Chapter 7
Over-Sampling and Multi-Rate
DSP Systems
Objectives
•
•
•
•
•
•
•
•
•
Describe the anti-aliasing problem and its solution with over-sampling.
Demonstrate the details of digital anti-aliasing and anti-imaging.
Demonstrate the processes of down-sampling (decimation) and upsampling (interpolation) to change the effective sampling rate of a digital
signal.
Show that interpolation is accomplished by up-sampling and a gain-adjusted
low-pass filter.
Demonstrate the process of rational rate conversion by cascaded
interpolation and decimation.
Derive the relationship between sampling rate and the spectral density of
random noise.
Demonstrate the de-noising of signals with over-sampling.
Describe the process of delta-sigma quantization and derive the transfer
functions for a first-order noise-shaping quantizer.
Demonstrate the process of delta-sigma quantization and the recovery of a
quantized signal by low-pass filtering
The Anti-Aliasing Problem
• If a signal contains a band-width of interest, W, then the
minimum sampling frequency is 2W (Nyquist
requirement)
• However, to prevent aliasing, the signal must be analog
pre-filtered to a band-width of fs/2 prior to sampling
• If the signal contains frequency components or noise
beyond W, the analog low-pass filter (“anti-aliasing” filter)
must be impractically sharp if sampling is done at 2W
• Solution: Multi-rate processing
– Over-sample the signal at a rate considerably higher than 2W
and band-width limit the signal with a sharp digital low-pass filter
– “Down-sample” the signal digitally to an effective rate 2W after
digital filtering
– This allows the use of a simple analog anti-aliasing filter
Why Does Over-Sampling Help
Anti-Aliasing?
Over-sampled
W
W
0fs
1fs
2fs
Nyquist-sampled
W
W
Magnitude Responses of Required AntiAliasing Analog Filters
Digital Anti-Aliasing and Anti-Imaging
Simple Analog
LP Filter
High Rate
Sampling (ADC)
Digital LP Filter
Down-sample to
Nyquist Rate
Decimation
Anti-Aliasing Step
Up-sample to
high rate
Digital LP Filter
DAC at High
Rate
Interpolation
Anti-Imaging Step
Simple Analog
LP Filter
Down-Sampling and Decimation
• Removal of samples is called “down-sampling”
• Down-sampling by a factor N reduces the
effective sampling frequency by the same factor.
That is, the new sampling frequency is fs/N
• To prevent aliasing, the down-sampled signal
should be band-width limited to fs/2N by lowpass filtering prior to sample removal
• Low-pass filtering followed by down-sampling is
called “decimation”
Down-Sampling a Sinusoid
>> [ysin,tsin]=analog(100,1,30,8000);
>> stem(ysin),title('100 Hz Sinusoid Sampled at 8 kHz')
>> xlabel('Sample')
>> dysin=downsample(ysin,4);
>> figure,stem(dysin);
>> title('100 Hz Sinusoid Sampled at 8 kHz, Down-Sampled by 4x')
>> xlabel('Sample')
Down-Sampling a Sinusoid
100 Hz Sinusoid Sampled at 8 kHz
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
0
50
100
150
Sample
200
250
Down-Sampling a Sinusoid
100 Hz Sinusoid Sampled at 8 kHz, Down-Sampled by 4x
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
0
10
20
30
40
Sample
50
60
70
Decimation
• With decimation, low-pass filtering precedes down-sampling to
prevent aliasing.
• Example: Decimating a 100 & 1200 Hz signal by a factor of 4 with
the MATLAB “decimate” command
>> x=analog([100,1200],[1,1],1000,8000);
>> xd4=downsample(x,4); % Effective sampling frequency is now 2000 Hz
>> xdecim4=decimate(x,4); % The LP filter will have a cut-off of 0.8*(8000/8) = 800 Hz
and will remove the 1200 Hz component
>> subplot(3,1,1),dtft_demof(x,0,1500,2024,8000);
>> title('Original Signal')
>> subplot(3,1,2),dtft_demof(xd4,0,1500,2024,2000);
>> title('Down-Sampled (Aliasing)')
>> subplot(3,1,3),dtft_demof(xdecim4,0,1500,2024,2000);
>> title('Decimated (No 1200 Hz Aliasing)')
Decimation Example
Original Signal
4000
2000
0
0
500
1000
1500
Hz
Down-Sampled (Aliasing)
1000
500
0
0
500
1000
Hz
Decimated (No 1200 Hz Aliasing)
1500
1000
500
0
0
500
1000
Hz
1500
Up-Sampling and Interpolation
• The process of adding zero-valued samples
between existing samples is called “upsampling”
• Up-sampling by a factor N effectively increases
the sampling frequency to Nfs
• Estimating the values between existing samples
is called “interpolation”
• Interpolation is accomplished by low-pass
filtering an up-sampled signal followed by an
amplitude correction to restore the signal power.
