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

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

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

📑 Table of Contents

• ➤ Introduction
• ➤ Statement of the Question
• ➤ What is ROC?
• ➤ Properties of ROC
• ➤ Key Takeaways
• ➤ Final Words
• ➤ Related Posts

Introduction

In this post, we will define the Region of Convergence (ROC) and explain its important properties in the context of the Z-Transform. Understanding ROC is crucial for analyzing signal behavior and system stability in the Z-domain. This concept is regularly asked in GTU Signals & Systems exams.

Statement of the Question

Define ROC and explain the properties of ROC with respect to the Z-Transform.

What is ROC?

The Region of Convergence (ROC) refers to the set of complex values of z for which the Z-Transform of a discrete-time signal x[n] converges (i.e., has a finite value).

Mathematically, the Z-Transform is given by:

X(z) = Σ x[n] · z-n (summed over all n)

The ROC is the range of z in the complex plane for which this infinite summation converges.

Properties of ROC (Region of Convergence)

The ROC has the following key properties:

1. The ROC is a ring or disk in the z-plane:
   It is centered around the origin and, depending on the type of signal, bounded by an an inner and/or outer radius.
2. ROC does not contain any poles:
   The ROC is always free of poles because the Z-transform diverges at the poles.
3. For right-sided sequences:
   If x[n] = 0 for n < n₀, the ROC is of the form |z| > r.
4. For left-sided sequences:
   If x[n] = 0 for n > n₁, the ROC is of the form |z| < r.
5. For two-sided (bilateral) sequences:
   The ROC lies between two circles, i.e., r₁ < |z| < r₂.
6. The ROC must be continuous:
   The ROC cannot have gaps; it is always a continuous annular region in the complex plane.
7. ROC determines system stability:
   For a system to be stable, the ROC must include the unit circle |z| = 1.
8. ROC of finite duration signals is the entire z-plane except for z = 0 or ∞:
   If the signal is of finite length, its ROC includes almost the entire z-plane.

Key Takeaways

• ROC helps in identifying where the Z-transform of a signal is valid and convergent.
• Different types of signals (right-sided, left-sided, or two-sided) lead to different ROC patterns.
• The system's stability and causality are often analyzed using the ROC.

Final Words

The Region of Convergence (ROC) is a fundamental concept in Z-Transform analysis. Knowing the ROC helps engineers and students understand where a Z-transform exists and how to determine system characteristics like causality and stability. This makes ROC an essential topic in the Signals and Systems subject, especially for GTU Semester 4 students.