---
title: "Stability and causality of LTI systems"
subject: "Digital Signal Analysis and Processing"
chapter: 1
key: "semester_7/digital_signal_analysis_and_processing:stability-causality-lti"
source: https://www.bctengineer.com/q/semester_7/digital_signal_analysis_and_processing/stability-causality-lti
site: BCT Engineering
---

# Stability and causality of LTI systems

Digital Signal Analysis and Processing · Chapter 1: Discrete Time Signals and Systems

- Asked in: 2072 Chaitra, 4 marks
- Video: [Signals & System Lect 18 | Stability and causality of LTI systems | Solved examples](https://www.youtube.com/watch?v=OjAbfBHSQaM) — IMPLearn
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_7/digital_signal_analysis_and_processing/stability-causality-lti
- Whole chapter: https://www.bctengineer.com/md/semester_7/digital_signal_analysis_and_processing/chapter-1

## Question

Describe stability and causality characteristics of LTI system in terms of impulse response and ROC of its transfer function with suitable examples.

## Short answer

An LTI system is causal iff h[n]=0 for n<0, equivalently ROC of H(z) is outside a circle; it is BIBO stable iff sum |h[n]| is finite, equivalently ROC includes the unit circle. For a causal rational system, stability requires all poles inside the unit circle; e.g. (0.5)^n u[n] is causal and stable, 2^n u[n] is causal but unstable.

## Answer

#### Causality
An LTI system is **causal** if its output at time n depends only on present and past inputs.
- Time domain: h[n] = 0 for n < 0.
- z-domain: H(z) is the z-transform of a right-sided sequence, so the **ROC is the exterior of a circle**, |z| > r (extending to ∞). (For a rational H(z), also the numerator degree must not exceed the denominator degree.)
- Example: h[n] = (0.5)ⁿ u[n] gives H(z) = 1/(1 − 0.5z⁻¹), ROC |z| > 0.5: causal. h[n] = (0.5)ⁿ u[−n−1] has ROC |z| < 0.5: not causal.

#### Stability (BIBO)
An LTI system is **stable** if every bounded input gives a bounded output.
- Time domain: Σ |h[n]| < ∞ (absolutely summable).
- z-domain: the **ROC of H(z) includes the unit circle** |z| = 1 (so the frequency response exists).
- Example: h[n] = (0.5)ⁿ u[n], Σ|h| = 2 < ∞, ROC |z| > 0.5 includes |z| = 1: stable. h[n] = 2ⁿ u[n] has ROC |z| > 2, which excludes the unit circle: unstable.

#### Both together
A causal LTI system with a rational H(z) is stable if and only if **all poles lie inside the unit circle**, because the ROC |z| > (largest pole magnitude) must include |z| = 1.

| h[n] | H(z) | ROC | Causal | Stable |
|---|---|---|---|---|
| (0.5)ⁿ u[n] | 1/(1−0.5z⁻¹) | \|z\| > 0.5 | Yes | Yes |
| 2ⁿ u[n] | 1/(1−2z⁻¹) | \|z\| > 2 | Yes | No |
| −2ⁿ u[−n−1] | 1/(1−2z⁻¹) | \|z\| < 2 | No | Yes |

The same H(z) can be stable or causal depending on the ROC chosen.

## Check yourself

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

1. For a causal system the ROC of H(z) is
   A. outside a circle |z| > r
   B. inside a circle
   C. an annulus only
   D. the whole plane
   Answer: A. Right-sided h gives an exterior ROC. (Key point: Causal ROC)

2. BIBO stability requires
   A. h[n]=0 for n<0
   B. sum h[n]=0
   C. h[n] bounded only
   D. sum|h[n]| < infinity
   Answer: D. Absolute summability. (Key point: Stability in time domain)

3. Stability requires the ROC to
   A. exclude the unit circle
   B. be inside all poles
   C. include the unit circle
   D. include z = 0
   Answer: C. Then the frequency response exists. (Key point: Stability in z domain)

4. A causal rational system is stable iff
   A. all poles outside the unit circle
   B. all poles are inside the unit circle
   C. all zeros inside the unit circle
   D. poles on unit circle
   Answer: B. ROC |z| > largest pole magnitude must include |z| = 1. (Key point: Causal and stable)

## Related in Digital Signal Analysis and Processing

- [Plot sequence; properties of LTI system](https://www.bctengineer.com/question.md?key=semester_7%2Fdigital_signal_analysis_and_processing%3Aplot-sequence-lti-properties) — Digital Signal Analysis and Processing, chapter 1
- [LTI output, h[n]=(1/2)^n window](https://www.bctengineer.com/question.md?key=semester_7%2Fdigital_signal_analysis_and_processing%3Alti-output-exp-window-1) — Digital Signal Analysis and Processing, chapter 1, asked 4×
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- [LTI output for h=2u[n]−2u[n−4]](https://www.bctengineer.com/question.md?key=semester_7%2Fdigital_signal_analysis_and_processing%3Alti-output-step-difference) — Digital Signal Analysis and Processing, chapter 1, asked 4×
- [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

## Related in other subjects

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