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

Determine the Z – Transform and ROC of the Sequence: x(n) = (3)ⁿu(n) – (4)ⁿu(n) | GTU Question 4(a) Solution

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

📘 Table of Contents

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

Introduction

In this blog post, we provide a detailed solution to a frequently asked GTU question from the subject Signals and Systems, specifically from the Summer 2021 examination – Question 4(a). The problem involves determining the Z-Transform and the Region of Convergence (ROC) for a discrete-time signal defined as:

x(n) = (3)nu(n) – (4)nu(n)

This type of question is typical of the GTU Semester 4 Signals and Systems curriculum and tests the student’s understanding of Z-transform properties, especially for exponentially growing sequences. The solution below includes both theoretical concepts and step-by-step evaluation, ensuring clarity for exam preparation.

Question Statement

GTU Summer 2021 | Signals and Systems | Question 4(a):
Determine the Z – Transform and Region of Convergence (ROC) of the following discrete-time sequence:

x(n) = (3)nu(n) – (4)nu(n)

Where u(n) is the unit step function. This is a classical Z-transform question frequently asked in GTU exams and is important for understanding the convergence behavior of exponential sequences in the Z-domain.

Theory Part Related to the Question

To solve this problem, we need to apply the fundamental concepts of the Z-Transform for discrete-time signals. The Z-transform of a signal x(n) is defined as:

X(z) = Σ [x(n)·z–n] from n = –∞ to ∞

For causal signals (i.e., signals multiplied with u(n)), the lower limit becomes n = 0 instead of –∞.

One of the standard Z-Transform pairs is:

• an·u(n) ⇨ Z-transform: 1 / (1 – a·z–1)
• Region of Convergence (ROC): |z| > |a|

In the given expression, we are dealing with the difference of two exponential sequences: x(n) = (3)n·u(n) – (4)n·u(n). We will compute the Z-Transform of each term separately using the formula above, and then subtract them accordingly.

Solution in Detail

We are given:

x(n) = (3)n·u(n) – (4)n·u(n)

Step 1: Apply Z-Transform Linearity

Use the linearity property of the Z-Transform:

X(z) = Z{(3)n·u(n)} – Z{(4)n·u(n)}

Step 2: Use the Standard Z-Transform Pair

Using the formula Z{an·u(n)} = 1 / (1 – a·z–1) with ROC: |z| > |a|:

• Z{(3)n·u(n)} = 1 / (1 – 3·z–1), ROC: |z| > 3
• Z{(4)n·u(n)} = 1 / (1 – 4·z–1), ROC: |z| > 4

Step 3: Final Z-Transform Expression

Subtract the results:

X(z) = [1 / (1 – 3·z–1)] – [1 / (1 – 4·z–1)]

Step 4: Determine the Region of Convergence (ROC)

For both terms to converge, the ROC must satisfy both conditions:

ROC: |z| > max(3, 4) ⇒ |z| > 4

Final Answer:

Z-Transform:
X(z) = [1 / (1 – 3·z–1)] – [1 / (1 – 4·z–1)]

X(z) = [z / (z – 3)] – [z / (z – 4)]

ROC: |z| > 4

Handwritten Solution Images

Key Takeaways

• The Z-transform of a causal exponential sequence an·u(n) is 1 / (1 – a·z–1), with ROC: |z| > |a|.
• Using linearity, the Z-transform of a sum or difference of signals is the sum or difference of their Z-transforms.
• For x(n) = (3)n·u(n) – (4)n·u(n), the final Z-Transform is:
  X(z) = 1 / (1 – 3·z–1) – 1 / (1 – 4·z–1)
• The Region of Convergence (ROC) is determined by the more dominant exponential term. Here, ROC: |z| > 4.
• This type of problem is standard in GTU Signals and Systems exams and helps reinforce a foundational understanding of discrete-time transforms.

Final Words

In this post, we thoroughly analyzed and solved a classic GTU Signals and Systems problem from the Summer 2021 paper – Question 4(a). By applying the standard Z-Transform formulas and understanding the Region of Convergence, we arrived at a complete and accurate solution for the sequence x(n) = (3)nu(n) – (4)nu(n).

This question not only reinforces key theoretical concepts but also prepares you for similar problems that frequently appear in GTU examinations. For more solutions to GTU PYQs with handwritten images and detailed explanations, explore other posts on Drk Knowledge 24.

Stay focused, and keep solving!

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