---
title: "Output y(n) for h={5,4,3,2}, x={1,0,3,2}"
subject: "Digital Signal Analysis and Processing"
chapter: 1
key: "semester_7/digital_signal_analysis_and_processing:lti-output-4-point-seq"
source: https://www.bctengineer.com/q/semester_7/digital_signal_analysis_and_processing/lti-output-4-point-seq
site: BCT Engineering
---

# Output y(n) for h={5,4,3,2}, x={1,0,3,2}

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

- Asked in: 2070 Chaitra, 6 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/lti-output-4-point-seq
- Whole chapter: https://www.bctengineer.com/md/semester_7/digital_signal_analysis_and_processing/chapter-1

## Question

Find output y(n) when h(n) = {5,4,3,2} and x(n) = {1,0,3,2}.

## Short answer

Linear convolution of h(n)={5,4,3,2} and x(n)={1,0,3,2} starting at n=0 gives y(n)={5,4,18,24,17,12,4} for n=0 to 6.

## Answer

y(n) = Σ x(k) h(n − k), with both sequences starting at n = 0:
h = {5, 4, 3, 2}, x = {1, 0, 3, 2}. The output length is 4 + 4 − 1 = 7.

#### Table method
Each entry is x(j)·h(i), summed along diagonals i + j = n.

| | x0=1 | x1=0 | x2=3 | x3=2 |
|---|---|---|---|---|
| h0=5 | 5 | 0 | 15 | 10 |
| h1=4 | 4 | 0 | 12 | 8 |
| h2=3 | 3 | 0 | 9 | 6 |
| h3=2 | 2 | 0 | 6 | 4 |

- y(0) = 5
- y(1) = 0 + 4 = 4
- y(2) = 15 + 0 + 3 = 18
- y(3) = 10 + 12 + 0 + 2 = 24
- y(4) = 8 + 9 + 0 = 17
- y(5) = 6 + 6 = 12
- y(6) = 4

**Answer:** y(n) = {5, 4, 18, 24, 17, 12, 4}, n = 0 … 6.

**Check:** Σy = 84 = (5 + 4 + 3 + 2)(1 + 0 + 3 + 2) = 14 × 6. ✓

## Check yourself

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

1. Convolution of two 4-sample sequences has length
   A. 8
   B. 4
   C. 7
   D. 16
   Answer: C. 4 + 4 - 1 = 7. (Key point: Length)

2. y(3) is the sum of products with indices
   A. i - j = 3
   B. i + j = 3
   C. i j = 3
   D. i = j = 3
   Answer: B. y(n) sums x(j) h(i) with i + j = n. (Key point: Diagonal rule)

3. y(3) for h = {5,4,3,2}, x = {1,0,3,2} is
   A. 24
   B. 18
   C. 17
   D. 12
   Answer: A. 10 + 12 + 0 + 2 = 24. (Key point: Value y(3))

4. sum y =
   A. 14 + 6 = 20
   B. 84 / 14
   C. 24
   D. 14 x 6 = 84
   Answer: D. Sum of h is 14, sum of x is 6. (Key point: Check)