How does the unit circle relate to discrete stability?

A discrete system is stable if all its poles lie strictly inside the unit circle (|z| < 1) in the complex z-plane.

Z-Transform Discrete Simulator - AI Math

System Analysis

Adjust discrete system poles to observe amplitude, frequency, and filter stability bounds.

Pole (Active): Determines damping & resonance frequency.

○

Zero (Inactive): Places notches to block specific frequencies (not present in this model).

Real Part (Re)0.60

Imag Part (Im)0.60j

Filter Stability Status

🟢 Stable (Impulse Decays)

Variable Adjuster

Magnitude (r)0.85

01.3

Angle (θ)45°

0180

Discrete Samples (N)16

1040

Auto-Sweep Engine

Target Param:

Sweep Speed: 2x

Z-Transform Explorer

ZTRAN

The Z-Transform maps discrete-time sequences to the z-domain. By graphing system poles relative to the Unit Circle (|z| = 1) in the complex z-plane, stability is evaluated: poles inside the circle (|z| < 1) yield stable geometrically decaying impulse responses, while poles outside cause unstable exponential expansion.

X(z) = \sum_{n=-\infty}^{\infty} x[n] z^{-n}

Whiteboard Solver Steps

Step 1

z-Domain System Transfer Function

Pole Locations: - Complex conjugate pole pair at $z = r e^{\pm j\theta} = 0.85 e^{\pm 45.0^\circ j}$ . - In rectangular coordinates: $z = 0.60 \pm 0.60j$ .

H(z) = \frac{z^2}{(z - re^{j\theta})(z - re^{-j\theta})} = \frac{z^2}{z^2 - 1.20z + 0.72}

Step 2

Discrete-Time Impulse Response Derivation

Discrete Stability: - **Magnitude ( $r = 0.85$ )**: Since $r < 1$ , the system impulse response sequence is stable (decaying). - Stems inside the unit circle ( $r < 1$ ) decay geometrically. Stems crossing outside ( $r > 1$ ) experience exponential runaway.

h[n] = \mathcal{Z}^{-1}\{ H(z) \} = r^n \cos(n \theta) = (0.85)^n \cos(n \cdot 45.0^\circ)

Step 3

Z-Transform Utility in Digital Filters & DSP

Real-World Utility: - Digital Filters (IIR): Audio equalizers, echo effects, and noise-canceling headphones model signal pathways using difference equations in the z-domain. - Stability Guarantees: A filter is only safe to deploy if all its feedforward/feedback poles lie strictly inside the z-plane unit circle ( $|z| < 1$ ) to prevent clipping/speaker damage.

y[n] = x[n] + b_1 y[n-1] + b_2 y[n-2] \implies \text{Poles Inside Unit Circle } (|z| < 1) \text{ is Stable}

RV Anveshana