1. Homework 3

2. Problem 3.7 The input to a causal, LTI system is:
The output z-transform is:
Determine:
(a) H(z) and ROC
(b) ROC of Y[z]
(c) y[n]

3. Problem 3.7
Solve X[z]

4. Problem 3.7
(a) Solve H[z]
Causal

5. Problem 3.7
(b) ROC of Y[z]
Possible ROCs: 𝑧 < 1 2 , 1 2 < 𝑧 <1, 1< 𝑧
Since one of the poles of X[z], which limited the roc OF x[Z] to be less than 1, is cancelled by the zero of H[z], the ROC of Y[z] is the region of the z-plane that satisfies the remaining two constraints 𝑧 >1 𝑎𝑛𝑑 𝑧 > Hence Y[z] converges on 𝑧 >1 .

6. Problem 3.7
Solve for y[n]

7. Problem 3.8 The causal system function is:
The input is: 𝑥 𝑛 = (1/3) 𝑛 𝑢 𝑛 +𝑢 −𝑛−1
(a) Find h[n]
(b) Find y[n]
(c) Is H stable, absolutely summable?

8. Problem 3.8 (a) The ROC is 𝑧 >3/4, since it is causal
First divide to get: H[z]=
another way:

9. Problem 3.8
(b) Find y[n]
First solve for X[z], then Y[z] =X[z]H[z]

10. Problem 3.8
(c) Stable and absolutely summable since ROC includes unit circle

11. Problem 3.17
An LTI system with input x[n] and output y[n] satisifes the difference equation:
Determine all possible values for the system’s impulse response h[n] at n=0

12. Problem 3.17
Solve for H[z]
3 possible ROCs: 𝑧 < 1 2 , 1 2 < 𝑧 <2, 2< 𝑧

13. Problem 3.17
For 𝑧 < 1 2
ℎ 𝑛 =− 𝑛 𝑢 −𝑛−1 − 𝑛 𝑢 −𝑛−1 , ℎ 0 =0

14. Problem 3.17
1 2 < 𝑧 <2

15. Problem 3.17
2< 𝑧

16. Problem 3.32 Determine inverse transform:
For the 3rd term use the identity:

17. Problem 3.32
and
3rd term continued
Let
Let and
Then therefore

18. Problem 3.32
The other terms are done by inspection and a stable sequence implies 2-sided sequence by pole observation

19. Problem 3.32 (b)
(b)

20. Problem 3.32 (c)
(c)