Down-Sampling and Up-Sampling
100 Hz + 200 Hz Signal
>> [x,ts]=analog([100,200],[1,1],20,2000); % Construct a
short version of the signal
>> xd4=downsample(x,4);
>> xu4=upsample(xd4,4);
>> subplot(3,1,1),stem(x),title('Original Signal')
>> subplot(3,1,2),stem(xd4),title('Down-Sampled by 4')
>> subplot(3,1,3),stem(xu4),title('Down-Sampled Signal
Up-Sampled by 4x')
>> xlabel('Sample')
Down-Sampling and Up-Sampling
Original Signal
2
0
-2
0
5
10
15
20
25
30
35
40
45
Down-Sampled by 4
2
0
-2
1
2
3
4
5
6
7
8
9
10
11
Down-Sampled Signal Up-Sampled by 4x
2
0
-2
0
5
10
15
20
25
Sample
30
35
40
45
Spectra of the Down-Sampled and
Up-Sampled Signals
Spectrum of Initial Signal
1
0.5
0
0
100
200
300
400
500
600
700
Hz
Spectrum of Down-Sampled Signal
800
900
1000
1
0.5
0
0
50
100
150
200
Hz
Spectrum of 4X Zero Insertion of Down-Sampled Signal
250
0.4
0.2
0
0
100
200
300
400
500
Hz
600
700
800
900
1000
Interpolation Process
• The spectrum of the up-sampled signal gives the
clue for the process
– Low-pass filter the up-sampled signal to remove all
frequency components beyond fs/2N, where fs =
effective sampling frequency after up-sampling and N
is the up-sampling factor.
– Multiply the result by N to restore the correct the
power of the up-sampled signal
• Typical filter order for the low-pass filter is 8N
• This process is implemented in the MATLAB
interp command, using a special symmetric FIR
filter to compensate for group delay
Recovery of a Down-Sampled
Signal by Interpolation Filtering
>> x=analog([100,200],[1,1],1000,2000); %Construct the signal
>> xd4=downsample(x,4);
>> xu4=upsample(xd4,4);
>> n=0:32; % Begin the design of an order 8N filter LP filter
>> fs=2000;
>> N=4;
>> cutoff=fs/(2*N);
>> omega=2*pi*cutoff/fs;
>> h=(omega/pi)*sinc(omega*(n-16)/pi).*hamming(33)'; % LP filter
>> x_recovered=4*filter(h,1,xu4); % LP filter and multiply by N=4
>> subplot(3,1,1);stem(x(1:100)),title('Original Signal')
>> subplot(3,1,2);stem(xu4(1:100)),title('Zero Insertion of DownSampled Signal')
>> subplot(3,1,3);stem(x_recovered(1:100)),title('Recovered (LP
Filtered) Signal')
Interpolation Recovery Results
Original Signal
2
0
-2
0
10
20
30
40
50
60
70
80
90
100
80
90
100
80
90
100
Zero Insertion of Down-Sampled Signal
2
0
-2
0
10
20
30
40
50
60
70
Recovered (LP Filtered) Signal
2
0
-2
0
10
20
30
40
50
60
70
Note that the group delay of the LP filter is 16 in this example
Interpolation with “interp”
>> x=analog([100,200],[1,1],1000,2000); %Construct the
signal
>> xd4=downsample(x,4);
>> xu4=upsample(xd4,4);
>> x_interp=interp(xd4,4);
>> subplot(3,1,1),stem(x(1:100)),title('Original Signal')
>> subplot(3,1,2),stem(xd4(1:25)),title('Down-Sampled
Signal')
>> subplot(3,1,3),stem(x_interp(1:100)),title('Recovered
Signal Using "interp"')
“Interp” Results
Original Signal
2
0
-2
0
10
20
30
40
50
60
70
80
90
100
Down-Sampled Signal
2
0
-2
0
5
10
15
20
25
Recovered Signal Using "interp"
2
0
-2
0
10
20
30
40
50
60
70
80
90
100
Sampling Rate Conversion by
Rational Factors
• The sampling frequency can be changed by rational factors L/M
where L is the interpolation factor and M is the decimation factor.
