GTU Solution | Z-Transform & ROC of x(n) = 3ⁿu(n) - 2ⁿu(–n–1) | Sem 4 Signals and Systems Summer 2021 Q.5(b)OR

GTU Sem 4 | Signals and Systems | Summer 2021 | Question 5(b) OR

📑 Table of Contents

introduction
Question Statement
Theory Behind the Question
Detailed Step-by-Step Solution
Handwritten Solution Images
Key Takeaways
Final Words
Related Posts

📖 Introduction

In this post, we delve into an important concept from the field of discrete-time signal analysis — the Z-Transform and Region of Convergence (ROC). This question was part of the GTU Summer 2021 exam for Semester 4 students under the subject Signals and Systems. Specifically, the problem is from Question 5(b) OR, and it requires a solid understanding of both right-sided and left-sided signals in the Z-domain.

The objective is to determine the Z-Transform and its corresponding ROC for the sequence:

GTU Sem 4 Signals and Systems Question 5(b) OR - Z-Transform of x(n) = 3ⁿu(n) - 2ⁿu(–n–1)

📘 Question Statement

Determine the Z – Transform & ROC of the following sequence:
x(n) = (3)ⁿu(n) - (2)ⁿu(–n–1)

📚 Theory Related to the Question

To evaluate the Z-Transform of a given sequence, we use the basic definition:

X(z) = Σ x(n)·z^–n, where the summation is over all integers n.

In this problem, the given signal x(n) is composed of two parts:

(3)ⁿu(n) – a right-sided exponential sequence.
(2)ⁿu(–n–1) – a left-sided exponential sequence.

Each part has its own region of convergence (ROC), and we must evaluate them separately before combining the final result. Understanding the behavior of unit step functions u(n) and u(–n–1) is crucial:

u(n) = 1 for n ≥ 0, 0 otherwise.
u(–n–1) = 1 for n ≤ –1, 0 otherwise.

🔁 Concept Refresher: Z-Transform of Exponential Sequences

Z-Transform of aⁿu(n): X(z) = 1 / (1 – az^–1), ROC: |z| > |a|
Z-Transform of aⁿu(–n–1): X(z) = –1 / (1 – az^–1), ROC: |z| < |a|

Key Insight: Always analyze the ROC separately for right-sided and left-sided sequences, as the overall ROC is the intersection of individual ROCs.

🧠 Solution in Detail

We are given:

x(n) = (3)ⁿu(n) – (2)ⁿu(–n–1)

➡️ Step 1: Z-Transform of (3)ⁿu(n)

This is a right-sided sequence. The standard Z-transform of aⁿ·u(n) is:

X₁(z) = 1 / (1 – a·z⁻¹), for |z| > |a|

So here:

X₁(z) = 1 / (1 – 3·z⁻¹), ROC: |z| > 3

➡️ Step 2: Z-Transform of (2)ⁿu(–n–1)

This is a left-sided sequence. The Z-transform of aⁿ·u(–n–1) is:

X₂(z) = –1 / (1 – a·z⁻¹), for |z| < |a|

So here:

X₂(z) = –1 / (1 – 2·z⁻¹), ROC: |z| < 2

➡️ Step 3: Combine the Transforms

Now we combine both parts:

X(z) = X₁(z) + X₂(z)

X(z) = [1 / (1 – 3·z⁻¹)] + [1 / (1 – 2·z⁻¹)]

➡️ Step 4: Determine the ROC

From earlier steps:

First part: ROC is |z| > 3
Second part: ROC is |z| < 2

Therefore, there is no common ROC. These two individual ROCs do not intersect.

Hence, the Z-transform exists only bilaterally (two-sided Z-transform) and has no ROC in the usual sense.

⚠️ Common Mistakes

Assuming the overall ROC is the union of individual ROCs — it must be their intersection.
Confusing u(–n–1) with u(–n) — they are not the same.
Skipping sign conventions when applying the left-sided Z-transform formula.
Forgetting that left-sided signals have ROCs |z| < |a| and not |z| > |a|.

📊 Summary Table: Z-Transform and ROC

Component	Expression	Z-Transform	ROC
(3)ⁿu(n)	Right-sided	1 / (1 – 3z⁻¹)	\|z\| > 3
(2)ⁿu(–n–1)	Left-sided	–1 / (1 – 2z⁻¹)	\|z\| < 2
Total x(n)	Two-sided	[1 / (1 – 3z⁻¹)] + [1 / (1 – 2z⁻¹)]	No common ROC

🖼️ Handwritten Solution Images

Handwritten Z-transform of 3^n u(n) - Signals and Systems GTU Q5(b) Summer 2021

Handwritten Z-transform of 2^n u(-n-1) - Signals and Systems GTU Sem 4 Solution

🔑 Key Takeaways

When a signal is a combination of left- and right-sided sequences, calculate the Z-transform of each separately.
Always analyze the ROC independently for each component.
If the ROCs do not overlap, the Z-transform exists only bilaterally and has no common ROC.
Remember the Z-transform formulas for aⁿ·u(n) and aⁿ·u(–n–1).

💬 Final Words

This question from the GTU Summer 2021 Signals and Systems paper emphasizes the importance of understanding both the algebra and convergence aspects of Z-transforms. Mastering such problems not only strengthens your concept of signal behavior in the Z-domain but also prepares you for advanced DSP topics. If you found this solution helpful, do share it with your classmates and bookmark it for future revision.

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