, second and third harmonics


- 55 -

XT4 800D Description 3-way vented-box system 3-way vented-box system

Height: 1138 mm (44.8 in) (not including feet)

Height: 1180 mm (46.5 in) (not including feet)

Width: 152 mm (6 in) Width: 450 mm (17.7 in) Dimensions

Depth: 200 mm (7.9 in) Depth: 645 mm (25.4 in)

Net Weight 24.5 kg 125kg

Freq. Response 40 Hz - 22 kHz ±3 dB on reference axis

32 Hz - 28 kHz ±3 dB on reference axis

Freq. Range -6dB at 34Hz and 50kHz -6dB at 25Hz and 33kHz

Sensitivity 86 dB spl (2.83 V, 1 m) 90 dB spl (2.83 V, 1 m)

Normal Impedance 8 Ohm (minimum 3.1 Ohm) 8 Ohm (minimum 3.1 Ohm)

Power Handling 50 W - 150 W into 8 Ohm on unclipped programme

50 W - 1000 W into 8 Ohm on unclipped programme

Unit 1: 1x 25 mm (1 in) aluminium dome high-frequency

Unit 1: 1x 25 mm (1 in) diamond dome high-frequency

Unit 2: 1x 130 mm (5 in) woven Kevlar® cone midrange

Unit 2: 1x 150 mm (6 in) woven Kevlar® cone FST midrange Drive Units

Unit 3: 2x 130 mm (5 in) Paper/Kevlar® cone bass

Unit 3: 2x 250 mm (10 in) Rohacell® cone bass

Finish Cabinet: Natural aluminium Grille: Black cloth

Cabinet: Real wood veneers of Cherrywood, Rosenut or Black Ash

Description: Within 2 dB of reference response

Description: Within 2 dB of reference response

Horizontal: over 60º arc Horizontal: over 60º arc Dispersion

Vertical: over 10º arc Vertical: over 10º arc

Harmonic Distortion 2nd and 3rd harmonics (90 dB, 1 m) < 1 % 120 Hz - 20 kHz

2nd and 3rd harmonics (90 dB, 1 m) < 1 % 45 Hz - 100 kHz, < 0.5 % 80 Hz - 100 kHz

Crossover Frequency 360 Hz, 3.5 kHz 350 Hz, 4 kHz

Max. Recommended Cable Impedance 0.1 Ohm 0.1 Ohm

Table 4.3. Manufacturer specifications corresponding to the two full-range system used for experiments

4.6 Transducer Arrays

Three main 6-bit transducer arrays were built. The first two arrays had one transducer per bit and a total of six transducers for each array, arranged in line (figure 4.21, left) and circular (figure 4.21, middle) geometries, and the B&W transducers were used. In the line array geometry, the minimum spacing of 5.5 cm allowed by the transducer size was employed, which imposed an upper frequency limit of near 3 kHz on-axis to avoid spatial aliasing distortion. The B&W transducer resonance frequency was measured to be near 850 Hz, therefore test signals within this range, i.e. 1 kHz and 2 kHz were used.


- 56 -

For the circular array geometry DTA acoustic response, a sweet-spot was expected on-axis for all listening distances. Non-symmetrical near-field polar responses were also expected, increasing their symmetry as the listening distance was increased.

The other type of array implemented was a bit-grouped topology (figure 4.21, right) which comprised a total of 63 transducers arranged in concentric circles, with binary-weighted transducer numbers for each bit (one transducer for the LSB and 32 transducers for the MSB). It was decided to build one bit-grouped DTA to set boundaries on its performance and obtain detailed experience on DTA implementation.

This particular bit-grouped array geometry was implemented since this was the one with which less THD on-axis was predicted in [4]. The array arrangement consisted of concentric circles with binary-weighted transducer numbers, with 32 transducers in the outermost ring assigned to the MSB. The transducers used in this prototype were the Harman Odyssey 2 transducers because of their reduced size and because of the larger number of transducers required for this topology. This way the outer array ring diameter was 26 cm.

Transducers were wired in series-parallel combinations (figure 4.20), allowing the same current to flow through each ring. The equivalent impedance of each ring was either 4 or 8 Ω, contrary to the case of previous experiments, in which all array transducers had the same nominal impedance. Therefore the main change made to the amplifier set was doubling the gains for those rings in which the transducer wiring caused the impedance to double.

Figure 4.20. Transducer wiring for teach bit of the bit-grouped DTA prototype


- 57 -

Figure 4.21. 6-bit prototype DTAs. Left: Line array one bit-per-transducer, 36 cm long; Middle: Circle array, one bit-per-transducer, 12 cm diameter; Right: bit-grouped concentric circle DTA, 26 cm outer ring diameter.

4.7 Summary

The main specifications of the measurement systems, DSP platform, amplifier set, transducers and transducer arrays used for DTA prototype implementation have been discussed. Three main transducer types were used, two of which presented similar electromechanical parameters although significantly different breakup peak characteristics which affected their transient responses. The other type was of smaller size and efficiency and presented a less damped step response. Based on these transducers, two main DTA topologies with different geometries were implemented: the one transducer-per-bit DTA and the bit-grouped DTA. In addition, the two full-range loudspeakers used for digital reconstruction have been described.

The effect on reconstruction of different transducer FRFs, step responses, transducer non-uniformities, mismatches and baffle will be seen in chapter 5.

Chapter 5 Experimental Results

- 58 -

5 Experimental Results

In this chapter, the effect of single and combined harmonic distortion sources in a direct acoustic DAC is quantified experimentally in terms of THD versus frequency. Results in the form of frequency response magnitudes, individual harmonic distortion components, near-field directivities and swept-sine responses are included as appropriate.

The main sources of harmonic distortion in a direct-acoustic converter arise from the transducer itself and from the array geometry in relation with the listener position. Harmonic distortion sources inherent to the transducer include transducer mismatches, transducer break-up and transducer FRF non-idealities. The most important THD factor due to the array is path-length difference induced distortion, though in practice this should be accounted for by assigning a different delay to each transducer, for each given listening position.

In section 5.1, each harmonic distortion source was isolated by selecting appropriate experimental conditions. By using one or several transducers at a series of fixed positions inside the anechoic chamber to record the bitstreams of a given signal, the effects of path-length difference distortion and transducer mismatches were artificially removed. Performing binary-weighting and adding bitstream responses in software removed any possible errors due to binary-weighting as well as any possible acoustic non-linearities due for instance to speed-of-sound fluctuations in the air. The obtained results under these “ideal” experimental conditions represent the best obtainable in a direct acoustic conversion with each kind of transducer at each particular sampling rate. Results may also be considered representative of DTA far field behaviour where path length differences are insignificant compared to the array-to-microphone distance. Results indicate that the particular transducer type, its bandwidth and transient response determine the amount of THD present in reconstructed signals.

In section 5.2, results are provided for the two implemented real-time DTA topologies: the one transducer-per-bit and the bit-grouped DTA. The obtained THD figures were generally higher than in section 5.1, suggesting that the effects of mismatches, path length differences, transducer break-up and FRF non-idealities now add up in the real-time prototypes. Factors intrinsic to the array such as array size, geometry and measurement distance were also seen to have an important effect on reconstruction. The effect of a baffle was analysed after the bit-grouped DTA topology. Possible gateways for harmonic cancellation and SNR enhancement are explored by means of a noise-shaping algorithm DSP implementation.

The DTA driving signals used for all multi-bit prototypes followed from [13,14] and were derived from PCM coded signals after converting serial digital words to parallel form. Parallel bitstreams corresponding to two cycles of a 1 kHz sinusoid input signal sampled at 96 kHz are shown in figure 5.1. Spectral analysis of these signals is shown in figure 5.2. These figures show that the harmonic content of the bitstreams is higher than that of their analogue counterpart. Harmonic reduction is therefore regarded as a fundamental problem to direct-converting multi-bit DACs, though the necessary transducer bandwidth limits and their relationship to the induced harmonic distortion remain the key issues to be explored.


- 59 -

0

2

4

Bit 1 (V

)

0

2

4B

it 2 (V)

0

2

4

Bit 3 (V

)

0

2

4

Bit 4 (V

)

0

2

4

Bit 5 (

V)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

2

4

Bit 6 (

V)

t (ms)

Figure 5.1. Parallel bitstreams corresponding to two cycles of a 1 kHz sine wave. Top: MSB; Bottom: LSB.

Figure 5.2. Spectral analysis for the bitstreams shown in figure 5.1. Top: MSB; Bottom: LSB.

5.1 Reconstruction with One Driver

In the following experiments, one transducer was set in an anechoic chamber and driven directly by a Rottel class AB voltage amplifier. The experimental set-up used is depicted in figure 5.3.


- 60 -

Postprocessing (binary weighting and adding) was then carried out in software. Software postprocessing also ensured that the error arising from binary-weighting and any non-linearities due to the acoustic medium were removed in all of the following experiments. The different harmonic distortion sources were experimentally isolated as follows:

The effect of sampling rate on reconstruction was revealed by comparing the THDs obtained with bitstream recordings of one transducer at a fixed position driven digital bitstreams generated at different sampling rates.

The effect of transducer break-up on reconstruction was revealed after comparing the THD obtained from bitstream recordings taken with the aluminium-dome transducer with the THD obtained from bitstream recordings taken with the diamond-dome transducer, both taken at an on-axis position.

The effect of path length differences was isolated by fixing the sampling rate and recording bitstream responses with the same transducer at different positions. At each transducer-microphone position, all six bitstream responses were recorded. Software postprocessing allowed reconstruction with any six different bit responses recorded at different positions.

Finally, the effect of transducer mismatches was isolated by recording bitstream responses with a set of 24 transducers nominally of the same type at a fixed on-axis position. With each transducer, all six bitstream responses were recorded. Software postprocessing then allowed reconstruction without path length differences and with any set of six transducers of the set.

Figure 5.3. Experimental set-up for one transducer bitstream response measurements

5.1.1 Individual Bitstream Responses

Six bitstreams obtained from a swept-sine extending from 200 Hz to 48 kHz were used as input signals for this experiment. The acoustic responses to the six equal-weight bitstreams were recorded one by one into six different wave files. These files were then post-processed in software, where binary-weighting and adding took place.


- 61 -

Results for individual transducer bitstream responses for a 1 kHz sine wave measured in an anechoic chamber at 1.5 m distance with the aluminium and diamond-dome tweeters are shown in figure 5.4 and figure 5.5.

Reconstruction with the aluminium-dome transducer led to individual transducer bitstream responses presenting a ripple at the break-up frequency of 30 kHz, which indicates that bitstream responses may in fact be thought of as consecutive step responses. This ripple was not present for reconstruction with the diamond-dome tweeter due to its smoother roll-off and less ringing step response (figure 4.7 and figure 4.10).

For a 1 kHz input sine wave at 96 kHz sampling rate, reconstructed signals resulted in a THD value of 15 % for the aluminium-dome transducer and a THD value of 5.8 % for the diamond-dome transducer. Therefore at a frequency of 1 kHz, THD was reduced by almost a factor of three by changing to the diamond-dome transducer, which presents a smoother break-up frequency and closely matched electromechanical parameters, from the aluminium-dome transducer. Thus with the only difference of the transducer type, a factor of three THD increase was obtained. Although the two transducers used presented slightly different electromechanical parameters, the main measured difference between the two transducer types was the shape of the frequency response magnitude, with a smooth roll-off in the case of the diamond-dome. This result constitutes evidence of the fact that a smooth transducer FRF roll-off improves reconstruction in direct-converting systems, although this is not an exclusive cause-effect phenomenon and the improved reconstruction may have possibly been attributed to a different cause such as efficiency or a given electromechanical parameter.

Figure 5.4. Aluminium-dome tweeter 1 kHz sine wave bitstream responses.


- 62 -

Figure 5.5. Diamond-dome tweeter 1 kHz sine wave bitstream responses.

5.1.2 Digital Reconstruction with Moving-Coil Tweeters

THD figures are now presented obtained for digital 6-bit reconstruction with each of the three tweeters described in section 4.4 used as transducers, on and off-axis and at different sampling rates. Figure 5.6 shows a THD vs. frequency plot for the aluminium-dome tweeter on-axis driven with swept-sine bitstreams sampled at 192 kHz and 96 kHz (in which cases the Nyquist frequency was beyond the transducer break-up frequency of 30 kHz) and at 48 kHz (whereby the Nyquist frequency was below the transducer break-up frequency). The input swept-sine extended from 200 Hz to 48 kHz and therefore aliasing from 24 kHz onwards was expected in the 48 kHz curve. Figure 5.7 and figure 5.8 show on-axis reconstruction THD versus frequency for the diamond-dome tweeter and the micro-tweeter, respectively.

The use of the aluminium-dome tweeter (with a prominent break-up peak near 30 kHz) employing a Nyquist frequency below transducer break-up resulted in improved reconstruction, as seen in the overall decreased THD curve for the 48 kHz case. On the other hand, by using the diamond-dome tweeter, which presented a smooth roll-off in its frequency response, the obtained reconstructed waveform THD was lowest with the highest sampling rate of 192 kHz. This result illustrates the benefit of having a smooth roll-off as the transducer upper frequency response magnitude trend.

With regards to off-axis responses, figure 5.9 shows THD versus frequency on each subplot where the four curves represent reconstruction THD at the on-axis, 15º, 30º and 45º off-axis positions. Data obtained for the 192 kHz, 96 kHz and 48 kHz sampling rates is shown in different columns and data obtained for the different transducer types is shown in rows. The diamond-dome tweeter driven by bitstreams at 192 KHz (upper left corner subplot) shows the lowest THD versus frequency at all positions over the 1 kHz to 20 kHz frequency band.


- 63 -

1e+002 1e+003 1e+0041

10

100

f (Hz)

THD (%

)

fs = 192 kHz

fs = 96 kHz

fs = 48 kHz

Figure 5.6. Reconstruction THD at different sampling rates with the aluminium-dome tweeter.

1e+002 1e+003 1e+0041

10

100

f (Hz)

THD (%

)

fs = 192 kHz

fs = 96 kHz

fs = 48 kHz

Figure 5.7. Reconstruction THD at different sampling rates with the diamond-dome tweeter.


- 64 -

1e+002 1e+003 1e+0041

10

100

f (Hz)

THD (%

)

fs = 192 kHz

fs = 96 kHz

fs = 48 kHz

Figure 5.8. Reconstruction THD at different sampling rates with the micro-tweeter.

Figure 5.9. THD versus frequencies at different off-axis positions obtained for the different transducer types and sampling rates. In columns, from left to right, decreasing sampling rates. In rows from top to bottom: diamond-

dome tweeter, aluminium-dome tweeter and micro tweeter.


