ECE 4680 DSP Laboratory 6: Signal Generation Using DDS

Lab6_12.fmDue 12:15 PM Friday, December 12, 2014
Introduction and Background Signal processing systems, in particular communications systems, need to generate signals in addition to processing them. Text Chapter 5, entitled Signal Generation, drills down on this important topic. Signal generation requirements might be for sinusoids, pulse type signals, pseudo-random data, or noise waveforms, to name a few. In this lab the focus will be on the generation of sinusoidal signals using what is known as direct digital synthesis (DDS) [1]–[3]. A application case study is described in Part II, which is a communications receiver for frequency modulation (FM). The FM receiver implements complex frequency translation of the input signal in order for demodulation to be performed at complex baseband.
Part I: Direct Digital Synthesis The text describes two means of sinusoidal signal generation: (1) direct digital synthesizer (DDS) and (2) the digital resonator. The digital resonator uses an IIR filter with poles located on the unit circle that is excited by impulse to start the oscillation. The focus here is the DDS technique, as it is quite popular in communications transmitter and receivers.
The Voltage Controlled Oscillator as Motivation The DDS is motivated by the voltage controlled oscillator (VCO), which is used as sinusoidal signal generator in analog electronics [1]. A VCO has output frequency, , that is proportional to the input control voltage, plus the quiescent frequency .Working from the VCO block diagram of Figure 1, you have
(1)
with VCO gain constant having units of Hz/v.The total phase of the VCO output, , is related to the instantaneous frequency of the VCO as
fi t( ) e t( ) f0
φ t( ) 2πKv e λ( ) λd ∞–
t

Kv θ t( )
(2)
where is the VCO quiescent frequency. From the above equations you can now draw the behav- ioral level block diagram of Figure 2.
Note when the instantaneous frequency becomes Hz.
Converting the VCO Model to the Discrete-Time Domain In the discrete-time domain consider a sampling rate of , with being the sampling period or spacing. You have
, (3)
where the discrete-time phase is .The integrator is replaced by an accumulator when you consider approximating the integral via rectangular areas. This is shown in Figure 3.
With the integration above understanding you can write
(4)
where rad/sample. The recursive form for in the last line of (4) motivates the discrete-time form of the VCO shown in Figure 4.
fi t( ) 1 2π ------ td
d = θ t( ) f0 Kve t( )+=
f0
2πKv ( )cos1 s---
fs 1 T⁄= T
φ n[ ] φ nT( )=
a n 1–( )T( )a t( )
1 s---
n 1–
=
Figure 3: Discrete-time integration approximation using the left endpoint for a.
φ n[ ] φ nT( ) 2πKvT e nT( ) k ∞–=
n 1–
kv 2πKvT= φ n[ ]
ECE 4680 DSP Laboratory 6: Signal Generation Using DDS
As a fixed frequency generator let and set or to the desired quiescent frequency . This behavior is similar to the analog VCO. Note the discrete-time VCO is sometimes referred to as a numerically controlled oscillator (NCO), but more typically is the idealized mathematical form of the DDS. The output equation for the NCO/DDS when is
(5)
(6)
C-Code Implementation Using Floats On the OMAP-L138 a near ideal DDS implementation is possible since float-point arithmetic is available and the math.h library can be used to for on-the-fly calculation of sine and cosine using sinf() and cosf(). The code snippets below produce an example of how this can be done:
#include <math.h> // added for trig function support
// DDS variables
float theta = 0;
...
/////////////////////////////////////////////////
2πf0T
ω0 n φ n[ ]+⋅
Can make output a mod(2π) value due to cos( )
e n[ ] 0= f0 ωˆ 0 2πf0 fs⁄= 0 f0 fs 2⁄< <
e n[ ] 0=
θ n[ ] ω0n=
θ n 1+[ ] ω0 n 1+( ) ω0n ω0+ θ n[ ] ω0+= = =
Part I: Direct Digital Synthesis 3
...
