---
title: "LTI output for h(t)=e^(-at)u(t), x(t)=u(t)-u(t-2)"
subject: "Data Communication"
chapter: 2
key: "semester_5/data_communication:lti-output-u-minus-u2"
source: https://www.bctengineer.com/q/semester_5/data_communication/lti-output-u-minus-u2
site: BCT Engineering
---

# LTI output for h(t)=e^(-at)u(t), x(t)=u(t)-u(t-2)

Data Communication · Chapter 2: Signals and Systems

- Asked in: 2069 Chaitra, 5 marks
- Video: [Time domain - tutorial 8: LTI systems, impulse response & convolution](https://www.youtube.com/watch?v=Y-h7qoTVQtw) — SP hacks with Iman
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/data_communication/lti-output-u-minus-u2
- Whole chapter: https://www.bctengineer.com/md/semester_5/data_communication/chapter-2

## Question

Sketch the output of Linear Time invariant system having impulse response h(t) = e^(-at)u(t), for a > 0 and input signal x(t) = u(t) - u(t-2).

## Short answer

Output: y(t)=0 (t<0), y(t)=(1-e^(-at))/a (0≤t<2), y(t)=(e^(2a)-1)e^(-at)/a (t≥2).

## Answer

h(t) = e^(-at) u(t), a > 0; x(t) = u(t) - u(t - 2), a rectangular pulse of unit height from 0 to 2.

By linearity and time invariance, the output is the step response s(t) minus its 2-second delayed copy. The step response is

s(t) = h(t) * u(t) = (1 - e^(-at))/a for t ≥ 0 (and 0 for t < 0).

y(t) = s(t) - s(t - 2)

#### Region 1: t < 0
y(t) = 0.

#### Region 2: 0 ≤ t < 2
y(t) = (1 - e^(-at))/a

#### Region 3: t ≥ 2
y(t) = (1 - e^(-at))/a - (1 - e^(-a(t-2)))/a = (e^(-a(t-2)) - e^(-at))/a = e^(-at)(e^(2a) - 1)/a

**Answer:**
```text
        0                          t < 0
y(t) =  (1 - e^(-at))/a            0 <= t < 2
        (e^(2a) - 1) e^(-at)/a     t >= 2
```

#### Sketch
```text
y(t)
 (1-e^-2a)/a |        .-'‾'-.
             |      /         '-.
             |    /               '-.__
             |  /                       '-.___
 ------------+------------+------------------> t
             0            2
```
The output charges exponentially while the pulse is on, peaks at t = 2 with y(2) = (1 - e^(-2a))/a, then decays exponentially.

## Check yourself

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

1. The output to u(t)-u(t-2) is found by
   A. s(t) + s(t-2)
   B. s(t)s(t-2)
   C. h(t) - h(t-2)
   D. s(t) - s(t-2)
   Answer: D. Linearity and time invariance. (Key point: Method)

2. For 0 <= t < 2, y(t) =
   A. (1 - e^(-at))/a
   B. 0
   C. e^(-at)(e^(2a)-1)/a
   D. 1/a
   Answer: A. Only s(t) contributes. (Key point: 0<=t<2)

3. For t >= 2, y(t) =
   A. (1 - e^(-at))/a
   B. (e^(2a) - 1)e^(-at)/a
   C. 0
   D. 1/a
   Answer: B. Subtract delayed step response. (Key point: t>=2)

4. The output peaks at
   A. t = 0
   B. t = 1
   C. t = 2
   D. t infinity
   Answer: C. Pulse ends there; afterwards it decays. (Key point: Peak)

## 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×
- [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×
- [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×
- [Unit step response for h(t) = exp(-at)](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Aunit-step-response-exp) — Data Communication, chapter 2
- [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

## Related in other subjects

- [LTI output of a WSS input is WSS](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Alti-wssp-output) — Communication Systems, chapter 9
- [LTI output for 2δ(n) − δ(n−1) input](https://www.bctengineer.com/question.md?key=semester_7%2Fdigital_signal_analysis_and_processing%3Alti-output-impulse-input) — Digital Signal Analysis and Processing, chapter 1
- [LTI system output in time domain](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Alti-output-time-domain) — Communication Systems, chapter 1
- [Parseval's theorem; output PSD of LTI system](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aparseval-power-lti-psd) — Communication Systems, chapter 1