12. Consider the second-order IIR filter
yn a
1
yn À 1 a
2
yn À 2 xn, n ! 0:
(a) Compute Hz.
(b) Discuss stability conditions related to coefficients a
1
and a
2
.
13. Determine the stability of the following IIR filters:
a Hz
zz À 1
z
2
À z 1z 0:8
:
b 3yn 3:7yn À 1 À 0:7yn À 2 xn À 1, n ! 0:
14. In Figure 4.4(b),let H
1
z and H
2
z are the transfer functions of the two first-order IIR
filters defined as
y
1
n xn À 0:5y
1
n À 1, n ! 0
y
2
n xn y
2
n À 1, n ! 0:
(a) Find the overall transfer function Hz H
1
z H
2
z.
(b) Find the output yn if the input xn À1
n
, n ! 0.
15. Consider the first-order IIR system
yn
1 À a
2
xn xn À 1 ayn À 1, n ! 0,
Find the squared-magnitude response jH!j
2
.
16. Consider a moving average filter defined in (3.2.1). Find the magnitude response jH!j and
the phase response f!.
17. Consider the FIR filter
yn xn 2xn À 1 4xn À 2 2xn À 3 xn À 4, n ! 0:
Find the transfer function Hz and magnitude response jH!j.
18. Derive Equation (4.4.18).
Part B
19. Compute c
k
given in (4.1.4) for A 1, T
0
0:1,and t 0:05,0:01,0:001,and 0.0001.
Using MATLAB function stem to plot c
k
for k 0, Æ 1, F F F Æ 20.
20. Repeat the Problem 19 for A 1, t 0:001 and T
0
0:005,0.001,and 0.01.
EXERCISES
179