---
title: "Output y[n] from G[k]=X[k]H[k]"
subject: "Digital Signal Analysis and Processing"
chapter: 7
key: "semester_7/digital_signal_analysis_and_processing:dft-product-x-h-71s"
source: https://www.bctengineer.com/q/semester_7/digital_signal_analysis_and_processing/dft-product-x-h-71s
site: BCT Engineering
---

# Output y[n] from G[k]=X[k]H[k]

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

- Asked in: 2071 Shrawan, 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-x-h-71s
- Whole chapter: https://www.bctengineer.com/md/semester_7/digital_signal_analysis_and_processing/chapter-7

## Question

A system has input signal x[n] = {1,2,3,4} and impulse response h[n] = {1,3,5,7} and the DFT of x[n] is X[k] and the DFT of h[n] is H[k]. Find the output of the system y[n] if G[k] = X[k].H[k].

## Short answer

The product of 4-point DFTs corresponds to 4-point circular convolution of x and h. Computing it gives y[n] = {42, 46, 42, 30}.

## Answer

Here G[k] = X[k]H[k] with 4-point DFTs, so the output y[n] = IDFT{G[k]} is the 4-point **circular convolution** of x[n] and h[n] (multiplication of DFTs ⇔ circular convolution). Both sequences already have N = 4 points, so no zero padding is needed:

- x[n] = {1, 2, 3, 4}
- h[n] = {1, 3, 5, 7}

y[n] = Σ (m = 0 to 3) x[m] · h[(n − m) mod 4]

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

```text
|    1    4    3    2 |   | 1 |
|    2    1    4    3 |   | 3 |
|    3    2    1    4 |   | 5 |
|    4    3    2    1 |   | 7 |
```

Row by row:

- y[0] = (1)(1) + (4)(3) + (3)(5) + (2)(7) = **42**
- y[1] = (2)(1) + (1)(3) + (4)(5) + (3)(7) = **46**
- y[2] = (3)(1) + (2)(3) + (1)(5) + (4)(7) = **42**
- y[3] = (4)(1) + (3)(3) + (2)(5) + (1)(7) = **30**

Check using DFTs: the inverse DFT of X(k)·H(k) for k = 0 … 3 is {42, 46, 42, 30}. Also, Σy[n] = 160 = X(0)·H(0) = 10 × 16 = 160.

**Answer: y[n] = {42, 46, 42, 30}, n = 0 … 3.**

## Check yourself

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

1. For G[k] = X[k]H[k] with 4-point DFTs, how is the output y[n] found?
   A. y[n] = linear convolution with 7 samples
   B. y[n] = DFT of x times n
   C. y[n] = IDFT of G[k], the 4-point circular convolution of x and h
   D. y[n] = x[n]h[n] sample by sample
   Answer: C. The inverse DFT of the product is the circular convolution. (Key point: System output)

2. Which expression is used?
   A. y[n] = sum x[m] h[m]
   B. y[n] = sum x[m] h[(n - m) mod 4]
   C. y[n] = sum x[m] h[n + m]
   D. y[n] = x[n] + h[n]
   Answer: B. Circular index (n - m) mod N. (Key point: Formula)

3. Which output is correct?
   A. {42, 46, 42, 30}
   B. {30, 42, 46, 42}
   C. {1, 6, 18, 40}
   D. {10, 16, 26, 160}
   Answer: A. Row by row: 42, 46, 42, 30. (Key point: Result)

4. What check confirms the output?
   A. Sum of y = X(0)H(0) = 10 x 16 = 160
   B. y[0] = 10
   C. Sum of y = 26
   D. y[3] = x[3]h[3]
   Answer: A. 42 + 46 + 42 + 30 = 160. (Key point: Check)

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