• Rate changes (to higher or lower rates) is always done with
interpolation (L) and decimation (M) in that order to prevent loss of
frequency components of interest in the signal.
• Example: Changing a music signal from the CD rate (44.1 kHz) to
the digital tape rate (48 kHz) is accomplished by interpolation by a
factor of 160 followed by decimation by a factor of 147
L
M

48
44.1

480
441

160
147
Rate Conversion Example
• Convert a 100 Hz sinusoidal signal sampled at
1200 Hz to a signal sampled at 1000 Hz
• L/M=1000/1200 = 5/6
>> x=analog(100,1,40,1200);
>> xL=interp(x,5); % The “interp” command carries out interpolation
>> xLM=decimate(xL,6); % “decimate” carries out decimation
>> subplot(2,1,1),stem(x),title(' Signal, fs = 1.2 kHz')
>> subplot(2,1,2),stem(xLM),title('Rate Conversion 1.2 kHz to 1 kHz')
>> axis([0,50,-1,1])
Rate Conversion Example
Results
Signal, fs = 1.2 kHz
1
0.5
0
-0.5
-1
0
5
10
15
20
25
30
35
40
45
50
40
45
50
Rate Conversion 1.2 kHz to 1 kHz
1
0.5
0
-0.5
-1
0
5
10
15
20
25
30
35
Note in both cases the signal is about 40 ms long.
Over-Sampling and Random Noise
• Gaussian noise power is uniformly
distributed in the frequency domain
• As a consequence, noise power density is
reduced by a factor of 2 (3 dB) for each
doubling of the sampling frequency.
Power Density in the Frequency
Domain
P 

1
2 N
d 
2

| X N ( ) | d 
2

df
fs
and
f  fs / 2
fo r   
so
f /2
s
 1   2 
2
P 
  | X N ( f ) | df

 2 N   f s   f s / 2
or
fs / 2
P 

| X(f)|
 fs / 2
p( f ) 
Nfs
| X(f)|
Nfs
2
2
df
Calculating Mean Noise Power Density for
Different Sampling Frequencies
>> [zero_sig,tt]=analog(1,0,1000,50000);
>> analog_noise=zero_sig+randn(size(zero_sig));
>> n1000=sample(tt,analog_noise,1000);
>> n2000=sample(tt,analog_noise,2000);
>> n4000=sample(tt,analog_noise,4000);
>> [N1,f1]=dtft_demof(n1000,0,500,512,1000);
>> [N2,f2]=dtft_demof(n2000,0,1000,512,2000);
>> [N4,f4]=dtft_demof(n4000,0,2000,512,4000);
>> pd1=(abs(N1)).^2/(length(n1000)*1000);
>> pd2=(abs(N2)).^2/(length(n2000)*2000);
>> pd4=(abs(N4)).^2/(length(n4000)*4000);
>> mean(pd1)
>> mean(pd2)
>> mean(pd4)
ans =
9.9750e-004
ans =
4.8431e-004
ans =
2.5061e-004
%Construct a psuedo-analog “zero” signal
% Construct a psuedo-analog noise signal
% Sample the noise signal at 1 kHz, 2 kHz, and 4 kHz
% Compute the DTFT for each digital noise signal
% Compute the spectral power density
% Compute the mean value of the spectral power density
Notice than the average power
density is reduced by a factor of 2
for each doubling of the sampling
frequency
De-Noising a Signal by Over-Sampling and
Low-Pass Filtering
• Over-sample a signal at a rate many times
higher than the band-width of interest W
• Over-sampling reduces the noise power in W by
3 dB for each doubling of the sampling
frequency
• Low-pass filter with a cut-off frequency near W.
