GTU Sem 4 Signals & Systems Summer 2021 PYQ Question 3(b) Solution

Determine Whether the Following System with Impulse Response h(n) = 2ⁿ u(-n) is Stable or Not

📘 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 problem, we are tasked with determining the stability of a system given its impulse response h(n) = 2ⁿ u(-n), where u(n) is the unit step function. Stability in discrete-time systems is determined by examining the absolute summability of the impulse response. We will analyze the system to check if it satisfies the condition for stability.

Let’s dive into the detailed solution to understand whether this system is stable or not.

Question Statement

Determine whether the system with impulse response h(n) = 2ⁿ u(-n) is stable or not.

Theory Part Related to the Question

In discrete-time systems, a system is considered stable if its impulse response h(n) is absolutely summable, meaning that the sum of the absolute values of the impulse response is finite:

Stability condition: A system is stable if

∑_n=-∞^∞ |h(n)| < ∞

In this case, we need to check if the sum of the absolute values of the impulse response is finite. To do this, we need to analyze the impulse response given as h(n) = 2ⁿ u(-n), where u(-n) is the time-reversed unit step function that is defined as:

u(-n) = 1 for n ≤ 0, and 

u(-n) = 0 for n > 0

Therefore, the impulse response becomes:

h(n) = 2ⁿ for n ≤ 0, and 

h(n) = 0 for n > 0.

Solution in Detail

Now, we will analyze the system's stability by checking whether the sum of the absolute values of the impulse response is finite.

We know that the impulse response is:

h(n) = 2ⁿ for n ≤ 0, and 

h(n) = 0 for n > 0.

Thus, the sum we need to evaluate for stability is:

∑_n=-∞⁰ |h(n)| = ∑_n=-∞⁰ |2ⁿ|

Since h(n) = 2ⁿ for n ≤ 0, we can rewrite the sum as:

∑_n=-∞⁰ |2ⁿ| = ∑_n=0^∞ (1/2)ⁿ

Now, we recognize this as a geometric series. The sum of a geometric series with first term a and common ratio r is given by:

Sum = a / (1 - r), where |r| < 1, for convergence.

In our case, the series is:

∑_n=0^∞ (1/2)ⁿ = (1/2)⁰ / (1 - 1/2) = 2

This sum converges because it results in a finite positive value. Therefore, the series is absolute summable, meaning the system is stable.

Handwritten Solution Images

Key Takeaways

• A discrete-time system is stable if its impulse response is absolutely summable.
• Given h(n) = 2ⁿ u(-n), the impulse response is nonzero only for n ≤ 0.
• The sum ∑_n=-∞⁰ 2ⁿ converges because 2ⁿ decays as n → -∞.
• Since the absolute sum is finite, the system is stable.

Final Words

In conclusion, the system with impulse response h(n) = 2ⁿ u(-n) is stable because the total absolute sum of its impulse response converges. Understanding the concept of absolute summability is key when analyzing the stability of discrete-time systems, especially when step functions or exponential terms are involved.

If you're preparing for GTU exams, make sure to practice similar problems where impulse responses are expressed with unit step functions or piecewise definitions. These questions are common and test your fundamental understanding of system properties.