In this problem you are to design a simple causal

1.10.4 In this problem you are to design a simple causal real discrete-time LTI system with the following properties:
(a) The system should kill the signals ( - 1)
n
and cos(0 . 5 ⇡ n )
(b) The system should have unity dc gain. That is, H f (0) = 1.
For the system you design:
(a) Find the di ↵ erence equation to implement the system.
(b) Sketch the impulse response of the system.
(c) Roughly sketch the frequency response magnitude | H
f
( ! ) | . Clearly show the nulls of the frequency response.
1.10.4) Solution
186

− π − 0.5 π 0 0.5 π π
0
5
10
15
FREQUENCY RESPONSE 2
ω
− π − 0.5 π 0 0.5 π π
0
2
4
6
8
FREQUENCY RESPONSE 4
ω
− π − 0.5 π 0 0.5 π π
0
2
4
6
8
FREQUENCY RESPONSE 3
ω
− π − 0.5 π 0 0.5 π π
0
2
4
6
FREQUENCY RESPONSE 1
ω
− 1 − 0.5 0 0.5 1
− 1
− 0.5
0
0.5
1
2
ZERO − POLE DIAGRAM 3
− 1 − 0.5 0 0.5 1
− 1
− 0.5
0
0.5
1
2
ZERO − POLE DIAGRAM 1
− 1 − 0.5 0 0.5 1
− 1
− 0.5
0
0.5
1
ZERO − POLE DIAGRAM 2
− 1 − 0.5 0 0.5 1
− 1
− 0.5
0
0.5
1
ZERO − POLE DIAGRAM 4
1.11.1) Solution
IMPULSE RESPONSE FREQUENCY RESPONSE POLE-ZERO DIAGRAM
1 2 3
2 4 1
3 3 2
4 1 4
190

1.12.4 Feedback Loop Consider the following connection of two discrete-time LTI systems.
- -
h
2
( n )
-
h
1
( n )
6
+
i
You are told that
h
1
( n ) = δ ( n ) + δ ( n - 1) , h
2
( n ) =
1
2
δ ( n - 1) .
(a) Find the impulse response h ( n ) of the total system.
(b) Sketch h
1
( n ), h
2
( n ), and h ( n ).
(c) Find the di ↵ erence equation describing the total system.
1.12.4)

0 10 20
− 0.5
0
0.5
1
1.5
2
h
1
(n) [IMPULSE RESPONSE]
n
0 10 20
− 0.5
0
0.5
1
1.5
2
h
2