---
title: "LTI output for 2δ(n) − δ(n−1) input"
subject: "Digital Signal Analysis and Processing"
chapter: 1
key: "semester_7/digital_signal_analysis_and_processing:lti-output-impulse-input"
source: https://www.bctengineer.com/q/semester_7/digital_signal_analysis_and_processing/lti-output-impulse-input
site: BCT Engineering
---

# LTI output for 2δ(n) − δ(n−1) input

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

- Asked in: 2071 Chaitra, 6 marks
- Video: [Discrete Time Convolution](https://www.youtube.com/watch?v=A6qR7UhLvng) — 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-impulse-input
- Whole chapter: https://www.bctengineer.com/md/semester_7/digital_signal_analysis_and_processing/chapter-1

## Question

A discrete time LTI system has impulse response h(n) = {1, 3, 2, −1, 1} for −1 ≤ n ≤ 3. Determine the system output y(n) if the input x(n) is given by x(n) = 2δ(n) − δ(n−1).

## Short answer

By linearity and time-invariance, y(n)=2h(n)-h(n-1): y(n)={2,5,1,-4,3,-1} for n=-1 to 4.

## Answer

**Given:** h(n) = {1, 3, 2, −1, 1} for n = −1 … 3, so h(−1) = 1, h(0) = 3, h(1) = 2, h(2) = −1, h(3) = 1.

Input x(n) = 2δ(n) − δ(n − 1).

Because the system is LTI, linearity and time-invariance give

y(n) = 2h(n) − h(n − 1)

(an impulse δ(n − k) produces h(n − k)).

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

**Answer:** y(n) = {2, 5, 1, −4, 3, −1} for n = −1 … 4.

**Check:** Σy = 6; (Σx)(Σh) = (2 − 1)(1 + 3 + 2 − 1 + 1) = 6. ✓

## Check yourself

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

1. An input delta(n - k) to an LTI system produces
   A. h(n) + k
   B. k h(n)
   C. h(n - k)
   D. h(k - n)
   Answer: C. Time invariance: a shifted impulse gives a shifted impulse response. (Key point: Impulse response meaning)

2. For x(n) = 2 delta(n) - delta(n-1), y(n) =
   A. 2h(n) + h(n-1)
   B. 2h(n) - h(n-1)
   C. h(2n) - h(n)
   D. 2h(n-1) - h(n)
   Answer: B. Linearity scales and adds the shifted responses. (Key point: Output expression)

3. The first output sample is at n =
   A. -1
   B. 0
   C. 1
   D. -2
   Answer: A. h starts at n = -1 and the second term only starts at n = 0. (Key point: Starting index)

4. Sum y must equal
   A. 2 + 6
   B. 3
   C. 7
   D. (2-1)(1+3+2-1+1) = 6
   Answer: D. Sum x = 1, sum h = 6, product 6 = sum y. (Key point: Check)

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