Some Considerations for Spectral Analysis of Delta-Sigma Data Converters

8/10/2019 Some Considerations for Spectral Analysis of Delta-Sigma Data Converters

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AbstractSpectral analysis with sinusoid input is a common

way to evaluate data converters, and is often used to specify the

signal-to-noise ratio, the harmonic distortion, and the effective

number of bits. If the input is arbitrarily sampled, the measured

output must be windowed to reduce DFT leakage effects that

obscure the spectrum. Alternatively, often preferably, coherent

sampling can be used to eliminate these leakage effects, hence

obliterating the need for windowing. In converters based on

delta-sigma modulation however, the noise shaping loop may

cause leakage effects to impair the measurement, even if input

leakage is eliminated. This paper demonstrates and explains this

effect, showing why coherent sampling does not guarantee useful

results when analyzing delta-sigma converters.

Index TermsData Conversion, Analog-to-Digital, Digital-to

Analog, Delta-Sigma, Sigma-Delta, Spectral Analysis, DFT

I. INTRODUCTION

IGH resolution data converters are increasingly being

based on delta-sigma modulation (DSM). Historicallymostly used for audio; DSM conversion has lately gained

foothold in higher bandwidth applications due to the increased

speed of modern integrated circuit (IC) processes. This paper

does not give a comprehensive review of the DSM and

assumes some beforehand knowledge. If confused the reader is

recommended to look in e.g. Schreiers textbook [1].

DSM conversion trades speed for resolution by combining

oversampling and loop filtering around the quantizer, pushing

the quantization noise out of the signal band. As such very

coarse quantization can be used while maintaining high

effective resolution. The input-output relation is given by:

Manuscript received July 29, 2008. This work was supported in part by the

Norwegian Research Council under Grant 162101 SPECK.

Ivar Lkken and Anders Vinje are with the Norwegian University of

Science and Technology, Department of Electronics and

Telecommunications, Trondheim, NO-7491, Norway (e-mail:

[email protected], [email protected]).

( ) ( )

( ) ( )

( ) ( )

( ) ( ) ( ) ( )

0

1 1

1

1 1

q

q

L zQ z X z E z

L z L z

STF z X z NTF z E z

= +

= +

(1)

STF and NTF are abbreviations for Signal Transfer

Function and Noise Transfer Function respectively, Eq is the

quantization error. Since having no delay-free loops is a

condition for realizability, it is given that l1[n]|n=0=0 or

ntf[n]|n=0=1. Thus the quantization error cannot be reduced in

total power, but given an appropriate loop filter it can be

shaped so that very little of it falls in-band.L0is chosen so that

the in-band STF approximates or equals unity.

II. THE DFT,LEAKAGE AND WINDOWING

The spectrum of a discrete-time sequence is defined through

a special case of the z-transform, the Discrete Time Fourier

Transform (DTFT):

{ } [ ]( )def

n

n

DT FT x X x n e

=

= !i (2)

To be computable the DTFT has to be of finite length so the

spectrum must be sampled onto a discrete frequency variable

and out of a finite length sequence. If being of length Nwith

equidistant sampling we get (3) which is the definition of the

N-point finite DTFT or Discrete Fourier Transform:

{ } [ ]21

0

( ) , 0,1, , 1knNdef

NN

n

DFT x X k x n e k N

=

= = !i

! (3)

In cases where the available signal sequence is shorter than

N it can be zero-padded to obtain an N-point DFT [2]. When

implemented for simulation, measurement, or other purposes,

the DFT is in most cases computed using a Fast Fourier

Transform (FFT) algorithm [3]. FFT algorithms often require

Nto be a power of two which if necessary can be ensured with

zero-padding. Note that zero-padding does not provide any

additional information about the spectrum.

The DFT will have incongruities compared to the DTFT of

a general function. If the input signal xin[n] is a function

defined in n"-!,!#, picking a limited sample set of length Nto obtain (3) equals the multiplication of this function with a

rectangular window w[n] of lengthN.

