zplanes

Zplanes refers to the complex planes used in the analysis of discrete-time systems through the Z-transform. In this context, a discrete-time signal x[n] is represented by a transfer function H(z) in the complex variable z, with H(z) often written as a ratio of polynomials in z or z^{-1}. The Z-transform X(z) = sum over n of x[n] z^{-n} maps time-domain sequences to points in the complex plane, and the region of convergence indicates where this series converges.

A common visualization is the pole-zero plot in the z-plane. Zeros are the roots of the numerator and poles are the roots of the denominator of the transfer function. The locations of zeros and poles on the complex plane determine the filter’s characteristics, including its frequency response and stability. Z-plane plots typically show the unit circle (|z| = 1), since evaluating H(z) on the unit circle (z = e^{jω}) yields the discrete-time frequency response.

Stability and causality are closely tied to the z-plane. For a causal, time-invariant discrete-time system, stability (BIBO stability) requires that all poles lie inside the unit circle (|z_p| < 1). Poles on or outside the unit circle indicate marginal or unstable behavior. The frequency response is obtained by sampling H(z) on the unit circle, linking the z-plane geometry to how the system reacts to different frequencies.

Relationships to other planes include the s-plane used for continuous-time systems. Transforms such as the bilinear transform map the s-plane to the z-plane, enabling design and analysis in one domain and implementation in another. Zplanes are widely used in digital filter design, distinguishing between FIR (typically zeros only) and IIR (poles and zeros) structures. Software tools often provide zplane plots to visualize pole-zero configurations quickly.