}
From the above code snippet you see that the DDS quiescent frequency is determined by choosing
(7)
Practical Implementation Considerations Most DDS implementations, in particular ASIC and FPGA forms, utilize fixed-point arithmetic. Finite precision impacts include
• Bit width of the accumulator; controls the ultimate frequency precision or smallest frequency step size via Hz, where is the sampling rate and is the accumulator bit width
• The size of the sine/cosine look-up-table (LUT); the bit width here is typically reduced from the accumulator bit width, that is , this the table contains at most entries
• The storage precision or bit width of the sine/cosine values:
A modified DDS block diagram, that includes finite precision attributes is shown in Figure 5. For
an accumulator input step size (an integer value), the DDS output frequency is
Hz (8)
. (9)
Note in [3] and elsewhere is referred to as the DDS clock frequency, .
A MATLAB simulation of this system is the following:
function [x,a_out,n] = DDS(f0,fs,N_samps,Bcos,Bacc,Bw)
fΔ fs 2Bacc⁄= fs Bacc
Bw Bacc< 2Bw
Bacc Bw Bcos
Input Output NΔ
% f0 = desired output frquency
% Bw = bit width of the LUT address
% //////////////// Outputs ///////////////////
% n = time index
if a >= 2^Bacc
a = a - 2^Bacc;
end
To see the DDS simulation in action suppose kHz and consider 32 bits (very large) for all of the bit widths and compare that to the case and :
>> [x,a_out,n] = DDS(14,48,2^16,32,32,32);
>> plot(n(1:50),a_out(1:50),'r.')
The idealized result (32 bits everywhere) is shown in Figure 6 and the reduced bit width results are shown in Figure 7. As the bit width is reduced spurious outputs (spurs) start to appear. This is a result of the finite precision arithmetic involved in the design.
fs 48= Bacc 32 Bw, 12= = Bcos 14=
Part I: Direct Digital Synthesis 5
Further discussion of DDS spurs can be found in the appendix of this document.
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Frequency (kHz)
Figure 6: DDS output at 14 kHz for kHz and 32-bits in three loca- tions.
fs 48=
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Frequency (kHz)
Figure 7: DDS output at 14 kHz for kHz, , and .fs 48= Bacc 32 Bw, 12= = Bcos 14=
spurious outputs
spurious outputs
The 32-bit accumulator output for kHz and kHz is shown in Figure 8.
What to Expect in Your OMAP-L138 Implementation When you implement the DDS using floats on the OMAP-L138 you will obtain results similar to
f0 14= fs 48=
Figure 8: DDS accumulator output for kHz for kHz, , and .
f0 14= fs 48= Bacc 32 Bw, 12= = Bcos 14=
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0.2
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1
those shown in.
Part II: FM Communications Receiver To put the DDS of Part I to good use, you now explore a communications receiver utilizing complex frequency translation. The sine and cosine signals of a DDS form a complex sinusoid that is used to complex frequency translate the input/received signal from 30 kHz to baseband ( ). Demodulation is then performed on the now complex signal to recover the message signal.
Receiving System Details The signal of interest is a frequency modulated (FM) carrier waveform given by
Figure 9: DDS output at 1 kHz from the OMAP-L138 with kHz using real-time floats.
fs 48=
About 88 dB to the noise floor
About 78 dB spurious-free dynamic range (SFDR)
f 0=
. (10)
You study this waveform in detail in ECE 4625/5620, Communications Systems I. The signal carrier of center frequency is Hz. The information or message carried by the signal is . The message signal can be analog voice or music, or a digitally encoded information. The approximate frequency spectrum is shown in Figure 10. Mathematically the spectrum of takes the form
(11)
where is the complex baseband spectrum corresponding to . Note is really just
. (12)
To get your hands on all you need to do is complex frequency translate (10) either to he left or right by Hz and lowpass filter to one half the FM RF bandwidth,
. (13)
I will later chose to make the translation frequency negative to move the spectrum to the left. If in doubt, recall from Fourier transform theory that
(14)
where . The second step is to demodulate the message from . In communication systems you learn that a frequency discriminator of some sort is required. Here I will use a DSP implementation fits well with the overall receiver architecture.
