---
title: "Convolution is commutative; find y[n]"
subject: "Signals and Systems"
chapter: 6
key: "semester_4/signals_and_systems:lti-commutative-convolution"
source: https://www.bctengineer.com/q/semester_4/signals_and_systems/lti-commutative-convolution
site: BCT Engineering
---

# Convolution is commutative; find y[n]

Signals and Systems · Chapter 6: Discrete-Time Systems

- Asked in: 2073 Bhadra, 2+3+5 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/signals_and_systems/lti-commutative-convolution
- Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-6

## Question

What is LTI system? In a LTI system show that convolution operation is commutative, find y[n] when x[n] = {1, 2, 3, 4} (↑ at 1, i.e. x[0] = 1) and h[n] = {2, 1, 2} (↑ at 1, i.e. h[0] = 1).

## Short answer

An LTI system is linear and time-invariant, fully described by its impulse response h[n]. Convolution is commutative because substituting m=n-k changes sum x[k]h[n-k] into sum h[m]x[n-m]. For x={1,2,3,4}, h={2,1,2}, the output is y={2,5,10,15,10,8} for n=-1 to 4, so y[0]=5.

## Answer

#### LTI system
An **LTI system** is one that is both linear (superposition holds) and time-invariant (a delay in the input gives the same delay in the output). It is completely described by its impulse response $h[n]$, and the output for any input is $y[n]=x[n]*h[n]=\sum_k x[k]h[n-k]$.

#### Commutative property
To show $x[n]*h[n]=h[n]*x[n]$:

$$
x[n]*h[n]=\sum_{k=-\infty}^{\infty}x[k]\,h[n-k]
$$

Put $m=n-k$, so $k=n-m$. As $k$ goes from $-\infty$ to $\infty$, $m$ also covers $-\infty$ to $\infty$:

$$
x[n]*h[n]=\sum_{m=-\infty}^{\infty}x[n-m]\,h[m]=\sum_{m=-\infty}^{\infty}h[m]\,x[n-m]=h[n]*x[n]
$$

So a system with impulse response $h$ driven by $x$ gives the same output as a system with impulse response $x$ driven by $h$.

#### Computing y[n]
$x[n]=\{1,2,3,4\}$ with $x[0]=1$ (so $n=0..3$); $h[n]=\{2,1,2\}$ with $h[0]=1$ (so $h[-1]=2,\ h[0]=1,\ h[1]=2$).

Start index $=0+(-1)=-1$, length $=4+3-1=6$, so $y[n]$ runs from $n=-1$ to $4$.

| | $n=-1$ | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|---|
| $x[0]h = 1\cdot h$ | 2 | 1 | 2 | | | |
| $x[1]h = 2\cdot h$ | | 4 | 2 | 4 | | |
| $x[2]h = 3\cdot h$ | | | 6 | 3 | 6 | |
| $x[3]h = 4\cdot h$ | | | | 8 | 4 | 8 |
| $y[n]$ | **2** | **5** | **10** | **15** | **10** | **8** |

For example, $y[1]=x[0]h[1]+x[1]h[0]+x[2]h[-1]=2+2+6=10$. Sum check: $\sum y=50=10\times5$.

**Answer: $y[n]=\{2,\ 5,\ 10,\ 15,\ 10,\ 8\}$ for $n=-1,\dots,4$, i.e. $y[0]=5$.**

Computing $h*x$ (each $h$ value times $x$) gives the same numbers, which verifies the commutative property.

## Check yourself

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

1. What defines an LTI system?
   A. Linear (superposition) and time-invariant (shifted input gives shifted output)
   B. Linear and causal
   C. Time-invariant and stable only
   D. Memoryless and linear
   Answer: A. LTI means both linearity and time-invariance; it is then fully described by h[n]. (Key point: LTI definition)

2. Which substitution proves x*h = h*x?
   A. m = n+k
   B. m = n-k in the convolution sum
   C. m = k^2
   D. m = x[k]
   Answer: B. With m=n-k the sum becomes sum of h[m]x[n-m], which is h*x. (Key point: Commutative proof)

3. x has 4 samples from n=0 and h has 3 samples from n=-1. Over what range does y[n] run?
   A. n=0 to 5 (6 samples)
   B. n=-1 to 3 (5 samples)
   C. n=-1 to 4 (6 samples)
   D. n=-1 to 5 (7 samples)
   Answer: C. Start index is 0+(-1)=-1 and length is 4+3-1=6, so it ends at n=4. (Key point: Output range)

4. What is y[n] for n=-1,...,4?
   A. {1, 5, 10, 15, 10, 8}
   B. {2, 5, 10, 15, 10, 4}
   C. {2, 4, 10, 15, 10, 8}
   D. {2, 5, 10, 15, 10, 8}
   Answer: D. Adding the shifted rows: 2, 1+4, 2+2+6, 4+3+8, 6+4, 8. The sum 50 = 10 x 5 checks it. (Key point: Result)

## Related in Signals and Systems

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- [Verify commutative law of convolution](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Averify-commutative-law) — Signals and Systems, chapter 5
- [Commutative, associative, distributive properties](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Alti-convolution-properties) — Signals and Systems, chapter 6
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