---
title: "Stability of h(t) = t·e^(-3t)u(t)"
subject: "Data Communication"
chapter: 2
key: "semester_5/data_communication:stability-te-3t"
source: https://www.bctengineer.com/q/semester_5/data_communication/stability-te-3t
site: BCT Engineering
---

# Stability of h(t) = t·e^(-3t)u(t)

Data Communication · Chapter 2: Signals and Systems

- Asked in: 2075 Chaitra, 8 marks
- Video: [Stable and Unstable Systems](https://www.youtube.com/watch?v=ngJ2QWvMPbI) — Neso Academy
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/data_communication/stability-te-3t
- Whole chapter: https://www.bctengineer.com/md/semester_5/data_communication/chapter-2

## Question

State the condition for the stability of an LTI system. Test the stability of the system whose impulse response is h(t) = t·e^(-3t)·u(t).

## Short answer

An LTI system is BIBO stable if its impulse response is absolutely integrable. For h(t)=t·e^(-3t)·u(t), ∫₀^∞ t e^(-3t) dt = 1/9 < ∞, so the system is stable.

## Answer

#### Condition for stability
An LTI system is BIBO stable if and only if its impulse response is absolutely integrable:

∫ |h(t)| dt < ∞, integrated over -∞ to ∞.

(Equivalently, all poles of H(s) lie in the left half of the s-plane and the ROC includes the jω axis.)

#### Test for h(t) = t·e^(-3t)·u(t)
Since h(t) ≥ 0 for t ≥ 0,

∫|h(t)| dt = ∫₀^∞ t e^(-3t) dt

Integrate by parts (u = t, dv = e^(-3t) dt):

= [-t e^(-3t)/3]₀^∞ + (1/3) ∫₀^∞ e^(-3t) dt
= 0 + (1/3)(1/3)
= 1/9

Standard result: ∫₀^∞ tⁿ e^(-at) dt = n!/a^(n+1), so with n = 1, a = 3 it gives 1/9.

The value is finite. (H(s) = 1/(s + 3)², a double pole at s = -3 in the left half-plane.)

**Answer: the system is BIBO stable (∫|h| dt = 1/9).**

## Check yourself

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

1. LTI stable iff
   A. h bounded
   B. h causal
   C. H(s) zero
   D. integral |h| < infinity
   Answer: D. Absolute integrability. (Key point: Condition)

2. Integral of t e^(-3t) from 0 is
   A. 1/9
   B. 1/3
   C. 3
   D. infinity
   Answer: A. n!/a^(n+1). (Key point: Integral)

3. Poles of H(s)=1/(s+3)^2
   A. at s=3
   B. double at s=-3, left half
   C. at origin
   D. right half
   Answer: B. Stable. (Key point: Poles)

4. System is
   A. unstable
   B. marginal
   C. BIBO stable
   D. non-causal
   Answer: C. Finite integral. (Key point: Conclusion)

## 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×
- [Stable/unstable systems; test stability](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Astability-define-test-72c) — Data Communication, chapter 2
- [Recursive/nonrecursive systems; stability](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Arecursive-nonrecursive-stability) — Data Communication, chapter 2
- [Range of a, b for stability](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Astability-range-ab) — 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×

## Related in other subjects

- [LTI properties; BIBO stability condition](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Alti-bibo-stability) — Signals and Systems, chapter 6
- [Condition for LTI system to be causal](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Alti-causality-condition) — Signals and Systems, chapter 6, asked 3×
- [Causality condition for CT LTI system](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Acausal-ct-lti-condition) — Signals and Systems, chapter 5
- [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×