DSP Receiver Implementation A DSP based receiver utilizing the capabilities of the Zoom OMAP-L138 board is shown in Fig-
xc t( ) Ac 2πfct 2πfd m λ( ) λd
t
+cos=
xc t( )
Xc f( ) 1 2--- XBB f fc–( ) XBB f fc+( )+[ ]=
XBB f( ) xc t( ) xBB t( )
xBB t( ) Ac 2πfd m λ( ) λd
t
exp=
⋅
=
X f( ) FT x t( ){ }= m t( ) xBB t( )
Part II: FM Communications Receiver 9
ure 11. The main signal flow passes from put to output using the left audio channel. The sampling rate is set to 96 ksps, which is the fastest rate supported by the audio codec.
Since the complex frequency translation is performed on the sampled input signal, , there are a some differences due to spectral images. Consider Figure 12 to help visualize how sampling followed by complex frequency translation and a lowpass filter achieves the intended result.
Note that the spectrum of a complex signal is not symmetrical (in magnitude) about . Com- munications applications of real-time DSP typically involve complex signals.
Complex Baseband Discriminator
The complex baseband signal for the case of an FM input signal, is of the form
. (15)
Note . The frequency discriminator seeks to find the derivative of the phase . In the continuous-time domain the derivative of the inverse tangent function is
ADC IIR LPF
N = 4 DAC
1 2 11
Figure 11: Zoom (also LCDK) OMAP-L138 FM receiver block diagram.
fs 96 kHz=
fs 96 kHz=
xc t( ) xc n[ ] x n[ ] y n[ ] z n[ ]
z t( )
Discrimin.
48-48-96 9630 66-30-66
Principle alias band
Figure 12: The spectrum (a) following sampling and (b) following frequency translation when kHz and kHz (shift left).
xc t( ) fs 96= fc 30=
f (kHz)
Xc e j2πf fs⁄
( )
f 0=
y n[ ]
y n[ ] Ace jφ n[ ] Ac φ n[ ]( )cos j φ n[ ]( )sin+{ } yI n[ ] jyQ n[ ]+= = =
φ n[ ] tan 1– xQ n[ ] xI n[ ]⁄( )= φ n[ ]
. (16)
. (17)
Here the denominator of (17) can be omitted since the magnitude squared of the FM signal is a constant.
MATLAB Simulation
To verify the receiver design before writing and C-code, I construct a MATLAB simulation and drive the input with a real FM signal. The message signal will be a 1 kHz sinusoid deviating the 30 kHz carrier by about 1.5 kHz peak. To generate an FM signal you can use a modified version of the DDS m-code, FM_gen.m:
function [x,a_out] = FM_gen(e,f0,fs)
The complex baseband discriminator is implemented in the function discrim.m:
function disdata = discrim(x)
% function disdata = discrimf(x)
% Mark Wickert
X=real(x); % X is the real part of the received signal
Y=imag(x); % Y is the imaginary part of the received signal
N=length(x);% N is the length of X and Y
b=[1 -1]; % filter coefficients for discrete derivative
a=[1 0];
yI t( )yQ' t( ) yQ t( )yI' t( )–
yI 2 t( ) yQ
2 t( )+ ---------------------------------------------------------= =
z n[ ] yI n[ ] yQ n[ ] yQ n 1–[ ]–( ) yQ n[ ] yI n[ ] yI n 1–[ ]–( )⋅–⋅
yI 2 n[ ] yQ
derY=filter(b,a,Y); % derivative of Y,
derX=filter(b,a,X); % derivative of X,
disdata=(X.*derY-Y.*derX)./(X.^2+Y.^2);
The function DDS.m is used to generate a complex sinusoid for frequency translation and the function butter() us used to generate the lowpass filter coefficients. The complete simulation is as follows:
>> [x,a_out,n] = DDS(30,96,10000,32,32,32); % Use a_out to form both cos & sin
>> cpx_car = cos(2*pi*a_out) - j*sin(2*pi*a_out); % form exp[-j*2*pi*30/96*n]
>> e = 2000*cos(2*pi*1000/96000*n); % The 1 KHz message signal, 2 kHz peak dev
>> xc = FM_gen(e,30000,96000); % The 30 kHz FM carrier
>> x = xc.*cpx_car; % The complex frequency translated signal
>> [b,a] = butter(4,2*10/96); % The 4th-order lowpass filter with fc = 10 kHz
>> y = filter(b,a,x); % The filtered complex signal, also complex
>> z = discrim(y); % The discriminator output
The input spectrum, the frequency translated spectrum, and the filtered spectrum are shown as subplots in Figure 13.
