ECE250 Assignment 3 Solution

Guidelines for submission
Theory Problems:
• Submit a hard copy of your solutions in the wooden box kept on the 3rd Floor of Old Academic Block
(right side of the lift).
• Do all questions in sequence.
• Use A4 sheets (Plain). Staple your sheets properly
Programming Problems:
• Use Matlab or python to solve the programming problems.
• For your solutions, you need to submit a zipped file on Google classroom with the following:
– program files (.m) or (.ipynb) with all dependencies.
– a report (.pdf) with your coding outputs and generated plots. The report should be self-complete with all your assumptions and inferences clearly specified.
• Before submission, please name your zipped file as: “A3 RollNo Name.zip”. • Codes/reports submitted without a zipped file or without following the naming convention will NOT be checked.

Theory Problems (35 points)
[CO3] Q1. Consider a causal continuous time LTI system whose input x(t) and output y(t) are related by the following differential equation:[2x5 Points]
) (1)
Find the Fourier series representation of the output y(t) for each of the following inputs

(b) x(t) = sin(4πt) + cos(6πt + π/4)
[CO3] Q2. Consider a periodic signal x(t) with fourier series coefficient ak given below with the condition that the fundamental angular frequency w0 is equal to π for x(t), then Determine the signal x(t).[6 Points]
1

Figure 1: Problem 2
[CO3] Q3. Determine the Fourier series coefficients for each of the following discrete-time periodic signa]s. Plot the
magnitude and phase of each set of coefficients ak.[24 Points]
(a) Each x[n] depicted in figure 2.[3x4 Points]
(b) x[n] = sin(2πn/3)cos(πn/2)[4 Points]
(c) x[n] is periodic with period 4 and[4 Points]
x[n] = 1 − sin(πn/4) for 0 ≤ n ≤ 3 (2)
(d) x[n] is periodic with period 12 and [4 Points]
x[n] = 1 − sin(πn/4) for 0 ≤ n ≤ 11 (3)
[CO3] Q4. The signal x(t) is defined as follows, with time period T = 6:
 t + 2 for − 2 < t < −1

x(t) = 1 for − 1 < t < 1 (4)
2 − t for 1 < t < 2
(a) Determine the Fourier series representation for the signal x(t). [4 Points] (b) Determine the ratio of power in the 7th harmonic to the power in the 5th harmonic. [1 Points]

Figure 2: Problem 3
Programming Problems (10 points)
[CO3] Q1.
1. Given a discrete-time signal x[n]:[6 Points]

1 for − N1 ≤ n ≤ N1,N1 = 2
x[n] = (5)
0 for
Compute the Fourier series for the following cases and give the inference.
(a) N =4N1 + 1
(b) N =8N1 + 1
(c) N =10N1 + 1
[CO3] Q2. A periodic function is defined by:[4 Points]
f(x) = x + π, -π ≤ x < π
f(x + 2π) = f(x)
(a) Sketch the graph of f(x) for three periods.
(b) Find the Fourier series of f(x) on the interval −π < x < π.
Hint: To simulate continuous signals use appropriate discretization (wherever required).