CMPE2322 Signals and Systems Page 1 of 2
Assignment #9 IIR Filters Spring 2009
Problem 33. You can recreate Figure 8-3 page 199 with the following matlab statements.
xn = zeros(1,20);
xn(1) = 2;
xn(2) = -3;
xn(4) = 2;
note the index of xn starts with 1 not zero as in the text so the response will be displaced
by 1. The IIR filter in the text a1 =0.8 b0 = 5 noting the sign reversal is implemented by
the matlab steps:
a =[ 1 -0.8];
b = 5;
and the response yn is calculated by the matlab filter() function:
yn = filter(b,a,xn);
To plot the variable versus n the statements needed are:
n = 0:1:19;
stem(n,yn)
This should result in Figure 8.3.
Plot the filter response for xn(1), xn(2), and xn(3) having your assigned values and the
rest zero up to xn(20) for the first order text example b = 5 a1 = 0.8 ( a = -0.8).
Problem 34. For the general first order system y[n] = a1*y[n-1]+b0*x[n]+b1*x[n-1] the
impulse response is shown in the text to be h[n] = b0*(a1^n)*u[n]+b1*(a1^(n-1))*u[n-1].
You can use the filter function to find h[n] by making x[n] = [1 0 0 0 …..] and using
filter() as in problem 33. For your values of coefficients find h[n] for n = 0 to 20 using
matlab by creating x[n] equal to the impulse function
xn = zeros(1,20);
xn(1) =1;
then
b = b0;
a = [1 -a1];
Then generate hn
hn = filter(b,a,xn);
n = 0:1;19;
stem(n,hn)
This should give you a graph of your hn.
Problem 35. Find h[n] for your assigned IIR a and b filter coefficients (a is a1 a2 a3), (b
is b1 b0) where y[n] = a1*y[n-1]+a2*y[n-2] +a3*y[n-4]+b0*x[n]+b1*x[n-1] the same as
in problem 34 by making xn = zeros(1,20) and xn(1) =1, a = [1 -a1 -a2 -a3] and b= [ b0
b1] and hn = filter(b,a,xn). Find H(z) for your IIR filter. Plot the poles and zeros of H(z)
using zplane(b,a). Do a partial fraction expansion of H(z) = r1/(z-p1)+r2/(z-p2)+r3/(z-
p3)+k1+k2*z-1 using [r,p,k] = residue(b,a);.
