ECE 310 Digital Signal Processing

Homework 6

1. A causal LSI system is described by the difference equation: y[n] − y[n − 1] = x[n].

(a) Determine the system’s transfer function H(z)

(b) Determine the system’s unit pulse response h[n]

(c) Determine the system’s frequency response Hd(ω); is Hd(ω) = H(z)|z=e

jω ? if not, explain why.

2. An LSI system is described by the difference equation

y[n] = x[n] + x[n − 10]

(a) Compute and sketch its magnitude and phase response

(b) Determine its output to inputs

i. x[n] = cos π

10n + 3 sin

π

3

n +

π

10

ii. x[n] = 10 + 5 cos

2π

5

n +

π

2

3. The frequency response of an LSI system is

Hd(ω) = ωej sin ω

, |ω| ≤ π .

Determine the system output y[n] for the following inputs:

(a) x[n] = 5 + 10e

j(

π

4

n+45◦) + j

n

(b) x[n] = 5 + 10 cos( π

4

n + 45◦

) + j

n

.

4. The difference equation of a causal LSI system is given by

y[n] −

1

√

4

y[n − 1] = x[n], −∞ < n < ∞ .

Determine y[n] for input x[n] = 10 + cos( π

4

) sin( π

2

n) + 2(−1)n

, −∞ < n < ∞.

5. The response of a real LSI system for input

x[n] = 3 + cos π

4

n + 10◦

+ sin π

3

n + 25◦

is

y[n] = 9 + 2 sin π

4

n + 10◦

.

Determine the system response ˜y[n] for input

x˜[n] = 5 + 2 sin π

4

n + 15◦

+ 10 cos

−

π

3

n + 25◦