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# Digital Signal Processing Homework 6

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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

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◦
 Digital Signal Processing Homework 6
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