ECE 4680 DSP Laboratory 6: Signal Generation Using DDS Introduction and Background

Transcription

ECE 4680 DSP Laboratory 6:
Signal Generation Using DDS
Due 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, f i ( t ) , that is proportional to
the input control voltage, e ( t ) plus the quiescent frequency f 0 .Working from the VCO block diagram of Figure 1, you have
t
φ ( t ) = 2πK v 
–∞
VCO
(1)
x ( t ) = A cos [ 2πf 0 t + φ ( t ) ]





e(t)
e ( λ ) dλ
θ(t)
Figure 1: VCO high level block diagram.
with VCO gain constant K v having units of Hz/v.The total phase of the VCO output, θ ( t ) , is
related to the instantaneous frequency of the VCO as
ECE 4680 DSP Laboratory 6: Signal Generation Using DDS
1 d
f i ( t ) = ------ θ ( t ) = f 0 + K v e ( t )
2π d t
(2)
where f 0 is the VCO quiescent frequency. From the above equations you can now draw the behavioral level block diagram of Figure 2.
θ(t)
1
--s
2πK v
cos ( )
x(t)



e(t)
bias term { 2πf 0
integrator
Figure 2: VCO behavioral level block diagram.
Note when e ( t ) = 0 the instantaneous frequency becomes f i ( t ) = f 0 Hz.
Converting the VCO Model to the Discrete-Time Domain
In the discrete-time domain consider a sampling rate of f s = 1 ⁄ T , with T being the sampling
period or spacing. You have
x [ n ] = x ( nT ) = cos [ 2πf 0 nT + φ ( nT ) ] ,
(3)
where the discrete-time phase is φ [ n ] = φ ( nT ) .The integrator is replaced by an accumulator
when you consider approximating the integral via rectangular areas. This is shown in Figure 3.
a(t)
a ( ( n – 1 )T )
a( t)
b ( nT ) = T
1
--s
n–1

b(t)
a ( kT )
( n – 2 )T
k = –∞
( n – 1 )T
t
nT
Figure 3: Discrete-time integration approximation using the left endpoint for a.
With the integration above understanding you can write
φ [ n ] = φ ( nT ) = 2πK v T
n–1

e ( nT )
k = –∞
= 2πK v T
n–1

(4)
e [ n ] = kv e [ n – 1 ] + φ [ n – 1 ]
k = –∞
where k v = 2πK v T rad/sample. The recursive form for φ [ n ] in the last line of (4) motivates the
discrete-time form of the VCO shown in Figure 4.
2
e[n]
ˆ 0 ⋅ n + φ[n]
ω
θ[n]
–1
z
cos ( )
kv



