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

Prove Commutative Property of Convolution

GTU Sem 4 | Signals and Systems | Summer 2021 | Question 3(b) OR

📘 Table of Contents

• Introduction
• Definition of Convolution
• Proof: Commutative Property
• Conclusion

Introduction

In discrete-time signal processing, convolution is a key operation used to determine the output of Linear Time-Invariant (LTI) systems. One of the fundamental properties of convolution is its commutative property. In this post, we will prove that convolution is commutative, i.e., for two signals x(n) and h(n):

x(n) * h(n) = h(n) * x(n)

Commutative Property of Convolution – Signals & Systems Summer 2021 Question 3(b)OR

Definition of Convolution

The convolution of two discrete-time signals x(n) and h(n) is defined as:

y(n) = x(n) * h(n) = Σk=-∞∞ x(k) · h(n - k)

Proof: Commutative Property

We start with the definition of convolution:

y(n) = x(n) * h(n) = Σk=-∞∞ x(k) · h(n - k)

Now, perform a change of variable:

Let m = n - k ⟹ k = n - m

As k goes from -∞ to ∞, m also goes from ∞ to -∞, so we reverse the limits:

y(n) = Σm=∞-∞ x(n - m) · h(m)

Rewriting with standard summation order:

y(n) = Σm=-∞∞ h(m) · x(n - m)

This is equal to:

y(n) = h(n) * x(n)

Hence,

x(n) * h(n) = h(n) * x(n)

Conclusion

We have successfully shown that convolution in discrete time is commutative. This means the order of signals does not affect the result of convolution:

x(n) * h(n) = h(n) * x(n)

This property is very useful in analyzing LTI systems, as it allows us to treat the system and the input interchangeably in the convolution operation.

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

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