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0
20
−40 −30 −20 −10 0 10 20 30 40 −40
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0
20
(a)
(b)
(c)
Figure 13: The spectra at (a) the input, (b) following complex frequency transla- tions, and (c) after the lowpass filter, in the MATLAB simulation of the FM receiver.
image spectrum
desired spectrum
received signal
The final discriminator output signal is given in Figure 13.
All is well, so now the system can be implemented on the Zoom OMAP-L138 with confidence.
OMAP-L138 Sample Outputs
A complete implementation on the OMAP-L138 is tested using the Analog Discovery for the FM signal source and the scope and spectrum analysis capabilities. An FM signal with a sinusoidal message at 1 kHz and 30 kHz carrier is configured as shown in Figure 15.
z n[ ]
−0.1
−0.05
0
0.05
0.1
0.15
0.2
The 1kHz message sinusoid
Figure 14: The 1 kHz message signal recovered at the output of the discriminator in the MATLAB simulation.
Filter transient
The signal captured at the output of the complex baseband discriminator is shown in Figure 16. The spectrum is also shown so that you can see that the 4th-order lowpass filter has not com- pletely removed the image signal at 36 kHz. The image signal in fact manages to make its way through the discriminator.
Figure 15: Setting up the function generator in Analog Explorer to produce an FM carrier at 30 kHz with a 1 kHz sinusoidal message at peak deviation of 0.05 x 30 kHz = 1.5 kHz.
Expectations When completed, submit a lab report which documents code you have written and a summary
of your results. Screen shots from the scope and any other instruments and software tools should be included as well. I expect lab demos of certain experiments to confirm that you are obtaining the expected results and knowledge of the tools and instruments.
The ZIP package Lab6.zip, explained in the appendix, contains m-code simulation func- tions and a CCS project entitled Zoom_DDS.
Problems 1. Implement the DDS code described at the bottom of p. 3 on the OMAP-L138. Choose the
sampling frequency to be 48 kHz. Modify this code so that you can control the output frequency of the sinusoids (cos/sin) via a GEL slider to step from 1 Hz to 20 kHz in 1 Hz steps. The GEL file can implement float calculations, so you can have the slider select the desired sinusoid frequency in Hz and have the GEL convert the value to a float .
1 kHz message
36 kHz leakage
Recovered 1 kHz sinusoid from
Figure 16: OMAP-L138 output for a 1 kHz sinusoidal message at 1.5 kHz peak frequency deviation and kHz.fs 96=
30 kHz FM carrier
ω0 2πf0 fs⁄=
I will ask you to demo this via the spectrum analyzer (Agilent 4395A in spectrum analyzer mode). Note that sinf()/cosf() is used instead of sin()/cos(), what is the differ- ence in the context of ANSI C?
2. Experimentally find the ISR service time of the simple DDS of Problem 1 using the digital I/ O technique first described in Lab 4. Initially assume no optimization then try -o3 optimization and note the improvement.
