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

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

📑 Table of Contents

• ➤ Introduction
• ➤ Statement of the Question
• ➤ Theory Part Related to the Question
• ➤ Solution in Detail
• ➤ Key Takeaways
• ➤ Final Words
• ➤ Related Posts

Introduction

In this post, we will explain the Time Shifting property of the Fourier Transform with respect to angular frequency ω. This is a key theoretical concept in Signals and Systems and appears frequently in GTU exams. It helps us understand how a shift in time affects a signal’s representation in the frequency domain.

Statement of the Question

The question asks to explain the Time Shifting Property of the Fourier Transform.

This property is stated as:

If x(t) ↔ X(Ꞷ), then x(t - t₀) ↔ X(Ꞷ) · e-jꞶt₀

That is, shifting a signal in time by t₀ results in multiplying its Fourier Transform by an exponential factor in the frequency domain.

Theory Part Related to the Question

In continuous-time Fourier Transform, if:

x(t) ↔ X(Ꞷ)

Then the Time Shifting Property states:

x(t - t₀) ↔ X(Ꞷ) · e-jꞶt₀

This means:

• A time shift of t₀ in the time domain results in a multiplication by e-jꞶt₀ in the frequency domain.
• The magnitude spectrum remains the same.
• The phase spectrum changes due to the exponential term.

Solution in Detail

Example:

Let x(t) = cos(Ꞷ₀t). The Fourier Transform of this is two impulses at +Ꞷ₀ and -Ꞷ₀.

If we time shift it by t₀, the new signal is x(t - t₀) = cos(Ꞷ₀(t - t₀)).

Using the Time Shifting property:

FT{x(t - t₀)} = X(Ꞷ) · e-jꞶt₀

The result is the same magnitude spectrum, but with a linear phase shift introduced by the exponential term. This helps analyze how delays affect signal behavior in the frequency domain.

Key Takeaways

• Time delay t₀ results in a phase shift  e-jꞶt₀ in the frequency domain.
• The shape and magnitude of the spectrum do not change.
• This property is widely used in signal analysis and filtering.

Final Words

The Time Shifting property of the Fourier Transform (with respect to Ꞷ) is a foundational tool in signal processing. It shows how time-domain manipulations reflect phase changes in the frequency domain. Mastering this property is crucial for understanding linear systems, modulation, and spectral analysis in Signals and Systems.