---
title: "Convolve {1, 2, 2, 1} and {1, 2, 2, −1}"
subject: "Signals and Systems"
chapter: 6
key: "semester_4/signals_and_systems:convolution-1221"
source: https://www.bctengineer.com/q/semester_4/signals_and_systems/convolution-1221
site: BCT Engineering
---

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

Signals and Systems · Chapter 6: Discrete-Time Systems

- Asked in: 2082 Chaitra, 6 marks
- Video: [Method to Find Discrete Convolution](https://www.youtube.com/watch?v=Nj8no2GMm8s) — 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/convolution-1221
- Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-6

## Question

Find the convolution between signals x[n] = {1, 2, 2, 1} and h[n] = {1, 2, 2, −1}. Choose your own origin.

## Short answer

With both sequences starting at n=0, tabular convolution gives y[n]={1,4,8,8,4,0,-1} and y[0]=1.

## Answer

**Chosen origin:** first sample of each sequence at $n = 0$:
$x[n] = \{\underset{\uparrow}{1}, 2, 2, 1\}$ and $h[n] = \{\underset{\uparrow}{1}, 2, 2, -1\}$, both for $0 \le n \le 3$.

$$
y[n] = \sum_k x[k]\,h[n-k]
$$

Output starts at $0 + 0 = 0$, length $4 + 4 - 1 = 7$, so $0 \le n \le 6$.

#### Tabular method

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

#### Direct working

- $y[0] = 1(1) = 1$
- $y[1] = 1(2) + 2(1) = 4$
- $y[2] = 1(2) + 2(2) + 2(1) = 8$
- $y[3] = 1(-1) + 2(2) + 2(2) + 1(1) = 8$
- $y[4] = 2(-1) + 2(2) + 1(2) = 4$
- $y[5] = 2(-1) + 1(2) = 0$
- $y[6] = 1(-1) = -1$

**Sum check:** $(\sum x)(\sum h) = 6 \times 4 = 24$, and $1 + 4 + 8 + 8 + 4 + 0 - 1 = 24$.

**Answer:**

$$
y[n] = \{\underset{\uparrow}{1},\ 4,\ 8,\ 8,\ 4,\ 0,\ -1\}
$$

Stem plot drawn sideways (each # = 0.5; left of | is negative):

```text
 n    y[n]
 0       1    |##
 1       4    |########
 2       8    |################
 3       8    |################
 4       4    |########
 5       0    |
 6      -1  ##|
```

## Check yourself

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

1. x and h each have 4 samples starting at n = 0. The output y has:
   A. 8 samples, 0 to 7
   B. 7 samples, 0 to 6
   C. 4 samples, 0 to 3
   D. 16 samples
   Answer: B. Length = 4 + 4 - 1 = 7. (Key point: Output length)

2. What is y[3] for x = {1,2,2,1}, h = {1,2,2,-1}?
   A. 8
   B. 7
   C. 9
   D. 6
   Answer: A. y[3] = 1(-1) + 2(2) + 2(2) + 1(1) = 8. (Key point: Value)

3. What does the sum of y equal?
   A. 10
   B. 24
   C. 12
   D. 4
   Answer: B. (sum x)(sum h) = 6 x 4 = 24. (Key point: Sum check)

4. What is the last sample y[6]?
   A. 1
   B. -1
   C. 0
   D. 2
   Answer: B. y[6] = x[3]h[3] = 1 x (-1) = -1. (Key point: Last sample)

## Close variants set in other exams

- [Convolve {1, 2, 3, 4} and {1, 2, −3}](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aconvolution-1234-12-3) — Signals and Systems, chapter 6, asked 2×

## Related in Signals and Systems

- [Convolve {2, 1, 5, 7} and {1, 3, 5, 9, 6}](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aconvolution-2157) — Signals and Systems, chapter 6, asked 2×
- [Convolve {2, 0, −1, 2} and {1, 2, −1, 3}](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aconvolution-20-12) — Signals and Systems, chapter 6, asked 2×
- [Convolve {3, 5, 6, 9} and {−1, 3, 5}](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aconvolution-3569) — 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 {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} and {1, 2, 3, 1}](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aconvolution-121-1231) — Signals and Systems, chapter 6

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