Chapter 7

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 up-

sampling (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?

1fs0fs 2fs

Over-sampled

W W

Nyquist-sampled

WW

Magnitude Responses of Required Anti-Aliasing Analog Filters

Aliasing Noise Example

Suppose a communications speech signal has a bandwidth of interest of 4 kHz. The signal is bandwidth-limited by a 2nd order Butterworth analog filter (fc = 4 kHz) prior to sampling. Compare the aliasing noise at 3 kHz for:

– fs = 8 kHz (Nyquist sampling)– fs = 128 kHz (16x over-sampling)

Aliasing Noise Example% Use M-file ALIASNOISE=alias_effect_mag(filterorder,fc,fs,f1

>> alias_effect_mag(2,4000,8000,3000); figure,

>> alias_effect_mag(2,4000,128e3,3000);

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Spectral Overlap for Order 2 Butterworth Pre-Filter: Fc = 4000 Fs = 8000

Aliasing Noise Level at 3000 Hz = 61.8482 percent

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Spectral Overlap for Order 2 Butterworth Pre-Filter: Fc = 4000 Fs = 128000

Aliasing Noise Level at 3000 Hz = 0.11749 percent

Digital Anti-Aliasing and Anti-Imaging

Simple Analog

LP Filter

High Rate Sampling (ADC)

Digital LP Filter Down-sample to Nyquist Rate

Up-sample to high rate

Digital LP Filter DAC at High Rate

Simple Analog LP Filter

Anti-Aliasing Step

Anti-Imaging Step

Decimation

Interpolation

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 low-pass 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

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Sample

Down-Sampling a Sinusoid

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Sample

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

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Up-Sampling and Interpolation

• The process of adding zero-valued samples between existing samples is called “up-sampling”

• 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

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Sample

Spectra of the Down-Sampled and Up-Sampled Signals

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Spectrum of Initial Signal

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Spectrum of Down-Sampled Signal

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Spectrum of 4X Zero Insertion of Down-Sampled Signal

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 Down-

Sampled Signal')>> subplot(3,1,3);stem(x_recovered(1:100)),title('Recovered (LP

Filtered) Signal')

Interpolation Recovery Results

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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

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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

48 480 160

44.1 441 147

L

M

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 ExampleResults

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1Rate Conversion 1.2 kHz to 1 kHz

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

2

/ 22

/ 2

/ 2 2

/ 2

2

1| ( ) |

2

2

/ 2 for

1 2| ( ) |

2

| ( ) |

| ( ) |( )

s

s

s

s

N

s

s

f

Ns f

f

sf

s

P X dN

d dff

and

f f

so

P X f dfN f

or

X fP df

Nf

X fp f

Nf

Calculating Mean Noise Power Density for Different Sampling Frequencies

>> [zero_sig,tt]=analog(1,0,1000,50000); %Construct a psuedo-analog “zero” signal>> analog_noise=zero_sig+randn(size(zero_sig)); % Construct a psuedo-analog noise signal>> n1000=sample(tt,analog_noise,1000); % Sample the noise signal at 1 kHz, 2 kHz, and 4 kHz>> n2000=sample(tt,analog_noise,2000);>> n4000=sample(tt,analog_noise,4000);>> [N1,f1]=dtft_demof(n1000,0,500,512,1000); % Compute the DTFT for each digital noise signal>> [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); % Compute the spectral power density>> pd2=(abs(N2)).^2/(length(n2000)*2000);>> pd4=(abs(N4)).^2/(length(n4000)*4000);>> mean(pd1) % Compute the mean value of the spectral power density>> mean(pd2)>> mean(pd4)ans = 9.9750e-004

ans = 4.8431e-004

ans = 2.5061e-004

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 ExampleResults

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De-Noising ExampleResults

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Noisy Signal Spectrum

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Filtered Noisy Signal Spectrum

Noise Power Uniformly Distributed

Noise Power Eliminated by LP Filter

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 QuantizerFunctional Diagram

Accumulator / Integrator Quantizer / Comparator

1-bit DAC

-+ 1-bit Output

-

Input

(1,-1)X DC

W

+V

-V

B

Delta-Sigma QuantizerDiscrete-Time Model

Z-1

Accumulator, H(z)

+ +-x[n] y[n]= -1,1

e[n]

v[n] w[n]

Quantizer

)()1()()( 11 zEzzXzzY

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

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Output Sample

8-bit Running Average of the Delta-Sigma Quantized Sine

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Delta-Sigma Quantization of a Sinusoid – Quantization Noise Shaping

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Spectrum of Delta-Sigma Output Signal

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

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Spectrum of Low-Pass Filtered Delta-Sigma Quantizer Output

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Milliseconds

Filter group delay

Processing a Δ-Σ Signal

Delta-Sigma Quantizer

Serial-to-N-bit Parallel Converter

Low-Pass Filter

Decimation to Nyquist Rate

Binary stream (fs)

N-bit numbers (fs/N)

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)

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Quanitzer output and sampled signal

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8-bit samples and sampled signal

>> 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

>> 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

Processing a Δ-Σ Signal MATLAB Simulation

Signals (10 ms period)

>> 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

>> 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

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Processing a Δ-Σ Signal MATLAB Simulation

Sampled Signal Spectrum

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Spectrum of the sampled signal

>> dtft_demof(x,0,1500,4000,24000); % Begin generating signal spectra>> legend('Spectrum of the sampled signal')

Processing a Δ-Σ Signal MATLAB Simulation

Quantizer and 8-Bit Signal Spectra (Showing Quantization Noise)

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Spectrum of the delta-sigma quantizer output

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Spectrum of the 8-bit serial-to-parallel signal

>> figure,dtft_demof(y,0,1500,4000,24000);>> legend('Spectrum of the delta-sigma quantizer output')

>> figure, dtft_demof(y8bits,0,1500,4000,3000);>> legend('Spectrum of the 8-bit serial-to-parallel signal')

Processing a Δ-Σ Signal MATLAB Simulation

8-Bit Signal LP Filtered and Decimated(Quantization Noise Minimized)

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Spectrum of the low-pass filtered 8-bit signal (recovered signal)

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Final signal decimated by 4 (fs = 750 Hz)

>> figure, dtft_demof(yfilt,0,1500,4000,3000);>> legend('Spectrum of the low-pass filtered 8-bit signal (recovered signal)')

>> figure,dtft_demof(ydec,0,1500,4000,750);>> legend('Final signal decimated by 4 (fs = 750 Hz)')

Reduction in Noise due to Over-sampling and Noise Shaping

Quantization

Reduced Noise due to M-times Oversampling

Low Pass Digital Filter

Further Reduced Noise due Noise Shaping

sf

2

sf

2sf

M2

sfM

2

sf

2

a.)

b.)

c.)

Summary• 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.