---
title: "Convolution of given sample values, verify"
subject: "Digital Signal Analysis and Processing"
chapter: 1
key: "semester_7/digital_signal_analysis_and_processing:lti-output-sparse-samples-2"
source: https://www.bctengineer.com/q/semester_7/digital_signal_analysis_and_processing/lti-output-sparse-samples-2
site: BCT Engineering
---

# Convolution of given sample values, verify

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

- Asked in: 2073 Chaitra, 6 marks
- Video: [Discrete Time Convolution (Tabular Method)](https://www.youtube.com/watch?v=C_kwjJLepLg) — Neso Academy
- 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/lti-output-sparse-samples-2
- Whole chapter: https://www.bctengineer.com/md/semester_7/digital_signal_analysis_and_processing/chapter-1

## Question

Calculate y[n], if x[n] is x[−2] = 0.5, x[0] = 1, x[1] = 0.75, x[3] = 0.5 and h[n] is h[0] = 1, h[1] = 0.75 and h[2] = 0.5, and verify your result.

## Short answer

Summing copies of h(n)={1,0.75,0.5} shifted by non-zero input samples yields y[n]={0.5,0.375,1.25,1.5,1.0625,0.875,0.375,0.25} for n=-2 to 5.

## Answer

**Samples (others are zero):**
- x[−2] = 0.5, x[0] = 1, x[1] = 0.75, x[3] = 0.5
- h[0] = 1, h[1] = 0.75, h[2] = 0.5

y[n] = Σ x[k] h[n − k] is the sum of copies of h, each shifted to the position of a sample of x and scaled by that sample. Here h = {1, 0.75, 0.5} starts at n = 0.

| n | −2 | −1 | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|---|---|
| 0.5·h[n+2] | 0.5 | 0.375 | 0.25 | 0 | 0 | 0 | 0 | 0 |
| 1·h[n] | 0 | 0 | 1 | 0.75 | 0.5 | 0 | 0 | 0 |
| 0.75·h[n−1] | 0 | 0 | 0 | 0.75 | 0.5625 | 0.375 | 0 | 0 |
| 0.5·h[n−3] | 0 | 0 | 0 | 0 | 0 | 0.5 | 0.375 | 0.25 |
| **y[n]** | **0.5** | **0.375** | **1.25** | **1.5** | **1.0625** | **0.875** | **0.375** | **0.25** |

**Answer:** y[n] = {0.5, 0.375, 1.25, 1.5, 1.0625, 0.875, 0.375, 0.25} for n = −2 … 5.

#### Verification
- Σx = 2.75, Σh = 2.25, product = 6.1875. Σy = 0.5 + 0.375 + 1.25 + 1.5 + 1.0625 + 0.875 + 0.375 + 0.25 = 6.1875. ✓
- Direct check of one point: y[2] = x[0]h[2] + x[1]h[1] = 0.5 + 0.5625 = 1.0625. ✓

## Check yourself

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

1. y[n] for sparse inputs is found by
   A. multiplying x and h sample by sample
   B. dividing X by H
   C. adding scaled, shifted copies of h at each non-zero x[k]
   D. differentiating x
   Answer: C. Each x[k] triggers x[k] h[n-k]; the copies are added. (Key point: Method)

2. The first non-zero output is at n =
   A. 0
   B. -2
   C. -1
   D. 1
   Answer: B. x starts at -2 and h at 0, so y starts at -2. (Key point: Start of the output)

3. y[2] equals
   A. x[0]h[2] + x[1]h[1] = 1.0625
   B. x[0]h[2] = 0.5
   C. x[1]h[2] = 0.375
   D. 1.25
   Answer: A. 0.5 + 0.75 x 0.75 = 0.5 + 0.5625 = 1.0625. (Key point: Value y[2])

4. Sum y must equal
   A. 2.75 + 2.25 = 5
   B. 2.75 / 2.25
   C. 2.25
   D. 2.75 x 2.25 = 6.1875
   Answer: D. Sum of x times sum of h. (Key point: Verification)

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