- 65 -

5.1.3 Digital Reconstruction with Full-Range Systems

Digital reconstruction quality was calculated with different transducers, including two full-range loudspeaker systems. Full-range systems present the advantage of a wider bandwidth which extends through the whole audio range (20 Hz to 20 kHz). Reconstruction was tested with six bits and one transducer on-axis, by recording the bitstream responses from a logarithmic sweep and performing binary-weighting and adding in software.

Figure 5.10 shows reconstruction spectrograms obtained for the two full-range systems of the study compared to those obtained with the two tweeters used in section 5.1.2. Figure 5.11 shows measured FRF magnitudes, second and third harmonic distortion components of the previously mentioned systems.

Figure 5.10. Reconstruction spectrograms obtained for the two full-range systems (top, left and right) and the two tweeters (bottom, left and right).

The spectrograms show qualitatively that by increasing the transducer bandwidth, extra bitstream harmonics cancel out, SNR increases and THD decreases. The reconstructed signal bandwidth is increased. Quantitatively, after deconvolving the obtained sweep responses with the input swept-sine, mean THD values of 4.2% and 3.1 % were obtained for the two full-range systems (Figure 5.12), which compare to the minimum THD values of 15% and 5.8 % obtained for reconstruction with the two tweeters. Moreover, the lowest minimum THD was obtained with the system where the FRF magnitude presented a smooth roll-off (figure 5.11, top right), a result that adds to the argument presented in section 5.1.2 about the benefit of a smooth roll-off in the FRF after comparing reconstruction THD obtained with the aluminium and the diamond dome tweeters.


- 66 -

The minimum THD obtainable with a perfect 6-bit system is due to quantisation only and has a value of 2.7 %. Results indicate that a transducer system which ranges the whole audio band should be enough to bring THD down to a value of the order of that set by quantisation distortion only, which can then be improved with dither. On the other hand, if the transducer used for reconstruction lacks the low and mid frequency bands, minimum THD values may be doubled with respect to the previous case (as evidenced with the diamond-dome tweeter) or even be multiplied by a factor of five as it happened with the aluminium-dome tweeter.

Figure 5.11. Transducer FRF, second and third harmonic distortion components measured for the two full-range systems (top, left and right) and the two tweeters (bottom, left and right).


- 67 -

Figure 5.12. THD versus frequency plot obtained for 6-bit reconstruction with the two full-range systems

5.1.4 Transducer Interspacing

The effect on reconstruction of the transducer interspacings was isolated from all the other effects as follows. One transducer of the aluminium-dome tweeter type was set on-axis. The responses to a 6-bit swept-sine were measured anechoically at a distance of 1.5 m. All the transducer responses were measured again after having raised the transducer vertically by 3 cm. The whole process was repeated at 3 cm steps up to a vertical displacement of 48 cm above the microphone (figure 5.13).

The effect of different transducer interspacings was revealed after performing the postprocessing of recorded bitstreams in software, with the added advantage of being able to change the transducer-to-bit-assignment. For a fixed 531246 transducer-to-bit-assignment (where 1 represents the MSB and 6 the LSB), spectrograms for reconstructed swept-sines with 0 cm, 3 cm, 6 cm and 9 cm interspacings are shown in figure 5.14.

The effect of increasing path length differences as seen in reconstructed spectrograms can be summarised qualitatively as increased background noise floor together with increased odd order harmonic distortion, i.e. decreased SNR and increased THD. The THDs calculated from the spectrograms shown in figure 5.14 are shown in figure 5.15. Quantitatively, the mean THDs observed in the 1 kHz to 20 kHz band were 12.9 % for d = 0 cm, 36 % for d = 3 cm, 75 % for d = 6 cm and 95.6 % for d = 9 cm. In conclusion, THD was increased by at least 20 % after increasing the interspacing d by 3 cm, keeping the measurement distance fixed at 1.5 m.


- 68 -

Figure 5.13. Path length differences digital reconstruction experiment

t (s)

f (H

z)

d = 0 cm

0 0.5 1 1.5 2 2.5 3 3.5 4 4.50.0e+000

5.0e+003

1.0e+004

1.5e+004

2.0e+004

2.5e+004

3.0e+004

3.5e+004

4.0e+004

4.5e+004

t (s)

f (H

z)

d = 3 cm

0 0.5 1 1.5 2 2.5 3 3.5 4 4.50.0e+000

5.0e+003

1.0e+004

1.5e+004

2.0e+004

2.5e+004

3.0e+004

3.5e+004

4.0e+004

4.5e+004

t (s)

f (H

z)

d = 6 cm

0 0.5 1 1.5 2 2.5 3 3.5 4 4.50.0e+000

5.0e+003

1.0e+004

1.5e+004

2.0e+004

2.5e+004

3.0e+004

3.5e+004

4.0e+004

4.5e+004

t (s)

f (H

z)

d = 9 cm

0 0.5 1 1.5 2 2.5 3 3.5 4 4.50.0e+000

5.0e+003

1.0e+004

1.5e+004

2.0e+004

2.5e+004

3.0e+004

3.5e+004

4.0e+004

4.5e+004

p (dB)

-50

-40

-30

-20

-10

0

10

20

30

Figure 5.14. Isolated effect of transducer interspacings or array size on reconstruction


- 69 -

102

103

104

101

102

f (Hz)

THD (%

)

0 cm3 cm6 cm9 cm

Figure 5.15. Calculated THD for the different spectrograms in figure 5.14


In order to isolate the effect of transducer mismatches on reconstruction, a large set of transducers of the same type was employed. In particular, the availability permitted 24 transducers of the aluminium-dome type. Bitstream responses were recorded with each transducer at the same microphone position. The measured bitstream responses for any set of six different transducers were then binary-weighted and added in software. Experimental conditions led to information on the effect of transducer mismatches only without any other apparent distortion causes, such as path length difference distortion. In addition, all impedances, FRFs and electromechanical parameters of the 24 transducer set were measured with the Klippel Analyser System in order to quantify the extent of the mismatches.

Reproducibility was tested for each electromechanical parameter by analysing the results distribution obtained after repeating the measurement 20 times with the same transducer and exactly the same measurement setting. Reproducibility errors ranging from 0.13 % to 18.8 % across the different electromechanical parameters were found, with an average value of 6.15 %. Table 5.1 shows the mismatches found in each electromechanical parameter within the 24 transducer set together with the experimentally determined reproducibility error.

With regards to measured impedance curves for the transducer set (figure 5.18), impedance peak values ranged from 4.84 Ω to 7.46 Ω with a mean of 6.49 Ω and a standard deviation of 0.66 Ω. Resonance frequencies ranged from 763.6 Hz to 1008.4 Hz with a mean of 838.6 Hz and a standard deviation of 41.8 Hz.

The variation found between transducers was rather significant although it is thought to be typical of current moving-coil transducer sets. Differences are due to the tolerance accepted in material properties during the fabrication process: the smaller the required tolerance the higher the overall material and manufacturing costs.

A different reason why significant differences were found between transducers is that the available Klippel system was not originally intended to perform small tweeter measurements; this is why


- 70 -

reproducibility had to be tested for each electromechanical parameter and a higher experimental error was found.

Parameter Mismatch (%) Reproducibility Error (%) Parameter Meaning

Re 19.8 0.4 electrical voice coil resistance at DC

Le 69.6 0.0 frequency independent part of voice coil inductance

L2 58.9 0.0 para-inductance of voice coil

R2 31.2 2.4 electrical resistance due to eddy current losses

Cmes 123.2 1.0 electrical capacitance representing moving mass

Lces 78.0 0.0 electrical inductance representing driver compliance

Res 92.4 1.5 resistance due to mechanical losses

fs 14.4 0.1 driver resonance frequency

Mms 91.1 8.2 mechanical mass of driver diaphragm assembly including air load and voice coil

Mmd (Sd) 91.3 8.3 mechanical mass of voice coil and diaphragm without air load

Rms 94.5 8.8 mechanical resistance of total-driver losses

Cms 88.9 18.8 mechanical compliance of driver suspension

Kms 92.9 8.2 mechanical stiffness of driver suspension

Bl 58.4 4.2 force factor (Bl product)

Table 5.1. Extent of mismatches for the different electromechanical parameters

The effect of a change in just a single given electromechanical parameter on reconstruction cannot be easily isolated as the electromechanical parameters are interrelated and changing one of them usually implies changing some others. However, useful statistics were drawn after calculating a large set of reconstruction THDs considering random transducers and therefore random mismatches (figure 5.16 and figure 5.17). The minimum in-band averaged THD value at a sampling frequency of 96 kHz was calculated to be 16.3 %, whereas the maximum in-band averaged THD found was 111 %. The maximum in-band averaged SNR value was obtained to be 28.2 dB and the minimum in-band averaged SNR found was 12.4 dB. The maximum THD differences found produced by any transducer mismatches occurred at frequencies higher than 4 kHz and were in excess of 100 %. In other words, for this particular type of transducer and sampling rate, there is less chance of obtaining a THD difference occurring from mismatches in the 1 kHz to 4 kHz band. A narrow band in the transducer mass-controlled region seems the least sensitive to transducer mismatches.


- 71 -

Figure 5.16. Reconstruction THD vs. frequency obtained for 70 runs with random transducer mismatches

Figure 5.17. Reconstruction SNR vs. frequency obtained for the same 70 runs as above with random transducer mismatches. X-axis scale is logarithmic.


- 72 -

An experiment was devised to isolate the effect of transducer sensitivity on reconstruction by connecting a resistor in series with the transducer. The impedance curve of any given bitstream was raised and therefore the sensitivity was lowered. Postprocessing then allowed reconstruction with any mismatched bitstream. A 0.1 Ω resistor in series with the MSB led to a sensitivity decrease of 0.25 dB at 1.5 m and a 12.5 % change in average THD over the 1 kHz to 20 kHz band. A sensitivity decrease of 1 dB in the MSB obtained with a 0.4 Ω series resistor produced a 23 % change in average THD over the 1 kHz to 20 kHz band. This illustrates the importance of equal transducer sensitivity across the elements of the array, as well as that of accurate binary-weighting.

2

3

4

5

6

7

8

100 1000 10000

f (Hz)

Mag

nitu

de (O

hm)

Figure 5.18. Measured impedance curves for the 24 transducer set

5.1.6 DTA Far-Field Directivity

In order to obtain experimental information on directivities when no mismatches are present, bitstream responses were measured separately for each bit at a series of angles ranging 180º in 5º steps, with a sampling rate of 96 kHz and just one aluminium-dome transducer. Bitstream responses were individually recorded and stored for postprocessing. At each transducer-microphone position, six bitstream responses were binary weighted and added in software, removing any possible errors arising from amplifier gains or out-of-phase acoustic addition by acoustic path length differences. Smoothing with a moving-average filter scheme was applied to the combined frequency response. This led to directivity plots at any desired frequency bin. Directivity plots at 1 kHz, 2 kHz, 4 kHz, 8 kHz and 16 kHz are shown in figure 5.19. Polar response curves present a similar aspect to the tweeter polar response, though the maximum number of bits in the case of the point DTA was 6-bits. This evidence suggests that in the far-field limit, where the array size is seen as that of one transducer, and with no transducer mismatching, the DTA polar response tends to the constituent transducer polar response.


- 73 -

-10 0

30

60

90-90

-60

-300


Figure 5.19. One transducer DTA post-processed directivities at a sampling frequency of 96 kHz and one aluminium-dome tweeter.

5.1.7 Multiple Frequency Excitation

A direct acoustic DAC will ultimately be employed to listen to music, voice or any kind of broadband signal. The swept-sine measurement technique [96-98] has proved to be accurate and yield enhanced SNR for the determination of impulse and FRFs, individual harmonic distortion components and total harmonic distortion. However this technique only uses single-frequency excitation at each instant and should be complemented with broadband excitation measurements to account for effects such as intermodulation distortion.

5.1.7.1 White Noise

The following results were obtained after exciting one transducer with the 6-bit parallel PCM bitstreams of a white noise sequence low-pass filtered to 10 kHz, recording bitstream responses and software postprocessing. The spectrum of the output sequence is shown in figure 5.20. To see how this output was correlated with the input, the reconstructed signal was cross-correlated with the input (figure 5.21). Results show cross-correlation only at zero lag values as expected, with a noise-floor due to the limited number of bits used in describing the input signal amplitude, as well as the transducer low-pass filtering action.

5.1.7.2 Two-Tone

A two-tone signal was designed with non-harmonically related tones (f1 = 1 kHz and f2 = 2.333 kHz) to test for intermodulation distortion. Its six bitstreams were computed and used to excite one transducer of the aluminium-dome type on-axis. Bitstream responses were recorded and postprocessed in software.

Spectral analysis of the resulting output sequence is shown in figure 5.22, together with a table showing the peak values and their meaning (table 5.4). The two-tone reconstructed signal showed equally spaced peaks extending up till the Nyquist limit. Peaks were spaced by an average frequency of 333 Hz, which is a linear combination of the two fundamental frequencies (f2 – 2f1). The most important peaks after the two fundamentals seen in figure 5.22, higher than any other harmonic peaks, occurred at frequencies of 2f2 + f1 and 2f2 - f1, which imply third order intermodulation distortion products.


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Figure 5.20. Reconstructed output sequence (6 bits) with a white noise input low-pass filtered to 10 kHz.

Figure 5.21. Input/output cross-correlation for low-pass filtered input white noise sequence.


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Figure 5.22. Intermodulation distortion products of a 6-bit direct acoustic DAC with 1 kHz and 2.333 kHz input signal frequencies.

Experimental f (Hz) Peak value (dB) Linear

combination Meaning

2339 146.1 f2 fundamental

1002 144.4 f1 fundamental

5664 113.7 2f2 + f1 third order intermodulation

3664 112.8 2f2 - f1 third order intermodulation

5995 112.0 6f1 harmonics of f1

3995 111.9 4f1 harmonics of f1

3333 111.4 f1 + f2 second order

intermodulation 7002 111.3 3f2 harmonics of f2

7664 111.3 2f2 + 3f1 higher order


6339 110.8 f2 + 4f1 higher order


5333 109.6 f2 + 3f1 higher order


6670 107.9 2f2 + 2f1 higher order


7333 106.8 4f2 - 2f1 higher order

intermodulation

Table 5.2. Peak values and meaning from output signal with two-tone input.


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5.2 Reconstruction with Real-Time Prototypes

In the previous experiments, the effects in isolation of several experimental factors (such as transducer mismatches, path length differences, and transducer break-up and its relationship with sampling rate) on reconstruction were seen, mainly through THD-versus-frequency figures. In real-time DTA systems some or all of these factors may combine, degrading the quality of the reconstructed response.

In this section, experimental results obtained with simultaneous distortion sources are provided. In order to obtain these conditions, both one bit-per-transducer DTAs and bit-grouped DTAs were implemented. A DSP platform was used to generate the bitstreams corresponding to any input waveform automatically, therefore implementing the real-time experimental system with which DTA prototypes were investigated. The experimental setup is depicted in figure 5.23.

