---
title: "Range of a, b for stability"
subject: "Data Communication"
chapter: 2
key: "semester_5/data_communication:stability-range-ab"
source: https://www.bctengineer.com/q/semester_5/data_communication/stability-range-ab
site: BCT Engineering
---

# Range of a, b for stability

Data Communication · Chapter 2: Signals and Systems

- Asked in: 2074 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/stability-range-ab
- Whole chapter: https://www.bctengineer.com/md/semester_5/data_communication/chapter-2

## Question

Define LTI system. Determine the range of values of "a" and "b" for the stability of LTI system with impulse response h(t) = e^(at)u(t) + e^(-bt)u(t).

## Short answer

An LTI system is linear and time-invariant. For h(t)=e^(at)u(t)+e^(-bt)u(t), BIBO stability requires a<0 and b>0 so both exponentials decay.

## Answer

**LTI system**: a system that is linear (superposition holds) and time-invariant (a shift in input shifts the output equally). It is completely described by its impulse response h(t), with output y(t) = x(t) * h(t).

#### Stability range
For BIBO stability, ∫|h(t)| dt must be finite.

h(t) = e^(at) u(t) + e^(-bt) u(t), for t ≥ 0.

∫₀^∞ |e^(at) + e^(-bt)| dt ≤ ∫₀^∞ e^(at) dt + ∫₀^∞ e^(-bt) dt

Each term must converge (both terms are positive, so the sum is finite exactly when each is finite):

- ∫₀^∞ e^(at) dt = [e^(at)/a]₀^∞ converges only if **a < 0**, giving -1/a.
- ∫₀^∞ e^(-bt) dt = [-e^(-bt)/b]₀^∞ converges only if **b > 0**, giving 1/b.

**Answer: the system is stable for a < 0 and b > 0** (any a < 0 and any b > 0, with ∫|h| dt = 1/b - 1/a). If a ≥ 0 or b ≤ 0, one of the exponentials does not decay and the system is unstable.

## Check yourself

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

1. What is the BIBO stability condition for an LTI system?
   A. h(t) is zero for t < 0
   B. h(t) is bounded by 1
   C. The integral of h(t) is zero
   D. The integral of |h(t)| is finite
   Answer: D. An LTI system is BIBO stable iff h is absolutely integrable. (Key point: Stability criterion)

2. For the e^(at)u(t) term to be absolutely integrable, what is needed?
   A. a < 0
   B. a > 0
   C. a = 0
   D. a can have any value
   Answer: A. Only a decaying exponential has a finite integral to infinity. (Key point: Condition on a)

3. For the e^(-bt)u(t) term, what is needed?
   A. b < 0
   B. b > 0
   C. b = 0
   D. b >= 0
   Answer: B. e^(-bt) decays only if b > 0; for b = 0 the integral diverges. (Key point: Condition on b)

4. When stable, what is the integral of |h|?
   A. 1/a + 1/b
   B. a + b
   C. 1/b - 1/a
   D. 1/(a b)
   Answer: C. The integrals give -1/a and 1/b, which sum to 1/b - 1/a. (Key point: Total integral)

## 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×
- [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
- [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×

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