Some Considerations for Spectral Analysis of

Delta-Sigma Data Converters

Ivar Lkken, Anders Vinje

H

Fig.1. Delta-Sigma Modulator

ISAST Transactions on Electronics and Signal Processing, No. 2, Vol. 3, 2008

Ivar Lkken and Anders Vinje: Some Considerations for Spectral Analysis of Delta-Sigma Data Converters19


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Shown in fig.2, this can generally be written as:

[ ] [ ] [ ] [ ]1 , 0 1

,0 , otherwise

def

in

n Nx n x n w n w n

$= = %

& (4)

Since time-domain multiplication as known is the dual of

spectral convolution, the equivalent DTFT becomes:

( ) ( ) ( ) ( )( 1)

2

sin2

,

sin2

N

in

N

X X W W e

' () *+ ,

= =' () *+ ,

i (5)

If the input is a sinusoid xin[n]=sin(!xn), the spectrum

should be zero everywhere except !=!x. But because a finite

length sample set of this function is spectrally convolved with

a window, the resulting spectrum is nonzero also elsewhere.

This is known as spectral leakage. The spectral leakage is

sampled in the DFT causing leakage effects that obscure the

simulated spectrum. Figure 3 illustrates leakage effects for a

DFT whereN=64. This DFT is not very usable.This situation can be improved by multiplying the signal

with a dedicated window function that is not rectangular,

known as windowing. The leakage effect can be understood as

an edge effect in that it is the sharp edges that cause the lobes

in the spectrum of the rectangular window, and hence it is the

sampled endpoint discontinuities of the convolution product

(see fig.2) that give rise to leakage effects. A tapered window

has smaller spectral side lobes and a broader main lobe. It

apodizes the signal by reducing the sharp edge discontinuities,

and thus makes the leakage more narrowband. Windowing is

conceptually illustrated in fig.4.

Figure 5 shows the spectrum of a sinewave convolved with a

hann window [4], perhaps the most popular window function.It is seen how side lobe leakage is suppressed at the cost of a

wider main lobe. The main lobe width limits the frequency

resolution of the DFT since it obscures a given frequency

range, whereas the side lobe height limits the dynamic range

since side lobes obscure spectral information below a given

amplitude. Many different window functions with different

properties exist [5], and in general the choice of window will

be a trade off between frequency resolution and dynamic

range. A longer window obviously improves this tradeoff, but

what window is best depends on the application of the DFT.

Fig.4. Acquisition of limited length sequence with windowingFig.2. Acquisition of limited length sequence

Fig.5. Sinewave convolved with hann windowFig.3. Spectral leakage in the sampled DFT




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III. COHERENT SAMPLING AND ELIMINATION OF DFT

LEAKAGE EFFECTS

Even though proper windowing can improve the usefulness

of the DFT a lot, the resolution of the sampled spectrum is still

limited. For good performance estimation of high accuracy

circuits, like hi-res data converters, the DFT has to be long

also when windowing. Fortunately leakage effects in the DFT

of a sinusoid can be eliminated with coherent sampling[6].

Coherent sampling ensures that the sinewave has an integer

number of periods within the acquisition. A sequence of length

N contains exactly K periods of a sinusoid function

xin[n]=sin(!xn) if its angular frequency is:

2x

K

N

= (6)

Setting Kan integer of choice, (6) can be solved for !x. The

physical frequencyfxfor a given sampling frequencyfscan be

found from the angular frequency definition, !=2"f/fs.

x s

Kf f

N= (7)

When periodic in N/K the waveform is also periodic in N,

meaning that the edges of the rectangular window do not cause

endpoint discontinuities, as illustrated in fig.6. This in turn

means there is no side lobe energy in the correspondingly

sampled spectrum, which can be shown by correlating the

sinewave with the basis functions of the DFT (3). For DFT

sample k=Kthe correlation between the basis function and thesignal is exactly one and otherwise it is zero.

As an example: Assume an ADC designer wants to simulate

the performance for 1MHz input and 100MHz sampling rate,

using a DFT of lengthN=214

. The closest integer is K=164 and

the closest rational frequencies for coherent sampling are then

fx=1.000.072Hz and fs=99.690.104Hz. An illustration with a

much shorter DFT ofN=64 to make its samples clearly visible

is shown in fig.7. One can see how leakage is not sampled in

the DFT and leakage effects are hence eliminated.