Figure 5.23. Experimental set-up for real time DTA prototypes.

In this set-up, the control PC was used for communication with the TMS320C6713DSK DSP evaluation board and algorithm downloading. A basic DSP application constantly read audio samples from the DSP codec analogue line input, put them in a buffer, quantised to 16-bits, and continuously wrote them to the digital parallel output. A more elaborate DSP application was subsequently written by adding triangular probability density function (PDF) dither before integer quantisation, thus reducing quantisation harmonics at the expense of a slightly increased noise floor. Following bitstream generation, power amplification was obtained through the set of class AB voltage power amplifiers described in the previous chapter. From the DSP input onwards, no standard ADC or DAC was present throughout the system, as in a direct acoustic DAC.

5.2.1 Electronic Binary-Weighting: the One Bit-Per-Transducer DTA

Implementing binary-weighted gains in the amplifier set and assigning one transducer to each bit of the signal led to this implementation approach. The line and circular array geometries were


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considered, implemented with both the B&W aluminium and the B&W diamond tweeters. Results are now presented showing FRFs, THD, near-field directivities and swept-sine responses for the different geometries and transducer types.

5.2.1.1 Frequency Responses and Total Harmonic Distortion

Results for near-field FRF and THD line and circular array geometries are shown in figure 5.24 and figure 5.25, respectively. Experimental array near-field FRFs for both geometries obtained at a 1.5 m distance from the array presented a similar aspect to the transducer frequency response. Third harmonic distortion component was found higher in the line array geometry because of the different acoustic path lengths. Second harmonic distortion component on the other hand was found similar regardless of geometry.

In the line array geometry case, the far field lies at a further distance in comparison to the circular array geometry on-axis, were path lengths are identical to each other to within the experimental error. In the 1 kHz to 3 kHz band (both within the mass-controlled region and before the spatial aliasing limit) an average difference between fundamental and third harmonic of 10 dB was found. The effect of different path lengths arising from array geometry, together with transducer mismatching, binary-weighting error and transducer non-linearities are believed to be the reasons which made the obtained SNR be less than the 36.7 dB value expected theoretically for a 6-bit quantisation.

The experimentally determined on-axis total harmonic distortion (figure 5.25) decreased as the transducer reached its mass-controlled region. The THD was found to reach a minimum between 1 kHz and 3 kHz, increasing thereafter for the line array geometry, as the spatial aliasing limit set by transducer interspacing was reached near 3 kHz. For the circular array geometry, on-axis THD decreased slightly with increasing frequency from 3 kHz to the Nyquist limit, owing to the increased harmonic cancellation as a consequence of nominally equal path-lengths.

5.2.1.2 Directivities

Figure 5.27, figure 5.28 and figure 5.29 show directivity polar plots obtained by measuring at 1.5 m from the line DTA. Polar plots at 1 kHz, 2 kHz, 4 kHz, 8 kHz and 16 kHz are shown in each figure, for which different transducer-to-bit assignments were used. By changing the transducer-to-bit assignment different radiation patterns were obtained. A more evenly spread radiated power across the array elements was sought with the different arrangements in order to minimise spurious directivity lobes. The obtained directivity presented irregularities due to the small measurement distance (1.5 m) compared to the length of the array (36 cm). At this distance there was not enough space for extra bit harmonics to cancel out. In other words, the three different path lengths present in the array produced phase differences in the radiated sound field which prevented high-frequency bitstream harmonics from cancelling. The sensitivity of an exponential to a change in its exponent is large and therefore the resultant sound field contains phase distortion which results in polar response irregularities.


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Figure 5.24. Near-field on-axis frequency responses, second and third harmonic distortion components for the line and circular 1 transducer-bit DTAs.

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Figure 5.25. THD comparison for the line and circular DTAs, aluminium-dome transducer in both cases.


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Figure 5.26. THD comparison for the aluminium and diamond-dome transducers, circular DTA geometry in both cases.

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Figure 5.27. One bit-per-transducer DTA directivities with a 123456-bit-transducer assignment (1 - MSB, numbering from top to bottom) directivity polars

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Figure 5.28. One bit-per-transducer DTA directivities with a 325164 bit-transducer assignment (1 - MSB, numbering from top to bottom) directivity polars


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Figure 5.29. One bit-per-transducer DTA directivities with a 531246-bit-transducer assignment (1 - MSB, numbering from top to bottom) directivity polars

5.2.1.3 Swept-sine Responses

A useful measurement used in audio to compute system FRFs is the swept-sine response. This measurement allows qualitative and quantitative analysis of individual harmonic distortion components. Qualitative analysis may be inferred from the extra harmonic lines present in the spectrogram of the recorded swept-sine response, whereas quantitative analysis is computed via deconvolution with the original swept-sine [97].

Measuring the DTA response to a logarithmic sweep from 200 Hz to 20 kHz with the line array DTA, the results shown in figure 5.30 (line DTA) and figure 5.31 (circular DTA) were obtained. A maximum DSP sampling rate of 96 kHz was used. Even if the sampling rate of the sound card was greater (192 kHz) than the DSP sampling rate, the minimum sampling rate of any of the elements in the system (96 kHz) determined the maximum bandwidth. It is convenient to ensure that the Nyquist limit is below transducer break-up in order to minimise non-linear radiation of frequency components.

Odd order harmonic distortion was present and is seen in figure 5.30 and figure 5.31 as the extra sweep lines giving the 3rd, 5th, 7th, 9th etc. harmonic lines. Higher order harmonic distortions are seen in the spectrogram for the line DTA (figure 5.30). The horizontal fringe observed in the spectrograms near 30 kHz is caused by transducer break-up. By changing the transducer type to the diamond-dome one, the break-up fringe disappeared in the spectrogram due to its smoother roll-off.


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t (s)

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Figure 5.30. Spectrogram of a logarithmic sweep (200 Hz to 20 kHz) response obtained at 1.5 m on-axis with the line one bit-per-transducer DTA (aluminium-dome tweeters).

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Figure 5.31. Spectrogram of a logarithmic sweep (200 Hz to 20 kHz) response obtained at 1.5 m on-axis with the circular one bit-per-transducer DTA (aluminium-dome tweeters).


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5.2.2 Acoustic Binary-Weighting: the Bit-Grouped DTA

Perhaps a closer concept to all-digital transduction, resembling the standard operation of a multi-bit DAC, is that implemented by the bit-grouped DTA. In this case, binary-weighting takes place in the acoustic domain via binary-weighted transducer numbers.

In this experiment the DSP platform was used to generate bitstreams for any input waveform in real-time. The experimental set-up is that given in figure 5.23 although with the different array and modified amplifier gains as described in the previous chapter. The transducers used for this array were the Harman Odyssey 2 due to their reduced size.

Available experimental conditions allowed measurement of the DTA anechoically at a maximum distance of 1.5 m, due to the anechoic chamber size. The anechoic chamber cut-off frequency was found lower than 100 Hz although measurements were not made in any case at frequencies lower than 200 Hz, due to the transfer characteristics of the transducers used. However, the limited chamber size constituted the experimental limit for mapping the DTA prototype sound field. A listening room allowing greater measurement distances was used to complement the results, corroborate harmonic cancellation and measure live THDs and transfer functions. The room properties are described later in this section.

The obtained anechoic DTA FRF, second and third harmonic distortion components at a 1.5 m microphone distance are shown in figure 5.32. Second and third harmonic distortion components were high and even reached the fundamental at certain frequencies. A detailed analysis of the bitstreams coming out of the DSP for a sine wave input, as well as amplifier input and outputs, array ring impedances and output acoustic signals led to the conclusion that the only element that could be compromising the measured result of the DTA was in the acoustic domain. Simulations with different microphone distances showed that by increasing the microphone distance the sweet-spot widened and THD reduced.

The next logical step was to measure the response at a further distance. The DTA was set in an 8 x 5 x 3 m (length x width x height) acoustically-treated listening room whose reverberation times were measured with the WinMLS system according to the ISO 3382 standard. A 10 s length sweep was used for the room acoustics measurement, together with a B&W 800D loudspeaker. Measured reverberation time parameters are displayed in table 5.3.

The sound field in the room was intended to be mainly direct although with the right proportion of reverberant sound field added in order to aim for clarity and transparency of sound during loudspeaker subjective tests.

Frequency (Hz) 63 125 250 500 1000 2000 4000 8000 16000

T20 (s) 0,22 0,30 0,20 0,23 0,27 0,29 0,29 0,26 0,20

T30 (s) 0,23 0,40 0,28 0,25 0,28 0,27 0,29 0,26 0,20

Table 5.3. Measured listening room reverberation time parameters


- 83 -

The obtained FRF, second and third harmonic distortion components measured in the room with the microphone set at a 5.5 m distance are shown in figure 5.32. The main effect attributable to the room is a high frequency ripple in the frequency response together with a high frequency roll-off of approximately 6 dB/oct equivalent to time integration. Although third harmonic distortion component remained at the same relative level from the fundamental, it can be seen that second harmonic distortion component decayed to around 40 dB from the fundamental.

5.2.2.1 Frequency Responses and Total Harmonic Distortion

Results for on-axis THD measured at 1.5 m and 5.5 m are shown in figure 5.33. The second harmonic fall seen previously from near to far measurements had a big impact on THD figures, and a 10% THD decrease was appreciated between near and far measurements.


- 84 -

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Figure 5.32. Experimental live 6-bit DTA FRF magnitude, second and third harmonic distortion components as a function of frequency. Each figure shows a measurement taken at a relevant microphone position specified in

polar coordinates by r and θ


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Figure 5.33. Experimental live 6-bit DTA THDs as a function of frequency at different positions. Top left: on-axis THDs at 1 m and 5.5 m. Top right: 15 º off-axis THDs at 1 m and 4 m. Bottom left: 30º off-axis THDs at 1 m and 3

m. Bottom right: 45º off-axis THDs at 1 m and 3 m

5.2.2.2 Directivities

Directivity polar plots measured anechoically at 1.5 m are shown in figure 5.34. The shape of these polar plots was even more irregular than in the line DTA case. This is due to increased path length differences at the 1.5 m measurement distance and to decreased transducer transient response performance.


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Nearfield bit-grouped DTA


Figure 5.34. Bit-grouped DTA anechoic directivities

5.2.2.3 Swept-Sine Responses

By using a swept-sine ranging logarithmically from 200 Hz to 20 kHz during 10 s, the results shown in figure 5.35 and figure 5.36 were obtained. The recorded swept-sine responses were previously normalised in order to obtain meaningful and comparable results. Results show significant harmonic cancellation in the band from 5 kHz to the Nyquist limit at all times. Second and third harmonic strengths are also lower by an average 10 dB through the whole sweep response.

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Figure 5.35. Bit-grouped DTA near-field spectrogram


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Figure 5.36. Bit-grouped DTA far-field spectrogram

5.2.2.4 Effect of the Baffle

The responses of one transducer on the baffle used for implementing the bit-grouped DTA system were measured individually and anechoically at different positions on the baffle and reported in [99]. It was seen that measured on-baffle transducer responses depend on the position of the transducer within the baffle. No diffraction step was appreciated since for this baffle size the diffraction step centre frequency lies below the transducer mass-controlled region. However, several dips and peaks were observed at different positions depending on where the transducer was placed.

Anechoic FRF measurements were taken with one driver set at 11 positions within the baffle, starting from the baffle centre and moving away towards the edges, first horizontally and then vertically. As the transducer position within the baffle was increased either horizontally or vertically, the frequency locations of the dips were also increased with respect to the measurement on the baffle centre, although the mean spectral distance between consecutive peaks and dips remained unchanged. Nevertheless no significant difference was found between the measurements where the transducer was placed horizontally or vertically on the baffle: the variation of baffle effect on transducer FRF was found to a good approximation equal as the transducer was placed radially outwards in any angular direction. Figure 5.37 shows the FRF magnitudes of three on-baffle measurements, one on the baffle centre and the other two at equidistant radially outward positions, superimposed to a free-field transducer FRF magnitude. The frequencies of dips and peaks for the on-axis transducer location were recorded and are shown in Table 5.4. By averaging the spectral differences between consecutive peaks and dips, a frequency of 1314 Hz (λ = 26.2 cm) was obtained, which corresponds to the baffle diameter of 26 cm. Together with a comparison to the free-field measurement this constitutes significant evidence of the fact that the peaks and dips at the frequencies shown in figure 5.37 were due to baffle diffraction.


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Leve

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CentreMiddleEndFree field

Figure 5.37. Experimental FRF magnitudes for one driver on the baffle centre and at two other outward positions compared to a measurement on free-field

Baffle peak / dip frequencies (Hz)

3856 10167 17340 26865

5175 11659 19004 27955

5806 13151 20955 28873

6610 14413 22848 31455

8044 15159 24168 8848 16249 25487

Table 5.4. Frequencies of peaks and dips corresponding to the on-axis transducer on-baffle measurement

5.3 SNR Enhancement with Noise-Shaping

A noise-shaping algorithm was implemented on the DSP platform to be used with different noise-shaping filters of different orders and shapes in conjunction with the DTA prototypes developed, in order to report on possible benefits brought to direct-acoustic converting systems by noise-shaping technology. The noise shapers tried derived from those given in [55-57,64], with modified coefficients due to oversampling up to 96 kHz, which was the maximum sampling rate allowed by the DSP platform. Detailed experimental results were presented in [67].

The effect of a 9th order noise shaper on the on-axis response of a one transducer-per-bit 6-bit line DTA was observed to be a general reduction in even order low frequency harmonics and in both even and odd order high frequency harmonics (figure 5.38 and figure 5.39). The on-axis DTA response


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tends to the noise shaped input and compensation for other effects such as baffle, transducer non-idealities or mismatches can be included in the noise-shaping filter response, in a similar manner that psychoacoustic shaping was included in non-oversampling converters [64], reducing the problem to designing a digital filter. Off-axis live DTA responses were seen in section 5.2.2.1. Further simulated results arising from noise shaping as applied to the DTA are provided in section 7.9.

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Figure 5.38. One transducer-per-bit line DTA without noise-shaping on-axis swept sine response spectrogram

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Figure 5.39. One transducer-per-bit line DTA with noise-shaping on-axis swept sine response spectrogram


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5.4 Summary

The main experimental sources of harmonic distortion arising in direct-converting systems have been isolated, quantified and discussed. These were, from highest to lowest importance: path length differences caused by transducer interspacing, transducer mismatches, and transducer break-up in relation to sampling rate. Experimental results for two types of real-time DTA implementations with moving-coil transducers (the one-transducer-per-bit and the bit-grouped DTA) comprising FRFs, second and third harmonic distortion components, THDs, swept-sine spectrograms and polar responses were then presented, showing high THD levels as a result of the presence of combined harmonic distortion sources.

