---
title: "Circular convolution of h and x, 5 and 4 points"
subject: "Digital Signal Analysis and Processing"
chapter: 7
key: "semester_7/digital_signal_analysis_and_processing:circular-conv-5-and-4"
source: https://www.bctengineer.com/q/semester_7/digital_signal_analysis_and_processing/circular-conv-5-and-4
site: BCT Engineering
---

# Circular convolution of h and x, 5 and 4 points

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

- Asked in: 2072 Chaitra, 7 marks
- Video: [Q4a Perform Circular Convolution of the sequences x1(n)={2,1,2,1} & x2(n)={1,2,3,4}](https://www.youtube.com/watch?v=zYAEuvCYTHc) — EnggClasses
- 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/circular-conv-5-and-4
- Whole chapter: https://www.bctengineer.com/md/semester_7/digital_signal_analysis_and_processing/chapter-7

## Question

Compute circular convolution of h(n) = {1, 2, 1, −1, 1} and x[n] = {1, 2, 3, 1}.

## Short answer

Pad x[n] to length N=5: {1,2,3,1,0}. Compute the 5-point circular convolution: y[0]=1, y[1]=6, y[2]=9, y[3]=8, y[4]=4. Result: {1,6,9,8,4}.

## Answer

h(n) has 5 points and x(n) has 4 points, so take N = 5 (the longer length) and pad x(n) with one zero. The 5-point **circular convolution** y[n] = h[n] ⊛ x[n] is the sequence whose DFT is Y(k) = H(k)·X(k):

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

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

Written as a matrix product (the first column of the circulant is h, and each later column is the previous one shifted down circularly):

```text
|    1    1   -1    1    2 |   | 1 |
|    2    1    1   -1    1 |   | 2 |
|    1    2    1    1   -1 |   | 3 |
|   -1    1    2    1    1 |   | 1 |
|    1   -1    1    2    1 |   | 0 |
```

Row by row:

- y[0] = (1)(1) + (1)(2) + (−1)(3) + (1)(1) + (2)(0) = **1**
- y[1] = (2)(1) + (1)(2) + (1)(3) + (−1)(1) + (1)(0) = **6**
- y[2] = (1)(1) + (2)(2) + (1)(3) + (1)(1) + (−1)(0) = **9**
- y[3] = (−1)(1) + (1)(2) + (2)(3) + (1)(1) + (1)(0) = **8**
- y[4] = (1)(1) + (−1)(2) + (1)(3) + (2)(1) + (1)(0) = **4**

Check using DFTs: the inverse DFT of H(k)·X(k), k = 0 … 4, is {1, 6, 9, 8, 4} (Σy = 28 = H(0)·X(0) = 4 × 7).

**Answer: y[n] = {1, 6, 9, 8, 4}, n = 0 … 4.**

(For comparison, the linear convolution has 5 + 4 − 1 = 8 points: {1, 4, 8, 8, 4, 0, 2, 1}; the 5-point circular result is obtained by adding the last 3 points back onto the first 3: {1, 6, 9, 8, 4}.)

## Check yourself

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

1. h(n) has 5 points and x(n) has 4 points. What circular convolution length is used?
   A. N = 4
   B. N = 5
   C. N = 8
   D. N = 9
   Answer: B. N is the longer length, 5, and the 4-point x(n) is padded with one zero. (Key point: Choice of N)

2. Why does the product H(k)X(k) correspond to circular convolution?
   A. Because the sequences are both real
   B. Multiplication of DFTs equals circular convolution in the time domain
   C. Because padding makes it linear convolution automatically
   D. Because DFT is linear only
   Answer: B. The DFT property: multiplying DFTs is equivalent to N-point circular convolution of the sequences. (Key point: DFT property)

3. Which formula gives the circular convolution?
   A. y[n] = sum h[m] x[(n - m) mod 5]
   B. y[n] = sum h[m] x[n + m]
   C. y[n] = sum h[m] x[n - m] with no wrap-around
   D. y[n] = h[n] x[n]
   Answer: A. The index n - m is taken modulo N so that it wraps around. (Key point: Formula)

4. The linear convolution has 8 points {1, 4, 8, 8, 4, 0, 2, 1}. How is the 5-point circular result obtained from it?
   A. Multiply the last 3 by the first 3
   B. Discard the last 3 points
   C. Average the two halves
   D. Add the last 3 points back onto the first 3 points
   Answer: D. Time aliasing: points 5 to 7 wrap around and add to points 0 to 2, giving {1, 6, 9, 8, 4}. (Key point: Linear vs circular)

## Related in Digital Signal Analysis and Processing

- [Significance of convolution; compute convolution](https://www.bctengineer.com/question.md?key=semester_7%2Fdigital_signal_analysis_and_processing%3Aconvolution-significance-compute) — Digital Signal Analysis and Processing, chapter 1
- [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
- [Linear convolution via circular convolution](https://www.bctengineer.com/question.md?key=semester_7%2Fdigital_signal_analysis_and_processing%3Alinear-conv-via-circular) — Digital Signal Analysis and Processing, chapter 7

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