GTU Sem 4 Signals & Systems Summer 2021 PYQ Question 3(c) Solution

Introduction

In this blog post, we’ll explore the Fourier series coefficients of a continuous-time periodic signal defined piecewise over the interval 0 ≤ t < 2. These types of questions are commonly asked in GTU exams under the Signals and Systems or related subjects, especially when testing a student's understanding of periodic signals and their frequency-domain representation.

Here, we’ll solve the question step by step using the standard Fourier series coefficient formula and provide a conceptual understanding of the approach. This will help you prepare for exams and reinforce your fundamentals.

Question Statement

Find the Fourier series coefficients for the following continuous-time periodic signal:

x(t) =  1.5     for   0 ≤ t < 1  
     = –1.5     for   1 ≤ t < 2

The fundamental angular frequency is given as ω₀ = π.

Fourier series coefficients for the continuous time periodic signal – GTU Signals and Systems Question 5(b)

Theory Part Related to the Question

To find the Fourier series coefficients of a continuous-time periodic signal, we use the formula for the complex exponential form of the Fourier series:

Cₙ = (1 / T) ∫₀ᵀ x(t) · e^{−j n ω₀ t} dt

Where:

Cₙ is the nth Fourier series coefficient
T is the period of the signal
ω₀ is the fundamental angular frequency = 2π / T

In this question, the signal is defined over the interval 0 ≤ t < 2, and ω₀ is given as π. That means the period T = 2.

Solution in Detail

Given:
x(t) = 1.5 for 0 ≤ t < 1

x(t) = -1.5 for 1 ≤ t < 2

Fundamental frequency ω₀ = π, so period T = 2

We use the general formula for the nth complex Fourier coefficient:

Cₙ = (1 / T) ∫₀ᵀ x(t) · e^{−j n ω₀ t} dt

Substitute T = 2 and split the integral into two parts based on the definition of x(t):

Cₙ = (1/2) [ ∫₀¹ (1.5) · e^{−j n π t} dt + ∫₁² (−1.5) · e^{−j n π t} dt ]

Evaluate each integral separately:

First integral:
= 1.5 ∫₀¹ e^{−j n π t} dt 
= 1.5 [ e^{−j n π t} / (−j n π) ] from t = 0 to 1
= 1.5 [ (e^{−j n π} − 1) / (−j n π) ]

Second integral:
= -1.5 ∫₁² e^{−j n π t} dt
= -1.5 [ e^{−j n π t} / (−j n π) ] from t = 1 to 2
= -1.5 [ (e^{−j 2n π} − e^{−j n π}) / (−j n π) ]

Combine both results:

Cₙ = (1/2) × [ 1.5 (e^{−j n π} − 1) / (−j n π) − 1.5 (e^{−j 2n π} − e^{−j n π}) / (−j n π) ]

Since e^{−j 2n π} = 1 for any integer n, simplify:

Cₙ = (1/2) × [ 1.5 (e^{−j n π} − 1) / (−j n π) − 1.5 (1 − e^{−j n π}) / (−j n π) ]
     = (1/2) × [ 1.5 (2e^{−j n π} − 2) / (−j n π) ]
     = (1.5 / 2) × (2e^{−j n π} − 2) / (−j n π)
     = 1.5 (e^{−j n π} − 1) / (−j n π)

Since e^{−j n π} = (-1)ⁿ for any integer n, simplify:

So, the final expression for the Fourier series coefficients is:

Cₙ = [1.5 (1 − (-1)ⁿ)] / (j n π)

Handwritten Solution Images

Below are the handwritten steps for solving the Fourier series coefficient problem from the Summer 2021 GTU exam, Signals and Systems subject.

Fourier series handwritten solution page 1 – Summer 2021 Signals and Systems

Fourier series handwritten solution page 2 – Summer 2021 Signals and Systems

Key Takeaways

The Fourier series helps represent periodic signals using complex exponentials.
Always check the piecewise definition of the signal before integrating.
For exponential Fourier series, the formula Cₙ = (1 / T) ∫ x(t) · e^{−j n ω₀ t} dt is fundamental.
Simplification using Euler’s identity or exponential properties can significantly reduce manual effort.
In this example, symmetry in the signal helped simplify the final expression of the coefficients.

Final Words

Understanding how to derive Fourier series coefficients is crucial for analyzing and interpreting periodic signals in the frequency domain. This problem from the Summer 2021 GTU Signals and Systems exam is a classic example that tests your fundamentals in integration and complex exponentials. Practice more variations like this to build confidence for exams.

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GTU Sem 4 Signals & Systems Summer 2021 PYQ Question 3(c) Solution