---
title: "Linear convolution via circular convolution"
subject: "Digital Signal Analysis and Processing"
chapter: 7
key: "semester_7/digital_signal_analysis_and_processing:linear-conv-via-circular"
source: https://www.bctengineer.com/q/semester_7/digital_signal_analysis_and_processing/linear-conv-via-circular
site: BCT Engineering
---

# Linear convolution via circular convolution

Digital Signal Analysis and Processing · Chapter 7: Discrete Fourier Transform

- Asked in: 2074 Chaitra, 5 marks
- Video: [Linear convolution using circular convolution | Solved problem | DSP Module 1 | Lecture 11](https://www.youtube.com/watch?v=NjQ5d3YfTGY) — IMPLearn
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_7/digital_signal_analysis_and_processing/linear-conv-via-circular
- Whole chapter: https://www.bctengineer.com/md/semester_7/digital_signal_analysis_and_processing/chapter-7

## Question

Find the linear convolution through circular convolution with padding of zeros for the following sequences: x[n] = {1,1,1,1} and h[n] = {2,3}.

## Short answer

Pad both sequences with zeros to N=5 so circular convolution equals linear convolution. Using the circulant method gives y[n] = {2, 5, 5, 5, 3}.

## Answer

The linear convolution of sequences of lengths L = 4 and M = 2 has L + M − 1 = **5** samples. A circular convolution equals the linear convolution if the DFT length is N ≥ 5. So pad both sequences with zeros to N = 5:

- x[n] = {1, 1, 1, 1, 0}
- h[n] = {2, 3, 0, 0, 0}

y[n] = Σ (m = 0 to 4) x[m] h[(n − m) mod 5]

Matrix form (circulant formed from x, multiplied by the column h):

```text
| 1 0 1 1 1 |   | 2 |   | 2 |
| 1 1 0 1 1 |   | 3 |   | 5 |
| 1 1 1 0 1 | · | 0 | = | 5 |
| 1 1 1 1 0 |   | 0 |   | 5 |
| 0 1 1 1 1 |   | 0 |   | 3 |
```
Row by row:

- y[0] = 1×2 + 0×3 = 2
- y[1] = 1×2 + 1×3 = 5
- y[2] = 1×2 + 1×3 = 5
- y[3] = 1×2 + 1×3 = 5
- y[4] = 0×2 + 1×3 = 3

Result: y[n] = {2, 5, 5, 5, 3}.

Check by direct linear convolution: y[n] = 2x[n] + 3x[n−1] gives {2, 5, 5, 5, 3}. The sum of the output (20) equals Σx × Σh = 4 × 5.

Method by DFT: y[n] = IDFT{ DFT₅(x) · DFT₅(h) } gives the same sequence.

**Answer: y[n] = {2, 5, 5, 5, 3}.**

## 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 2 samples. How many samples does their linear convolution have?
   A. 6
   B. 8
   C. 5
   D. 4
   Answer: C. L + M - 1 = 4 + 2 - 1 = 5. (Key point: Length of linear convolution)

2. What condition on the DFT length N makes circular convolution equal linear convolution?
   A. N must be a power of 10
   B. N >= L + M - 1
   C. N = max(L, M)
   D. N <= L + M - 1
   Answer: B. With N at least L + M - 1 there is no time aliasing, so the circular result equals the linear one. (Key point: Condition on N)

3. How are the sequences prepared here?
   A. No padding is needed
   B. Both are padded with zeros to length 5
   C. Only x is padded to length 8
   D. Both are truncated to length 2
   Answer: B. x = {1,1,1,1,0} and h = {2,3,0,0,0}. (Key point: Zero padding)

4. Which check confirms y[n] = {2, 5, 5, 5, 3}?
   A. The sum of y equals 4 + 5
   B. y[0] must equal sum of x
   C. The sum of y equals (sum of x)(sum of h) = 4 x 5 = 20
   D. y must have 4 samples
   Answer: C. Sum of outputs = sum x times sum h, and 2+5+5+5+3 = 20. (Key point: Result check)

## Related in Digital Signal Analysis and Processing

- [Circular convolution of {1,2} and u[n]−u[n−4]](https://www.bctengineer.com/question.md?key=semester_7%2Fdigital_signal_analysis_and_processing%3Acircular-conv-1-2-step) — Digital Signal Analysis and Processing, chapter 7, asked 2×
- [State and prove circular convolution property](https://www.bctengineer.com/question.md?key=semester_7%2Fdigital_signal_analysis_and_processing%3Acircular-conv-property-prove) — Digital Signal Analysis and Processing, chapter 7
- [Circular convolution of h and x, 5 and 4 points](https://www.bctengineer.com/question.md?key=semester_7%2Fdigital_signal_analysis_and_processing%3Acircular-conv-5-and-4) — Digital Signal Analysis and Processing, chapter 7
- [Define zero padding](https://www.bctengineer.com/question.md?key=semester_7%2Fdigital_signal_analysis_and_processing%3Adefine-zero-padding) — Digital Signal Analysis and Processing, chapter 7, asked 3×

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