---
title: "Convolution of e^(-at)u(t) with u(t)"
subject: "Data Communication"
chapter: 2
key: "semester_5/data_communication:convolution-exp-u-step"
source: https://www.bctengineer.com/q/semester_5/data_communication/convolution-exp-u-step
site: BCT Engineering
---

# Convolution of e^(-at)u(t) with u(t)

Data Communication · Chapter 2: Signals and Systems

- Asked in: 2070 Chaitra, 3+5 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/data_communication/convolution-exp-u-step
- Whole chapter: https://www.bctengineer.com/md/semester_5/data_communication/chapter-2

## Question

Define LTI system. Find the convolution between x(t) = e^(-at)u(t), (a > 0) and h(t) = u(t).

## Short answer

An LTI system is linear and time-invariant; its output is the convolution of input and impulse response. For x(t)=e^(-at)u(t) and h(t)=u(t), y(t) = (1/a)(1 - e^(-at)) u(t).

## Answer

**LTI system**: a system that is linear and time-invariant; its output is y(t) = x(t) * h(t) = ∫ x(τ) h(t - τ) dτ, where h(t) is its impulse response.

#### Convolution
x(t) = e^(-at) u(t), h(t) = u(t), a > 0.

y(t) = ∫ x(τ) h(t - τ) dτ = ∫ e^(-aτ) u(τ) u(t - τ) dτ

- u(τ) restricts τ ≥ 0.
- u(t - τ) restricts τ ≤ t.

**t < 0**: no overlap, y(t) = 0.

**t ≥ 0**:
y(t) = ∫₀^t e^(-aτ) dτ = [-e^(-aτ)/a]₀^t = (1 - e^(-at))/a

**Answer: y(t) = (1/a)(1 - e^(-at)) u(t)**

```text
y(t)
1/a |       __________
    |    _-'
    |  /
    | /
----+------------------> t
```
The output rises from 0 and settles at 1/a as t → ∞ (the step response of the system, i.e. the integral of the exponential).

## Check yourself

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

1. For t >= 0 the limits of integration are
   A. -infinity to infinity
   B. 0 to infinity
   C. t to infinity
   D. 0 to t
   Answer: D. The step restricts tau to 0..t. (Key point: Overlap)

2. The convolution result is
   A. (1/a)(1 - e^(-at))u(t)
   B. e^(-at)u(t)
   C. (1/a)e^(-at)u(t)
   D. t e^(-at)u(t)
   Answer: A. Integral of e^(-a tau) from 0 to t. (Key point: Result)

3. As t grows, y(t) approaches
   A. 0
   B. 1/a
   C. 1
   D. Infinity
   Answer: B. The exponential term vanishes. (Key point: Final value)

4. Convolving with u(t) corresponds to
   A. Differentiating it
   B. Squaring it
   C. Integrating the signal
   D. Delaying it
   Answer: C. u(t) is an integrator. (Key point: Meaning)

## Related in Data Communication

- [Define LTI system and impulse response](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Alti-impulse-response-define) — Data Communication, chapter 2, asked 2×
- [Convolution of u(t) with u(t)](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Aconvolution-u-u) — Data Communication, chapter 2
- [Output of LTI with h(t) = e^(-at)u(t) and e^(at)u(-t)](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Aconvolution-exp-u-and-u-neg) — Data Communication, chapter 2, asked 3×
- [LTI output for h(t)=e^(-2t), rectangular input](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Alti-output-rect-input) — Data Communication, chapter 2, asked 3×
- [Check LTI: y(t) = 2x(t) + 3](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Alti-check-2x-plus-3) — Data Communication, chapter 2
- [LTI output for h(t)=e^(-at)u(t), x(t)=u(t)-u(t-2)](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Alti-output-u-minus-u2) — Data Communication, chapter 2

## Related in other subjects

- [Periodic and non-periodic signals; LTI output is convolution](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aperiodic-nonperiodic-convolution) — Communication Systems, chapter 1
- [Frequency response of a CT LTI system](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Act-frequency-response) — Signals and Systems, chapter 5, asked 3×
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