Individual bitstream outputs reflected the ringing nature of the transducer step response and improved with a stiffer diaphragm material, as reflected in the results obtained with the diamond-dome tweeter. By not considering the effects of transducer mismatches or path length differences, it was shown that THD was decreased in the audio frequency band due to non radiation of the highest break-up frequency components by using a lower Nyquist frequency than the transducer break-up frequency. The effect of path length differences only was isolated and observed as increased total harmonic distortion.

It was seen that reductions in THD towards values of the order of the THD determined by quantisation are possible in DTA prototypes by:

Having a smooth transducer roll-off and a sampling rate with a Nyquist frequency at least of 192 kHz (sections 5.1.1, 5.1.2 and 5.1.3).

Increasing the bandwidth of each transducer system to include all audio frequencies (from 20 Hz to 20 kHz) (section 5.1.3).

Having no path length differences (section 5.1.4), a fact that is assumed to have a possible practical solution by assigning digital delays to each transducer, although this was not implemented throughout the thesis.

Reducing mismatches, especially in the transducers associated to the most significant bits (section 5.1.5).

Polar results obtained after digital reconstruction with one transducer suggested that without the effect of transducer mismatches, the DTA polar response in the far-field tends to that of the transducer polar response (section 5.1.6).

Reconstruction with random noise low-passed to 10 kHz yielded good correlation with the input with six bits, 96 kHz and the aluminium-dome tweeter (section 5.1.7.1). Third order intermodulation distortion was seen to be of the order of the resulting third harmonic distortion as reconstruction was limited to six bits and the transducer used was a 1 kHz to 30 kHz tweeter (section 5.1.7.2).

The one transducer-per-bit topology is more compact in terms of transducer numbers than the bit-grouped DTA topology, and this was reflected in lower THD figures (section 5.2.1). The obtained directivity plots for measurement distances comparable to the array size were highly irregular, although remained constant over a wider angle as the listening distance was increased, indicating that the sweet-spot widens for increased listening distances. Swept-sine response spectrograms obtained on-axis showed high values of THD which resulted from bitstream binary-weighting amplitude departures from nominal values. THD figures were higher in the line DTA geometry due to different


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path lengths and transducer mismatches. A lower SNR than that theoretically expected for 6-bits was observed.

In the case of the bit-grouped DTA, higher path length differences compromised harmonic cancellation and this fact was reflected in high second harmonic distortion component which decreased as the listening distance was increased (section 5.2.2). The sensitivity of the transducers used to implement the bit-grouped DTA prototype was lower than that of the one bit-per-transducer prototype transducers. The measured anechoic directivity polars at a 1.5 m distance as in the previous case of one bit-per-transducer topologies were found more irregular for the bit-grouped topology, providing evidence of the fact that the sweet-spot lies at a higher listening distance.

Finally, the introduction of a dithered noise shaper in the DSP stage was seen to result in reduced even order harmonic distortion up to 10 kHz and reduced even and odd order harmonic distortion from 10 kHz to 20 kHz. In a more general case, the noise shaper order and frequency response will determine the band and the order of the harmonics cancelled.

Chapter 6 Experimental Results Discussion

- 92 -

6 Experimental Results Discussion

This chapter discusses and analyses in-depth issues arising from experimental results and looks for the main causes of THD generation. To conclude, the main sources of experimental errors are discussed.

Whether by firing one transducer with bitstreams, recording acoustic responses and postprocessing; by firing all transducers simultaneously previous electronic binary-weighting, or by firing all transducers simultaneously implementing acoustic binary-weighting, experimental results showed higher levels of THD and lower SNR from the theoretical level expected for the number of bits employed, when the transducer used was a moving-coil tweeter. However, reconstruction with full-range systems showed that THD values could be of the order of that expected from quantisation only as long as path length differences were compensated for and mismatches were controlled.

Reconstruction quality was affected by different transducer properties measured in transducer FRFs, transient responses and mismatches. Array FRFs presented a similar aspect to those of the constituent transducers, with increased directionality in the near-field due to the array arrangement.

6.1 Reconstruction with One Driver

In this experiment, the conditions were altered to artificially remove the experimental effects of acoustic path-length differences and transducer mismatches. Different driving signals, sampling rates and transducer types were tried. Reconstruction was measured on-axis and at different off-axis positions. THD figures were derived from the obtained reconstruction spectrograms. Results for different sampling rates and transducer characteristics led to the assessment of reconstruction quality evaluated through mean THD figures and SNR values.

Out of the three tweeters tried, the least efficient and least sensitive one, with the weakest motor strength, led to the highest and most variable THD with poorest SNR at all sampling rates and all off-axis angles. The mean THD values over the transducer operating frequency band showed inverse correlation with transducer sensitivity and efficiency. By increasing the sampling rate to a Nyquist frequency higher than the transducer bandwidth, the THD was reduced only when the roll-off decreased smoothly and no significant break-up peak was present.

It was interesting to study off-axis DTA responses in order to quantify sweet-spot area or volume at any given listening distance and thus obtain any possible correlation or causality with sampling rates or transducer characteristics. For this purpose, similar swept-sine bitstream response measurements were taken at different off-axis positions, namely at 15º, 30º and 45º. Results shown in table 6.1 apply to the on-axis DTA response as well as the 15º, 30º and 45º off-axis responses.

On-axis, at a sampling rate of 192 kHz, in the 1 kHz to 20 kHz band, the mean reconstruction THD for the different transducer types resulted in 13.2 % for the aluminium-dome tweeter, 4.0 % for the diamond tweeter and 72.8 % for the micro tweeter (table 6.1).


- 93 -

fs = 48 kHz fs = 96 kHz fs = 192 kHz

B&W Dia 9.20 9.10 4.00

B&W Ali 10.4 12.5 13.2 θ

= 0º

Harman 54.2 49.7 72.8

B&W Dia 18.5 19.3 5.50

B&W Ali 8.20 14.7 15.7

θ =

15º

Harman 100 61.7 67.0

B&W Dia 7.80 11.6 6.10

B&W Ali 9.90 10.0 10.5

θ =

30º

Harman 65.5 52.1 42.3

B&W Dia 12.0 10.6 8.60

B&W Ali 10.0 9.90 6.50

θ =

45º

Harman 69.2 43.9 71.4

Table 6.1. Mean reconstruction THD (%) in the 1 kHz to 20 kHz band at different microphone positions for different sampling rates and transducer types

On-axis, by increasing the sampling rate from 48 kHz to 192 kHz whilst keeping the same transducer type, the mean THD was increased by 27 % for the aluminium-dome tweeter and decreased by 57 % for the diamond-dome tweeter. The main difference between the FRF magnitudes in these two transducer types was a prominent break-up frequency peak in the case of the aluminium-dome tweeter, not present in the diamond-dome transducer. Bearing in mind that the 48 kHz sampling rate is below the break-up frequency in both cases, the evidence suggests that reconstruction can be improved at sampling rates higher than twice the transducer bandwidth and also with smooth transducer roll-offs without breakup peak. The transducer acts as a desampling filter and reproduction of bitstream ultrasonic frequencies is not necessary as long as the whole audio band is covered by the transducer or transducer system.

At any sampling rate and an on-axis position the average THD was highest for the micro tweeter with a minimum mean value of 49.7 % at 96 kHz sampling rate, followed by the aluminium-dome tweeter (minimum mean value of 10.4 % at 48 kHz) and closely followed by the diamond-dome tweeter with a minimum mean value of 4.0 % at 192 kHz.

At any listening position, the THD was found to increase with increasing frequency, in occasions presenting peaks with sudden increases higher than 100 % which increased its mean value. The THD of the reconstructed signal increased generally as the off-axis angle was increased, consistent with the results numerically predicted in [13]. The THD also increased for reconstruction with the transducer type which presented lowest efficiency. Efficiency and sensitivity can therefore be thought of as the most significant parameters which correlate inversely with reconstruction THD.

The overall lowest THD versus frequency at all angles was obtained with the diamond-dome tweeter at 192 kHz. Analysis of the off-axis reconstruction THD revealed overall higher values than on-axis reconstruction THD. These values were of the same order with aluminium and diamond-dome transducers and approximately a factor of five higher with the micro transducer.

From the experimental data it can be seen that the different transducer types have a crucial impact on the reconstruction quality measured by THD and SNR figures. In all cases when a tweeter was used, the reconstruction THD was significantly higher than the 2.7 % value set by quantisation. By increasing the transducer bandwidth, extra harmonic distortion was decreased. In particular, the results


- 94 -

obtained with a full-range audio system showed that the on-axis THD values obtained reached those set by quantisation only as long as the path length differences and transducer mismatches were accounted for.

6.2 Electronic Binary Weighting: One Transducer-Per-Bit DTA

In this experiment, the main difference in experimental conditions from those present in the case of reconstruction with one driver were the following:

Bitstreams, generated in the DSP, were read from the GPIO port, which introduced high frequency noise in the driving signal, increasing the noise floor and subsequently decreasing SNR in the reconstructed signal.

Binary-weighting was subject to an error of around 2 % arising from the amplifier gains, which was partly responsible for the extra harmonic lines seen in spectrograms. If the harmonics corresponding to a given bitstream did not cancel out due to different amplitude from its nominal value, these particular bitstream harmonics were present in the reconstruction spectrogram.

The DTA presented in this case transducer mismatches up to 5 % and a maximum path length difference of 1.6 % at 1 m for the line array geometry.

Considering the THD measured at a distance of 1 m on-axis with the line and the circular geometries (aluminium-dome transducer), the mean values were increased from 14 % to 26 % over the 1 kHz to 20 kHz band with a sampling rate of 96 kHz. In other words, the change in DTA geometry measuring at 1 m on-axis resulted in an 86 % mean THD increase. At a 1 m distance with a 36 cm long line array, the maximum path length is 1.6 % higher than the minimum path length. Therefore the effect of a 1.6 % maximum path length resulted in an 86 % mean THD increase over the 1 kHz to 20 kHz band.

By comparing the THDs obtained at 1 m on-axis for the circular geometries with the aluminium and diamond-dome transducers, the mean THD value decreased from 14 % to 11.7 %. Therefore a 16.4 % THD decrease was obtained by changing the transducer type, at 96 kHz sampling rate.

The DTA FRF magnitudes presented a similar shape to that of the constituent transducer FRF magnitude, with break-up peak values decreasing as the off-axis angle was increased. This is similar to what happens in analogue arrays, which present higher directivities than its constituent transducers. This fact has traditionally been exploited to produce directional sound beams in highly reverberant spaces.

The directivities were measured at a distance of 1.5 m, which for the line array implies a maximum path length difference of 1.1 %. Their polar shape presented more prominent irregularities than the correspondent analogue directivity. At this distance, the signal amplitudes set by binary-weighting and each particular transducer response as well as transducer-to-bit assignment play a major role in the DTA polar response. In the case of the circular array, directivities were also conditioned by the transducer-to-bit assignment and the measurement distance.

Swept-sine responses showed significant spectral harmonic cancellation by comparing results obtained with the line array to the circular array.


- 95 -

6.3 Acoustic Binary-Weighting: the Bit-Grouped DTA

With the adopted implementation approach, the main differences in experimental conditions from those set in the previous case now were:

The transducer used in the array was the micro tweeter. As it was seen previously (section 4.4) from the transducer electromechanical parameters and step responses, the efficiency and sensitivity of this transducer was lower than that of the B&W tweeters due to a weaker magnet and a wider magnet gap. This fact was seen to result in compromised reconstruction quality through increased THD figures. In particular it was seen that reconstruction THDs with one transducer increased from typical values of 10 % to typical values of 50 %. However, in a very narrow frequency band ranging from 3 kHz to 5 kHz obtained THDs were of the order of 10 %.

The presence of a baffle in this array influenced obtained array FRFs. In the previous cases of the one transducer-per-bit arrays, transducers were clamped to a metal rod (line array) and onto a PVC ring (circular array) whilst kept in individual back radiation isolating structures specifically designed for these transducer types. However, in the case of bit-grouped DTAs transducers had to be held together in a rigid structure and it was thought that the best and simplest would be an MDF baffle.

The series-parallel transducer wirings made in order to obtain either 4 or 8 Ω impedances in order not to overload the amplifiers led to a higher variability of bit responses which will surely influence obtained results as discussed in the transducers mismatches section. Implementation of future direct-converting systems will need individualised and identical amplifier and DSP platform for each array element.

The effect of the listening room where far-field measurements were taken was considered not significant due to the low reverberation time of the room. For the purpose of comparisons between near and far-field measurements the assumption of homogeneity within the room is considered valid though for comparison with anechoic data experimental conditions were different and can not be compared to each other. However, as it was suggested in [98], the swept-sine measurement technique in the presence of full live rooms loses accuracy.

Live experimental DTA FRFs (on and off-axis) as well as THDs were presented in section 5.2.2.1. Results showed generally high second and third harmonic distortion components and an array frequency response roll-off of approximately 6 dB / octave. Possible reasons for this roll-off were different transducer wirings for the different bits and / or non linear acoustic addition of the driver responses on the baffle. It is believed that this roll-off would not have been as prominent with a more efficient transducer.

The effect of harmonic cancellation was seen as lower second and third harmonic distortion component values for all frequencies for θ = 0º as r was increased. For angles other than θ = 0º, harmonic cancellation was not evident since the measurement was not taken in the sweet-spot. Possible factors which compromised harmonic cancellation were identified as transducer mismatches, binary-weighting errors, path length difference distortion, transducer wiring, baffle diffraction, driver interferences and room acoustics.

As the off-axis angle was increased at a constant radial distance, acoustic interference manifested in the FRF magnitudes as a comb filter response, with increasing number of lobes and nulls in measured FRFs. These can be seen in figure 5.32 by reading across columns. Interference effects were further corroborated with FRF simulations, which did not take into account any modelling of baffle effects.


- 96 -

At an on-axis position, live THD presented a minimum of 2 % at 1 m only, at the frequency where the FRF magnitude was a maximum (3 kHz). As the listening distance was increased, irregularities occurred in the measured THD. However, in this case, the minimum THD was extended through a wider frequency band on-axis and at higher distances, with values under 20 % from 3 to 8 kHz.

For the off-axis measurements, the difference in THD between the near and far radial distances was not as significant for two reasons: firstly, a 5.5 m radial distance could not be reached because of the available room size; secondly, the measurement was taken outside the sweet-spot and the distance to walls or objects in the room was smaller, giving rise to irregularities in measured THDs.

At a frequency of 3 kHz significant spectral harmonic cancellation was seen between the near and the far fields, by comparing sine wave DTA outputs. Swept-sine responses measured in the near and far fields showed harmonic cancellation in the 2 kHz to 5 kHz band. A decrease in high-frequency harmonics was seen in swept-sine output spectrograms as the measurement distance from the array was increased.

