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
2
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)
3
Z-Transform Pairs: X(z) =
∑∞
n= ∞ x[n]z
n and x[n] = 1
2pi j
∮
ccw
X(z)zn 1dz
Z-Transform Properties
4
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.
5
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) =
