---
title: "Significance of convolution; compute convolution"
subject: "Digital Signal Analysis and Processing"
chapter: 1
key: "semester_7/digital_signal_analysis_and_processing:convolution-significance-compute"
source: https://www.bctengineer.com/q/semester_7/digital_signal_analysis_and_processing/convolution-significance-compute
site: BCT Engineering
---

# Significance of convolution; compute convolution

Digital Signal Analysis and Processing · Chapter 1: Discrete Time Signals and Systems

- Asked in: 2070 Asar, 2+4 marks
- Video: [Convolution in 5 Easy Steps](https://www.youtube.com/watch?v=aMaw4EumwyE) — Iain Explains Signals, Systems, and Digital Comms
- 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/convolution-significance-compute
- Whole chapter: https://www.bctengineer.com/md/semester_7/digital_signal_analysis_and_processing/chapter-1

## Question

Illustrate the significance of convolution summation in digital signal analysis. Compute the convolution of the following signals: h(n) = {1,0,1} and x(n) = {1,−2,−2,3,4}.

## Short answer

Convolution models LTI output as sum of scaled shifted impulses, essential for time-domain filtering. For h(n)={1,0,1}, x(n)={1,-2,-2,3,4}, y(n)=x(n)+x(n-2)={1,-2,-1,1,2,3,4} for n=0 to 6.

## Answer

#### Significance of convolution
- The output of any LTI system is the convolution of the input with the impulse response: y[n] = x[n] * h[n] = Σ x[k] h[n − k]. So the single sequence h[n] fully characterises the system.
- Any input is a sum of scaled, shifted impulses; the output is the sum of correspondingly scaled, shifted impulse responses (superposition).
- It is how digital filtering is done in the time domain (FIR filtering is convolution with the filter coefficients).
- Its properties (commutative, associative, distributive) let us combine cascaded and parallel systems: cascade h = h₁ * h₂, parallel h = h₁ + h₂.
- Convolution in time becomes multiplication in the z or frequency domain: Y(z) = X(z)H(z). This is the basis of fast convolution using the FFT.
- System properties such as causality and stability can be read directly from h[n].

#### Computation
h(n) = {1, 0, 1}, x(n) = {1, −2, −2, 3, 4}, both starting at n = 0. Output length = 3 + 5 − 1 = 7.

Since h has taps 1, 0, 1: y(n) = x(n) + x(n − 2).

| n | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|---|
| x(n) | 1 | −2 | −2 | 3 | 4 | 0 | 0 |
| x(n−2) | 0 | 0 | 1 | −2 | −2 | 3 | 4 |
| **y(n)** | **1** | **−2** | **−1** | **1** | **2** | **3** | **4** |

**Answer:** y(n) = {1, −2, −1, 1, 2, 3, 4}, n = 0 … 6.

**Check:** Σy = 8 = (Σx)(Σh) = 4 × 2. ✓

## Check yourself

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

1. The output of an LTI system equals
   A. the input multiplied by h[n]
   B. h[n] differentiated
   C. the input convolved with h[n]
   D. x[n] + h[n]
   Answer: C. h[n] fully characterises the system. (Key point: LTI output)

2. Cascade of systems has h =, parallel has h =
   A. h1 + h2; h1 * h2
   B. h1 * h2; h1 + h2
   C. h1 h2; h1 - h2
   D. h1 + h2; h1 + h2
   Answer: B. Convolution for cascade, addition for parallel. (Key point: Cascade/parallel)

3. Convolution in time corresponds to
   A. multiplication in z domain
   B. addition in z domain
   C. convolution in z domain
   D. division in z domain
   Answer: A. Y(z) = X(z)H(z), enabling fast FFT convolution. (Key point: Frequency domain)

4. For h = {1,0,1}, x = {1,-2,-2,3,4}, y(n) =
   A. x(n) + x(n-1)
   B. x(n) - x(n-2)
   C. x(n-2) only
   D. x(n) + x(n-2) giving {1,-2,-1,1,2,3,4}
   Answer: D. Taps 1, 0, 1 mean y(n) = x(n) + x(n-2); length 7. (Key point: Computation)

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