GTU Signals and Systems PYQ | DTFT of x(n) = {1, 0, 4, 2} | Sem 4 Summer 2021 Q5(a)

GTU Signals and Systems PYQ | DTFT of x(n) = {1, 0, 4, 2} | Sem 4 Summer 2021 Q5(a)

GTU Sem 4 | Signals and Systems | Summer 2021 | Question 5(a) Solution

📑 Table of Contents

• ❓ 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 derive the Discrete-Time Fourier Transform (DTFT) of the given discrete sequence x(n) = {1, 0, 4, 2}. This question appeared in the GTU Summer 2021 examination for Signals and Systems, Semester 4 as Question 5(a). We will begin by restating the question, and then provide the relevant theory, followed by a step-by-step analytical solution and a handwritten solution image.

This topic is an important part of GTU’s Unit 3: Fourier Transform for Discrete-Time Signals and is frequently asked in university exams to evaluate students’ understanding of frequency domain analysis for finite sequences.

❓ Question Statement

GTU Summer 2021 | Semester 4 | Signals and Systems | Question 5(a):
Find the Discrete-Time Fourier Transform (DTFT) of the sequence:

x(n) = {1, 0, 4, 2}

In this problem, we are given a finite-length discrete-time signal and asked to find its DTFT expression. The goal is to derive the frequency domain representation of this time-domain sequence using standard DTFT formulas.

This type of numerical problem is frequently asked in GTU exams and is considered essential for mastering Unit 4 concepts.

📚 Theory Part Related to the Question

The Discrete-Time Fourier Transform (DTFT) is used to analyze discrete-time signals in the frequency domain. For a discrete sequence x(n), the DTFT is defined as:

X(e^jω) = ∑ x(n) · e^−jωn

Where:

• x(n) is the input discrete-time signal.
• X(e^jω) is the DTFT of x(n), a continuous function of frequency ω (in radians/sample).
• The summation runs over all values of n for which x(n) is defined.

In the given question, the sequence is finite and non-periodic, so its length determines the summation limits. This makes the DTFT calculation straightforward using direct substitution.

Also note:

• DTFT is generally complex-valued and depends on ω ∈ [−π, π]
• The result can be expressed in exponential or trigonometric form as needed

🔄 Concept Refresher

• DTFT Formula: X(e^jω) = ∑ x(n) · e^−jωn
• Frequency Domain: The result X(e^jω) is a continuous function of ω ∈ [−π, π]
• Finite Sequence: For finite signals, the DTFT is computed by substituting each value of x(n) and its index n.
• Complex Result: DTFT results are usually complex, combining both magnitude and phase information.
• Use in Analysis: DTFT helps analyze how a discrete-time signal behaves in the frequency domain.

🧮 Solution in Detail

We are given a discrete-time sequence:

x(n) = {1, 0, 4, 2}

This implies:

• x(0) = 1
• x(1) = 0
• x(2) = 4
• x(3) = 2

We will apply the DTFT formula:

X(e^jω) = ∑ x(n) · e^−jωn

Substituting the known values of x(n):

X(e^jω) = x(0)·e^−jω·0 + x(1)·e^−jω·1 + x(2)·e^−jω·2 + x(3)·e^−jω·3

= 1·e⁰ + 0·e^−jω + 4·e^−j2ω + 2·e^−j3ω

= 1 + 0 + 4·e^−j2ω + 2·e^−j3ω

Final DTFT Expression:

X(e^jω) = 1 + 4·e^−j2ω + 2·e^−j3ω

This is the required Discrete-Time Fourier Transform of the given sequence. It is a complex-valued expression that varies continuously with frequency ω.

✍️ Handwritten Solution Image

Below is the handwritten solution for the DTFT of the sequence x(n) = {1, 0, 4, 2}. This image provides a visual representation of the calculation steps and final result.

🔑 Key Takeaways

• The DTFT is used to transform discrete-time signals from the time domain to the frequency domain.
• For finite sequences, the DTFT is calculated by substituting the sequence values into the DTFT formula.
• The result of the DTFT is generally a complex function that represents the frequency content of the signal.
• The DTFT of the sequence x(n) = {1, 0, 4, 2} is a combination of exponential terms.
• In practice, DTFT helps to understand how discrete signals behave across different frequencies.

📝 Final Words

In this post, we calculated the Discrete-Time Fourier Transform (DTFT) of the sequence x(n) = {1, 0, 4, 2}. The DTFT formula helped us derive the frequency-domain representation of the signal. This type of problem is fundamental for understanding how discrete-time signals behave in the frequency domain and is essential for your preparation in GTU Sem 4 Signals and Systems.

If you have any doubts or need further clarification on any step, feel free to leave a comment below. We're always here to help!

💬 Your Thoughts

We would love to hear your thoughts or questions about this solution. Did this explanation help you understand the DTFT better? Feel free to share your queries or feedback in the comments section below. We encourage discussions and will make sure to respond promptly.

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