Poles, Region of Convergence, and Stability in Discrete-Time LTI Systems

April 10, 2026 · Luciano Muratore

Having examined the asymptotic behavior of LSI systems in the time domain, we now transition to the Z-domain to analyze system stability and frequency response through the lens of transfer functions and their singularities.

1. Transfer Function in the Z-Domain

A discrete-time Linear Time-Invariant (LTI) system is fully characterized by its transfer function, $H(z)$, which is the Z-transform of its impulse response $h[n]$:

$H(z) = \sum_{n=-\infty}^{\infty} h[n] z^{-n}$

This representation is a complex power series that converges only within a specific region of the z-plane known as the Region of Convergence (ROC): $ROC = \{z \in \mathbb{C} \mid \sum_{n=-\infty}^{\infty} h[n] z^{-n} \text{ converges}\}$

2. Poles as Singularities

Poles are the values of $z$ where the transfer function $H(z)$ becomes unbounded. If the transfer function is expressed as a ratio of polynomials, $H(z) = \frac{B(z)}{A(z)}$, the poles are the roots of the denominator $A(z) = 0$.

Key Properties of Poles:

• They represent the system’s intrinsic modes or exponential components.
• The ROC is always an open set that excludes all poles.
• Poles effectively define the boundaries of the ROC.

3. Frequency Response and the Unit Circle

The frequency response of a system, $H(e^{j\omega})$, is obtained by evaluating the transfer function $H(z)$ specifically on the unit circle ($|z| = 1$): $H(e^{j\omega}) = H(z) \big|_{z=e^{j\omega}}$

Existence Condition:

The frequency response exists if and only if the unit circle is contained within the ROC. If the unit circle is outside the ROC:

• The series $H(e^{j\omega})$ diverges.
• The system cannot react predictably to sinusoidal inputs.

4. Stability Criterion

For a causal discrete-time LTI system, stability is directly linked to the position of the ROC relative to the unit circle.

BIBO Stability Condition:

A system is Bounded-Input Bounded-Output (BIBO) stable if and only if the unit circle lies inside the ROC.

• Stable System: If $|z|=1$ is in the ROC, the impulse response $h[n]$ decays over time.
• Unstable System: If $|z|=1$ is not in the ROC, the impulse response grows without bound.

Consequences of Instability:

When the unit circle is outside the ROC, the system is unstable, leading to:

• Impulse and bounded-input responses that “blow up”.
• Exponentially growing outputs even for bounded inputs.
• The inability to define a meaningful frequency response.

5. Summary Table