---
title: "Convolve {1, 2, 1} and {1, 2, 3, 1}"
subject: "Signals and Systems"
chapter: 6
key: "semester_4/signals_and_systems:convolution-121-1231"
source: https://www.bctengineer.com/q/semester_4/signals_and_systems/convolution-121-1231
site: BCT Engineering
---

# Convolve {1, 2, 1} and {1, 2, 3, 1}

Signals and Systems · Chapter 6: Discrete-Time Systems

- Asked in: 2071 Bhadra, 6 marks
- Video: [Discrete Time Convolution (Tabular Method)](https://www.youtube.com/watch?v=C_kwjJLepLg) — Neso Academy
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/signals_and_systems/convolution-121-1231
- Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-6

## Question

Convolve the signals x[n] = {1, 2, 1} and h[n] = {1, 2, 3, 1} (↑ at the first 1, i.e. h[0] = 1).

## Short answer

Use y[n]=x[n]*h[n] with x[n]={1,2,1}, h[n]={1,2,3,1} both origin at first sample. The convolution yields y[n]={1, 4, 8, 9, 5, 1} for n=0..5.

## Answer

Given $h[n]=\{1,2,3,1\}$ with $h[0]$ = first 1, so $n=0..3$. No origin is marked for $x[n]$, so take $x[0]$ = first sample: $x[n]=\{1,2,1\}$, $n=0..2$.

Output: $y[n]=\sum_k x[k]h[n-k]$, from $n=0$ to $5$ (length $3+4-1=6$).
Since $x[n]=\delta[n]+2\delta[n-1]+\delta[n-2]$, $y[n]=h[n]+2h[n-1]+h[n-2]$.

#### Tabular method

| | $n=0$ | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| $h[n]$ | 1 | 2 | 3 | 1 | | |
| $2h[n-1]$ | | 2 | 4 | 6 | 2 | |
| $h[n-2]$ | | | 1 | 2 | 3 | 1 |
| $y[n]$ | **1** | **4** | **8** | **9** | **5** | **1** |

$$
\begin{aligned}
y[2]&=x[0]h[2]+x[1]h[1]+x[2]h[0]=3+4+1=8\\
y[3]&=x[0]h[3]+x[1]h[2]+x[2]h[1]=1+6+2=9
\end{aligned}
$$

Sum check: $\sum y=28=(\sum x)(\sum h)=4\times7$.

**Answer: $y[n]=\{1,\ 4,\ 8,\ 9,\ 5,\ 1\}$ for $n=0,\dots,5$** (↑ at the first 1). If $x[n]$ were taken with origin at its middle sample (2), the same values would simply start at $n=-1$.

## Check yourself

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

1. x={1,2,1} from n=0 means y[n] equals what?
   A. h[n]+2h[n-1]+h[n-2]
   B. h[n]+2h[n+1]+h[n+2]
   C. 2h[n]+h[n-1]+2h[n-2]
   D. h[n]*2h[n-1]*h[n-2]
   Answer: A. x=delta[n]+2delta[n-1]+delta[n-2]. (Key point: Decomposition)

2. x has 3 samples and h has 4 samples. How long is y?
   A. 6 samples, n=0..5
   B. 7 samples
   C. 12 samples
   D. 4 samples
   Answer: A. Length is 3+4-1=6. (Key point: Output length)

3. What is y[2]?
   A. 6
   B. 3
   C. 8
   D. 9
   Answer: C. y[2]=x[0]h[2]+x[1]h[1]+x[2]h[0]=3+4+1=8. (Key point: A middle value)

4. What is y[n]?
   A. {1, 4, 8, 9, 5, 2}
   B. {1, 2, 8, 9, 5, 1}
   C. {1, 4, 7, 9, 5, 1}
   D. {1, 4, 8, 9, 5, 1}
   Answer: D. Sum 28 = 4 x 7 confirms it. (Key point: Final result)

## Related in Signals and Systems

- [Convolve {3, 1, 5} and {1, 4, −2, 3}](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aconvolution-315) — Signals and Systems, chapter 6
- [Convolve {1,2,1,−1} and {1,2,3,1}](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aconvolution-origin-given) — Signals and Systems, chapter 6, asked 2×
- [Convolve {1, 1, 2, −1} and {1, 2}](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aconvolution-1121-sketch) — Signals and Systems, chapter 6
- [Convolve {1, 2, 2, 3} and {1, 4, 3}](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aconvolution-1223-origin) — Signals and Systems, chapter 6
- [First-order system impulse and step response](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Afirst-order-system-response) — Signals and Systems, chapter 5
- [Convolve {1, 2, 0, −1} and {2, 0, 2}](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aconvolution-1-2-0-minus1) — Signals and Systems, chapter 6, asked 2×