---
title: "Find x3[n] from X3=X1X2 (5-point)"
subject: "Digital Signal Analysis and Processing"
chapter: 7
key: "semester_7/digital_signal_analysis_and_processing:dft-product-x3-74a"
source: https://www.bctengineer.com/q/semester_7/digital_signal_analysis_and_processing/dft-product-x3-74a
site: BCT Engineering
---

# Find x3[n] from X3=X1X2 (5-point)

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

- Asked in: 2074 Asoj, 7 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-x3-74a
- Whole chapter: https://www.bctengineer.com/md/semester_7/digital_signal_analysis_and_processing/chapter-7

## Question

If X1(k) and X2(k) are DFT of sequences x1[n] = {1, 2, 0, 1, −2} and x2[n] = {1, 0, 1, 1, 2} respectively, find the sequence x3[n], if DFT of x3[n] is given by X3(k) = X1(k) X2(k).

## Short answer

X3(k)=X1(k)X2(k) means x3[n] is the 5-point circular convolution of x1 and x2. Computing it gives x3[n] = {6, 1, 1, 0, 2}.

## Answer

Since X3(k) = X1(k)·X2(k), x3[n] is the 5-point **circular convolution** of x1[n] and x2[n] (multiplication of DFTs ⇔ circular convolution). Both sequences already have 5 points, so no padding is needed:

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

x3[n] = Σ (m = 0 to 4) x1[m] · x2[(n − m) mod 5]

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

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

Row by row:

- x3[0] = (1)(1) + (−2)(0) + (1)(1) + (0)(1) + (2)(2) = **6**
- x3[1] = (2)(1) + (1)(0) + (−2)(1) + (1)(1) + (0)(2) = **1**
- x3[2] = (0)(1) + (2)(0) + (1)(1) + (−2)(1) + (1)(2) = **1**
- x3[3] = (1)(1) + (0)(0) + (2)(1) + (1)(1) + (−2)(2) = **0**
- x3[4] = (−2)(1) + (1)(0) + (0)(1) + (2)(1) + (1)(2) = **2**

Check using DFTs: the inverse DFT of X1(k)·X2(k), k = 0 … 4, is {6, 1, 1, 0, 2} (Σx3 = 10 = X1(0)·X2(0) = 2 × 5).

**Answer: x3[n] = {6, 1, 1, 0, 2}, n = 0 … 4.**

## 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. Cross-correlation
   C. N-point circular convolution
   D. Sample-by-sample product
   Answer: C. Multiplication of DFTs in frequency corresponds to circular convolution in time. (Key point: DFT product property)

2. Which quick check should hold for the result x3[n]?
   A. Number of samples equals the sum of lengths
   B. Sum of x3 equals X1(0)·X2(0), the product of the two sequence sums
   C. Sum of x3 is zero
   D. x3[0] equals X1(0)
   Answer: B. X3(0) = sum of x3, and X3(0) = X1(0)X2(0). Here the sum is 10. (Key point: Check)

## Close variants set in other exams

- [Find x3[n] from X3=X1X2 (1,2,4)](https://www.bctengineer.com/question.md?key=semester_7%2Fdigital_signal_analysis_and_processing%3Adft-product-x3-72k) — Digital Signal Analysis and Processing, chapter 7, asked 2×

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