To maximize the probability of detecting local integral non-

linearities (INL) in the converter and see them as distortion, it

has been recommended to useprime sampling [7]. This simply

means coherent sampling where Kis a prime number. Then amaximum number of converter codes are used by the sequence

since its periodicity is irreducible.

IV. COHERENT SAMPLING AND DELTA SIGMA MODULATORS

Having explained coherent sampling as an alternative to

windowing we will proceed to investigate the special case of

delta-sigma modulators. Unfortunately a sinewave analysis of

a DSM converter can be impaired by leakage effects even if

coherent sampling is used for the input signal. Spectral

leakage from the modulators powerful out-of-band noise may

cause leakage effects in the DFT, which could lead a data

converter designer relying on coherent sampling to think there

are errors in the circuit or simulation setup. Additionally a very

small change of input amplitude or frequency can make these

artifacts vanish and they may appear in a seemingly randomway. This could cause confusion if the circuit designer is not

aware of the issues to be presented. To the authors knowledge

these considerations of DFT analysis and coherent sampling

specifically with regards to DSM converters, have not been

seen in any previous literature.

If we consider again the DSM of fig.1; its output signal Q,

which is what we want to analyze, consists of two components

as seen in (1). One is the signal component Xweighted by the

STF, the other is the quantization error Eq weighted by the

NTF, or in other words the shaped quantization noise.

Fig.7 Elimination of leakage effects in DFT with coherent sampling

Fig.6. Acquisition with coherent sampling Fig.8. Output amplitude spectrum of DSM designed for 16 times OSR.




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For convenience the STF is assumed to be unity in the signal

band. Even though the quantization error is a deterministic

function of the quantizer input, it is normal to approximate it

as an independent white noise source [8] to enable relativelystraightforward loop filter design. The amplitude response of

the DTFT will look something like fig.8, showing the output of

a DSM designed for 16 times oversampling (OSR). It has a

third order loop filter and a four bit quantizer.

It is desirable to analyze the performance, in particular the

signal-to-noise ratio (SNR) and distortion in the baseband. In

the baseband the shaped quantization noise is very small and

the DFT will need high dynamic range since the DSM has high

dynamic range. In addition the baseband is only a small part of

the Nyquist range meaning the frequency resolution also needs

to be high. To just capture the output and do a DFT is

pointless; fig.9 shows the simulated spectrum withN=214

. It is

completely corrupted by signal leakage, for decent resolution a

rectangular window would have to be enormously long. With

hann windowing the situation is a lot better, but from fig.10 it

is still seen that leakage to some extent obscures baseband

information. For this combination of N, OSR and SNR the

DFT is probably useful for error analysis, but in e.g. an audio

ADC the resolution will be much higher and with higher OSR

fewer DFT samples will be inside the baseband. For low-level

simulations, in particular transistor level, N=214

is also high

and simulation time may mandate a shorter DFT.

Considerations for DFT analysis of ADCs using short

windows has been explored in previous publications [9]-[10].

For simulations on a regular Nyquist ADC, these window

issues need not be a concern since one can use coherent

sampling and eliminate leakage effects, allowing for shorter

simulations and higher resolution. However in a DSM theremight be problems even if doing so. Figure 11 shows a

simulation for the same conditions as fig.10, but with the input

frequency moved to the nearest coherent sampling value. As

seen signal leakage is gone, but the simulation is still corrupted

by another leakage effect causing a white in-band error.

To explain this we first revisit the edge effect: In a Nyquist

converter the quantizer floors or rounds the input signal,

meaning that if an input sinewave is periodic in N, the

quantizedoutput is also periodic inN. Coherent sampling thus

works fine. In a delta-sigma modulator on the other hand, the

quantization error varies quickly and seemingly randomly

around the signal. The error is of course not really random

since the quantizer is a deterministic nonlinear function, and it

often has signal correlation artifacts such as limit cycles and

idle tones [11]. But since nonlinear feedback loops are

extremely difficult or impossible to predict analytically the

random quantizer model is used for most practical purposes.

