---
title: "Convolve {1/4, −1, 2} and {2, 2/4, −2}"
subject: "Signals and Systems"
chapter: 6
key: "semester_4/signals_and_systems:convolution-fractional"
source: https://www.bctengineer.com/q/semester_4/signals_and_systems/convolution-fractional
site: BCT Engineering
---

# Convolve {1/4, −1, 2} and {2, 2/4, −2}

Signals and Systems · Chapter 6: Discrete-Time Systems

- Asked in: 2081 Asoj, 6 marks
- Video: [#110 Solved problems -2 on Convolution Sum || EC Academy](https://www.youtube.com/watch?v=hl2T2adZWFI) — EC Academy
- 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-fractional
- Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-6

## Question

Find the output of an LTI system, for input x[n] = {1/4, −1, 2} and h[n] = {2, 2/4, −2}.

## Short answer

Taking origins at n=0, the convolution y[n]=x[n]∗h[n] results in y[n]={0.5, -1.875, 3, 3, -4} (i.e., 1/2, -15/8, 3, 3, -4) with y[0]=0.5.

## Answer

No origin is marked, so take the first sample at $n = 0$:
$x[n] = \{\underset{\uparrow}{0.25}, -1, 2\}$ and $h[n] = \{\underset{\uparrow}{2}, 0.5, -2\}$ (note $2/4 = 0.5$).

The output of an LTI system is $y[n] = x[n]*h[n] = \sum_k x[k]\,h[n-k]$.

Output starts at $n = 0$, length $3 + 3 - 1 = 5$, so $0 \le n \le 4$.

#### Tabular method

| Row | $n=0$ | $n=1$ | $n=2$ | $n=3$ | $n=4$ |
|---|---|---|---|---|---|
| $x[0]\,h[n]$ | 0.5 | 0.125 | -0.5 |  |  |
| $x[1]\,h[n-1]$ |  | -2 | -0.5 | 2 |  |
| $x[2]\,h[n-2]$ |  |  | 4 | 1 | -4 |
| $y[n]$ | **0.5** | **-1.875** | **3** | **3** | **-4** |

#### Direct working

- $y[0] = 0.25(2) = 0.5$
- $y[1] = 0.25(0.5) + (-1)(2) = 0.125 - 2 = -1.875$
- $y[2] = 0.25(-2) + (-1)(0.5) + 2(2) = -0.5 - 0.5 + 4 = 3$
- $y[3] = (-1)(-2) + 2(0.5) = 2 + 1 = 3$
- $y[4] = 2(-2) = -4$

**Sum check:** $(\sum x)(\sum h) = 1.25 \times 0.5 = 0.625$, and $0.5 - 1.875 + 3 + 3 - 4 = 0.625$.

**Answer:**

$$
y[n] = \left\{\underset{\uparrow}{\tfrac{1}{2}},\ -\tfrac{15}{8},\ 3,\ 3,\ -4\right\} = \{\underset{\uparrow}{0.5},\ -1.875,\ 3,\ 3,\ -4\}
$$

## Check yourself

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

1. x and h each have 3 samples starting at 0. The output y has:
   A. 6 samples
   B. 5 samples, n = 0 to 4
   C. 3 samples
   D. 9 samples
   Answer: B. Length = 3 + 3 - 1 = 5. (Key point: Output length)

2. What is y[1]?
   A. 0.25(0.5) + (-1)(2) = -1.875
   B. 0.25(2) + (-1)(0.5) = 0
   C. 0.25(0.5) = 0.125
   D. -1(2) = -2
   Answer: A. y[1] = x[0]h[1] + x[1]h[0] = 0.125 - 2 = -1.875. (Key point: Value y[1])

3. The sum of y must be:
   A. 1.25
   B. 0.625
   C. 0.5
   D. 1.75
   Answer: B. (0.25 - 1 + 2)(2 + 0.5 - 2) = 1.25 x 0.5 = 0.625. (Key point: Sum check)

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