ECE 310 Digital Signal Processing

Homework 2

1. Two systems are specified by the following input-output relations, where x[n] is the input and y[n]

the output:

(a) y[n] = y[n − 5] + x[n] + 10x[n − 1]

(b) y[n − 2] + 2y[n] = cos

π

6

n

x[n]

For each system, determine if it is: (a) linear or non-linear, (b) time-invariant or time-varying.

Justify your answers with proofs or counter-examples.

2. The output, y[n], of a given system is always related to its input x[n] by y[n] = h[n] ∗ x[n], where

h[n] is the system’s unit pulse response. Show that the system must be LSI.

3. Express the output y[n] of an LSI system with unit pulse response h[n] in terms of its step response

g[n] = h[n] ∗ u[n] and the input x[n]. Hint: Try to represent δ[n] in terms of shifted versions of u[n].

4. Assume that the zero-state response of an LSI system to input x[n] = 3(−n)u[n] is y[n] = 1

5n u[n − 1].

Use the system’s properties (linearity and shift-invariance) to find h[n], the system’s unit pulse

response.

5. Compute the convolution x[n] ∗ h[n] for the x[n] and h[n] given below. Note: The arrow indicates

n = 0.

(a) x[n] = {−1, 0

↑

, 1}, h[n] = {1

↑

, 2, 3, 4, 5}

(b) x[n] = 3(−n)u[n], h[n] = {0

↑

, 1, 2}

(c) x[n] = u[n], h[n] = n(u[n] − u[n − 4])

(d) x[n] = (−1)(−n)u[n], h[n] = e

(−n)u[n