Z-Domain Stability Criteria

In the world of digital signal processing and control systems, every filter, controller, or predictive algorithm is built upon a discrete-time system. The fundamental question is whether that system will behave predictably or spiral into uncontrollable oscillation. Stability analysis in the z-domain provides the definitive answer, transforming a complex dynamical question into a straightforward geometric and algebraic inspection of a system's poles. Mastering these criteria allows you to design robust digital systems that perform correctly under real-world conditions.

Defining BIBO Stability for Discrete-Time LTI Systems

A discrete-time Linear Time-Invariant (LTI) system is defined as BIBO (Bounded-Input, Bounded-Output) stable if every bounded input sequence produces a bounded output sequence. This is the practical definition of stability: if you feed the system a signal that doesn't blow up, its response won't blow up either. For an LTI system characterized by its transfer function $H(z)$, this property has a powerful and elegant test.

The system's transfer function, $H(z)$, is a rational function of the complex variable $z$, typically expressed as the ratio of the $z$-transforms of the output and input. The poles of $H(z)$ are the values of $z$ that make the denominator polynomial equal to zero. The core stability theorem states: *A causal discrete-time LTI system is BIBO stable if and only if all poles of its transfer function $H(z)$ lie strictly inside the unit circle in the complex z-plane.* The unit circle is defined by $|z|=1$. A pole on the unit circle leads to a marginally stable system (sustained oscillation), while any pole outside the unit circle results in an unstable, growing response. This geometric rule is the cornerstone of z-domain analysis.

The Jury Stability Test: An Algebraic Criterion

For low-order systems, finding the poles directly is straightforward. However, for higher-order systems, factoring the denominator polynomial can be computationally difficult. The Jury stability test provides an algebraic procedure to determine if all roots of a polynomial lie inside the unit circle, without explicitly computing them. It is the discrete-time analogue of the continuous-time Routh-Hurwitz criterion.

The test is applied to the denominator polynomial of $H(z)$, which we can denote as $D(z)=a_n{z}^n+a_{n-1}{z}^{n-1}+...+a_{1}z+a_0$, with $a_n>0$. You construct a "Jury table" or "Jury array" from the coefficients. The first row contains the coefficients in order from $a_0$ to $a_n$. The second row is the reverse of the first. From these, you compute new rows using determinant-like calculations. The stability conditions are twofold: (1) $D(1)>0$ and $(-1{)}^{n}D(-1)>0$, and (2) specific constraints derived from the elements of the Jury table must be satisfied (e.g., $|a_0|<a_n$, and other inner determinants are positive).

Consider a simple example: $D(z)=z^2-1.2z+0.35$. The direct pole calculation gives $z=0.6\pm j0.1$, both with magnitude $\sqrt{0.36+0.01}\approx 0.608$, inside the unit circle. The Jury test would confirm this: $D(1)=0.15>0$, $(-1{)}^{2}D(-1)=2.55>0$, and $|a_0|=0.35<a_2=1$. Both methods confirm stability.

Interpreting Pole Radius: The Decay Rate

The location of a pole within the unit circle is not just a binary indicator of stability; it precisely determines the character of the system's transient response. A pole at location $z=p$ in the complex plane contributes a time-domain mode of the form $p^k$ (for a real pole) or $r^k\cos (\omega k+\varphi )$ (for a complex conjugate pair with magnitude $r$ and angle $\omega$).

The pole radius, which is its magnitude $|p|=r$, directly dictates the decay rate of its corresponding mode. A pole at the origin ($r=0$) corresponds to a finite-duration impulse response. A pole with $0<r<1$ contributes a term that decays exponentially as $k$ increases; the closer $r$ is to 0, the faster the decay. For example, a pole at $z=0.5$ decays as $(0.5{)}^k$, while a pole at $z=0.9$ decays as $(0.9{)}^k$, a much slower decay that results in a longer "ringing" transient. Understanding this allows you to design systems with desired settling times by strategically placing poles at appropriate radii inside the unit circle.

Interpreting Pole Angle: The Oscillation Frequency

While the radius determines decay, the pole angle, $\angle p=\omega$, determines the oscillation frequency of the response. For a complex conjugate pole pair $p=r{e}^{\pm j\omega }$, the discrete-time oscillation frequency in radians/sample is exactly $\omega$. The relationship to a continuous-time frequency $f$ (in Hz) depends on the sampling period $T_s$: $f=\frac{\omega }{2\pi T_s}$.

A pole on the positive real axis ($\omega =0$) contributes a purely decaying (or growing) exponential with no oscillation. As the angle increases from 0 to $\pi$, the oscillation frequency increases. A pole at $z=-r$ ($\omega =\pi$) corresponds to the highest possible discrete-time frequency, an oscillation that alternates sign every sample period (the digital equivalent of the Nyquist frequency). Therefore, by examining the angles of complex poles, you can immediately diagnose the oscillatory components in a system's step or impulse response.

Common Pitfalls

Misinterpreting the Unit Circle Boundary: A common error is treating a pole on the unit circle as stable. It is not BIBO stable; it is marginally stable. For example, a pole at $z=1$ (integrator) or $z=-1$ will produce an output that does not decay, and for some bounded inputs (like a step for $z=1$), the output will be unbounded. True BIBO stability requires poles to be strictly inside ($|z|<1$).

Incorrectly Applying the Jury Test Conditions: When constructing the Jury table, it's easy to mishandle the recursive calculations or misapply the necessary and sufficient conditions. Remember that the primary conditions ($D(1)>0$, $(-1{)}^{n}D(-1)>0$) are necessary but not sufficient on their own. You must also check all the constraints generated from the table entries (e.g., $|b_0|>|b_{n-1}|$, $|c_0|>|c_{n-2}|$, etc., depending on your table notation). Omitting these leads to false stability conclusions.

Confusing Continuous and Discrete Stability Regions: Engineers familiar with the s-plane (where stability is a left-half-plane condition) sometimes mistakenly apply that geometry to the z-plane. Remember, the left-half s-plane maps to the interior of the unit circle in the z-plane via the mapping $z=e^{sT}$. A stable continuous-time pole at $s=-a$ maps to a z-plane pole at $z=e^{-aT}$, which indeed has magnitude less than 1.

Summary

• A causal discrete-time LTI system is BIBO stable if and only if all poles of its transfer function $H(z)$ lie strictly inside the unit circle ($|z|<1$) in the complex z-plane.
• The Jury stability test is an algebraic procedure, analogous to Routh-Hurwitz, used to determine if all polynomial roots are inside the unit circle without explicit factorization.
• The pole radius (magnitude) determines the exponential decay rate of the corresponding time-domain mode; poles closer to the origin decay faster.
• The pole angle determines the discrete-time oscillation frequency (in radians/sample) of the system's response.
• Always distinguish between true BIBO stability (poles inside the circle) and marginal stability (poles on the circle), as the latter can produce unbounded outputs for bounded inputs.