This eliminates the noise power at frequencies
beyond W.
De-Noising Example
>> [asig,tt]=analog(100,1,40,40000);
>> asign=asig+randn(size(asig));
>> [d8,t8]=sample(tt,asig,8000);
>> [d8n,t8]=sample(tt,asign,8000);
>> n=0:100;
>> omega=2*pi*125/8000;
>> hw=(omega/pi)*sinc(omega*(n-50)/pi).*blackman(101)';
>> d8nf=filter(hw,1,d8n);
>> subplot(3,1,1),plot(t8,d8),title('Clean Signal')
>> subplot(3,1,2),plot(t8,d8n),title('Noisy Signal')
>> subplot(3,1,3),plot(t8,d8nf),title('Filtered Noisy Signal')
>> subplot(2,1,1),fft_plot(d8n,8000);title('Noisy Signal Spectrum')
>> subplot(2,1,2),fft_plot(d8nf,8000);title('Filtered Noisy Signal
Spectrum')
De-Noising Example
Results
Clean Signal
1
0
-1
0
5
10
15
20
25
30
35
40
25
30
35
40
30
35
40
Noisy Signal
5
0
-5
0
5
10
15
20
Filtered Noisy Signal
1
0
-1
0
5
10
15
20
25
De-Noising Example
Results
Noisy Signal Spectrum
1
Noise Power Uniformly Distributed
0.5
0
0
500
1000
1500
2000
2500
3000
Hz
Filtered Noisy Signal Spectrum
3500
4000
3500
4000
0.8
0.6
Noise Power Eliminated by LP Filter
0.4
0.2
0
0
500
1000
1500
2000
Hz
2500
3000
Delta-Sigma (ΔΣ) Quantization
• The processing speed of a DSP system could be
measured by bits/sec in which case the speed is
Nfs, where N is the number of quantization bits
(ADC) and fs is the sampling frequency.
• For a given system speed, the maximum fs
requires quantization with one bit, which
appears impossible because quantization with
two levels implies impractically high quantization
noise.
• ΔΣ Quantization can do just that: quantize with 1
bit and still control the quantization noise.
Delta-Sigma Quantizer
Functional Diagram
A c cu m u la to r / In te g ra to r
In p u t
X
Q u a n tize r / C o m p a ra to r
1 -b it O u tp u t
+
-

B
(1 ,-1 )
C
-
+V
W
1 -b it D A C
-V
D
Delta-Sigma Quantizer
Discrete-Time Model
e [n ]
A c c u m u la to r, H (z )
x [n ]
-
v [n ]
y[n ]= -1 ,1
w [n ]
+
+
Z -1
Q u an tize r
Y (z)  z
1
X ( z )  (1  z
1
)E (z)
Output = input x delay + noise x high-pass filter
(e.g, “Noise-shaping” one-bit quantization)
Delta-Sigma Quantization of a Sinusoid
>> [x,ts]=analog(100,2,500,10000);
>> stem(ts(1:100),x(1:100))
>> title('High-Rate Sampled 100 Hz Sinusoid')
>> xlabel('milliseconds')
>> [x,ts]=analog(100,2,500,10000);
>> stem(ts(1:100),x(1:100))
>> title('High-Rate Sampled 100 Hz Sinusoid')
>> xlabel('milliseconds')
>> y=deltasigma(x); % This is the delta-sigma quantizer algorithm
>> figure,stem(y(1:100));
>> hold
>> plot(x(1:100),'k')
>> title('Delta-Sigma Quantizer Output for 100 Hz Sinusoid')
>> xlabel('Output Sample')
>> hold off
Delta-Sigma Quantization of a Sinusoid
High-Rate Sampled 100 Hz Sinusoid
Delta-Sigma Quantizer Output for 100 Hz Sinusoid
2
2
1.5
1.5
1
1
0.5
0.5
0
0
-0.5
-0.5
-1
-1
-1.5
-1.5
-2
0
1
2
3
4
5
6
milliseconds
7
8
9
10
-2
0
10
20
30
40
50
60
Output Sample
70
80
90
100
Delta-Sigma Quantization of a Sinusoid –
Quantization Noise Shaping
Spectrum of Delta-Sigma Output Signal
2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0
500
1000
1500
2000
2500
Hz
3000
3500
4000
4500
5000
Low-pass filtering would eliminate the quantization noise
Low-Pass Filtering Δ-Σ Sinusoid
>> n=0:500;
>> omega=2*pi*200/10000;
>> h=(omega/pi)*sinc(omega*(n-250)/pi);
