Stable and Unstable LTI Systems: BIBO Stability and Pole Analysis

Stable and Unstable LTI Systems

An In-depth Engineering Guide to BIBO Stability and Impulse Response Analysis

In signals and systems analysis, a Linear Time-Invariant (LTI) system is classified as stable if every bounded input results in a bounded output. This fundamental property ensures that the system does not produce divergent or uncontrollable signals under normal operating conditions. Conversely, if a system produces an unbounded output for at least one bounded input, it is classified as unstable.

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1. BIBO Stability (Bounded Input Bounded Output)

The BIBO Condition:

A system is BIBO Stable if for every input signal $x(t)$ (or $x[n]$) that is bounded:

|x(t)| ≤ M_x < ∞

The resulting output $y(t)$ (or $y[n]$) is also guaranteed to be bounded:

|y(t)| ≤ M_y < ∞

where $M_x$ and $M_y$ are finite constants.

Simply put: Bounded Input ⇒ Bounded Output. If a system’s internal energy grows without bound despite a finite input, it fails this criteria.

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2. Stability Condition for Continuous-Time LTI Systems

For a continuous-time system with impulse response h(t), the output is determined by the convolution integral:

y(t) = x(t) * h(t) = ∫_-∞^∞ h(τ) x(t - τ) dτ

The necessary and sufficient condition for BIBO stability is that the impulse response must be absolutely integrable:

∫_-∞^∞ |h(t)| dt < ∞

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3. Stability Condition for Discrete-Time LTI Systems

For a discrete-time system with impulse response h[n], the output is the convolution sum. The system is stable if and only if the impulse response is absolutely summable:

Σ_n=-∞^∞ |h[n]| < ∞

If the sum diverges to infinity, the discrete-time LTI system is unstable.

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4. Characteristics of Unstable LTI Systems

An unstable system is one where a finite input causes the system to "run away." Mathematically, this occurs when:

• The impulse response h(t) grows exponentially or remains constant (non-zero) as $t \to \infty$.
• The integral (or sum) of the absolute value of the impulse response is infinite.
• In physical terms, the system contains internal feedback that reinforces the signal indefinitely.

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5. Stability Analysis via Pole Criterion

In the frequency domain, the stability of a causal LTI system is determined by the location of the poles of its transfer function H(s) or H(z).

Continuous-Time Systems (S-Plane)

Pole Location (Real Part of s)	Stability Status
All poles in the Left Half Plane (Re{s} < 0)	Stable
Any pole in the Right Half Plane (Re{s} > 0)	Unstable
Simple poles on the Imaginary Axis (Re{s} = 0)	Marginally Stable

Discrete-Time Systems (Z-Plane)

• Stable: All poles lie inside the unit circle (|z| < 1).
• Unstable: Any pole lies outside the unit circle (|z| > 1).
• Marginally Stable: Simple poles lie on the unit circle (|z| = 1).

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6. Illustrative Examples

Example 1: Stable System

Consider a system with impulse response: h(t) = e−2t u(t)

Calculating the integral:

∫₀^∞ |e^−2t| dt = [−1/2 e^−2t]₀^∞ = 0 − (−1/2) = 1/2

Since 1/2 is finite, the system is Stable.

Example 2: Unstable System

Consider a system with impulse response: h(t) = e2t u(t)

Calculating the integral:

∫₀^∞ e^2t dt = [1/2 e^2t]₀^∞ = ∞

Because the integral diverges, the system is Unstable.

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7. Quick Summary Table

Feature	Stable System	Unstable System
BIBO Result	Output remains bounded	Output can grow to infinity
CT Condition	∫ |h(t)| dt < ∞	∫ |h(t)| dt = ∞
DT Condition	Σ |h[n]| < ∞	Σ |h[n]| = ∞
Pole Location (CT)	All poles in Left Half Plane	Any pole in Right Half Plane

Summary: For any LTI system, stability is an inherent property of the system itself, specifically its impulse response. It does not depend on the specific input signal used, provided that the input is bounded.

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