Attributing it to the random quantizer assumption or not, the

acquisition of a limited length sample set from a DSM may

have an edge error even if the input is coherently sampled.

Figure 12 shows such an occurrence: The input is periodic in

N=100 but due to an additional quantization error the DSM

output is not. Observe that q[0]=0 whereas q[N]=-1, meaning

that the output is clearly not periodic inN.

Fig.12. Endpoint error at DSM output

Fig.11. Simulated DFT of DSM output, coherent sampling,N=214Fig.9. Simulated DFT of DSM output, no windowing,N=214

Fig.10. Simulated DFT of DSM output, hann windowing,N=214




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That q[N] is nonzero even thoughx[N] is zero is as noted not

a result of random quantization but of the DSM having

memory. If unity STF, the inversez-transform of (1) yields:

[ ] [ ] [ ] [ ]0

q

k

q n x n ntf k e n k

=

= + ! (8)

This means that q[N] with coherent sampling (x[N]=0 and

eq[N]=0) will be:

[ ] [ ] [ ]1

q

k

q N ntf k e N k

=

= ! (9)

The DSM may have many samples memory meaning more of

the shaping response is lost in the acquisition than just the

zeroing of q[N]. This can be revealed by zero-padding the

input (x[n]|n#N=0 and eq[n]|n#N=0) and observe the outputovershoot as in fig.13. It is seen that the output takes a while to

settle; meaning quite a lot of the NTF response is cut off by

rectangular windowing. Denoting it the window error ew:

[ ] [ ] [ ]1

w q

k n

e n ntf k e N n k

= +

= + ! (10)

Extending N to 214

again, fig.14 shows the same DFT as

fig.11 together with the spectrum of the window error, or

overshoot beyondNfound by zero-padding the input.

It is seen that the window error or loss of filter response

samples, indeed isthe leakage effect impairing the DFT.

Since the quantization error sequence superimposed on the

input is random and generally cant be derived analytically,

the overshoot cannot be known a priori. Assuming the

quantization error eqin (10) is random can enable a prediction:

1 2 3

2 3

3

[1] [0]

0 [2] [1]

0 0 [3] [2]

q q q w

q q w

q w

e e e ntf e

e e ntf e

e ntf e

- . - . - ./ 0 / 0 / 0/ 0 / 0 / 0 =/ 0 / 0 / 0/ 0 / 0 / 01 2 1 2 1 2

!

!

!

" " " # " "

(11)

Where eqis a random variable with eq=0 and $eq2=1/12 [8].

Note that this is just a crude estimate. Simulations reveal that

the error will change if the input is changed, for instance some

input signals will result in the modulator not giving overshoot

at all in which case the DFT spectrum turns out correct. A

slight change of the input amplitude may cause the DSM to

take a completely different trajectory in its loop state space

(see e.g. [1], [11] or [12] for details), and since it is not

possible to know what error will occur it is thus recommended

as a safeguard to use both coherent sampling and windowing

when doing spectral analysis of a DSM converter.

The filter overshoot dies away quite rapidly, meaning that

the whole window error is near the end values of a periodic

extension of the DFT (although not merely an instantaneous

discontinuity as for a non-coherently sampled sinewave). In

other words the noise leakage is very effectively suppressed by

windowing. Figure 15 shows the spectral amplitude response

of the exact same output sequence as fig.11, but now

multiplied with a hann-window before doing the DFT. Then

the noise leakage is suppressed by the apodization that the

tapered window performs on the sequence, to the point of

being invisible since it is far below the in-band quantization

noise. The spectrum then as expected looks correct.

V. CONCLUSION

In spectral analysis of data converters, leakage effects in the

DFT may obscure the spectrum and mask the errors it is

desired for the simulation or measurement to reveal. This must

Fig.15. Simulated DFT of the same output sequence as fig.11, but now

hann-windowed

Fig.14. DSM output and filter overshoot spectra

Fig.13. Output overshoot caused by DSM memory




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be alleviated with windowing or coherent sampling. Since a

windowed DFT in high resolution applications still has to be

long to give reliable results, coherent sampling has become the

preferred method in ADC design. This is especially the case

for low-level analog or mixed-mode circuit simulations wherethe sequence length must be very limited to prevent excessive

simulation time.