>> hb200=h.*blackman(501)';
>> [x,ts]=analog(100,2,500,10000);
>> y=deltasigma(x); % This is the delta-sigma quantizer algorithm
>> y_lpf=filter(hb200,1,y);
>> fft_plot(y_lpf,10000);
>> title('Spectrum of Low-Pass Filtered Delta-Sigma Quantizer Output')
>> figure,plot(ts(1:500),y_lpf(1:500))
>> title('Time-Domain of the Filtered Quantizer Output')
>> xlabel('Milliseconds')
Low-Pass Filtering Δ-Σ Sinusoids
Spectrum of Low-Pass Filtered Delta-Sigma Quantizer Output
Time-Domain of the Filtered Quantizer Output
2
2
1.8
1.5
1.6
1
1.4
0.5
1.2
0
1
-0.5
0.8
-1
0.6
0.4
-1.5
0.2
-2
0
0
500
1000
1500
2000
2500
Hz
3000
3500
4000
4500
5000
-2.5
Filter group delay
0
5
10
15
20
25
30
Milliseconds
35
40
45
50
Processing a Δ-Σ Signal
Serial-to-N-bit
Parallel Converter
Delta-Sigma
Quantizer
Binary stream
(fs)
Low-Pass
Filter
N-bit numbers
(fs/N)
Decimation to
Nyquist Rate
Recovered Signal
Continue Processing
Processing a Δ-Σ Signal
• The high-rate sampling frequency of the Δ-Σ quantizer is
fs.
• Over N samples, the bit-stream of the Δ-Σ quantizer
output is the average signal value.
• Serial-to-N-bit parallel conversion creates N-bit numbers
equal to the average value. The effective sampling
frequency is fs/N.
• The N-bit numbers carry the quantization noise which is
eliminated by digital LP filtering. The filtered signal
represents the recovered signal.
• After filtering the signal can be decimated to the Nyquist
rate for further processing.
Processing a Δ-Σ Signal
MATLAB Simulation
% Create and Delta-Sigma Quantize a Signal with 100, 200, and 300 Hz
Components
>> x=analog([100,200,300],[1,1,1],1000,24000); % Sampled signal
>> y=deltasigma(x); % Delta-sigma quantization of the signal
%
% Simulate the 8-Bit Serial to Parallel Conversion of the Quantizer Stream
>> y8bits=decimate_and_average(y,8); % M-file to block process the signal y in
blocks of 8
%
% Low-Pass Filter the 8-Bit Signal with Cutoff of 350 Hz
>> n=0:100; % Start low-pass filter design
>> fs=24000/8; % The signal has been effectively decimated by a factor of 8
>> omega=2*pi*350/fs;
>> h=(omega/pi)*sinc(omega*(n-50)/pi).*hamming(101)'; % Windowed LP filter
>> yfilt=filter(h,1,y8bits); % Low-pass filter the 8-bit re-quantized signal
%
% Decimate the filtered signal to the Nyquist Rate
>> ydec=decimate(yfilt,4); % Decimate the filtered signal by a factor of 4 (fs = 750
Hz)
Processing a Δ-Σ Signal
MATLAB Simulation
Signals (10 ms period)
2.5
2.5
Quanitzer output and sampled signal
8-bit samples and sampled signal
2
2
1.5
1.5
1
1
0.5
0.5
0
0
-0.5
-0.5
-1
-1
-1.5
-1.5
-2
-2
-2.5
0
1
2
3
4
5
6
7
>> t24=0:1000/24000:10;
>> stem(t24(1:240),y(1:240))
>> hold
%Current plot held
>> plot(t24(1:240),x(1:240),'k')
>> legend('Quanitzer output and
sampled signal')
>> hold off
8
9
10
-2.5
0
1
2
3
4
5
6
7
8
>> t3k=0:(1/3):10;
>> stem(t3k(1:30),y8bits(1:30))
>> hold
%Current plot held
>> plot(t24(1:240),x(1:240),'k')
>> legend('8-bit samples and sampled
signal')
>> hold off
9
10
Processing a Δ-Σ Signal
MATLAB Simulation
Signals (10 ms period)
2.5
2.5
Decimated final signal and sampled signal
LP filtered 8-bit samples and sampled signal
2
2
1.5
1.5
1
1
0.5
0.5
0
0
-0.5
-0.5
-1
-1
-1.5
-1.5
