---
title: "Convolution of δ-sums with h[n] = 2δ[n] + 2δ[n−1]"
subject: "Signals and Systems"
chapter: 6
key: "semester_4/signals_and_systems:convolution-delta-two-taps"
source: https://www.bctengineer.com/q/semester_4/signals_and_systems/convolution-delta-two-taps
site: BCT Engineering
---

# Convolution of δ-sums with h[n] = 2δ[n] + 2δ[n−1]

Signals and Systems · Chapter 6: Discrete-Time Systems

- Asked in: 2083 Bhadra (new course), 5 marks
- Video: [Convolution between two finite sequences.](https://www.youtube.com/watch?v=IaUz9bafIPE) — Engg-Course-Made-Easy
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/signals_and_systems/convolution-delta-two-taps
- Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-6

## Question

Find the convolution sum for a given signal x[n] = δ[n] + 2δ[n−1] − δ[n−3] and h[n] = 2δ[n] + 2δ[n−1]

## Short answer

Convolution yields y[n]=2δ[n]+6δ[n-1]+4δ[n-2]-2δ[n-3]-2δ[n-4], or sample values {2,6,4,-2,-2} for n=0..4. Use y[n]=2x[n]+2x[n-1].

## Answer

Write the sequences as sample values:

$$
x[n]=\delta[n]+2\delta[n-1]-\delta[n-3]\ \Rightarrow\ x[n]=\{1,\ 2,\ 0,\ -1\},\ n=0..3
$$

$$
h[n]=2\delta[n]+2\delta[n-1]\ \Rightarrow\ h[n]=\{2,\ 2\},\ n=0,1
$$

#### Method 1: using $x[n]*\delta[n-n_0]=x[n-n_0]$

$$
\begin{aligned}
y[n]&=x[n]*\big(2\delta[n]+2\delta[n-1]\big)=2x[n]+2x[n-1]\\
&=2\delta[n]+4\delta[n-1]-2\delta[n-3]+2\delta[n-1]+4\delta[n-2]-2\delta[n-4]\\
&=2\delta[n]+6\delta[n-1]+4\delta[n-2]-2\delta[n-3]-2\delta[n-4]
\end{aligned}
$$

#### Method 2: tabular check

| | $n=0$ | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| $2x[n]$ | 2 | 4 | 0 | −2 | |
| $2x[n-1]$ | | 2 | 4 | 0 | −2 |
| $y[n]$ | **2** | **6** | **4** | **−2** | **−2** |

Sum check: $\sum y=8=(\sum x)(\sum h)=2\times4$.

**Answer: $y[n]=\{2,\ 6,\ 4,\ -2,\ -2\}$ for $n=0,\dots,4$, i.e. $y[n]=2\delta[n]+6\delta[n-1]+4\delta[n-2]-2\delta[n-3]-2\delta[n-4]$.**

## Check yourself

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

1. What is x[n]=delta[n]+2delta[n-1]-delta[n-3] as a sequence?
   A. {1, 2, 0, -1}
   B. {1, 2, -1}
   C. {1, 2, 0, 1}
   D. {1, 0, 2, -1}
   Answer: A. Delta at n=2 is missing so that sample is 0. (Key point: Sample form)

2. With h=2delta[n]+2delta[n-1], what is y[n] in terms of x?
   A. 2x[n]+2x[n+1]
   B. 4x[n]
   C. 2x[n]+2x[n-1]
   D. x[n]+x[n-1]
   Answer: C. x*delta[n-n0]=x[n-n0]. (Key point: Shifting property)

3. How many samples does y have, and over what range?
   A. 4 samples, n=0 to 3
   B. 5 samples, n=0 to 4
   C. 6 samples, n=0 to 5
   D. 5 samples, n=1 to 5
   Answer: B. Length 4+2-1=5. (Key point: Output length)

4. What is y[n]?
   A. {2, 4, 4, -2, -2}
   B. {2, 6, 4, 2, -2}
   C. {2, 6, 0, -2, -2}
   D. {2, 6, 4, -2, -2}
   Answer: D. 2x gives {2,4,0,-2}, 2x delayed gives {0,2,4,0,-2}, summed {2,6,4,-2,-2}. (Key point: Result)

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