ELEC3305/ELEC9305
Digital Signal Processing (Sem. 1 – 2022) Unit Coordinator: Craig Jin
Tutorial Assignment 1
Please attempt all questions and show all work. Some standard DSP tables
are included below.
Student ID:
1
Fourier Transform Pairs: X(ej ω) =
∑∞
k=−∞ x[k] e
−j ω k and x[n] = 1
2pi
∫ pi
−piX(e
j ω)ej ω ndω
Fourier Transform Symmetry
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Fourier Transform Theorems
Equations Related to Wide-Sense Stationary Random Processes
y[n] =
∞∑
k=−∞
x[k]h[n− k]
chh[l] =
∞∑
k=−∞
h[k]h[l + k]
Chh(e
j ω) = H(ej ω)H∗(ej ω) = |H(ej ω)|2
my = mx
∞∑
k=−∞
h[k]
φyy[m] =
∞∑
k=−∞
φxx[m− k]chh[k]
Φyy(e
j ω) = Chh(e
j ω) Φxx(e
j ω)
φyx[m] =
∞∑
−∞
h[k]φxx[m− k]
Φyx(e
j ω) = H(ej ω)Φxx(e
j ω)
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Z-Transform Pairs: X(z) =
∑∞
n=−∞ x[n]z
−n and x[n] = 1
2pi j
∮
ccw
X(z)zn−1dz
Z-Transform Properties
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1. (6 points) What is the period (in samples) of the discrete signal, x[n], shown below? Show
all working.
x[n] = cos
(
7
17
pi n
)
+ cos
(pi
5
n
)
2. (10 points) A sequence y[n] is the linear convolution of the two sequences x[n] and h[n] as
shown below. Values not shown are zero. Determine the values of y[n]. When giving the
answer specify the value of n for each y[n]. Please show the complete calculation using one
of the methods taught in class.
Figure 1: x[n] = 1 −2 2 3 2 4 for n = −3 −2 −1 0 1 2.
Figure 2: h[n] = 2 −2 1 −1 1 for n = −1 0 1 2 3.
3. (6 points) For this question only, you are required to write your solution using Latex.
Upload your latex file along with your solutions. A discrete-time system produces an
output signal, y[n], from the input signal, x[n], as follows:
y[n] =
x[2n] + x[−2n]
2
.
(a) (3 points) Is the system linear? Please show the complete working and proof to sup-
port your conclusion.
(b) (3 points) Is the system time-invariant? Please show the complete working and proof
to support your conclusion.
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4. (12 points) A system is specified by the following equation:
y[n] = cos(5
√
|n|)x[n] .
Determine if the system is (1) linear, (2) causal, (3) shift-invariant, and (4) BIBO stable.
Please provide a complete proof and working for your conclusions.
• (3 points) Linear
• (3 point) Causal
• (3 point) Shift-invariant
• (3 point) BIBO stable
5. (12 points) Let X(ej ω) denote the discrete-time Fourier transform of the signal x[n] shown
below.
Figure 3: x[n] = 2 1 −3 −1 2 0 2 −1 −3 1 2 for n = −2 −1 0 1 2 3 4 5 6 7 8.
(a) (4 points) Evaluate X(ej ω) at ω = pi. Show all calculations and working and describe
your reasoning.
(b) (4 points) Find ∠X(ej ω). Show all calculations and working and describe your rea-
soning.
(c) (4 points) Sketch the stem plot (or alternatively list the sequence values and sequence
indices) of the sequence whose Fourier transform is Re [X(ej ω)]. Describe your rea-
soning and show all working.
6. (12 points) Derive the formula for the discrete sequence, h[n], corresponding to an ideal
band-pass filter based on the frequency response shown below. Simplify your maths so that
there are no complex exponential functions.
H(ej ω) =

0, |ω| ≥ ωd
1 · e−j αω, ωc < |ω| ≤ ωd
0, |ω| ≤ ωc
7. (10 points) We generate a discrete-time random process, x[n], by drawing a sequence of
independent and identically distributed numbers from a random number generator with
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a uniform distribution in [−1, 1]. The random process, x[n], is then processed by an LTI
system with an impulse response, h[n], given by: h[n] = δ[n]+δ[n−1] to obtain the random
process y[n] = x[n] + x[n− 1]. Please answer the questions below and show all working.
(a) (2 points) What is the mean my[n] of y[n]?
(b) (4 points) What is the autocorrelation of φyy[m] of y[n]?
(c) (4 points) What is the power spectral density of Pyy(e
j ω) of y[n]?
8. (12 points) Consider a stable discrete-time LTI system described by the following frequency
response:
H(ej ω) =
1 + e−2 j ω
1− 3
2
e−j ω
.
(a) (4 points) Draw the pole-zero diagram.
(b) (4 points) Is the system causal ?
(c) (4 points) Write down a linear constant-coefficient difference equation that corre-
sponds to this system frequency response.
9. (20 points) You are told the Z-transform, X(z), of an absolutely summable sequence, x[n],
is given by:
X(z) =
1 + z−1 − 2z−2
1− 13
6
z−1 + z−2
.
Determine the sequence x[n]. Show all working.
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