Difference Equations and Impulse Responses
This project will help you to become more familiar with difference equations by
exploring their characteristics in both the time and frequency domains.
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8.
Find filter ................................................................................................................... 1
Calculate h[n] ............................................................................................................. 1
Find h[n] by filtering an impulse ............................................................................... 1
Filter pulse by filter and by conv ............................................................................... 2
Frequency response .................................................................................................... 4
Plot mag/phase response ............................................................................................ 4
Filter two cosines ....................................................................................................... 5
Demonstrate differentiation-in-frequency property of DTFT.................................... 6
1. Find filter
Let
This difference equation can be implemented using the filter command. Find the vectors
a and b that represent the difference equation above for the filter command.
a = [1 -0.5];
b = [0.1 0.8 0.8];
2. Calculate h[n]
Calculate h[n] analytically for the difference equation above. Hint: You may find the
residuez function useful.
3. Find h[n] by filtering an impulse
impulse = [1 zeros(1,99)];
h = filter(b,a,impulse);
stem([0:length(h)-1],h)
4. Filter pulse by filter and by conv
pulse = [ones(1,9) zeros(1,91)];
pulse_filt = filter(b,a,pulse);
pulse_conv = conv(h,pulse);
stem([0:length(pulse_filt)-1],pulse_filt)
stem([0:length(pulse_conv)-1],pulse_conv)
Visible difference is only in length. However, there are very tiny differences due to
truncating the IIR response with filter
stem([0:length(pulse_filt)-1],pulse_filt-pulse_conv(1:length(pulse_filt)))
5. Frequency response
% See #2.
6. Plot mag/phase response
freqz(b,a)
The system is lowpass.
7. Filter two cosines
x1 = cos(0.1*pi*[0:99]);
x2 = cos(0.8*pi*[0:99]);
x1filt = filter(b,a,x1);
x2filt = filter(b,a,x2);
stem([0:length(x1filt)-1],x1filt)
stem([0:length(x2filt)-1],x2filt)
x1 is amplified, but x2 is attenuated
8. Demonstrate differentiation-in-frequency property of DTFT
% generate sequence
n = [0:100];
z = 0.5.^n;
zn = n.*z; % time-domain effect
Zf = fft(z); % sampled DTFT of z, where $$\omega = \frac{2\pi}{101}$$
Znf = fft(zn); % sampled DTFT of zn
Zfdiff = j*diff([Zf(end) Zf]); % approximating derivative by difference
Zfdiff = Zfdiff*101/(2*pi); % scaling due to omega vs. k
subplot(2,1,1)
plot(abs(Znf))
subplot(2,1,2)
plot(abs(Zfdiff))
These are approximately the same, as the theory tells us