6.4 Sources of Experimental Error

The following section analyses and quantifies the potential sources of error incurred in the measurements. The achieved SNR for all implemented real-time DTA topologies did not reach the theoretical 37.06 dB value expected for a 6-bit quantisation, with SFDR values of the order of 25 dB and SNR values of the order of 20 dB, due to a variety of factors:

6.4.1 Measurement Distance Compared to Main Array Dimension

Due to the finite transducer size, the minimum array interspacings set the minimum array dimension (36 cm length for the line array, 26 cm diameter for the concentric-circle array). In relation to the available anechoic maximum measurement distance (1.5 m), the minimum array dimension was only a few times smaller. When this is the case, harmonic cancellation cannot occur. Acoustic path lengths must be much larger (at least a factor of 10) than the array dimension. This is achieved either by reducing the transducer size whilst keeping the same sensitivity (transducer technology developments) or by increasing the measurement distance.


Differences in transducer impedance value and resonance frequency location cause differences in the FRF responses of the different transducers in the array. Ultimately these differences are caused by the tolerances in material properties set in the transducer manufacturing process. The resulting different magnitudes and phases cause bitstreams harmonic components to be radiated unequally, compromising the in-air harmonic cancellation and affecting the quality of the direct-conversion. Sensitivity mismatches up to 5 % were present in the line and circular arrays, higher in the concentric circle array.


- 97 -

6.4.3 Hypersensitivity to Sweet-Spot Misalignment

At small microphone distances, for instance at 1.5 m for a 36 cm long line DTA, a source of experimental error arises from inaccurate microphone positioning. The on-axis microphone setting was performed manually and although great care was taken, there is always a possible human error. Although the error in angle positioning was estimated to be less than 5º, vertical maps showed that by moving the microphone location by only one degree off-axis, a THD increase of 20 % was incurred.

6.4.4 Binary-Weighting Error

Amplifier gains were set through two 1 % tolerance resistors, whose highest associated error is 2 %. However, a 5 % deviation in amplifier gains is a reasonable assumption since gains were set to specification and according to available resistor values. With different amplifier gains, reconstruction is especially compromised for the bits with highest significance. In simulations, a 10 % (with respect to nominal value) gain in the MSB resulted in a THD of 4.9 %, which represents a THD increase of more than 100 %. However, by applying a 10 % gain increase to the LSB in a 6-bit system, no significant THD increase was found.

6.4.5 Bitstream Noise

High-frequency random noise can be accidentally introduced in the bitstreams or at any other point in the circuit, a fact that was sometimes observed on the oscilloscope. In simulations, noise was modelled by adding Gaussian distributed random numbers with zero average and a percentage of the bitstream nominal amplitude to perfect bitstreams. The result on reconstruction was the addition of background noise and consequent decrease in SNR.

6.4.6 Transducer Non-Linearities

It is a well known fact that moving-coil transducers behave non-linearly. As a first approximation they may be considered linear although this approximation does not hold for more demanding applications when high accuracy is required. In section 7.1.4 a thorough comparison of linear convolution with experimental data was presented which accounts for non-linear effects in moving-coil transducers.

Chapter 7 Simulation Results

- 98 -

7 Simulation Results

Simulations of multi-bit DTA systems under realistic assumptions are presented in this chapter, with the aims of increasing the understanding of multi-bit direct acoustic conversions and extend the influence of system parameters on overall performance. For this purpose, different DTA topologies and geometries were considered. The 3D space was characterised in terms of THD and acoustic pressure. Effects arising in experimental implementations related to the real performance of moving-coil transducers such as mismatches, non-uniform FRFs, break-up frequency peak and its relationship to sampling rate were further modelled and simulated.

A simulation approach was developed based on bitstream convolution with transducer impulse responses. Impulse responses were simulated with a variety of assumptions. Convolutions were performed as frequency domain multiplications for ease of implementation and computational efficiency. In the simplest case, an idealised FRF with equal magnitude over the whole band and perfectly linear phase was considered. The FRF was subsequently modified to include the characteristic 12 dB / octave rise of moving-coil transducers before the mass-controlled region and a uniform magnitude thereafter, with linear phase over the whole pass band. In the most realistic case, transducer FRFs were derived from an on-axis experimental FRF, measured with the swept-sine technique.

The DTA sound field was mapped with both of these approaches by calculating acoustic pressures and THDs for different array geometries and driving strategies. Convolution with the experimentally derived FRFs resulted in higher THD figures over the free-field space, indicating the detrimental effect of transducer FRF non-uniformities. In addition, other real DTA effects such as transducer mismatches, number of bits increase, sampling rate or array size reduction and binary-weighting error were modelled.

The intention of mapped sound field simulation results was to obtain a clearer picture of the DTA spatial sound field obtained under different transducer FRFs, for different array geometries, bit-to-transducer assignments and DTA topologies, as well as obtaining useful relationships between array size and sweet-spot location and quality, without the need of building arrays experimentally.

The main limitation of the adopted simulation approach was the assumption of linearity implicit to the convolution: in practice, there is a difference between convolved and experimental responses due to the non-linear behaviour of the transducer. Second harmonic distortion was therefore not present in the results although third harmonic distortion was accurately modelled. A more accurate simulation would include higher order terms from the Volterra series expansion [100-103] and would lead to results including both even and odd order harmonic distortions.

7.1 Simulation Approach

The simulation used in this work was based on convolution of the discrete transducer impulse responses with the driving bitstreams, calculated as a frequency-domain multiplication for computational efficiency.


- 99 -

As a first approach, simulated transducer FRFs were obtained from a lumped-parameter description of the moving-coil transducer driver [104,105]. The behaviour of the transducer in the frequency domain was described by a +12 dB / octave roll-off before resonance and a perfectly uniform radiated pressure response over the mass-controlled region. A more accurate description was employed thereafter by using a semi-empirical simulation approach, where transducer FRFs were derived from an on-axis experimental frequency response and modified by a 1/r factor and a directivity function according to the relative transducer - microphone location.

7.1.1 Idealised Transducer FRF

The frequency response of the complex pressure field produced by one tweeter at a vector distance from the transducer centre r at a complex frequency s, is described through the expression:

( )( )

reGD

RMBlSe

,sHkrtj

ETMS

go−

⋅⋅⋅⎟⎟⎠

⎞⎜⎜⎝

⎛=

ω

ωθπρ

)()(2

r

(7.1)

where:

ρ0 – air density

eg - Thevenin equivalent source voltage generator

MMS = mechanical mass of the diaphragm and air load

Bl = force factor

S = diaphragm area

RET = sum of resistances of generator and voice coil

QTS= total quality factor of the driver unit

ωs = angular mechanical resonance frequency given by:

MSMSs CM

1=ω

(7.2)

where CMS represents the suspension mechanical compliance.

The transducer directivity is that of a planar piston of radius a and is given by:

θθθ

sin)sin(2)( 1

kakaJD =

(7.3)


- 100 -

where J1 denotes the first-order Bessel function of the first kind and k is the wave number.

A 12 dB/Octave rise was modelled with a standard second-order high-pass filter of the form:

11)( 2

2

+⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛

=

sTSs

s

sQ

s

s

sG

ωω

ω

(7.4)

where s = jω. Expression (7.1) yields a computationally efficient calculation of sound pressure at every point in space. A large number of drivers of the same type can be considered with relatively low computational expense and the overall DTA pressure at a given listening point is obtained as the sum of individual pressures produced by all transducers at that point. Figure 7.1 shows the FRF magnitude obtained with equation (7.1).

7.1.2 Semi-Experimental Transducer FRF

A more accurate simulation was developed by deriving transducer FRFs from an on-axis experimental frequency response, modified by a 1/r distance factor and a simulated directivity function derived for a planar piston [43,50,51]. In this way, the irregularities in transducer FRFs as well as transducer diaphragm break-up were taken into account.

Semiexperimental FRFs were derived by using an experimentally determined on-axis transducer FRF measured at 1.5 m which was modified by a theoretical circular piston directivity function given by equation (7.3) and a distance factor. Taking into account just the experimental transducer FRF magnitude and assigning a linear phase:

( ) ( ) ( )( )

i

krtj

ii reHDC,H

i−

⋅⋅⋅=ω

ωθω ~r

(7.5)

Considering both experimental FRF magnitude and phase:

( ) ( ) ( )( )

i

j

ii reHDC,H

ωφ

ωθω~

~ ⋅⋅⋅=r

(7.6)

Figure 7.1 shows a comparison of the FRF magnitudes obtained with the two FRFs described in equation (7.1) and equation (7.6).


- 101 -

Figure 7.1. Comparison of experimental and simulated transducer FRF magnitudes on-axis at 1.5 m distance.

7.1.3 Time-Domain DTA Response

The frequency response functions of each DTA array element were simulated with any one of the two approaches described in section 7.1.1 and section 7.1.2 and then multiplied in the frequency domain by the Fourier transform of each driving bitstream. The different frequency-domain bitstream responses were added and the time-domain DTA acoustic output was obtained through an inverse Fourier transform:

( ) ( ) ( )⎥⎦

⎤⎢⎣

⎡⋅ℑ= ∑

=

− ωω B,H,tyN

iii

1

1 rr

(7.7)

where B(ω) represents the bit spectrum as a function of frequency ω.

Time-domain DTA responses were computed with this approach at any point in space. The difference between the DTA responses calculated at any two given points in space was given by the different distance factors ri present in the transducer frequency response, which determined the FRF magnitude, and the angular information from the point to the transducer centre which determined the value of the directivity function D(ө,φ).


- 102 -

7.1.4 Limitations

The main differences between a simulation based on linear convolution and the experimental DTA output were the following:

Convolution only accounts for purely linear systems and therefore transducer non-linear behaviour was not included. The predicted harmonic content of the DTA output was less in simulations than in experiments, mainly because of the value of second harmonic distortion component. As a result, simulated THD figures were less than experimental ones and can be considered optimistic. Nevertheless the simulation was useful in order to find qualitative trends in spatial THD and evaluate the effects arising from transducer non-linearities.

Transducer responses were assumed identical, changing as 1/r with the source-to-receiver distance and slightly with the directivity function D(ө,φ). Transducer mismatches were in principle not present in simulations, though they were further modelled under different assumptions (results are presented in section 7.4).

The dips and peaks introduced in the frequency response by the presence of a baffle were not considered, and it is assumed that these could be compensated for by suitable noise-shaping transfer functions.

To obtain a clearer picture of how simulated data via convolution compares with experimental results, a logarithmic sweep of 2.5 s from 200 Hz to 20 kHz was convolved with a measured transducer impulse response and compared with the measured transducer sweep response, for the aluminium-dome tweeter. Five times averaging was employed to check the consistency of the results versus time. Figure 7.2 and figure 7.3 show the sweep response spectrograms obtained via convolution and experimentally, respectively. Results show that harmonic strength was stronger in the case of experimental results (figure 7.3), where both even and odd-order harmonics were present and where especially even-order harmonics can be more clearly seen. The mean THD calculated from the spectrograms in the 200 Hz to 20 kHz band was 2.1 % in the case of simulated convolution versus a 4.1 % in the case of purely experimental results.

Therefore THD figures calculated from data obtained after a convolution approach may be considered a best-case scenario approximation to experimental data since transducer non-linear effects were not taken into account in simulations. Nevertheless simulation results indicate qualitative trends in DTA spatial responses.


- 103 -

t (s)

f (H

z)

0 2 4 6 8 10 120

2000

4000

6000

8000

10000

12000

14000

16000

18000

20000p (dB)

-50

-40

-30

-20

-10

0

10

20

30

Figure 7.2. Sweep response spectrogram obtained after convolution of the sweep with an on-axis experimental impulse response of the aluminium-dome tweeter.

t (s)

f (H

z)

0 2 4 6 8 10 120

2000

4000

6000

8000

10000

12000

14000

16000

18000

20000p (dB)

-50

-40

-30

-20

-10

0

10

20

30

Figure 7.3. On-axis experimental sweep response spectrogram of the aluminium-dome tweeter.


- 104 -

7.2 DTA Sound Field Characterisation

By calculating the sound pressure level due to the DTA with a single frequency input at a fine grid of points set on given spatial regions which will be described next, with a technique resembling that of noise mapping, snapshots of the spatial acoustic pressure field were obtained. Movies resulted by sequencing several of these snapshots one after another. Horizontal and vertical sound pressure level and THD maps were obtained for arbitrarily defined near and far regions, defined in section 7.2.1. Sound pressure level and THD maps obtained with the convolution simulation approach provided a detailed picture of the spatial sound field and regions of interest that could be obtainable with multi-bit direct-converting loudspeaker arrays.

7.2.1 Coordinate System and Mapping Parameters

Throughout these DTA simulations the coordinate system shown in figure 7.4 was used and referred to. The positive z-axis was used as the acoustic axis of the array, and the array was positioned on the xy plane. When spherical coordinates were needed, the standard spherical set of coordinates (r, θ, Φ) was used. Their relationship to Cartesian coordinates is given by (7.8).

To define the near and far regions relative to the array dimensions, two planes were used as shown in figure 7.5 for the horizontal maps and figure 7.6 for the vertical maps.

The horizontal near region considered was a 1 m2 area on the yz plane directly in front of the array (figure 7.5). The horizontal far region considered was a 100 m2 area on the yz plane directly in front of the array. Both regions were sampled with a 500 x 500 point grid.

The vertical near region considered was a 1 m2 area on the xy plane at z0 = 1.6 m (figure 7.6). The vertical far region considered was a 100 m2 area on the xy plane at z0 = 5.5 m. The two vertical regions were also sampled with a 500 x 500 measurement point grid.

The z0 distances chosen were exactly the same as those available for experimental results (1.5 m for the near region and 5.5 m for the far region) for comparison purposes. Region dimensions and positions were chosen in this manner in order to show possible result differences between qualitatively small and big areas.

Figure 7.4.Simulation coordinates. z: acoustic axis. xy plane: array plane.


- 105 -

φθφθφ

cossinsincossin

rzryrx

===

(7.8)

Figure 7.5. Horizontal map near and far regions; dimensions and number of points

Figure 7.6. Vertical map near and far regions; dimensions and number of points

7.2.2 Simulated DTA Types

Two bit-grouped multi-bit DTA topologies were studied: the one transducer-per-bit DTA and the bit-grouped DTA. Results are shown for 6-bits and a sampling rate of 96 kHz: In the one transducer-per-bit DTA, the electronic binary-weighting was simulated with binary-weighted bitstreams, and each


- 106 -

binary-weighted signal was convolved with the impulse response of one transducer in the array. In the bit-grouped DTA, a total of 2N-1 different impulse responses were simulated, and a total of 2N-1 convolutions were performed for each measurement point, with equally-weighted driving bitstreams. This process accounted for the acoustic binary-weighting implemented by the bit-grouped DTA.

Two geometries were chosen for each topology as representative examples, although a variety of geometries is possible. Line and circular arrays of the one transducer-per-bit type were considered (figure 7.7, top). Square matrix honeycomb and concentric circle arrays were considered in the case of bit-grouped DTAs (figure 7.7, bottom).

