556 spring 2011 final

8/3/2019 556 Spring 2011 Final

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EE-556 Spring 2011

LENGTHY BUT EASY TAKE HOME FINAL

1. A common filtering task raises the sample rate of an input signal by a factor of four.This can be accomplished by zero packing the input signal with three zeros betweeneach input data signal and then filtering with a low pass filter. The zero packingraises the sample rate of the series so that four spectral copies can be observed inthe spectrum of the time series. The filter removes three of the copies. This processis shown in the figure below.

H ( Z )

1 : 4

f s

4 f s

4 f s

I n p

S p e c

F i l t e r

S p e c t

F i l t e r

1 - t o -P a c

S p e c

x ( n ) x ( 4 n ) y ( n )

A. The Input signal bandwidth has a nominal bandwidth of ± 10 kHz and is sampled at24 kHz. To approximate a signal spanning this bandwidth form an input signalcontaining 600 samples of a sum of 6-sinusoids at frequencies 0, 2, 4, 6, 8, & 10kHz. The sine waves have amplitudes of 0.2 and have uniform random phase. Plot50 samples of the time series and the windowed spectrum of 256 samples of thetime series. Display and label time axis in milliseconds and frequency axis in kHz.

B. Zero-pack the time series 1-to-4 and plot 200 samples of the time series and thewindowed spectrum of 1024 samples of the zero packed time series. Display andlabel time axis in milliseconds and frequency axis in kHz.

C. Use the Remez algorithm to design a FIR filter for sample rate of 96 KHz, with pass

band ± 10 kHz and stop band selected to reject the spectral replicates. (Transitionbandwidth ∆ f is 4 KHz). In-band ripple must be less than 0.1 dB, and out-of-bandattenuation must be 4 times your age (in years converted to dB) or if that number exceeds 100 dB replace with 100 dB. Plot time response and frequency response of your filter design.

D. Filter the 1-to-4 zero packed time series with the filter normalized to unity peakamplitude and plot 200 samples of the time series (after the filter length number of samples to avoid the filter transient) and the windowed spectrum of 1024 samples of

8/3/2019 556 Spring 2011 Final


the filtered time series. Display and label time axis in milliseconds and frequency axisin kHz

E. Zero pack the up-sampled sequence by another factor of 8 and plot 1600 samples of the time series and the windowed spectrum of 8192 samples of the zero packed timeseries. Display and label time axis in milliseconds and frequency axis in kHz

F. Use the Remez algorithm to design a second FIR filter for sample rate of 96*8 KHz,with pass band ± 10 kHz and stop band selected to reject the spectral replicates.

(Transition bandwidth ∆ f is 76 KHz) with don’t care regions between the stop bands.In-band ripple must be less than 0.1 dB, and out-of-band attenuation equal to 4 timesyour age (in years converted to dB) or if that number is larger than 100 db, replacewith 100 dB. Plot time response and frequency response of your filter design.

G. Filter the 1-to-8 zero packed time series with the second filter normalized to unitypeak amplitude and plot 1600 samples of the time series the filter length number of samples to avoid the filter transient) and the windowed spectrum of 8192 samples of the filtered time series. Display and label time axis in milliseconds and frequency axisin kHz.

H. Process the up-sampled time series by a two-loop sigma-delta modulator shownbelow.

Z- 1

Z- 1

x ( n )

y ( n )

a 1

b 1

w 1 w 2

b 2

Q

The Matlab code to implement this block diagram with a 4-bit quantizer is presentedbelow:

w1=0; w2=0; b1=1.0; b2=0.5; a1=1/128;

for n = 1:N_lengths3 = b1*w1 + b2*w2;y(n) = round(4*s3)/4;

s0 = x(n) -y(n);s1 = s0 + (1-a1)*w1 - a1*w2;s2 = w1 + w2;

w1 = s1; w2 = s2;end

8/3/2019 556 Spring 2011 Final


Plot 1600 samples of the sigma-delta modulated time series and the windowed spectrumof 8192 samples of the zero packed time series. Display and label time axis inmilliseconds and frequency axis in kHz

I. Use the Remez algorithm to design a third FIR filter for sample rate of 96*8 KHz, with

pass band ± 10 kHz and stop band selected to reject the quantizing noise formed by

the sigma-delta converter. The filter stop band can be made to fall at rate 1/f byreplacing the end points of the filter with their immediate neighbor (h(1)=h(2),,h(101)=h(1)). Plot the time response and frequency response of your filter design.

J. Filter the sigma-delta modulated time series with the third filter and plot 1600samples of the time series (the filter length number of samples to avoid the filter transient) and the windowed spectrum of 8192 samples of the filtered time series.Display and label time axis in milliseconds and frequency axis in kHz.

K. Repeat I and J above with a recursive filter. Use the cheby2 filter and show itsimpulse response and frequency response and then write the code to implement thefilter and filter the output of the sigma delta modulator.

Oh Boy, a DSP Exam!

I’ve got to get going!

556 spring 2011 final

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