---
title: "Commutative property of CT LTI systems"
subject: "Signals and Systems"
chapter: 5
key: "semester_4/signals_and_systems:commutative-property-ct"
source: https://www.bctengineer.com/q/semester_4/signals_and_systems/commutative-property-ct
site: BCT Engineering
---

# Commutative property of CT LTI systems

Signals and Systems · Chapter 5: Continuous-Time Systems

- Asked in: 2078 Baisakh, 3 marks
- Video: [Properties of Convolution](https://www.youtube.com/watch?v=mzYbd44X8Z0) — TutorialsPoint
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/signals_and_systems/commutative-property-ct
- Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-5

## Question

State and prove commutative property of continuous time LTI system.

## Short answer

Convolution is commutative: x(t)*h(t) = h(t)*x(t). Proof: substitute λ=t-τ in the convolution integral, yielding ∫ x(τ)h(t-τ)dτ = ∫ h(λ)x(t-λ)dλ. So the order of systems can be swapped.

## Answer

**Statement:** Convolution of continuous-time signals is commutative:
$$
x(t) * h(t) = h(t) * x(t)
$$
For LTI systems, this means the output is the same whether the input is $x(t)$ applied to a system with impulse response $h(t)$, or $h(t)$ applied to a system with impulse response $x(t)$.

**Proof:**
$$
x(t) * h(t) = \int_{-\infty}^{\infty} x(\tau)\,h(t - \tau)\,d\tau
$$
Substitute $\lambda = t - \tau$, so $\tau = t - \lambda$ and $d\tau = -d\lambda$. When $\tau \to -\infty$, $\lambda \to \infty$; when $\tau \to \infty$, $\lambda \to -\infty$:
$$
\begin{aligned}
x(t) * h(t) &= \int_{\infty}^{-\infty} x(t - \lambda)\,h(\lambda)\,(-d\lambda) \\
&= \int_{-\infty}^{\infty} h(\lambda)\,x(t - \lambda)\,d\lambda = h(t) * x(t)
\end{aligned}
$$
$\blacksquare$

```text
x(t) --> [ h(t) ] --> y(t)   ==   h(t) --> [ x(t) ] --> y(t)
```
**Use:** we can flip whichever signal is simpler when evaluating the integral; and for cascaded LTI systems, the order of the systems can be swapped.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What does the commutative property of CT convolution say?
   A. x(t)*h(t) = h(t)*x(t)
   B. x(t)*h(t) = x(-t)*h(t)
   C. x(t)*h(t) = x(t)h(t)
   D. x(t)*h(t) = h(t)*x(-t)
   Answer: A. Convolution is symmetric in its two arguments. (Key point: Statement)

2. Which substitution proves it from the integral of x(τ)h(t-τ)?
   A. λ = t + τ
   B. λ = t - τ
   C. λ = τ^2
   D. λ = -t
   Answer: B. With λ=t-τ, τ=t-λ, dτ=-dλ, and the limits swap, giving the integral of h(λ)x(t-λ). (Key point: Proof substitution)

3. What is a practical consequence for cascaded LTI systems?
   A. The cascade becomes non-linear
   B. The systems must be identical
   C. Their order can be swapped without changing the overall response
   D. Only the first system matters
   Answer: C. Since h1*h2 = h2*h1, the order of cascaded LTI systems does not matter. (Key point: Use)

## Related in Signals and Systems

- [Frequency response of a CT LTI system](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Act-frequency-response) — Signals and Systems, chapter 5, asked 3×
- [Causality condition for CT LTI system](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Acausal-ct-lti-condition) — Signals and Systems, chapter 5
- [Cascade of two CT LTI systems](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Acascade-impulse-response) — Signals and Systems, chapter 5
- [Properties of continuous time systems](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Act-system-properties) — Signals and Systems, chapter 5, asked 2×
- [Time shifting property of CTFT](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Actft-time-shifting) — Signals and Systems, chapter 3, asked 5×
- [Time-scaling property of CTFS](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Actfs-time-scaling) — Signals and Systems, chapter 2, asked 4×

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