The motivation behind these particular array geometries was the following:

The line array was chosen for its simplicity and for analytical analysis [37,38,106,107].

The circular array was chosen because, as opposed to the line array, the acoustic path length from all transducers to all on-axis points is nominally equal; the circular DTA has a well-defined sweet spot and it was interesting to investigate sweet-spot degradation with increasing off-axis angle (section 7.2.3).

Possible extensions of these geometries to two dimensional arrays were the square matrix DTA and the concentric circle DTA. However, the advantage of the honeycomb DTA was thought of as offering a better packing of transducers for the same radiating area [2].

The concentric circle DTA was chosen as it was the bit-grouped geometry where less on-axis THD was predicted in [13].


- 107 -

Figure 7.7. Simulated arrays. Top left: one bit-per-transducer, line array geometry. Top right: one bit-per-transducer, circular array geometry. Bottom left: bit-grouped, honeycomb geometry. Bottom right: bit-

grouped, concentric circle geometry.

7.2.3 Sound pressure level and THD Maps

For horizontal map calculations, the listening space was set on the zy plane directly in front of the array. The simulation parameters shown in table 7.1 were used. The one transducer-per-bit with line and circular geometries and the bit-grouped DTA with honeycomb and concentric circle geometries were considered. The horizontal and vertical near and far regions were considered. Taking into account all these options yields eight pressure maps and eight THD maps for each topology. The maps corresponding to the one bit-per-transducer DTA are shown in figure 7.8, where to top four maps are horizontal and the bottom four are vertical both in the near (left column) and far (right column) regions. The resulting acoustic response is not symmetric for not symmetric geometries i.e. the line DTA. Figure 7.9 shows THD maps for the above mentioned geometries in the same configuration. Figure 7.10 and figure 7.11 correspond to sound pressure level and THD maps for the bit-grouped DTA topology.


- 108 -

SIMULATION PARAMETERS

Input sine wave frequency f0 1 kHz Element spacing d 0.06 m

Transducer radius a 0.013 m Number of bits N 6

Sampling frequency fs 96 kHz Far-field y-axis sampling (-5, 5) m in 0.02 steps (501 points) Far-field z-axis sampling (0.02, 10) m in 0.02 steps (500 points)

Near-field y-axis sampling (-.5, .5) m in 0.002 steps (501 points) Near-field z-axis sampling (0.002, 1) m in 0.002 steps (500 points)

Table 7.1. Horizontal pressure map simulation parameters used with the one transducer-per-bit line and circular geometries and the bit-grouped DTA honeycomb and concentric circle geometries


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Figure 7.8. One transducer-per-bit DTA sound pressure level maps for 96 kHz sampling rate and 1 kHz input sinusoid. Left column, from top to bottom: horizontal map, circular geometry, near region; horizontal map, line geometry, near region; vertical map, circular geometry, near region; vertical map, line geometry, near region.

Right column, from top to bottom: horizontal map, circular geometry, far region; horizontal map, line geometry, far region; vertical map, circular geometry, far region; vertical map, line geometry, far region.


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Figure 7.9. One transducer-per-bit DTA THD maps for 96 kHz sampling rate and 1 kHz input sinusoid. Left column, from top to bottom: horizontal map, circular geometry, near region; horizontal map, line geometry, near region; vertical map, circular geometry, near region; vertical map, line geometry, near region. Right column, from

top to bottom: horizontal map, circular geometry, far region; horizontal map, line geometry, far region; vertical map, circular geometry, far region; vertical map, line geometry, far region.


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Figure 7.10. Bit-grouped DTA sound pressure level maps for 96 kHz sampling rate and 1 kHz input sinusoid. Left column, from top to bottom: horizontal map, concentric circle geometry, near region; horizontal map, honeycomb geometry, near region; vertical map, concentric circle geometry, near region; vertical map, honeycomb geometry, near region. Right column, from top to bottom: horizontal map, concentric circle geometry, far region; horizontal

map, honeycomb geometry, far region; vertical map, concentric circle geometry, far region; vertical map, honeycomb geometry, far region.


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Figure 7.11. Bit-grouped DTA THD maps for 96 kHz sampling rate and 1 kHz input sinusoid. Left column, from top to bottom: horizontal map, concentric circle geometry, near region; horizontal map, honeycomb geometry, near region; vertical map, concentric circle geometry, near region; vertical map, honeycomb geometry, near region.

Right column, from top to bottom: horizontal map, concentric circle geometry, far region; horizontal map, honeycomb geometry, far region; vertical map, concentric circle geometry, far region; vertical map, honeycomb

geometry, far region.


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Results show irregularities in the mapped sound field, especially in the near-field. For both one bit-per-transducer topologies, once the far-field is reached, the sweet-spot appears on-axis and is shown as the region of lowest THD, where the only remaining THD can be considered to be due only to quantisation.

7.3 Transducer FRF Non-Uniformities

Comparing the THD and pressure maps obtained by considering perfectly uniform FRFs derived from equation (7.1) with those obtained by considering semi-experimental transfer functions as in equation (7.6) (figure 7.1 shows the shape of the transducer FRF magnitude with each of the assumptions), the effect of transducer FRF non-uniformity was isolated.

Figure 7.12 shows the far-region horizontal THD map obtained under the assumption of equation (7.1) and figure 7.13 shows the far-region horizontal THD map obtained under the assumption of equation (7.6). Figure 7.14 and figure 7.15 show far-region horizontal pressure maps obtained under each of the two assumptions.

Results show degradation of the sound field quality at all distances from the array, especially off-axis, with a maximum THD increase of 20 % in the near region (figure 7.12 compared to figure 7.13). The quality loss was manifested as irregularities in the sound field and overall higher THD values regardless of array geometry (figure 7.14 compared to figure 7.15).

The maximum uniformity departure of ±3 dB from 200 Hz to 20 kHz together with the break-up peak arising in semi-experimental FRFs produced a mean THD increase of 12 % over the free-field space, leaving the extent of the sweet-spot region unaffected in the far-field.

By computing image histograms of the images shown in figure 7.12 and figure 7.13, a clearer picture of the amount of pixels in each image with a given THD value was obtained. Image histograms are shown in figure 7.16 and figure 7.17. The idealised FRF assumption resulted in a minimum THD of 8 % with a count higher than 500 points and a maximum THD value of 55 % with a count of less than 500 points (figure 7.16). On the other hand, the semi-experimental FRF assumption led to a minimum THD value of 20 % with a count higher than 500 points, all other THD values up to 100 % having a count higher than 500 points.


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Figure 7.12. Horizontal THD map for the far region and the concentric circle bit-grouped DTA considering idealised transducer frequency responses.

Figure 7.13. Horizontal THD map for the far region and the concentric circle bit-grouped DTA considering experimentally-derived transducer frequency responses with the aluminium-dome tweeter transfer function.


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Figure 7.14. Horizontal sound pressure level map for the near region and the concentric circle bit-grouped DTA considering idealised transducer frequency responses.

Figure 7.15. Horizontal sound pressure level map for the near region and the concentric circle bit-grouped DTA considering experimentally-derived transducer frequency responses with the aluminium-dome tweeter transfer

function.


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0 10 20 30 40 50 60 70 80 90 1000

500

1000

1500

2000

2500

3000C

ount

s

THD (%)

Figure 7.16. Image histogram of THD values present in map obtained with idealised FRF.

0 10 20 30 40 50 60 70 80 90 1000

500

1000

1500

2000

2500

3000

Cou

nts

THD (%)

Figure 7.17. Image histogram of THD values present in map obtained with experimentally derived FRF (aluminium-dome tweeter transfer function)


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7.4 Transducer Mismatches

The effect of mismatches in the different array transducer FRFs was modelled by assuming a random fluctuation around the mean value in each transducer frequency response magnitude or phase. In real loudspeaker arrays this effect arises from small manufacturing differences in the transducer assembly which give rise to differences in impedance values and main resonance frequency location. In practice even the best transducers present a 5 to 10 % variation in impedance value and resonance frequency location, which lead to uneven FRFs. Differences in the complex frequency response imply differences in the transducer phase response, which in a direct acoustic DAC affect the reconstructed signal by delaying the bitstream components by different amounts, resulting in spectral leakage in the spectral lines of the reconstructed waveforms.

t (s)

f (H

z)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.50

0.5

1

1.5

2

2.5

3

3.5

4

4.5

x 104

-35

-30

-25

-20

-15

-10

-5

0

5

10

15

t (s)

f (H

z)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.50

0.5

1

1.5

2

2.5

3

3.5

4

4.5

x 104

-35

-30

-25

-20

-15

-10

-5

0

5

10

15

Figure 7.18. Effect of transducer mismatching on output swept-sine spectrograms, aluminium-dome tweeter. Left: all transducers with matched FRF phase. Right: introduction of a random FRF phase fluctuation in each transducer.

7.5 Sampling Rate

With a transducer which presents a prominent break-up peak, increasing the sampling rate in a DTA system was seen to have either beneficial or detrimental consequences on the quality of the radiated sound field depending on whether the Nyquist frequency was below or above the transducer break-up frequency, respectively.

On the other hand, with a transducer presenting a smooth roll-off or under the assumption of flat transducer FRFs without break-up, by increasing the sampling rate, the quality of the radiated sound field should in principle improve on the basis of transients being resolved up to a higher sampling time. A digital delay line with a finer time resolution might also be implemented in each channel and closer listening distances might be achieved. The extra bandwidth also allows for more frequency bins in the ultrasonic range in which to shift unwanted frequency components with a noise shaping approach. However, as ultrasonic components are radiated they can fold back into the audio band and produce an increase in high-frequency components in the resulting waveform

In the case of experimental transducer FRFs, by increasing the sampling rate, the increased benefits were only seen for Nyquist frequencies below transducer break-up frequency; thereafter, because of the presence of the prominent break-up peak, no benefits were seen on reconstructed waveforms.


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However, by reducing the Nyquist frequency to before transducer break-up, the bandwidth was reduced and not enough bins were present for noise shaping. A non-oversampling noise shaping approach would then be required [64]. Furthermore, considering realistic transducer break-up peaks, high oversampling is recommended only on the DSP stage in order to noise shape the digitised signal and therefore produce noise shaped bitstreams; after this, a decimation stage would be needed in order to leave the Nyquist frequency just below transducer break-up.

Figure 7.19 shows the 1 kHz output of a six-bit one transducer-bit line DTA on-axis operating at 48 kHz in the time and frequency domains. In figure 7.20, the sampling rate was increased to 192 kHz with every other parameter fixed. A THD value over the audio band of 19.3 % was obtained at 48 kHz whereas 20.6 % was obtained at 192 kHz, both with comparable SFDRs.

Figure 7.21 and figure 7.22 show equivalent results under the consideration of experimental transducer FRFs. In this case a THD value over the audio band of 20.5 % was obtained at 48 kHz whereas 21.2 % was obtained at 192 kHz with comparable SFDRs.

Figure 7.19. 1 kHz sine wave one-transducer-bit line DTA on-axis output in time (top) and frequency (bottom) domains at a sampling rate of 48 kHz under idealised transducer FRF consideration.


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Figure 7.20. 1 kHz sine wave DTA on-axis output in time (top) and frequency (bottom) domains at a sampling rate of 192 kHz under idealised transducer FRF consideration.

Figure 7.21. 1 kHz sine wave one-transducer-bit line DTA on-axis output in time (top) and frequency (bottom) domains at a sampling rate of 48 kHz under experimental transducer FRF consideration.


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Figure 7.22. 1 kHz sine wave one-transducer-bit line DTA on-axis output in time (top) and frequency (bottom) domains at a sampling rate of 192 kHz under experimental transducer FRF consideration.

Idealised FRF Experimental FRF

fs = 48 kHz fs = 192 kHz fs = 48 kHz fs = 192

kHz f (kHz) L (dB) L (dB) L (dB) L (dB)

1 75 82 82 84

3 36 45 45 49

5 33 42 42 46

7 39 46 46 51

9 41 49 49 54

11 39 48 48 51

13 34 45 44 47

15 29 42 42 45

17 37 48 48 52

19 36 49 49 52

THD (%) 19.3 20.6 20.5 21.2

SFDR (dB) 34 33 33 30

Table 7.2. Peak values in dB re 20 µPa for the fundamental (1 kHz) and the different harmonics in one-transducer-bit line DTA reconstructed sine wave at different sampling rates under different transducer FRF

assumptions, calculated from figure 7.19 to figure 7.22. Calculated THD and SFDR values included for each case.


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7.6 Number of Bits

In normal digital systems, increasing the number of bits by one implies a 6 dB SNR increase [65]. However, in a bit-grouped DTA system, as the number of bits is increased by one the number of transducers is doubled, which implies doubling the array size if the transducer interspacing is kept constant and therefore increasing the distance to the sweet-spot. According to [1], array length determines the lower polar bandwidth and transducer interspacing determines the upper polar bandwidth. Therefore in order to increase array resolution by increasing the number of bits without affecting the lower polar bandwidth the interspacing would have to be reduced, affecting the upper polar bandwidth. Therefore in any case, increasing the resolution by increasing the number of bits will affect the polar bandwidth.

The suggested approach to increase bit resolution is to use a noise shaping system implemented in DSP. In our case, a bit reduction of 2 bits was obtained [67] by implementing a 9th order noise shaper with a FIR noise shaping filter whose coefficients were given in [64]. Hayama et al. [8] on their side reported a bit reduction from 16 to 12 bits using a noise shaper implemented in a multiple voice coil digital loudspeaker system.

7.7 Array Size Reduction

Array and transducer size in relation to the listening distance determine the path lengths and the extent to which they will differ from each other.

By reducing the transducer and array size whilst keeping the experimentally derived FRFs, sound pressure level maps were improved over the free-field space, due to reduced phase differences originating from reduced path length differences. Starting from a driver diameter of 26 mm and a spacing of 60 mm for the concentric circle bit-grouped DTA geometry, a mean THD value of 64.1 % was predicted over a 100 m2 map area, with a minimum THD of 7.8 %. The driver size and interspacing were then reduced and the resulting THD was measured.

Table 7.3 shows a summary of the findings. For a factor of 10 array and driver size reduction, the mean THD value over the map area was reduced to 32.8 %, (approximately half the previous mean THD), with a minimum value of 6.2 %. For a factor of 100 array and driver size reduction, a mean THD of 2.8 % with a minimum of 2.2 % was predicted. Moreover, the standard deviation for the THD values was reduced by 99 % implying a significant improvement in audio quality over the whole region, with smaller differences between the on-axis THD and that obtained at other off-axis regions. All cases considered a 96 kHz sampling rate and a 1 kHz sine wave input.

