Concordia UniversityDepartment of Electrical and Computer Engineering

Real–Time Computer Control SystemsS. Hashtrudi Zad

Assignment 2 (two pages)

Due: Jan. 28, 2014

1. Determine the z-transform for each of the following sequences. Indicate the region ofconvergence. Comment on whether the Fourier transform of the sequence exists.

(a) e[n] = n(12)n1[n]

(b) e[n] = 1[n]− 1[n− 2]

(c) e[n] = sin(ω0n)1[n]

2. Consider a signal with the transform

H(z) =z − 1

(z − 12)(z − 2)

.

What are the possible ROCs. In each case, find the inverse z-transform, h[n], andindicate whether or not h[n] can be the impulse response of a stable causal LTI system.

3. (Prob.4.22 of the textbook) Compute the inverse transform, f [n], for each of the fol-lowing transforms:

(a) F (z) = 11+z

−2 , |z| > 1;

(b) F (z) = z(z−1)z2−1.25z+0.25

, |z| > 1;

(c) F (z) = z

z2−2z+1

, |z| > 1;

(d) F (z) = z

(z−0.5)(z−2), 1

2< |z| < 2.

4. (Prob.4.20 of the textbook) Consider a signal with the transform (which converges for|z| > 2)

U(z) =z

(z − 1)(z − 2)

(a) What value is given by the formula of the Final Value Theorem applied to thisU(z)?

(b) Find the final value of u[k] by taking the inverse transform of U(z), using partialfraction expansion.

(c) Explain why the two results of (a) and (b) differ.

1

5. Consider an LTI system for which

u[n]− 2u[n− 1] = e[n− 1],e[n] = 4n, for n ≥ 0u[0] = 1.

Determine the response, u[n], for n ≥ 0.

2