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

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

GTU Sem 4 | Signals and Systems | Summer 2021 | Question 3(c) OR

📑 Table of Contents

• 🔍 Introduction
• 🧾 Question Statement
• 📘 Theory Related to the Question
• 🧮 Solution in Detail
• ✍️ Handwritten Solution Images
• ✅ Key Takeaways
• 📝 Final Words
• 📚 Related Posts

🔍 Introduction

In this blog post, we’ll solve GTU Semester 4 Signals and Systems Summer 2021 Question 3(c) OR, where we're given a periodic signal x(t) in the form of a rectangular waveform. The goal is to determine its complex exponential Fourier series representation.

This type of problem is commonly asked in GTU exams to test understanding of periodic signals and Fourier series analysis — a fundamental concept in the field of signal processing and communication systems.

🧾 Question Statement

Question 3(c) OR – GTU Summer 2021 (Signals and Systems):
Consider the periodic signal x(t) shown below. Determine its complex exponential Fourier series representation.

📘 Theory Related to the Question

To find the complex exponential Fourier series representation of a periodic signal x(t), we use the general formula:

x(t) = Σ_{n = −∞}^∞ C_n · e^{j·n·ω₀·t}

Where:

• C_n is the complex Fourier series coefficient
• ω₀ = 2π / T₀ is the fundamental angular frequency
• T₀ is the time period of the signal

The formula to compute each coefficient Cn is:

C_n = (1 / T₀) · ∫ x(t) · e^{−j·n·ω₀·t} dt

In this question, the signal x(t) is a periodic rectangular waveform with a specific amplitude and width, repeating every T₀. The Fourier series coefficients will be calculated by integrating over one period of the waveform where x(t) is non-zero.

🧮 Solution in Detail

The given signal x(t) is a periodic rectangular waveform of amplitude A and time period T₀. The signal is non-zero between 0 and T₀/2, and zero elsewhere within each period.

The fundamental angular frequency is:

ω₀ = 2π / T₀

The complex exponential Fourier series coefficient Cn is given by:

C_n = (1 / T₀) · ∫₀^T₀/2 A · e^{−j·n·ω₀·t} dt

Since A is a constant, we can take it outside the integral:

C_n = (A / T₀) · ∫₀^T₀/2 e^{−j·n·ω₀·t} dt

Solving the integral:

C_n = (A / T₀) · [ e^{−j·n·ω₀·t} / (−j·n·ω₀) ] from 0 to T₀/2

= (A / T₀) · [ e^{−j·n·ω₀·T₀/2} − 1 ] / (−j·n·ω₀)

Now plug in ω₀ = 2π / T₀:

C_n = (A / T₀) · [ e^−j·n·π − 1 ] / (−j·n·(2π / T₀))

Simplifying:

e−j·n·π = (−1)n, so:

C_n = (A / T₀) · [ (−1)ⁿ − 1 ] / (−j·n·(2π / T₀))

Canceling T₀:

C_n = A · [ (−1)ⁿ − 1 ] / (−j·2π·n)

This gives:

• If n is even: (−1)ⁿ − 1 = 1 − 1 = 0 → C_n = 0
• If n is odd: (−1)ⁿ − 1 = −1 − 1 = −2 → C_n = −j · A / (n · π)

For n = 0 (DC component):

C₀ = (1 / T₀) · ∫₀^T₀/2 A dt = A / 2

Final Result:

• C₀ = A / 2
• C_n = 0 for even n
• C_n = −j · A / (n · π) for odd n

Hence, the complex exponential Fourier series representation of x(t) is:

x(t) = Σ_{n = −∞}^∞ C_n · e^{j·n·ω₀·t}

✍️ Handwritten Solution Images

Below are the handwritten steps for solving the complex exponential Fourier series coefficients of the given signal from GTU Sem 4 | Signals and Systems | Summer 2021 | Question 3(c) OR. These images show all detailed derivations and final answers.

✅ Key Takeaways

• This question involves applying the complex exponential Fourier series to a time-domain signal that is non-zero only between 0 to T₀/2.
• We assumed the signal to be of amplitude A (a constant), and period T₀.
• By evaluating the integral, we found:
  • C₀ = A / 2 – this is the DC component.
  • C_n = 0 for even n values due to symmetry.
  • C_n = −j · A / (n · π) for odd n values.
• The final answer is presented in the standard form of the complex exponential Fourier series:

x(t) = Σ_{n = −∞}^∞ C_n · e^{j·n·ω₀·t}

• This is a frequently asked type of Fourier series problem in GTU exams, so practicing this step-by-step is highly recommended.

📝 Final Words

Understanding the derivation of complex exponential Fourier series coefficients is essential for mastering signal analysis in the Signals and Systems subject. This question from GTU Sem 4 | Summer 2021 | Question 3(c) OR perfectly illustrates the process of calculating Fourier coefficients for a periodic signal that is zero over half the interval.

Make sure to remember the conditions for odd and even harmonics, and always substitute ω₀ = 2π / T₀ properly when simplifying. For more such step-by-step solutions, explore other questions from previous GTU papers below.

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