---
title: "Find x3[n] from X3=X1X2, 4-point"
subject: "Digital Signal Analysis and Processing"
chapter: 7
key: "semester_7/digital_signal_analysis_and_processing:dft-product-x3-69c"
source: https://www.bctengineer.com/q/semester_7/digital_signal_analysis_and_processing/dft-product-x3-69c
site: BCT Engineering
---

# Find x3[n] from X3=X1X2, 4-point

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

- Asked in: 2069 Chaitra, 6 marks
- Video: [Compute the circular convolution using DFT and IDFT method](https://www.youtube.com/watch?v=7aFkjvC3R0M) — Engg-Course-Made-Easy
- 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/dft-product-x3-69c
- Whole chapter: https://www.bctengineer.com/md/semester_7/digital_signal_analysis_and_processing/chapter-7

## Question

Find x3[n] if DFT of x3[n] is given by X3(k) = X1(k) X2(k) where X1(k) and X2(k) are 4-point DFT of x1[n] = {1, 2, −2} and x2[n] = {1, 2, 3, −1} respectively.

## Short answer

Pad x1 to four points and perform 4-point circular convolution with x2. The result is x3[n] = {-7, 6, 5, 1}.

## Answer

Since X3(k) = X1(k)·X2(k), x3[n] is the 4-point **circular convolution** of x1[n] and x2[n] (multiplication of DFTs ⇔ circular convolution). x1[n] has only 3 points, so it is padded with one zero to N = 4:

- x1[n] = {1, 2, -2, 0}
- x2[n] = {1, 2, 3, -1}

x3[n] = Σ (m = 0 to 3) x1[m] · x2[(n − m) mod 4]

Written as a matrix product (the first column of the circulant matrix is x1, and each later column is the previous one shifted down circularly):

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

Row by row:

- x3[0] = (1)(1) + (0)(2) + (-2)(3) + (2)(-1) = **-7**
- x3[1] = (2)(1) + (1)(2) + (0)(3) + (-2)(-1) = **6**
- x3[2] = (-2)(1) + (2)(2) + (1)(3) + (0)(-1) = **5**
- x3[3] = (0)(1) + (-2)(2) + (2)(3) + (1)(-1) = **1**

Check using DFTs: the inverse DFT of X1(k)·X2(k) for k = 0 … 3 is {-7, 6, 5, 1}. Also, Σx3[n] = 5 = X1(0)·X2(0) = 1 × 5 = 5.

**Answer: x3[n] = {-7, 6, 5, 1}, n = 0 … 3.**

## Check yourself

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

1. X3(k) = X1(k)·X2(k) means x3[n] is which operation on x1 and x2?
   A. Linear convolution of unlimited length
   B. N-point circular convolution
   C. Sample-by-sample product
   D. Cross-correlation
   Answer: B. Multiplication of DFTs in frequency corresponds to circular convolution in time. (Key point: DFT product property)

2. x1[n] = {1, 2, -2} has 3 points but the DFTs are 4-point. What is done?
   A. Drop the last sample
   B. Pad with one zero to {1, 2, -2, 0}
   C. Use N = 3
   D. Pad at the front
   Answer: B. Zeros are appended so the length equals N = 4. (Key point: Zero padding)

3. Which quick check should hold for the result x3[n]?
   A. x3[0] equals X1(0)
   B. Number of samples equals the sum of lengths
   C. Sum of x3 is zero
   D. Sum of x3 equals X1(0)·X2(0), the product of the two sequence sums
   Answer: D. X3(0) = sum of x3, and X3(0) = X1(0)X2(0). Here the sum is 5. (Key point: Check)

## Close variants set in other exams

- [Find x3[n] from X3=X1X2, x1 5-point, x2 4-point](https://www.bctengineer.com/question.md?key=semester_7%2Fdigital_signal_analysis_and_processing%3Adft-product-x3-73s) — Digital Signal Analysis and Processing, chapter 7, asked 3×
- [Find x3[n] from X3=X1X2 (1,-2,5,1,2)](https://www.bctengineer.com/question.md?key=semester_7%2Fdigital_signal_analysis_and_processing%3Adft-product-x3-73c) — Digital Signal Analysis and Processing, chapter 7, asked 3×

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