---
title: "Find x3[n] from X3=X1X2 (1,-2,5,1,2)"
subject: "Digital Signal Analysis and Processing"
chapter: 7
key: "semester_7/digital_signal_analysis_and_processing:dft-product-x3-73c"
source: https://www.bctengineer.com/q/semester_7/digital_signal_analysis_and_processing/dft-product-x3-73c
site: BCT Engineering
---

# Find x3[n] from X3=X1X2 (1,-2,5,1,2)

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

- Asked in: 2073 Chaitra, 7 marks
- 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-x3-73c
- 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 5-point DFT of x1[n] = {1, −2, 5, 1, 2} and x2[n] = {1, 2, −3, −2} respectively.

## Short answer

Pad x2 to five points, then take the 5-point circular convolution of x1 and x2. The result is x3[n] = {-8, -8, -6, 15, -7}.

## Answer

x2[n] has 4 points; for the 5-point DFTs it is padded with one zero. 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):

- x1[n] = {1, −2, 5, 1, 2}
- x2[n] = {1, 2, −3, −2, 0}

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    5   -2 |   |  1 |
|   -2    1    2    1    5 |   |  2 |
|    5   -2    1    2    1 |   | -3 |
|    1    5   -2    1    2 |   | -2 |
|    2    1    5   -2    1 |   |  0 |
```

Row by row:

- x3[0] = (1)(1) + (2)(2) + (1)(−3) + (5)(−2) + (−2)(0) = **−8**
- x3[1] = (−2)(1) + (1)(2) + (2)(−3) + (1)(−2) + (5)(0) = **−8**
- x3[2] = (5)(1) + (−2)(2) + (1)(−3) + (2)(−2) + (1)(0) = **−6**
- x3[3] = (1)(1) + (5)(2) + (−2)(−3) + (1)(−2) + (2)(0) = **15**
- x3[4] = (2)(1) + (1)(2) + (5)(−3) + (−2)(−2) + (1)(0) = **−7**

Check using DFTs: the inverse DFT of X1(k)·X2(k), k = 0 … 4, is {−8, −8, −6, 15, −7} (Σx3 = −14 = X1(0)·X2(0) = 7 × (−2)).

**Answer: x3[n] = {−8, −8, −6, 15, −7}, 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. Sample-by-sample product
   D. N-point circular convolution
   Answer: D. Multiplication of DFTs in frequency corresponds to circular convolution in time. (Key point: DFT product property)

2. x2[n] has 4 points but the DFTs are 5-point. What is done?
   A. x2 is repeated periodically
   B. x2 is padded with one zero to {1,2,-3,-2,0}
   C. x1 is cut to 4 points
   D. N is changed to 4
   Answer: B. Both sequences must have length N = 5, so one zero is appended to x2. (Key point: Zero padding)

3. 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 -14. (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, 4-point](https://www.bctengineer.com/question.md?key=semester_7%2Fdigital_signal_analysis_and_processing%3Adft-product-x3-69c) — Digital Signal Analysis and Processing, chapter 7, asked 3×

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