Homework 3

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]

Problem 3.7 Solve X[z]

Problem 3.7 (a) Solve H[z] Causal

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 > 1 2 . Hence Y[z] converges on 𝑧 >1 .

Problem 3.7 Solve for y[n]

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?

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

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

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

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

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

Problem 3.17 For 𝑧 < 1 2 ℎ 𝑛 =− 2 3 2 𝑛 𝑢 −𝑛−1 − 1 3 1 2 𝑛 𝑢 −𝑛−1 , ℎ 0 =0

Problem 3.17 1 2 < 𝑧 <2

Problem 3.17 2< 𝑧

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

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

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

Problem 3.32 (b) (b)

Problem 3.32 (c) (c)