---
title: "Convolve {1, 2, 3, 4} and {1, 2, −3}"
subject: "Signals and Systems"
chapter: 6
key: "semester_4/signals_and_systems:convolution-1234-12-3"
source: https://www.bctengineer.com/q/semester_4/signals_and_systems/convolution-1234-12-3
site: BCT Engineering
---

# Convolve {1, 2, 3, 4} and {1, 2, −3}

Signals and Systems · Chapter 6: Discrete-Time Systems

- Asked in: 2083 Baisakh (new course), 4 marks
- Video: [#110 Solved problems -2 on Convolution Sum || EC Academy](https://www.youtube.com/watch?v=hl2T2adZWFI) — EC 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-1234-12-3
- Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-6

## Question

Find the convolution sum of x[n] = {1, 2, 3, 4} and h[n] = {1, 2, −3}. Choose your own origin.

## Short answer

y[n] = {1, 4, 4, 4, -1, -12} for n=0..5. Tabular method with both origins at index 0. Sum check: ∑y=0.

## Answer

Choose the origin at the first sample of each sequence:
$x[n]=\{1,2,3,4\}$, $n=0..3$ and $h[n]=\{1,2,-3\}$, $n=0..2$.

$y[n]=\sum_k x[k]h[n-k]$ has $4+3-1=6$ samples, $n=0..5$.

| | $n=0$ | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| $1\cdot h$ | 1 | 2 | −3 | | | |
| $2\cdot h$ | | 2 | 4 | −6 | | |
| $3\cdot h$ | | | 3 | 6 | −9 | |
| $4\cdot h$ | | | | 4 | 8 | −12 |
| $y[n]$ | **1** | **4** | **4** | **4** | **−1** | **−12** |

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

Sum check: $\sum y=0=(\sum x)(\sum h)=10\times0$.

**Answer: $y[n]=\{1,\ 4,\ 4,\ 4,\ -1,\ -12\}$ for $n=0,\dots,5$** (↑ at the first 1). With another origin the values are the same, only shifted.

## Check yourself

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

1. x has 4 samples and h has 3. How many samples in y?
   A. 7
   B. 6
   C. 5
   D. 12
   Answer: B. 4+3-1=6. (Key point: Output length)

2. In the tabular method, what does each row represent?
   A. One h[k] squared
   B. The sum of x and h
   C. One x[k] multiplied by h and shifted by k
   D. The derivative of h
   Answer: C. y is the sum of scaled, shifted copies of h. (Key point: Method)

3. What is y[2] for x={1,2,3,4}, h={1,2,-3}?
   A. 3
   B. -3
   C. 6
   D. 4
   Answer: D. y[2]=x0h2+x1h1+x2h0=-3+4+3=4. (Key point: A middle value)

4. Why is sum of y equal to 0 here?
   A. sum y = (sum x)(sum h) and sum h = 1+2-3 = 0
   B. Because x has a negative sample
   C. Because the system is unstable
   D. Because convolution always sums to zero
   Answer: A. The product of the sums is 10 x 0. (Key point: Sum check)

## Close variants set in other exams

- [Convolve {1, 2, 2, 1} and {1, 2, 2, −1}](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aconvolution-1221) — Signals and Systems, chapter 6, asked 2×

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