2πf 0 T
bias
term ω
ˆ 0 = 2πf ⁄ f
0 s
phase accumulator
x[n]
Can make output
a mod(2π) value
due to cos( )
Figure 4: Discrete-time VCO block diagram.
ˆ = 2πf ⁄ f to the desired quiescent
As a fixed frequency generator let e [ n ] = 0 and set f 0 or ω
0
0 s
frequency 0 < f 0 < f s ⁄ 2 . 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 e [ n ] = 0
is
ˆ 0n)
x [ n ] = cos ( 2πf 0 ⁄ f s ⋅ n ) = cos ( ω
(5)
where you can also write that
ˆ 0n
θ[n] = ω
ˆ 0(n + 1) = ω
ˆ 0n + ω
ˆ 0 = θ[n] + ω
ˆ0
θ[n + 1] = ω
(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
#define two_pi 6.283185307179586
float f0 = 1000; // desired DDS quiescent frequency
float theta = 0;
float w0 = 0.130899693899575; //2*pi*1000/48000, use GEL to compute more vals
...
interrupt void Codec_ISR()
{
...
// DDS
codecOutLeft = 32000*cosf(theta);
theta += w0;
if (theta >= two_pi) theta -= two_pi; // wrap accumulator output
/////////////////////////////////////////////////
3
...
}
From the above code snippet you see that the DDS quiescent frequency is determined by choosing
f
ˆ 0 = 2π ⋅ ---0-, 0 < f < f ⁄ 2
ω
0
s
fs
(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 freB
quency step size via Δf = f s ⁄ 2 acc Hz, where f s is the sampling rate and B acc 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 B w < B acc , this the table contains at most 2 B w entries
•
The storage precision or bit width of the sine/cosine values: B cos
A modified DDS block diagram, that includes finite precision attributes is shown in Figure 5. For
B acc
z
Input
ΔN
–1
Bw
Q( )
B cos
sin/cos
LUT
Output
B w < B acc
Bacc bit phase
accumulator
Figure 5: Finite precision DDS block diagram.
an accumulator input step size ΔN (an integer value), the DDS output frequency is
fs
- ⋅ ΔN Hz
f 0 = --------B
2 acc
(8)
or from a design standpoint
B acc
f0 ⋅ 2
ΔN = -----------------.
fs
(9)
Note in [3] and elsewhere f s is referred to as the DDS clock frequency, f clk .
A MATLAB simulation of this system is the following:
function [x,a_out,n] = DDS(f0,fs,N_samps,Bcos,Bacc,Bw)
% [x,a_out,n] = DDS(f0,fs,N_samps,Bcos,Bacc,Bw)
% //////////////// Inputs ////////////////////
4
%
f0 = desired output frquency
%
fs = sampling frequency
% N_samps = number of samples to simulate
%
Bcos = bit width of cos/sin values
%
Bacc = bit width of the accumulator
%
Bw = bit width of the LUT address
% //////////////// Outputs ///////////////////
%
x = output signal
%
a_out = accumulator normalized to a [0,1) float value
%
n = time index
%
% Mark Wickert November 2013
n = [0:N_samps-1];
x = zeros(1,N_samps);
a_out = zeros(1,N_samps);
a = 0;
w = 0;
theta = 0;
for k=1:N_samps
%x(k) = cos(2*pi*a);
x(k) = simpleQuant(cos(2*pi*w/2^Bw),Bcos,1,'none');
a = a + round(f0/fs*2^Bacc);
if a >= 2^Bacc
a = a - 2^Bacc;
end
w = round(a/2^(Bacc-Bw));
a_out(k) = w/2^Bw;
end
To see the DDS simulation in action suppose f s = 48 kHz and consider 32 bits (very large) for all
of the bit widths and compare that to the case B acc = 32, B w = 12 and B cos = 14 :
>> [x,a_out,n] = DDS(14,48,2^16,32,32,32);
>> simpleSA(x,2^14,48,-120,5,1);
>> [x,a_out,n] = DDS(14,48,2^16,14,32,32);
>> simpleSA(x,2^14,48,-120,5,1);
>> plot(n(1:50),a_out(1:50))
>> hold
Current plot held
>> 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.
5
Normalized Power Spectrum in dB
0
−20
−40
−60
−80
−100
−120
0
5
10
15
Frequency (kHz)
20
Figure 6: DDS output at 14 kHz for f s = 48 kHz and 32-bits in three locations.
0
−20
−40
−60
spurious outputs
−80
spurious outputs
−100
−120
0
5
10
15
Frequency (kHz)
20
Figure 7: DDS output at
14 kHz for f s = 48 kHz, B acc = 32, B w = 12 , and B cos = 14 .
Further discussion of DDS spurs can be found in the appendix of this document.
6
The 32-bit accumulator output for f 0 = 14 kHz and f s = 48 kHz is shown in Figure 8.
1
Normalized Accumulator Output
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
5
10
15
20
25
30
Sample Index n
35
40
45
50
Figure 8: DDS accumulator output for f 0 = 14 kHz for f s = 48 kHz,
B acc = 32, B w = 12 , and B cos = 14 .
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
7
those shown in.
Exponential averaging reveals
spurs spaced every 1 kHz
About 88 dB About 78 dB
to the noise spurious-free
floor
dynamic range
(SFDR)
Figure 9: DDS output at 1 kHz from the OMAP-L138 with f s = 48 kHz using
real-time floats.
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 ( f = 0 ).
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
8
t
x c ( t ) = A c cos 2πf c t + 2πf d  m ( λ ) dλ .
(10)
You study this waveform in detail in ECE 4625/5620, Communications Systems I. The signal carrier of center frequency is f c Hz. The information or message carried by the signal is m ( t ) . 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 x c ( t ) takes the form
BFM controlled
by m(t)
–fc
fc
0
f
Figure 10: Spectrum of FM carrier centered at f c Hz.
1
X c ( f ) = --- [ X BB ( f – f c ) + X BB ( f + f c ) ]
2
(11)
where X BB ( f ) is the complex baseband spectrum corresponding to x c ( t ) . Note x BB ( t ) is really
just
t
x BB ( t ) = A c exp 2πf d  m ( λ ) dλ .
(12)
To get your hands on x BB ( t ) all you need to do is complex frequency translate (10) either to he
left or right by f c Hz and lowpass filter to one half the FM RF bandwidth, B FM

± j2πf c t 
x BB ( t ) = LP  x c ( t ) ⋅ e
.


