---
title: "Unit step response for h(t) = exp(-at)"
subject: "Data Communication"
chapter: 2
key: "semester_5/data_communication:unit-step-response-exp"
source: https://www.bctengineer.com/q/semester_5/data_communication/unit-step-response-exp
site: BCT Engineering
---

# Unit step response for h(t) = exp(-at)

Data Communication · Chapter 2: Signals and Systems

- Asked in: 2069 Asar, 5 marks
- Video: [Linear Time-Invariant(LTI) system- concept, convolution, properties, deconvolution, identity system](https://www.youtube.com/watch?v=m_FgpCpGYNc) — Shrenik Jain
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/data_communication/unit-step-response-exp
- Whole chapter: https://www.bctengineer.com/md/semester_5/data_communication/chapter-2

## Question

Find the unit step response of a linear time invariant system having impulse response h(t) = exp(-at), for a>0.

## Short answer

The step response is the integral of the impulse response: s(t) = (1/a)(1 - e^(-at)) u(t).

## Answer

The step response is the output when x(t) = u(t):

s(t) = h(t) * u(t) = ∫ h(τ) u(t - τ) dτ = ∫_{-∞}^{t} h(τ) dτ

The given impulse response is e^(-at) for a > 0. For a physical (causal) system, this means h(t) = e^(-at) u(t), which I assume; if it were non-zero for t < 0, the integral would diverge.

**t < 0**: s(t) = 0.

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

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

```text
s(t)
1/a |        ________
    |     _-'
    |   /
    +-----------------> t
```
The response starts at 0 and approaches 1/a, the DC gain H(0) = ∫h dt = 1/a, with a time constant 1/a.

## Check yourself

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

1. The step response is
   A. The derivative of h
   B. h(t) squared
   C. h(t) times u(t) pointwise
   D. The integral of h(tau) from -infinity to t
   Answer: D. s = h * u. (Key point: Step response)

2. For h = e^(-at)u(t), s(t) =
   A. (1/a)(1 - e^(-at))u(t)
   B. e^(-at)u(t)
   C. (1/a)e^(-at)
   D. 1 - e^(at)
   Answer: A. Integrate from 0 to t. (Key point: Result)

3. The response settles at
   A. a
   B. 1/a, the DC gain
   C. 0
   D. 1
   Answer: B. Integral of h. (Key point: Final value)

4. Why assume h(t) = e^(-at)u(t)?
   A. It is simpler
   B. a is negative
   C. e^(-at) for t < 0 would make the integral diverge
   D. u(t) is unit
   Answer: C. Physical systems are causal. (Key point: Causal assumption)

## Related in Data Communication

- [Define unit impulse and unit step](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Aunit-impulse-step-define) — Data Communication, chapter 2
- [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×
- [Energy or power: unit step, impulse](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Aenergy-power-check-signals) — Data Communication, chapter 2, asked 2×
- [Unit impulse function and its significance](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Aunit-impulse-significance) — Data Communication, chapter 2
- [Linear/TI/stable/memoryless: y(t), y[k]](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Acheck-1-minus-e-4t-and-sin) — 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

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