Diaphragm Diameter (mm) 26 13 2.6 0.26

Transducer interspacing (mm) 60 30 6.0 0.60 Mean THD (%) 147 93.8 32.8 2.80

Std. dev. (%) 338 164 10.9 0.79 Min. THD (%) 7.76 3.79 6.17 2.15 Max. THD (%) 4715 806.2 108.8 4.70

Table 7.3. THD map statistics for different array sizes corresponding to the concentric circle bit-grouped DTA geometry


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Figure 7.23. Sound pressure level (left column) and THD (right column) maps for the one bit-transducer line DTA topology with 36 cm length (top row), x10 array and transducer size reduction (middle row) and x100 array and

transducer size reduction (bottom row).

7.8 Binary-Weighting Error

Direct-converting systems present a high sensitivity to bitstream amplitude: binary-weighting must be set to within 1 % to allow bitstream harmonics to cancel out fully, being left only with the signal resolved up to the number of bits considered and its images. Desampling, performed by the transducer


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low-pass action or by the DSP Nyquist frequency then eliminates extra signal images created by sampling.

The digital bitstreams of a swept-sine were created in software. A perfect binary-weight of 2-n with n = 1 for the MSB was first applied to the bitstreams, which were then added. Any harmonics present in the output signal were quantified from a signal spectrogram. A bitstream weighting gain error relative to the nominal binary-weighted value was then assigned to any desired bit separately. After addition, the THD of the output signal was calculated from the spectrogram (table 7.4). A 5 % gain error in the MSB led to almost 100 % increase in THD, revealing the MSB harmonics in the spectrogram. However, a 100 % error in the LSB did not affect THD significantly. The accuracy of the most significant bits binary weights must be set to a maximum of 2 % to incur a THD gain of less than 2 %.

The value of the fundamental, third and fifth harmonic peaks at a frequency of 3 kHz was measured from the reconstruction spectrograms. Even order harmonics were not significant in the measured data. It was observed that the harmonics present in a direct conversion with a given binary-weighting error in bit X are those present in bit X only and the rest of them will cancel out.

The spurious-free dynamic range (SFDR) was calculated from this data set as the difference between the fundamental and the highest spur in the signal. Figure 7.24 shows the measured values of the peaks corresponding to the fundamental, third and fifth harmonics versus a deliberate binary-weighting error in the MSB ranging from 1 % to 20 %. Figure 7.25 shows the calculated SFDR for binary-weighting errors in the three most significant bits. For an error affecting only the MSB, the SFDR decreases more rapidly than for an error affecting the bits of lower significance. Therefore it is most important to minimise any binary-weighting errors in the bits of higher significance.

THD (%) Amplitude

increase (%) MSB Bit 2 Bit 3 LSB 0 1.77 1.83 2.04 1.80 1 1.80 1.89 1.85 1.84 2 1.87 1.94 1.80 1.84 5 3.10 2.00 1.87 1.84 10 4.98 2.81 1.87 2.10 20 10.1 6.00 2.26 1.83 40 19.2 11.3 3.00 1.81

100 34.3 26.7 8.18 2.07

Table 7.4. THD increases for different gain errors in the bitstreams relative to the binary-weighted values after summing the altered weighted bitstreams.


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0 2 4 6 8 10 12 14 16 18 20-50

0

50

100

150

200

250

300

350

400

δ (%)

Pea

k Le

vel (

dB)

FundamentalThird harmonicFifth harmonic

Figure 7.24. Measured peak values at a fundamental frequency of 3 kHz for reconstruction under different bitstream binary-weighting error δ affecting the MSB.

0 2 4 6 8 10 12 14 16 18 200

10

20

30

40

50

60

δ (%)

SFD

R (d

B)

MSBBit 2Bit 3

Figure 7.25. Calculated SFDR from measured peaks at a fundamental frequency of 3 kHz for reconstruction under different bitstream binary-weighting error δ affecting the three most significant bits.


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7.9 Noise Shaping

A dithered noise shaper was simulated before the DTA input in order to shift the noise produced by the DTA into inaudible frequency bands. A 9th order noise shaper at 96 kHz sampling rate was implemented and extended from the previously known coefficients given by Wannamaker [64], in order to compare with a well known noise shaping transfer function. The one transducer-per-bit line array DTA with 6-bits and 2.6 mm diameter transducers and experimentally derived FRF (aluminium dome tweeter) was considered, at a 10 m listening distance on-axis.

As a result, the waveforms seen in figure 7.26 (without noise shaping) and figure 7.27 (with noise shaping) were obtained. Figure 7.28 and figure 7.29 show the DTA output waveform spectrograms without and with the noise shaper in place, respectively. The noise-shaped DTA binary input reduced the in-band noise floor by an average of 6 dB and increased the noise floor consequently at frequencies above 25 kHz (following the Gerzon-Craven theorem [53] the amount by which noise is reduced in band is the equivalent to that by which noise is increased out of band.). Other noise shaper orders and coefficients were used, specifically those suggested in [64] and in [55-57]. In every case results show that the output DTA waveform takes the shape of the noise-shaped input waveform.

Figure 7.26. DTA output waveform in time (top) and frequency (bottom) domains assuming experimental transducer FRF, a one transducer-per-bit line array topology and a transducer diameter of 2.6 mm and 10 m

listening distance. No noise shaping applied.


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Figure 7.27. DTA output waveform in time (top) and frequency (bottom) domains with a 9th order noise shaper and same assumptions as in figure 7.26.

Figure 7.28. DTA output waveform spectrogram without noise shaping, one transducer-per-bit line array topology, transducer diameter of 2.6 mm and 10 m listening distance.


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Figure 7.29. DTA output waveform spectrogram with a 9th order noise shaper, one transducer-per-bit line array topology, transducer diameter of 2.6 mm and 10 m listening distance.

As an arbitrary noise shaper frequency shape is theoretically possible by considering a high enough number of filter taps [53], any array FRF may be obtained, at least on-axis. Compensation for effects such as the dips and peaks introduced in the FRF by the baffle can therefore be incorporated advantageously in the noise shaper FRF.

7.10 Summary

Two approaches have been presented for the simulation of DTA sound fields. Near and far regions were mapped with horizontal and vertical maps and the DTA sound fields were thus characterised.

The most realistic simulation approach considered all the irregularities present in practice in a transducer FRF magnitude and resulted in increased THD over the free-field space. The effect of transducer mismatches was modelled as random fluctuations in the FRF magnitude and phase and it was seen that phase fluctuations resulted in spectral leakage which was also present in the experimental results. The error in binary-weighting was assessed and it was concluded that the accuracy of the MSB weights must be set to a maximum of 1 % for THD not to increase drastically.

The quality of the output in direct-converting systems was shown to be dependent on the practical properties of the transducers considered as well as on key digital system parameters such as sampling rate and number of bits. SNR was improved with the addition of a dithered noise shaper around the DSP quantizer. In the presence of a noise shaper, the on-axis DTA output waveform tends to the noise shaped input. Off-axis responses are determined by the path length differences in play at each particular listening position.


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The information obtained by sound pressure level and THD maps can be summarised as follows. In the near field, the spatial acoustic DTA response is a function of the array geometry. In the far field, array geometry does not influence the spatial response significantly. Vertical maps showed the sensitivity of the measurement point to elevation variations.

Chapter 8 Conclusions and Future Work

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8 Conclusions and Future Work

This work has provided novel results on the implementation and performance of multi-bit digital transducer array prototypes made with moving-coil transducers. Results on the first implementations of one transducer-per-bit and bit-grouped DTA prototypes have been reported and are very important to the audio field, in which direct-converting arrays have been discussed and simulated for years, although no such practical implementations had ever been reported, at least of multi-bit direct converters with moving-coil transducers. The quality of the radiated sound field has been explored through THD, FRF, directivities and swept-sine responses, and related to the fundamental properties of the constituent transducers.

Several implementations were carried out with moving-coil transducers presenting different pass bands, efficiencies and electromechanical properties. Reconstruction with tweeters showed that when no path-length differences and transducer mismatches were present, the minimum THD value present in the reconstructed signal was higher than the theoretical value set by quantisation for the number of bits employed by excess of 100 %. Reconstruction with a full-range system showed that the minimum THD value can in practice be reduced to the order of the theoretical value set by the number of bits as transducer bandwidth was increased to cover the full audio range and as transducer FRFs presented a smooth roll-off. Comparison of the input bitstream spectra with the output signal spectrum showed that the low-pass filter action of the transducer performs the task of a desampling filter and no acoustic desampling filter is thus required.

On the basis of obtained experimental results, certain effects arising in an experimental implementation such as transducer interspacing, transducer mismatches and the presence of a baffle as well as possible solutions by means of noise shaping technology were isolated and quantified experimentally:

Transducer interspacing. By increasing transducer interspacing, the acoustic path-length differences to a common listening point were increased and the result was a net increase in THD. It is assumed that the practical solution to this problem is the implementation of a digital delay associated to each transducer.

Transducer mismatches. Transducer sensitivities must be matched to less than 1 %, especially in the MSB, in order to reduce extra bitstream harmonics not cancelling out completely in the output signal. Measurements with a transducer set representative of the best-quality moving-coil transducers available in the market showed impedance mismatches of the order of 30 % which produced typical output SNR mismatches of 6 dB and typical THD variations higher than 30 % in the reconstructed sound field.

Baffle effect. The presence of a baffle introduces peaks and dips in the transducer responses which change with transducer positioning within the baffle as well as with array size. Average baffle response compensations might be incorporated in the noise shaper transfer function.

Noise shaping. The net effect introduced by noise shaping is a gain of in-band SNR. This benefit is accompanied in direct-acoustic conversions by higher bitstream harmonic cancellation. Radiation of the noise-shaped bitstreams with moving-coil transducers performs a low-pass filtering effect which suppresses the extra high-frequency noise by-product.


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With the help of simulations, the effects of transducer FRF irregularities, array size reduction and binary-weighting error were quantified:

FRF irregularities. It was seen that the mean THD was increased by 12 % over the free-field space by having a transducer FRF with magnitude variations of ±3 dB and a break-up peak in comparison to a transducer with 0 dB magnitude variations, linear phase and a similar 12 dB /octave rise.

Array size reduction. By reducing the transducer size to 0.6 mm diameter, reconstruction on the nearest square metre from the array was improved and resulted in a mean THD of 2.8 %, which is of the order of the theoretical value set by six-bit quantisation.

Binary-weighting errors. The deliberate introduction of errors in the amplitude of the MSB introduced harmonics in the reconstructed signal which resulted in an overall mean THD increase. In particular, after changing the MSB amplitude by 10 %, THD increased to almost 5 %, in the absence of any other practical effects set by the transducer.

A multi-bit direct acoustic DAC with moving-coil transducer arrays is possible over a finite bandwidth and finite spatial region. The quality and bandwidth of the conversion is improved as the following implementation criteria are met:

Transducer size of the order of 1 mm for listening distances of the order of 1 m. Alternatively, digital delays implemented in each transducer should aim for these values.

An oversampling dithered noise shaper accompanying the quantisation stage which can include peaks and dips in its transfer function to compensate for an average baffle response.

Smooth transducer roll-off without break-up peak and a bandwidth covering the full audio range.

Binary-weighting accurate to less than 1 % in the most significant bits.

The different real-time multi-bit DAC topologies implemented with moving-coil transducers and simplified electronic topologies served as proof-of-concept systems and helped realised which practical parameters impinge a key signature in direct DAC systems: transducer mismatches, path length differences, transducer step response, break-up peak and its relationship to the sampling rate.

Oversampling bitstreams by a high order such as 16 or 32 increases the bandwidth and presents several advantages: the ultrasonic band is extended providing extra headroom into which high-frequency bitstream noise can be shifted; time delays implemented by a digital delay line could be resolved up to finer sampling times, which would allow listening at closer distances, and bitstream discontinuities are sampled at closer time intervals, which improves transient radiation. After noise-shaping, the transducer acts as a desampling filter and its finite bandwidth limits the highest frequency in the conversion bandwidth due to spectral harmonic cancellation. High bandwidth transducers with a smooth roll-off peak are therefore recommended.

Simulated vertical and horizontal sound pressure level and THD maps for DTA reproduction in anechoic listening spaces were derived for two possible DTA topologies. Transducer FRFs which were derived from experimental data led to more realistic results than those obtained on the basis of purely ideal transducer performance. However all spatial maps presented may be considered best-case scenario since they were derived with a linear convolution simulation approach.


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The implementation of all-digital direct-converting systems currently needs of transducer technology to evolve to produce smaller and more efficient transducers if bitstream harmonics are to be radiated. A possible figure of merit to aim for derived from simulations is 1 mm diameter with 86 dB / 1 W / 1 m sensitivity. Acoustic attenuation at spatial regions outside the sweet-spot may lead to highly directive all-digital steerable audio systems. Room response corrections can be implemented through active attenuators and arrays can be made steerable by phasing transducer groups.

8.1 Future Work

The development of this work led to possible gateways on other projects which could possibly lead to contributions to knowledge and subsequent possible product developments integrating the direct converting technology. The main ideas are summarised in this section.

8.1.1 Transducer Development

Faster, smaller, more powerful, transducers are needed for the direct-converting technology to evolve. In fact different transducer types are advised other than moving-coil, such as piezoelectric [31,34] or MEMS membrane transducers [14,15], although highly efficient moving-coil micro-transducers may also perform satisfactorily. Integration with mass-production techniques such as integrated circuits [4] is a sensible direction for industrial applications.

8.1.2 Integration with Spatial Audio

An important result presented in this work resulting from simulations was spatial sound pressure level and THD maps. The sound field and its distortion products produced by one loudspeaker array only were presented. Direct converting technology will be more mature once it is integrated with spatial audio techniques [2], to incorporate several audio channels, increase the effective listening area and attenuate regions of high distortion. Either active or passive sound attenuation outside the sweet-spot could lead to the implementation of highly directive direct-converting systems.

8.1.3 Room Acoustics and Influence on Reconstruction

Although some measurements taken in a reverberant space were presented in this work, the effect of room acoustics was considered negligible in the measurements due to low reverberation times over the whole audio band and to its effect not being relevant for the harmonic distortion components. However, the influence of room acoustics on reconstruction should be tested thoroughly in more reverberant spaces and constitutes the idea for a future research project. Calibration for each room spot could be performed before hand. The presented simulations in free-field could be extended with a ray-tracing technique initially considering hard-walls and subsequently developing to include different wall materials and absorptions. Sound pressure level maps should then change depending on the amount of reflexion present in the simulated rooms and the amount of reverberation in place.


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8.1.4 DSP Algorithm Development

Many DSP algorithms represent promising potential applications in direct-converting systems, as it has been discussed that perhaps the main advantage of the DTA resides in that all new application development would be carried out in software packages, alleviating development and production times for the manufacturer.

Copyright information could be encrypted in the signal and be sent to a switch-type receiver which would then allow switching the DTA system on / off depending on the listener copyright status. Adaptive beamforming, directivity control, changing perceptual attributes, digital equalization, baffle compensation and noise-shaping represent the most significant DSP algorithm implementation alternatives. Integration with internet and mobile phone audio would be required to converge with current technological trends.