In delta-sigma based data converters, which are becoming

increasingly popular for a variety of applications, a DFT may

exhibit leakage effects that compromise the simulation results

even if coherent sampling is used. The shaped DSM

quantization noise has low baseband power but high total

power, and abruptly cutting off the loop output before its

response has died away may cause a leakage of shaped noise

into the baseband that obscures the actual performance.

As a safeguard it is therefore strongly recommended to use

both coherent sampling to prevent signal leakage, and

windowing to prevent quantization noise leakage, when doing

DFT-analysis of delta-sigma data converters.

REFERENCES

[1] R. Schreier, Understanding Delta-Sigma Data Converters, Wiley &

Sons, IEEE Press, ISBN 0-471-46585-2, 2005.

[2] A. V. Oppenheim, R. W. Schafer, Discrete-Time Signal Processing,

Second Edition, Prentice Hall Inc., ISBN 0-13-754920-2.

[3] P. Duhamel, M. Vetterli, Fast Fourier Transforms: A Tutorial Review

and State of the Art, J. Signal Processing, vol.19, no.4, pp.259-299,

Apr. 1990.

[4] R. B. Blackman and J. W. Tukey: "Particular Pairs of Windows." In The

Measurement of Power Spectra, From the Point of View of

Communications Engineering.New York: Dover, 1959.

[5] F. J. Harris, On the use of Windows for Harmonic Analysis with the

Discrete Fourier Transform, Proc. of the IEEE, vol.66, no.1, pp. 51-83,Jan. 1978.

[6]

Maxim Application Note APP1040, Coherent Sampling vs. Window

Sampling, http://pdfserv.maxim-ic.com/en/an/AN1040.pdf, Mar. 2002.

[7]

J. Blair, Histogram Measurement of ADC Nonlinearities Using Sine

Waves, IEEE Trans. Instrumentation and Measurement, vol.43,

pp.373-383, June 1994.

[8] R. M. Gray, Quantization Noise Spectra,IEEE Trans. Inform. Theory,

vol.36, Nov. 1990.

[9] O. M. Solomon, The use of DFT windows in signal-to-noise ratio and

harmonic distortion computations, IEEE Trans. Instrum. Meas.,

vol.43, pp. 194-199, April1994.

[10] P. Carbone, E. Nunzi and D. Petri, Windows for ADC dynamic testing

via frequency-domain analysis,IEEE Trans. Instrum. Meas., vol.50,

pp. 1571-1575, December 2001.

[11] J.Reiss, Understanding Sigma-Delta Modulation: The Solved and

Unsolved Issues,J. Audio Eng. Soc., vol.56, no.1/2, pp.49-64, January

2008[12] H.Wang, A Study of Sigma Delta Modulations as Dynamical

Systems, PhD Thesis, Colombia University, New York, AAT 9333879,

1993

Ivar Lkkenwas born in Lillehammer, Norway, in 1979. He received a B.Sc.

degree in electrical engineering from Sr- Trndelag University College,

Trondheim, Norway, in 2002 and an M.Sc degree in electrical engineering

from the Norwegian University of Science and Technology, Trondheim, in

2004. He is presently with the Circuit and Systems Group at the Norwegian

University of Science and Technology, working toward a Ph.D. degree on

high-resolution audio digital-to-analog converters. His research interests

include audio data converters and audio signal processing.

Anders Vinje was born in Trondheim, Norway, in 1979. He received an

M.Sc. degree in electrical engineering from the Norwegian University of

Science and Technology, Trondheim, Norway, in 2004. He is presently with

the Circuit and Systems Group at the Norwegian University of Science and

Technology, working toward a Ph.D. degree on high-speed, high-resolution

analog-to-digital converters. His main research interests include analog-to-digital converters and related signal processing.



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Some Considerations for Spectral Analysis of Delta-Sigma Data Converters