ECE250 Assignment 4 Solution

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
• Clarifications:
– Symbols have their usual meaning. Assume the missing information & mention it in the report. Use Google Classroom for any queries. In order to keep it fair for all, no email queries will be entertained.
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: “A4 RollNo Name.zip”. • Codes/reports submitted without a zipped file or without following the naming convention will NOT be checked.
• Important Note: Do not use inbuilt functions in MATLAB or PYTHON. Use mathematical equations/derivations to solve the required.

Theory Problems (60 points)
[CO4] Q1. A signal x(t) is represented by, ). Determine the following
), then determine g(t). [4 points]
(b) Prove X(jω) is periodic signal. Specify X(jω) over one time period. [3 Points].
[CO4] Q2. Determine which, if any, of the real signals given in the following figure-1 have Fourier Transforms that satisfy each of the following conditions: [6x3 Points].

Figure 1: Figure-Q1 (a) Re{X(jω)} = 0
(b) Img{X(jω)} = 0
(c) exp(jαω)X(jω) is real (for real value of α)
(d) R−∞∞ X(jω)dω = 0
(e) R−∞∞ ωX(jω)dω = 0
(f) X(jω) is periodic
[CO4] Q3. Consider the LTI system “S” with impulse response . Determine the output of “S” for each of the following inputs: [4x4 Points]


[CO4] Q4. The input and output of a causal LTI system are related by the differential equation-
) (1)
(a) Find the impulse response of the system. [2 Points]
(b) What is the response of the system if x(t) = t.e(−2t)u(t) [4 Points]
(c) Repeat part (a) for the causal LTI system described by the following equation [4 Points]
) (2)
[CO4] Q5. Let [5 Points]
(3)
where “*” denotes convolution and |ωc| ≤ π. Determine the stricter constraint on “ωc”, which ensure that-
(4)
[CO4] Q6. An input x[n] with length “3” is applied to an LTI system having an impulse response h[n] of length “5”. The output is y[n]. [4 points]
y[n] ↔ Y (ejω) (5)
|h[n]| ≤ L, |x[n]| ≤ B (6)
Find the maximum value of Y (ej0).
Programming Problems (10 points)
[CO4] Q1. Let X(jω) be the Fourier transform of the signal x(t) given in Fig.2.
(a) Determine and Plot the frequency domain signal X(jω). [1 Point]

Figure 2: Q1
(b) Plot the magnitude spectrum of the frequency domain signal X(jω). [1 Point]
(c) Plot the phase spectrum of the frequency domain signal X(jω). [1 Point]
(d) Plot the inverse Fourier transform of real part of {X(jω)}. [1 Point]
[CO4] Q2. Let x[n] be a discrete-time signal with Fourier Transform X(ejω), which is the given Fig. 3. Plot the frequency response, magnitude spectrum and phase spectrum of w[n] = x[n]p[n], for these p[n]

Figure 3: Q2
(a) p[n] = cos(πn) [2 Points]
(b) p[n] = sin(πn/2) [2 Points]
(c) p[n] = P∞k=−∞ δ(n − 2k) [2 Points]