---
title: "Stable/unstable systems; test stability"
subject: "Data Communication"
chapter: 2
key: "semester_5/data_communication:stability-define-test-72c"
source: https://www.bctengineer.com/q/semester_5/data_communication/stability-define-test-72c
site: BCT Engineering
---

# Stable/unstable systems; test stability

Data Communication · Chapter 2: Signals and Systems

- Asked in: 2072 Chaitra, 2+3+3 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-define-test-72c
- Whole chapter: https://www.bctengineer.com/md/semester_5/data_communication/chapter-2

## Question

Define stable and unstable systems. Test the stability of the LTI systems whose impulse responses are given as (i) h(t) = e^(ψt)u(t) (ii) h(t) = e^(-ψt)u(t) [exponent as printed, unclear in scan]

## Short answer

A BIBO stable LTI system satisfies ∫|h| dt < ∞; otherwise unstable. For ψ>0: (i) h(t)=e^(ψt)u(t) has infinite integral → unstable; (ii) h(t)=e^(-ψt)u(t) has ∫=1/ψ < ∞ → stable.

## Answer

#### Definitions
- **Stable system (BIBO)**: every bounded input |x(t)| ≤ M₁ produces a bounded output |y(t)| ≤ M₂. For an LTI system, this holds if and only if ∫|h(t)| dt < ∞.
- **Unstable system**: some bounded input produces an unbounded output (∫|h| dt = ∞). Example: an ideal integrator, h(t) = u(t).

#### Tests
The exponent printed as ψ is unclear in the scan; I treat it as a real constant ψ > 0 (for example 2) and give the general condition.

**(i) h(t) = e^(ψt) u(t)**

∫₀^∞ e^(ψt) dt = [e^(ψt)/ψ]₀^∞ = ∞ for ψ > 0

The integral diverges, so the system is **unstable**. (It would be stable only if ψ were negative, ψ < 0, giving 1/|ψ|.)

**(ii) h(t) = e^(-ψt) u(t)**

∫₀^∞ e^(-ψt) dt = [-e^(-ψt)/ψ]₀^∞ = 1/ψ < ∞ for ψ > 0

The system is **stable**. Its pole s = -ψ lies in the left half-plane.

**Answer: for ψ > 0, (i) is unstable and (ii) is stable with ∫|h| dt = 1/ψ.**

## Check yourself

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

1. What is a BIBO stable system?
   A. Output is always zero
   B. Output equals input
   C. Output is bounded for some input
   D. Every bounded input gives a bounded output
   Answer: D. This is the definition of BIBO stability. (Key point: BIBO definition)

2. For an LTI system, stability holds if and only if
   A. The integral of |h(t)| is finite
   B. h(t) is zero for t < 0
   C. h(t) is periodic
   D. The integral of h(t) is zero
   Answer: A. Absolute integrability of the impulse response. (Key point: LTI test)

3. h(t) = e^(psi t)u(t) with psi > 0 is
   A. Stable
   B. Unstable, the integral diverges
   C. Stable with integral 1/psi
   D. Non-causal
   Answer: B. A growing exponential has an infinite integral. (Key point: Growing exponential)

4. h(t) = e^(-psi t)u(t), psi > 0, has integral of |h| equal to
   A. psi
   B. Infinity
   C. 1/psi, so it is stable
   D. 0
   Answer: C. Its pole lies in the left half-plane. (Key point: Decaying exponential)

## Related in Data Communication

- [Recursive/nonrecursive systems; stability](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Arecursive-nonrecursive-stability) — 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×
- [Stability of h(t) = t·e^(-3t)u(t)](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Astability-te-3t) — 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
- [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×

## Related in other subjects

- [Stability and causality of LTI systems](https://www.bctengineer.com/question.md?key=semester_7%2Fdigital_signal_analysis_and_processing%3Astability-causality-lti) — Digital Signal Analysis and Processing, chapter 1
- [Stable vs unstable sorting; heap](https://www.bctengineer.com/question.md?key=semester_4%2Fdata_structure_and_algorithm%3Astable-unstable-heap) — Data Structure and Algorithm, chapter 7
- [Cascade of two CT LTI systems](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Acascade-impulse-response) — Signals and Systems, chapter 5
- [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