ASICs developed should include all DSP and FPGA needs for direct-converting systems, as well as an efficient software communication platform for the user via high level drag and drop functions.

References

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9 References

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2. Troughton P.T., Hooley A., Goudie A.G., Easton M.G., Bienek I.A., Davies J., Ryan D.T., and Windle P.R. "Digital Loudspeaker System", WO 03/059005A2, 2003.

3. Porrazzo E.M., Orell-Porrazzo, and Pamela K. "Multiple Voice Coil, Multiple Function Loudspeaker", [US Patent 5872855], 16-2-1999.

4. Busbridge S.C., Fryer P.A., and Huang Y., "Digital Loudspeaker Technology: Current State and Future Developments," 112th AES Convention, Munich, Germany, 2002.

5. Busbridge S.C., Huang Y, and Fryer P.A., "Crossover Systems in Digital Loudspeakers," 110th AES Convention, Amsterdam, The Nederlands, 2001.

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38. Smith D.L., "Discrete-Element Line Arrays - Their Modelling and Optimization," J. Audio Eng. Soc. 45, pp 949-964, 1997.

39. McGehee D. and Jaffe J.S., "Beamforming With Dense Random Arrays: The Development of a Spatially Shaded Polyvinylidene Fluoride Acoustic Transducer," J. Acoust. Soc. Am. 95, pp 318-323, 1993.

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41. Leahy R.M. and Jeffs B.D., "On the Design of Maximally Sparse Beamforming Arrays," IEEE Transactions on Antennas And Propagation 39, pp 1178-1188, 1991.

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45. Henwood D.J., "The Boundary Element Method And Horn Design," J. Audio Eng. Soc. 41, pp 485-496, 1993.

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48. Morse I., "Vibration and Sound", Acoust.Soc.America, 1981.

49. Wright J.R., "Fundamentals of Diffraction," 100th AES Convention, Copenhagen, 1997.

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References

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51. Meyer D.G., "Digital Control of Loudspeaker Array Directivity," J. Audio Eng. Soc. 32, pp 747-754, 1984.

52. Lipshitz S.P., Wannamaker R.A., and Vanderkooy J., "Quantization and Dither: A Theoretical Survey," AES 91st convention, New York, 2005.

53. Gerzon M. and Craven P.G., "Optimal Noise Shaping and Dither of Digital Signals," 87th AES Convention, New York, U.S.A., 1989.

54. Lipshitz S.P., Wannamaker R.A., and Vanderkooy J., "Quantization and Dither: A Theoretical Survey," J. Audio Eng. Soc. 40, pp 355-375, 1992.

55. Stuart J.R. and Wilson R.J., "Dynamic Range Enhancement Using Noise-Shaped Dither at 44.1, 48 and 96 kHz," 100th AES Convention, ed., Copenhagen, Denmark, 1996.

56. Stuart J.R. and Wilson R.J., "Dynamic Range Enhancement Using Noise-Shaped Dither Applied to Signals with and without Pre-emphasis," 96th AES Convention, ed., Amsterdam, The Netherlands, 1994.

57. Stuart J.R. and Wilson R.J., "A Search For Efficient Dither for DSP Applications," 92nd AES Convention, ed., Vienna, Austria, 1992.

58. Vanderkooy J. and Lipshitz S.P., "Digital Dither: Signal Processing with Resolution Far Below the Least Significant Bit," AES 7th International Conference, Toronto, 1989.

59. Vanderkooy J. and Lipshitz S.P., "Dither in Digital Audio," J. Audio Eng. Soc. 35, pp 966-975, 1987.

60. Adams R., "Unusual Applications of Noise Shaping Principles," 101st AES Convention, ed., Los Angeles, U.S.A., 1996.

61. Hawksford M.O.J., "Chaos, Oversampling and Noise Shaping in Digital-to-Analog Conversion," J. Audio Eng. Soc. 37, pp 980-1001, 1989.

62. Lipshitz S.P., Vanderkooy J., and Wannamaker R.A., "Minimally Audible Noise Shaping," J. Audio Eng. Soc. 39, pp 836-852, 1991.

63. Lipshitz S.P., Vanderkooy J., and Semyonov E.V., "Noise Shaping in Digital Test-Signal Generation," 113th AES Convention, Los Angeles, U.S.A., 2002.

64. Wannamaker R.A., "Psychoacoustically Optimal Noise Shaping," J. Audio Eng. Soc. 40, pp 611-620, 1992.

65. Watkinson J., "The Art of Digital Audio", Focal Press, 2001.

66. Pohlmann K.C., "Principles of Digital Audio", McGraw-Hill Professional, 2002.

67. Mendoza-López J., Yong A.W., and Busbridge S.C., "Digital Transducer Array Loudspeakers With Dithering And Noise Shaping," European DSP Education And Research Symposium, Munich, Germany, 2006.

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68. Berkhout A.J., De Vries D., and Vogel P., "Acoustic Control By Wave Field Synthesis," J. Acoust. Soc. Am. 93, pp 2764-2778, 1993.

69. Boone M.M., Verheijen E.N.G., and Jansen G., "Spatial Sound-Field Reproduction by Wave-Field Synthesis," J. Audio Eng. Soc. 43, pp 1003-1012, 1995.

70. Daniel J., Nicol R., and Moreau S., "Further Investigations of High Order Ambisonics and Wavefield Synthesis for Holophonic Sound Imaging," 114th AES Convention, Amsterdam, The Netherlands, 2003.

71. De Vries D. and Boone M.M., "Wave Field Synthesis and Analysis Using Array Technology," Proc. 1999 IEEE Workshop on Applications of Signal Processing to Audio and Acoustics, New Paltz, New York, 1999.

72. Furness R.K., "Ambisonics: An Overview," AES 8th International Conference, ed., Washington, USA, 1990.

73. Gerzon M.A., "Periphony: With-Height Sound Reproduction," J. Audio Eng. Soc. 21, pp 2-10, 1973.

74. Gerzon M.A., "Ambisonics in Multichannel Broadcasting and Video," J. Audio Eng. Soc. 33, pp 859-871, 1985.

75. Rimell A. and Hollier M., "Reproduction of Spatialised Audio in Immersive Environments with Non-Ideal Acoustic Conditions," AES 103rd Convention, ed., New York, USA, 1997.

76. Kirkeby O., Nelson P.A., and Hamada H., "Virtual Source Imaging Using the Stereo Dipole," AES 103rd Convention, ed., New York, USA, 1997.

77. Nelson P.A. "A Review of Some Inverse Problems in Acoustics", 6[3], pp 118-134, 2001.

78. Nelson P.A., Orduna-Bustamante F., Engler E., and Hamada H., "Experiments on a System for the Synthesis of Virtual Acoustic Sources," J. Audio Eng. Soc. 44, pp 990-1007, 1996.

79. Poletti M.A., "Three-Dimensional Surround Sound Systems Based on Spherical Harmonics," J. Audio Eng. Soc. 53, pp 1004-1025, 2005.

80. Rumsey F., "Spatial Audio", Focal Press, 2001.

81. Ward D.B. and Abhayapala T.D., "Reproduction of a Plane-Wave Sound Field Using an Array of Loudspeakers," IEEE Transactions on Speech and Audio Processing 9, pp 697-707, 2001.

82. Nicol R. and Emerit M., "3D-Sound Reproduction Over An Extensive Listening Area: A Hybrid Method Derived From Holophony and Ambisonic," AES 16th International Conference, Rovaniemi, Finland, 1999.

83. Morfey C.L., "Dictionary of Acoustics", Academic Press, 2001.

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85. Heil C. and Urban M., "Sound Fields Radiated by Multiple Sound Sources Arrays," 92nd AES Convention, Vienna, 1992.

86. Toole F.E., "Loudspeakers and Rooms for Sound Reproduction - A Scientific Review," J. Audio Eng. Soc. 54, pp 451-476, 2006.

87. Morse P.M. and Ingard K.U., "Theoretical Acoustics", McGraw-Hill, 1968.

88. Freedman A., "Transient Fields of Acoustic Radiators," J. Acoust. Soc. Am. 48, pp 135-138, 1969.

89. Oberhettinger, "On Transient Solutions of the "Baffled Piston" Problem," Journal of Research of the National Bureau of Standards -B- Mathematics and Mathematical Physics 65B, pp 1-6, 1960.

90. Reibold R. and Kazys R., "Radiation Of A Rectangular Strip-Like Focussing Transducer : Part 2: Transient Excitation," Ultrasonics 30, pp 56-59, 1992.

91. Sherman C.H. and Moran D.A. "Transient Sound Field of Simple Arrays of Circular Pistons", 1, pp 1-33, Parke Mathematical Laboratories, Inc., 1966.

92. Stepanishen P.R., "Transient Radiation From Pistons In An Infinite Planar Baffle," J. Acoust. Soc. Am. 49, pp 1629-1638, 1970.

93. Muller G.G., Black R., and Davis T.E., "The Diffraction Produced by Cylindrical and Cubical Obstacles and by Circular and Square Plates," J. Acoust. Soc. Am. 10, pp 6-13, 1938.

94. Bews R.M. and Hawksford M.O.J., "Application of the Geometric Theory of Diffraction (GTD) to Diffraction at the Edges of Loudspeaker Baffles," J. Audio Eng. Soc. 34, pp 771-779, 1986.

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Appendix I DSK6713 GPIO Use

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Appendix I – DSK6713 GPIO Use

The DSK6713 DSP platform was used as a real-time bitstream generator in the implemented real-time DTA systems. Parallel bitstreams were output through the GPIO port. Its details of use are described here.

The DSK can be programmed in assembly or C / C++. Controlling ports and peripherals in the DSK is a relatively standard procedure: include the relevant header files, configure the peripheral, declare a handle to the peripheral, open the port and enable the pins. Relevant commands are provided to perform these tasks:

//IncludeGPIO header

#include "csl_gpio.h"

//Configure the GPIO

GPIO_Config MyConfig = 0x0000, /* gpgc */

0xFFFF, /* gpen */

0xFFFF, /* gdir */

0xFFFF, /* gpval */

0xFFFF, /* gphm */

0x0000, /* gplm */

0x0000 /* gppol */;

// Declare GPIO Handle:

GPIO_Handle hGpio;

//Open the GPIO

hGpio = GPIO_open(GPIO_DEV0,GPIO_OPEN_RESET);

//Enable GPIO pins

GPIO_pinEnable(hGpio,PinID);

//Set all pins as output

Current_dir = GPIO_pinDirection(hGpio,PinID,GPIO_OUTPUT);

Appendix I DSK6713 GPIO Use

- 141 -

//Send one sample to the GPIO

GPIO_write(hGpio,PinID, mysample); //Parallel digital output

/*////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////

The following program sends a square wave (0 to 3.6V) to GPIO pin 8:

////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////*/

//Include header files as indicated above

//Configure the GPIO as indicated above

int main()

GPIO_Handle hGpio;

CSL_init();

hGpio = GPIO_open(GPIO_DEV0,GPIO_OPEN_RESET);

GPIO_config(hGpio,&MyConfig);

GPIO_pinEnable(hGpio, GPIO_PIN8);

while (1)

GPIO_pinWrite (hGpio,GPIO_PIN8,0); //write 0 to GPIO

GPIO_pinWrite (hGpio,GPIO_PIN8,1); //write 1 to GPIO

Appendix II The Swept-Sine Technique

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Appendix II – The Swept-Sine Technique

The logarithmic swept-sine technique offers a very important advantage with respect to other existing electroacoustic measurement techniques: individual harmonic distortion components and total harmonic distortion can be accurately determined [96,97]. In addition, SNR is increased and obtained results are reliable at least for measurements in quiet and empty rooms [98]. This appendix will comment on the basics of this technique, which was employed throughout the thesis to derive THD figures and impulse response measurements.

Suppose a given device under test (DUT) which is excited by a logarithmic swept-sine going from f0 to f1 in T seconds. Calling the input sweep x(t) and the output sweep response y(t), a deconvolution is required to extract the DUT properties from the sweep response. This deconvolution can be made as the inverse Fourier Transform of the divided Fourier Transforms of output and input:

( )( )⎥⎦

⎤⎢⎣

⎡ℑℑ

ℑ= −

)()()( 1

txtytd

(0.1)

Optionally, the Fourier transforms of the input and output can be zero-padded to double their original length previous their division.

The result d(t) of the deconvolution is a sequence that contains the main impulse response (IR) of the system and the individual harmonic impulse responses (HIRs) to the left of the main IR peak. The distance in samples between the main peaks of two given harmonic orders A and B is given by:

sAB fTffBAsamplesD ⋅⋅=

)/(log)/(log)(

012

2

(0.2)

where f1 is the final frequency of the sweep, f0 is the initial sweep frequency, T is the length of the sweep in seconds and fs is the sampling rate in Hz [96].


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Figure 0.1. Input x(t) logarithmic sweep ranging 20 Hz to 24 kHz in 2 s with a 0.1 s fade-in and fade-out.

Figure 0.2. Input x(t) logarithmic sweep spectrogram


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Figure 0.3. Output y(t) logarithmic sweep response measured from a highly distorted system

Figure 0.4. Output y(t) sweep response spectrogram


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Figure 0.5. Deconvolved IR and HIRs, d(t).

Figure 0.6. Deconvolved IR and HIRs spectrogram [ d(t) ]

Once the HIR peaks are determined, all HIRs are windowed with a rectangular window of the same length. An FFT is taken for each HIR with a number of points much bigger than the original window size. A spectral shift is then required to compare the contributions of the different HIRs and thus obtain the individual harmonic distortion components. The spectral shift is equivalent to a division of the frequency vector by integer n for harmonic order n. In order to obtain say the third harmonic distortion component and plot it on the same axis as the second harmonic distortion component, the frequency vector must be divided by three and by two, respectively.

Appendix III Publications

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Appendix III – Publications

“Direct Acoustic Digital to Analogue Conversion with Digital Transducer Array Loudspeakers”, Journal of The Audio Engineering Society 55 (6), pp472-502, 2007.

“Sound Field Characterisation in Audio Reproduction With The Bit-Grouped Digital Transducer Array”, 120th A.E.S. Convention, Paris, France, May 06.

“Digital Transducer Array with Oversampling and Noise Shaping”, T.I. European D.S.P. Education and Research Symposium, Munich, Germany, Apr 06.

“Direct Acoustic Digital to Analogue Conversion with Digital Transducer Array Loudspeakers”, 118th A.E.S. Convention, Barcelona, Spain, May 05.

“Engineering Research in Action Conference Digest”, University of Brighton 2006, ISBN-13: 978-1905593002.

“Engineering Research in Action Conference Digest”, University of Brighton 2005, ISBN-10: 1905593007.

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