(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
FT { x ( t )e
j2πf 0 t
} = X ( f – f0 )
(14)
where X ( f ) = FT { x ( t ) } . The second step is to demodulate the message m ( t ) from x BB ( t ) . 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-
9
xc ( t )
1
1
2
ADC
xc [ n ]
f s = 96 kHz
x[n]
(LEFT)
1
2
IIR
LPF
N=4
c s
e
– j2π ( 30 ⁄ 96 )n
= e
= complex
1
z(t)
DAC
(LEFT)
1
choose re{}
DAC
(RIGHT)
test point or im{}
options
f s = 96 kHz
– j2π ( f ⁄ f )n
= real
Complex 1
Baseband
y [ n ] Discrimin. z [ n ]
2
Figure 11: Zoom (also LCDK) OMAP-L138 FM receiver block diagram.
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, x c [ n ] , 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.
Xc ( e
Spectrum Before
j2πf ⁄ f s
f s = 96 kHz
f c = 30 kHz
)
...
...
(a)
-66
-96
-48
Spectrum After
(b)
0
-30
X( e
...
j ( 2πf ) ⁄ f s
30
move left
30 kHz
)
-60 -48
-30
0
36
f (kHz)
96
Important image
for the LPF to
suppress
ideal
lowpass
-96
66
48
48
66
...
f (kHz)
96
Principle alias band
Figure 12: The x c ( t ) spectrum (a) following sampling and (b) following frequency translation when f s = 96 kHz and f c = 30 kHz (shift left).
Note that the spectrum of a complex signal is not symmetrical (in magnitude) about f = 0 . Communications applications of real-time DSP typically involve complex signals.
Complex Baseband Discriminator
The complex baseband signal y [ n ] for the case of an FM input signal, is of the form
y [ n ] = Ac e
jφ [ n ]
= A c { cos ( φ [ n ] ) + j sin ( φ [ n ] ) } = y I [ n ] + jy Q [ n ] .
(15)
–1
Note φ [ n ] = tan ( x Q [ n ] ⁄ x I [ n ] ) . The frequency discriminator seeks to find the derivative of
the phase φ [ n ] . In the continuous-time domain the derivative of the inverse tangent function is
10
y I ( t )y Q' ( t ) – y Q ( t )y I' ( t )
dφ ( t )
z ( t ) = ------------- = --------------------------------------------------------.
2
2
dt
y (t) + y (t)
I
(16)
Q
A discrete-time approximation to the above derivative is
yI [ n ] ⋅ ( yQ [ n ] – yQ [ n – 1 ] ) – yQ [ n ] ⋅ ( yI [ n ] – yI [ n – 1 ] )
z [ n ] = --------------------------------------------------------------------------------------------------------------------------------------.
2
2
yI [ n ] + yQ [ n ]
(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)
% [x,a_out] = FM_gen(e,f0,fs)
%
x = zeros(size(e));
a_out = zeros(size(e));
a = 0;
w = 0;
theta = 0;
for k=1:length(e)
x(k) = cos(2*pi*a);
a = a + f0/fs + e(k)/fs;
a = mod(a,1);
a_out(k) = a;
end
The complex baseband discriminator is implemented in the function discrim.m:
function disdata = discrim(x)
% function disdata = discrimf(x)
% x is the received signal in complex baseband form
% Mark Wickert
X=real(x); %
Y=imag(x); %
N=length(x);%
b=[1 -1];
%
a=[1 0];
X is the real part of the received signal
Y is the imaginary part of the received signal
N is the length of X and Y
filter coefficients for discrete derivative
11
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
{ S xc ( f ) } dB
PSD (dB)
The input spectrum, the frequency translated spectrum, and the filtered spectrum are shown as
subplots in Figure 13.
20
0
−20
−40
{ S x ( f ) } dB
PSD (dB)
(a)
20
0
−20
−40
{ S y ( f ) } dB
(c)
PSD (dB)
(b)
20
0
−20
−40
received
signal
0
5
10
15
20
25
30
Frequency (Hz)
desired
spectrum
peak dev.
of 2 kHz
35
40
45
image
spectrum
−40
−30
−20
−10
0
10
Frequency (Hz)
20
30
40
−40
−30
−20
−10
0
10
Frequency (Hz)
20
30
40
Figure 13: The spectra at (a) the input, (b) following complex frequency translations, and (c) after the lowpass filter, in the MATLAB simulation of the FM
receiver.
12
The final discriminator output signal z [ n ] is given in Figure 13.
0.2
Filter
transient
Discriminator Output Amplitude
0.15
The 1kHz message
sinusoid
0.1
0.05
0
−0.05
−0.1
−0.15
0
1
2
3
Time (ms)
4
5
6
Figure 14: The 1 kHz message signal recovered at the output of the discriminator in the MATLAB simulation.
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.
13
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.
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 completely removed the image signal at 36 kHz. The image signal in fact manages to make its way
