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Problem 1. (25 points) (Use MATLAB for part (c) of this problem) Consider the DTLTI
system shown in Fig. 1:
Figure 1: System Block diagram for Problem 5
(a) (5 points) Express the impulse response of the system as a function of the impulse responses
of the subsystems.
(b) (10 points) Let
h1[n] = e
−0.1nu[n]
h2[n] = h3[n] = u[n] − u[n − 3]
and
h4[n] = δ[n − 2]
Determine the impulse response heq[n] of the equivalent system. Note that you need to do
the convolution on paper, but are free to use MATLAB (“conv” command) to verify if your
calculations are correct.
(c) (10 points) Let the input signal be a unit-step, that is, x[n] = u[n]. Determine and plot y[n]
in MATLAB.
(Include the MATLAB plots for n = 0, 1, . . . , 10 samples only, and use “stem” for the plot
command)
Problem 2. (20 points) Impulse response problems:
(a) (10 points) The response of a linear time-invariant system to x[n] = u[n] is y[n] = 0.5
nu[n].
Find its response to δ[n].
(b) (10 points) True or False: If both the input sequence to an LTI system and its impulse
response sequence are even sequences, then the output sequence is also even.
1
Problem 3. (20 points) Evaluate the convolutions:
(a) (10 points) u[n] ∗
1
2
n
u[n − 1].
(b) (10 points) u[−n] ∗
1
2
n
u[n − 1].
Problem 4. (15 points) Determine the DTFS representation of the signal
x[n] = 1 + cos(0.24πn) + 3 sin(0.56πn)
Assume that the period of x[n] is the LCM of the periods of the sinusoidal terms. Sketch the DTFS
spectrum. Note that if the DTFS coefficients are complex, you need to plot the magnitude and the
phase of the coefficients separately.
Problem 5. (20 points) Consider the periodic signal g˜[n] in the following figure. g˜[n] is a delayed
version of the signal x˜[n] also shown below.
(a) (10 points) Determine the DTFS coefficients of g˜[n], denoted by ˜dk, k = 0, …, 4 directly from
the DTFS analysis equation:
˜dk =
1
N
N
X−1
n=0
g˜[n]e
−j(2π/N)kn
.
You are free to use MATLAB for calculations (addition/multiplication of complex numbers,
for example). Alternately you can also leave the final expression as a summation of complex
exponential terms.
(b) (10 points) Determine the DTFS coefficients ˜dk, k = 0, …, 4 by applying the time shifting
property to the coefficients of x˜[n]. The coefficients of x˜[n], denoted by c˜k are given in the
table below. You are free to use MATLAB for calculations (addition/multiplication of complex
numbers, for example).
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