-2
-2
-2.5
-2.5
0
1
2
3
4
5
6
7
>> stem(t3k(1:30),yfilt(81:110))
>> hold
%Current plot held
>> plot(t24(1:240),x(245:484),'k')
>> legend('LP filtered 8-bit
samples and sampled signal')
>> hold off
8
9
10
0
1
2
3
4
5
6
7
8
>> tf=downsample(t3k(1:30),4);
>> yfiltf=downsample(yfilt(81:110),4);
>> stem(tf,yfiltf)
>> hold
%Current plot held
>> plot(t24(1:240),x(245:484),'k')
>> legend('Decimated final signal and
sampled signal')
>> hold off
9
10
Processing a Δ-Σ Signal
MATLAB Simulation
Sampled Signal Spectrum
Discrete Time Fourier Transform
12000
Spectrum of the sampled signal
10000
8000
6000
4000
2000
0
0
500
1000
Hz
>> dtft_demof(x,0,1500,4000,24000); % Begin
generating signal spectra
>> legend('Spectrum of the sampled signal')
1500
Processing a Δ-Σ Signal
MATLAB Simulation
Quantizer and 8-Bit Signal Spectra
(Showing Quantization Noise)
Discrete Time Fourier Transform
Discrete Time Fourier Transform
12000
1500
Spectrum of the delta-sigma quantizer output
Spectrum of the 8-bit serial-to-parallel signal
10000
8000
1000
6000
4000
500
2000
0
0
500
1000
Hz
>> figure,dtft_demof(y,0,1500,4000,24000);
>> legend('Spectrum of the delta-sigma
quantizer output')
1500
0
0
500
1000
Hz
>> figure, dtft_demof(y8bits,0,1500,4000,3000);
>> legend('Spectrum of the 8-bit serial-to-parallel
signal')
1500
Processing a Δ-Σ Signal
MATLAB Simulation
8-Bit Signal LP Filtered and Decimated
(Quantization Noise Minimized)
Discrete Time Fourier Transform
Discrete Time Fourier Transform
1500
400
Spectrum of the low-pass filtered 8-bit signal (recovered signal)
Final signal decimated by 4 (fs = 750 Hz)
350
300
1000
250
200
150
500
100
50
0
0
500
1000
Hz
>> figure, dtft_demof(yfilt,0,1500,4000,3000);
>> legend('Spectrum of the low-pass filtered
8-bit signal (recovered signal)')
1500
0
0
500
1000
Hz
>> figure,dtft_demof(ydec,0,1500,4000,750);
>> legend('Final signal decimated by 4 (fs = 750
Hz)')
1500
Reduction in Noise due to Over-sampling and
Noise Shaping
a.)
Quantization
fs
2
Low Pass Digital Filter
b.)
Reduced Noise due to M-times Oversampling
fs
M
2
fs
2
c.)
Further Reduced Noise due Noise
Shaping
fs
2
M
fs
2
Summary
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The spectral content of a digital signal is replicated at integer multiples of
the sampling frequency.
Oversampling (sampling well beyond the Nyquist rate) spreads the
frequency content over a wider frequency range.
The requirements on the analog anti-aliasing filter can be relaxed by
oversampling followed by digital low-pass filtering.
The extra samples in an oversampled and bandwidth-limited signal can be
removed by downsampling or decimation, thereby reducing the effective
sampling frequency.
A signal can be restored to a higher sampling frequency by the processes of
upsampling and interpolation.
Since Gaussian noise is uniformly distributed in the frequency domain, the
combination of oversampling and filtering can effectively de-noise a signal in
a bandwidth of interest.
Oversampling can be maximized by delta-sigma quantization, which is
quantization with one bit.
Delta-sigma quantizers have the property of noise-shaping, which allows
the elimination of quantization noise by low-pass filtering.