3. In Problem 1 the basic DDS calculation is of the form
ADC_output = (short) 32000*cosf(theta);
An efficient DDS uses a LUT in place of this on-the-fly cosine calculation. In this problem, rather than implementing an actual LUT, you will emulate one by doing some fixed-point conversions in the argument of cosf(). To start with note that the 32000 scales the cosf() value to lie near the max amplitude range of a short signed integer (recall [- 32768, 32767]). The only fixed-point quantization that is taking place presently is casting the cosine values from float to short as they are output to the audio codec (specifically the DAC).
To emulate the LUT consider a rework of the original code to set up a float accumulator run- ning from [0, 1), but quantized in the argument of cosf(). The code below increments the accumulator a by (in code f0_fs), for . The argument of cosf() is , where is a W bit quantizer implemented as
. (18)
Note Short[ ] represents casting a float value to a Short integer. The returned float value of (18) is again float because is represented as a float constant. The returned value is how- ever quantized. The modified DDS c-code is given below:
... // DDS variables #define two_pi 6.283185307179586 #define W 16 float a = 0; float f0_fs = 0.020833333333333; // f0/fs = 1000/48000 ...
interrupt void Codec_ISR() {
/* add any local variables here */ WriteDigitalOutputs(1); // Write to GPIO J15, pin 6; begin ISR timing pulse float codecInLeft, codecInRight, codecOutLeft, codecOutRight; float a_scale_p = pow(2,W); // emulate a table size of 2^W entries float a_scale_m = pow(2,-W); // using these scaling constants short a_short;
if(CheckForOverrun())// overrun error occurred (i.e. halted DSP) return; // so serial port is reset to recover
0 f0 fs⁄ 1< < f0 fs⁄ 1000 48000⁄= 2π QW f0 fs⁄( )⋅ QW ( )
QW x( ) Short x 2W⋅[ ] 2 W–⋅=
2 W–
Problems 16
CodecDataIn.UINT = ReadCodecData();// get input data samples
/* add your code starting here */ codecInLeft = CodecDataIn.Channel[ LEFT]; codecInRight = CodecDataIn.Channel[ RIGHT]; codecOutRight = codecInRight; // This input channel will not be used
///////////////////////////////////////////////// // DDS a_short = (short)(a*a_scale_p+ 0.5); codecOutLeft = 32000*cosf(two_pi*a_short*a_scale_m); a += f0_fs; if (a >= 1) a -= 1.0; /////////////////////////////////////////////////
CodecDataOut.Channel[ LEFT] = (short) codecOutLeft; CodecDataOut.Channel[RIGHT] = (short) codecOutRight; /* end your code here */
WriteCodecData(CodecDataOut.UINT);// send output data to port WriteDigitalOutputs(0); // Write to GPIO J15, pin 6; end ISR timing pulse
}
a) Using the spectrum analyzer mode of the Agilent 4395A, characterize the spectrum quality of the 1 kHz output signal for W=16. The quantity of interest is the spurious free dynamic range (SFDR) as shown in Figure 9. The SFDR measures the maximum dynamic range between the signal of interest and any adjacent spurs. The ISRs code DDS_ISRs_p3.c has the frequency set to 1 kHz. Keep that setting for all measure- ments.
b) Repeat part (a) for W=12.
4. In this problem you will explore the FM receiver design of Part II. As an FM signal source you will use the internal FM capability of the Agilent 33250 function generator found on your lab bench. You lab instructor will you with the set-up of the generator.
a) Write C code to implement the FM receiver described in Figure 11. As a starting point find the file FM_Demod_ISRs.c as the starting point.
b) Configure the Agilent 33250 to produce a sinusoidal FM signal having kHz, a modulation frequency of 1 kHz, and a peak frequency deviation in the range of 1 to 2 kHz.
c) Experiment with mistuning the frequency of the DDS relative to the known FM signal carrier at 30 kHz. How far above and below 30 kHz can you tune the DDS without resulting in a heavily distorted demodulated 1 kHz sinusoid?
fc 30=
Problems 17
sey, 2009.