through the discriminator.
14
Recovered
1 kHz
sinusoid from
30 kHz FM
carrier
about 55 dB down
message harmonics (2 & 3 kHz)
1 kHz
message
36 kHz
leakage
Figure 16: OMAP-L138 output for a 1 kHz sinusoidal message at 1.5 kHz peak
frequency deviation and f s = 96 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 functions 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
ˆ = 2πf ⁄ f .
sinusoid frequency in Hz and have the GEL convert the value to a float ω
0
0 s
Expectations
15
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 difference 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 running from [0, 1), but quantized in the argument of cosf(). The code below increments the
accumulator a by 0 < f 0 ⁄ f s < 1 (in code f0_fs), for f 0 ⁄ f s = 1000 ⁄ 48000 . The argument
of cosf() is 2π ⋅ Q W ( f 0 ⁄ f s ) , where Q W ( ) is a W bit quantizer implemented as
W
Q W ( x ) = Short [ x ⋅ 2 ] ⋅ 2
–W
.
(18)
Note Short[ ] represents casting a float value to a Short integer. The returned float value of
–W
(18) is again float because 2 is represented as a float constant. The returned value is however 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
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 measurements.
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 f c = 30 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?
Problems
17
References
[1] Michael Rice, Digital Communications: A Discrete-Time Approach, Prentice Hall, New Jersey, 2009.
[2] Analog Devices MT-85 Tutorial, Fundamentals of Direct Digital Synthesis (DDS), 2009,
http://www.analog.com/static/imported-files/tutorials/MT085.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 Q ( ) introduces
phase error
δθ [ n ] = θˆ [ n ] – θ [ n ] .
(19)
The accumulator output driving the LUTs is of the form
θˆ [ n ] = θ [ n ] + δθ [ n ] .
In the generation of a complex sinusoid (sin and cos), i.e., e
can write
e
ˆ
jθ [ n ]
= e
= e
jθ [ n ]
⋅e
j δθ [ n ]
jθ [ n ]
⋅ { 1 + j δθ [ n ] }
= e
jθ [ n ]
ˆ
jθ [ n ]
(20)
ˆ
ˆ
= cos ( θ [ n ] ) + j sin ( θ [ n ] ) , you
{ cos ( δθ [ n ] ) + j sin ( δθ [ n ] ) }
(21)
assuming the error is small. The error phase is also a periodic ramp signal so it effectively modujθ [ n ]
lates θ [ n ] via the term j δθ [ n ]e
. 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
References
18
with a dominant random phase error [1]. The corresponding error spectrum is transformed from
B acc
Input
ΔN
z
–1
Q( )
θ[n]
Bacc bit phase
accumulator
Bw
B acc
dither
signal
d[ n]
θˆ [ n ]
B cos
sin/cos
LUT
Output
B w < B acc
pseudo-random sequence
toggling one or more LSBs
Figure 17: Finite precision DDS with dithering to mitigate spurs.
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)
% [x,a_out,n] = DDS_dither(f0,fs,N_samps,Bcos,Bacc,Bw)
%
n = [0:N_samps-1];
x = zeros(1,N_samps);
a_out = zeros(1,N_samps);
a = 0;
w = 0;
theta = 0;
for k=1:N_samps
%x(k) = cos(2*pi*a);
x(k) = simpleQuant(cos(2*pi*w/2^Bw),Bcos,1,'none');
a = a + round(f0/fs*2^Bacc);
if a >= 2^Bacc
a = a - 2^Bacc;
end
a = a + randn(1,1)*2^(Bacc - Bw - 3); % Dithering added here
w = round(a/2^(Bacc-Bw));
a_out(k) = w/2^Bw;
end
19
Reworking the results of Figure 6, you now have the results shown in Figure 18.
0
−20
−40
Phase noise added
around the carrier
Noise floor
−60
−80
−100
−120
no
dither
0
5
10
15
Frequency (kHz)
20
Figure 18: DDS output at 14 kHz for f s = 48 kHz, B acc = 32, B w = 12 , and
B cos = 14 with (red) and without (blue) dithering.
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.
Use the CCS exclude from build
feature to toggle ISRs files on and
off in the project. Use DSP_Config.h
Presently excluded
to change the sampling rate.
from build
Active DDS ISR that
is incomplete for Problem 1
Use this file to change between
48 and 96 kHz sampling rates
Gel file for DDS
frequency tuning
at fs = 48 and 98 kHz
Figure 19: Contents
of CCS project Zoom
_DDS.
ISR Omitted from ZIP
Complete ISR for Problem 3
ISR Omitted from ZIP
Incomplete ISR for Problem 4
20

ECE 4680 DSP Laboratory 6: Signal Generation Using DDS Introduction and Background

Transcription

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