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

Prove that the DT LTI system is causal if and only if h(n) = 0 for n < 0

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

Causality Condition in DT LTI System – Signals & Systems Summer 2021 Question 3(a)OR

Introduction

In this post, we will prove an important property of Discrete-Time Linear Time-Invariant (DT LTI) systems. Specifically, we will show that a DT LTI system is causal if and only if its impulse response h(n) is zero for all n < 0.

📘 Table of Contents

• Introduction
• What is Causality?
• Impulse Response and LTI Systems
• Proof (⇒) If System is Causal, Then h(n) = 0 for n < 0
• Proof (⇐) If h(n) = 0 for n < 0, Then System is Causal
• Conclusion

What is Causality?

A system is said to be causal if its output at any time n depends only on the present and past input values (i.e., x(n), x(n-1), x(n-2), ...), and not on future inputs like x(n+1), x(n+2), etc.

Impulse Response and LTI Systems

For an LTI system, the output y(n) is given by the convolution of the input x(n) with the impulse response h(n):

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

Proof (⇒) If System is Causal, Then h(n) = 0 for n < 0

Assume the system is causal. Apply the unit impulse input x(n) = δ(n). Then the output is:

y(n) = h(n)

Since the system is causal, it cannot respond before the input is applied. Therefore, for n < 0, the output must be zero:

h(n) = 0 for n < 0

Proof (⇐) If h(n) = 0 for n < 0, Then System is Causal

Assume that h(n) = 0 for all n < 0. The output of the system is given by:

y(n) = Σk = -∞∞ x(k) · h(n - k)

For values of k > n, n - k < 0, so h(n - k) = 0. Hence, only values of k ≤ n contribute to the output.

This means y(n) depends only on the present and past values of the input x(k), and not on any future values. Therefore, the system is causal.

Conclusion

We have proven both directions:

• If the DT LTI system is causal, then h(n) = 0 for n < 0
• If h(n) = 0 for n < 0, then the system is causal

Therefore, a DT LTI system is causal if and only if h(n) = 0 for all n < 0.

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

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