[2] Analog Devices MT-85 Tutorial, Fundamentals of Direct Digital Synthesis (DDS), 2009, http://www.analog.com/static/imported-files/tutorials/MT-
085.pdf.
[3] Xilinx LogiCore, DS246 Product Specification v5.0, April 28, 2005. http://
www.xilinx.com/support/documentation/ip_documentation/dds.pdf.
Appendix
Dealing with Spurs Spurs are a known fact of DDS implementations. Design techniques to mitigate spurs are described in [1],[2], and [3]. The root cause of spurs is the fact that the quantizer introduces phase error
. (19)
The accumulator output driving the LUTs is of the form
. (20)
In the generation of a complex sinusoid (sin and cos), i.e., , you can write
(21)
assuming the error is small. The error phase is also a periodic ramp signal so it effectively modu- lates via the term . This action produces sidebands at frequency offsets from the fundamental frequency (4 kHz in the case of Figure 7).
One mitigation approach is to inject a small random dithering signal at the input to the quantizer as shown in Figure 17. The idea is that the dithering signal disrupts the periodicity and replaces it
Q ( )
ejθ ˆ n[ ] θ n[ ]( )cos j θ n[ ]( )sin+=
ejθ ˆ n[ ] ejθ n[ ] ej θ n[ ]δ⋅ ejθ n[ ] θ n[ ]δ( )cos j θ n[ ]δ( )sin+{ }= =
ejθ n[ ] 1 j θ n[ ]δ+{ }⋅=
θ n[ ] j θ n[ ]ejθ n[ ]δ
References 18
with a dominant random phase error [1]. The corresponding error spectrum is transformed from
spectral lines to a flat noise-like spectrum across the entire spectrum. The spectrum now has a noise floor, but the spurs are gone and/or reduced.
A MATLAB model that includes dithering is
function [x,a_out,n] = DDS_dither(f0,fs,N_samps,Bcos,Bacc,Bw)
if a >= 2^Bacc
a = a - 2^Bacc;
a = a + randn(1,1)*2^(Bacc - Bw - 3); % Dithering added here
w = round(a/2^(Bacc-Bw));
a_out(k) = w/2^Bw;
Bacc Bw Bcos
Figure 17: Finite precision DDS with dithering to mitigate spurs.
Input Output NΔ
θ n[ ] θ n[ ]
Reworking the results of Figure 6, you now have the results shown in Figure 18.
LAB 6 ZIP Contents The file Lab6.zip contins MATLAB simulation code and the CCS project Zoom_DDS. The simulation code is discussed earlier in this lab document. In the CCS project shown in Figure 19 I make use of the CCS feature exclude from build to effectively turn on and off ISR files.
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Frequency (kHz)
Figure 18: DDS output at 14 kHz for kHz, , and with (red) and without (blue) dithering.
fs 48= Bacc 32 Bw, 12= = Bcos 14=
Noise floor Phase noise added around the carrier
no dither
Presently excluded from build
ISR Omitted from ZIP
ISR Omitted from ZIP
Incomplete ISR for Problem 4
Use the CCS exclude from build feature to toggle ISRs files on and off in the project. Use DSP_Config.h to change the sampling rate.
Gel file for DDS frequency tuning at fs = 48 and 98 kHz
Use this file to change between 48 and 96 kHz sampling rates
A MATLAB model that includes dithering is 20
Introduction and Background
The Voltage Controlled Oscillator as Motivation
Converting the VCO Model to the Discrete-Time Domain
C-Code Implementation Using Floats
Part II: FM Communications Receiver
Receiving System Details
DSP Receiver Implementation
LAB 6 ZIP Contents

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ECE 4680 DSP Laboratory 6: Signal Generation Using DDS