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

 📌 Condition for Stability of Discrete-Time LTI System – Explained with Proof

📚 Table of Contents

• 🔍 Introduction
• ❓ Question Statement
• ✅ Complete Answer – Condition for Stability of Discrete-Time LTI System
• 📎 Key Takeaways
• 📝 Final Words
• 📌 Related Posts

🔍 Introduction

In this post, we’ll dive into GTU Sem 4 Signals & Systems Summer 2021 PYQ Question 3(a), which asks us to state and prove a condition for a discrete-time Linear Time-Invariant (LTI) system to be stable. Stability is one of the most crucial characteristics of a system because it tells us whether the output of the system will remain bounded for any bounded input.

This concept is particularly important for engineering students studying signal processing and system analysis. In this solution, we'll clearly state the condition required for stability in discrete-time LTI systems and then walk through a detailed proof to help you understand it thoroughly.

❓ Question Statement

GTU Sem 4 Signals & Systems – Summer 2021 – Question 3(a):
State and prove a condition for a discrete-time LTI system to be stable.

This question is frequently asked in GTU exams and forms the foundation for understanding system behavior in the time domain. Below, we’ll first state the mathematical condition required for stability, followed by a clear and step-by-step proof.

✅ Complete Answer – Condition for Stability of Discrete-Time LTI System

For a discrete-time Linear Time-Invariant (LTI) system to be Bounded-Input Bounded-Output (BIBO) stable, the system's impulse response must be absolutely summable.

📌 Condition for Stability:

A discrete-time LTI system is BIBO stable if and only if:

∑_n=-∞^∞ |h(n)| < ∞

where h(n) is the impulse response of the system.

🧾 Proof:

Let x(n) be the input signal and y(n) be the output. For an LTI system, the output is given by the convolution of x(n) and h(n):

y(n) = ∑_k=-∞^∞ x(k) · h(n - k)

To check for BIBO stability, we assume the input x(n) is bounded. That means:

|x(n)| ≤ B    for all n, where B is a finite positive constant.

Then the magnitude of the output is:

|y(n)| = |∑_k=-∞^∞ x(k) · h(n - k)| ≤ ∑_k=-∞^∞ |x(k)| · |h(n - k)|

Since |x(k)| ≤ B, we get:

|y(n)| ≤ B · ∑_{k = -∞}^∞ |h(n - k)| = B · ∑_{k = -∞}^∞ |h(k)|

So, if ∑ |h(k)| is finite, the output y(n) will also be bounded, which means the system is BIBO stable.

✔️ Conclusion:

A discrete-time LTI system is stable if and only if its impulse response h(n) is absolutely summable:

∑_n=-∞^∞ |h(n)| < ∞

This condition ensures that for any bounded input, the output remains bounded.

📎 Key Takeaways

• Stability in discrete-time LTI systems means that the output remains bounded for any bounded input (BIBO Stability).
• The mathematical condition for stability is that the impulse response h(n) must be absolutely summable.
• This is expressed as: ∑_{n = -∞}^∞ |h(n)| < ∞.
• This condition ensures the system’s output will not grow unbounded even with continuous input signals.

📝 Final Words

This was the complete answer for GTU Sem 4 Signals & Systems Summer 2021 PYQ Question 3(a), where we stated and proved the condition required for the stability of a discrete-time LTI system. Make sure to understand not just the condition but also how it's derived, as this is a fundamental concept often repeated in exams.

If you found this explanation helpful, feel free to check out more solved PYQs on our blog!