GTU Signals and Systems PYQ | Fourier Transform of x(t) = e^(-at)u(t)

GTU Signals and Systems PYQ | Fourier Transform of x(t) = e^(-at)u(t) | Summer 2021 Sem 4 Q4(a)OR

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

📘 Table of Contents

🔹 Introduction
🔹 Question Statement
🔹 Theory Part Related to the Question
🔹 Solution in Detail
🔹 Handwritten Solution Image
🔹 Key Takeaways
🔹 Final Words
🔹 Related Posts

Introduction

In this post, we will explore the step-by-step solution to a frequently asked GTU examination problem from the Signals and Systems subject, specifically from the Summer 2021 exam paper – Question 4(a) OR. The question asks us to find the Fourier Transform of the signal defined as:

x(t) = e^−at u(t)

This type of question is common in the Signals and Systems course and is particularly important for students who aim to build a solid foundation in signal analysis and system behavior in the frequency domain. The exponential signal multiplied by the unit step function is a classic case that introduces us to the Fourier Transform properties of causal signals.

In this article, we will go through the theoretical concepts, followed by a detailed step-by-step solution, a Concept Refresher Box, common mistakes to avoid, and a helpful flowchart-style summary. A high-quality handwritten solution image is also provided for better understanding and reference.

GTU Signals and Systems Summer 2021 Question 4(a)OR Solution – Fourier-Transform problem

Question Statement

GTU Summer 2021 – Signals and Systems – Question 4(a) OR:
Find the Fourier Transform of the signal x(t) = e^−at u(t), where a > 0.

This question requires applying the standard definition of the Fourier Transform and evaluating it for an exponentially decaying causal signal. The presence of the unit step function u(t) ensures that the signal exists only for t ≥ 0, making it suitable for unilateral transforms.

We will now revisit the theoretical background related to this question, including key formulas and properties necessary to solve the problem accurately.

Theory Part Related to the Question

To find the Fourier Transform of x(t) = e^−at u(t), we need to recall the fundamental definition of the Fourier Transform for a continuous-time signal x(t):

X(ω) = ∫_−∞^∞ x(t) · e^−jωt dt

Since u(t) is the unit step function, the signal becomes:

x(t) = e^−at for t ≥ 0 and x(t) = 0 for t < 0

This simplifies the limits of integration to:

X(ω) = ∫₀^∞ e^−at · e^−jωt dt

Combining the exponentials, we get:

X(ω) = ∫₀^∞ e^{−(a + jω)t} dt

This is a standard integral that converges when Re(a) > 0, and its result leads to a rational function of ω. The result is an essential part of Fourier analysis and frequently appears in control systems, communication theory, and signal processing.

📘 Concept Refresher Box

Fourier Transform of e^−at u(t): This is a standard transform and is defined for a > 0.
Unit Step Function (u(t)): Ensures that the signal is zero for t < 0, making the signal causal.
Convergence: The integral converges only if the real part of (a + jω) is positive.
Useful Property: Fourier transform of exponential signals with u(t) typically results in a rational function of frequency ω.

Solution in Detail

We are given the signal:

x(t) = e^−at u(t), a > 0

We apply the Fourier Transform definition:

X(ω) = ∫_−∞^∞ x(t) · e^−jωt dt

Due to u(t), the signal exists only for t ≥ 0. Hence, the integral becomes:

X(ω) = ∫₀^∞ e^−at · e^−jωt dt

Combine the exponential terms:

X(ω) = ∫₀^∞ e^{−(a + jω)t} dt

Let s = a + jω. Then the integral becomes:

X(ω) = ∫₀^∞ e^−st dt

This is a standard integral with the solution:

∫₀^∞ e^−st dt = 1/s, Re(s) > 0

Substitute back:

X(ω) = 1 / (a + jω)

Final Answer:

Fourier Transform of x(t) = e^−at u(t) is X(ω) = 1 / (a + jω)

🚫 Common Mistakes to Avoid

Forgetting to apply the unit step function and using limits from −∞ to ∞.
Not combining exponentials properly before integration.
Using the Laplace transform formula instead of Fourier when jω is expected.
Missing the condition a > 0 required for convergence.

📊 Step-by-Step Summary Table

Step	Description
1	Write the expression: `x(t) = e^−at u(t)`
2	Apply Fourier Transform definition: `X(ω) = ∫₀^∞ e^−(a+jω)t dt`
3	Use standard integral: `∫ e^−st dt = 1/s`
4	Substitute `s = a + jω` to get the final answer
5	Final Result: `X(ω) = 1 / (a + jω)`

Handwritten Solution Image

Visual Reference: Below is the handwritten solution for better clarity and exam-style presentation. This is especially useful for students preparing for GTU exams.

Fourier Transform of x(t) = e^-at u(t) - Signals and Systems - GTU PYQ Handwritten Solution

Key Takeaways

The given signal x(t) = e^−at u(t) is causal and exponentially decaying, making it suitable for Fourier analysis.
The presence of the unit step function u(t) limits the integration to t ≥ 0, which simplifies the evaluation process.
Using the Fourier Transform definition and solving the integral leads to a standard result: X(ω) = 1 / (a + jω).
This result is frequently used in signal processing, control systems, and communication systems for analyzing system behavior in the frequency domain.
Always remember the condition a > 0 to ensure the convergence of the integral.

Final Words

Understanding the Fourier Transform of x(t) = e^−at u(t) is a fundamental skill for GTU Signals and Systems students. This problem is a standard question that not only tests conceptual clarity but also helps reinforce integration techniques and exponential signal behavior in the frequency domain.

If you're preparing for GTU Semester 4 Signals and Systems exams, ensure you practice such standard transforms thoroughly. They often appear in both theoretical and application-based questions.

For more handwritten and explained solutions like this, keep exploring Drk Knowledge 24. Stay consistent, and success will follow.

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GTU Signals and Systems PYQ | Fourier Transform of x(t) = e^(-at)u(t) | Summer 2021 Sem 4 Q4(a)OR