---
title: "Convolve {2, 1, 5, 7} and {1, 3, 5, 9, 6}"
subject: "Signals and Systems"
chapter: 6
key: "semester_4/signals_and_systems:convolution-2157"
source: https://www.bctengineer.com/q/semester_4/signals_and_systems/convolution-2157
site: BCT Engineering
---

# Convolve {2, 1, 5, 7} and {1, 3, 5, 9, 6}

Signals and Systems · Chapter 6: Discrete-Time Systems

- Asked in: 2078 Baisakh, 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-2157
- Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-6

## Question

Find the convolution between the following sequences. h[n] = {2, 1, 5, 7}, x[n] = {1, 3, 5, 9, 6}. Choose origin as per your choice.

## Short answer

By convolution sum (tabular method), the output sequence is y[n] = {2, 7, 18, 45, 67, 86, 93, 42} for n = 0 to 7.

## Answer

**Chosen origin:** first sample at $n = 0$ for both:
$h[n] = \{\underset{\uparrow}{2}, 1, 5, 7\}$ ($0 \le n \le 3$) and $x[n] = \{\underset{\uparrow}{1}, 3, 5, 9, 6\}$ ($0 \le n \le 4$).

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

Output starts at $n = 0$; length $5 + 4 - 1 = 8$, so $0 \le n \le 7$.

#### Tabular method

| Row | $n=0$ | $n=1$ | $n=2$ | $n=3$ | $n=4$ | $n=5$ | $n=6$ | $n=7$ |
|---|---|---|---|---|---|---|---|---|
| $x[0]\,h[n]$ | 2 | 1 | 5 | 7 |  |  |  |  |
| $x[1]\,h[n-1]$ |  | 6 | 3 | 15 | 21 |  |  |  |
| $x[2]\,h[n-2]$ |  |  | 10 | 5 | 25 | 35 |  |  |
| $x[3]\,h[n-3]$ |  |  |  | 18 | 9 | 45 | 63 |  |
| $x[4]\,h[n-4]$ |  |  |  |  | 12 | 6 | 30 | 42 |
| $y[n]$ | **2** | **7** | **18** | **45** | **67** | **86** | **93** | **42** |

#### Direct working

- $y[0] = 1(2) = 2$
- $y[1] = 1(1) + 3(2) = 7$
- $y[2] = 1(5) + 3(1) + 5(2) = 18$
- $y[3] = 1(7) + 3(5) + 5(1) + 9(2) = 45$
- $y[4] = 3(7) + 5(5) + 9(1) + 6(2) = 67$
- $y[5] = 5(7) + 9(5) + 6(1) = 86$
- $y[6] = 9(7) + 6(5) = 93$
- $y[7] = 6(7) = 42$

**Sum check:** $(\sum x)(\sum h) = 24 \times 15 = 360$, and $2 + 7 + 18 + 45 + 67 + 86 + 93 + 42 = 360$.

**Answer:**

$$
y[n] = \{\underset{\uparrow}{2},\ 7,\ 18,\ 45,\ 67,\ 86,\ 93,\ 42\}
$$

## Check yourself

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

1. h has 4 samples and x has 5. The output has:
   A. 9 samples
   B. 8 samples, n = 0 to 7
   C. 20 samples
   D. 5 samples
   Answer: B. Length = 5 + 4 - 1 = 8. (Key point: Output length)

2. What is y[3]?
   A. 45
   B. 40
   C. 47
   D. 18
   Answer: A. y[3] = 1(7) + 3(5) + 5(1) + 9(2) = 45. (Key point: Value y[3])

3. The sum of y should be:
   A. 39
   B. 360
   C. 24
   D. 15
   Answer: B. (1+3+5+9+6)(2+1+5+7) = 24 x 15 = 360. (Key point: Sum check)

4. y[7] equals:
   A. 42
   B. 6
   C. 7
   D. 49
   Answer: A. Only x[4]h[3] = 6 x 7 = 42 contributes. (Key point: Last sample)

## Close variants set in other exams

- [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×

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- [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×

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