Definition
A pole–zero plot is a graphical representation used in signal processing, control theory, and related fields to illustrate the locations of the poles and zeros of a system’s transfer function in the complex plane.
Overview
The transfer function of a linear time-invariant (LTI) system, expressed as a ratio of two polynomials in the complex frequency variable $s$ (continuous-time) or $z$ (discrete-time), can be factored into its constituent zeros (roots of the numerator) and poles (roots of the denominator). Plotting these points on the complex plane provides insight into the system’s frequency response, stability, and temporal behavior. In continuous‑time analysis, the horizontal axis represents the real part of $s$ (damping), while the vertical axis represents the imaginary part (oscillation frequency). In discrete‑time analysis, the unit circle in the $z$-plane delineates the boundary between stable and unstable regions.
Etymology/Origin
The term combines “pole” and “zero,” mathematical concepts originating from complex analysis. The practice of visualizing these points on a complex plane emerged in the mid‑20th century alongside the development of classical control theory and the Z‑transform for digital signal processing. Early textbooks on analog filter design (e.g., Bode, 1945) and later digital filter literature (e.g., Oppenheim & Schafer, 1975) popularized the pole–zero diagram as a design and analysis tool.
Characteristics
| Aspect | Description |
|---|---|
| Axes | Real and imaginary components of the complex frequency variable ($s$ for continuous‑time, $z$ for discrete‑time). |
| Symbols | Zeros are typically marked with “o” (circle) and poles with “×” (cross). |
| Stability Criteria | • Continuous‑time: All poles must lie in the left half‑plane (Re$s$ < 0). • Discrete‑time: All poles must lie inside the unit circle ( |
| Frequency Response Insight | Points near the imaginary axis (continuous) or the unit circle (discrete) strongly affect magnitude and phase at corresponding frequencies. |
| Design Utility | Allows engineers to shape filter characteristics (e.g., low‑pass, band‑pass) by placing poles and zeros at desired locations. |
| Transformations | Mapping between $s$- and $z$-domains (e.g., bilinear transform) can be visualized by tracking how poles and zeros move between the two planes. |
| Limitations | The plot conveys only linear, time‑invariant behavior; nonlinear or time‑varying systems require different analysis tools. |
Related Topics
- Transfer function
- Bode plot
- Nyquist plot
- Z‑transform and Laplace transform
- Stability analysis (Routh–Hurwitz criterion, Jury test)
- Digital filter design (FIR, IIR)
- Control system design (root locus)
Note: Information presented reflects the consensus of standard engineering literature up to the knowledge cutoff date.