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Math concept
Z-Transform
Core equation
$$X(z) = \sum_{n} x[n]\,z^{-n}$$
The Z-transform converts a discrete-time sequence into a function of a complex variable $z$, playing the role for discrete systems that the Laplace transform plays for continuous ones. It enables analysis of linear time-invariant (LTI) systems through algebraic manipulation of transfer functions and characterization of stability via pole locations.
Definition and region of convergence
The bilateral Z-transform of a sequence $x[n]$ is:
\[X(z) = \sum_{n=-\infty}^{\infty} x[n]\,z^{-n}\]| where $z \in \mathbb{C}$. The region of convergence (ROC) is the annulus $r_1 < | z | < r_2$ where the sum converges absolutely. Causality and stability impose constraints: a causal system’s ROC is the exterior of a disk, and stability requires the ROC to include the unit circle $ | z | = 1$. |
Poles, zeros, and transfer functions
For an LTI system with impulse response $h[n]$, the transfer function $H(z) = \mathcal{Z}{h[n]}$ is a rational function of $z$. Poles determine stability: all poles inside the unit circle guarantee bounded-input bounded-output (BIBO) stability. The inverse Z-transform recovers $x[n]$ via partial fractions or contour integration:
\[x[n] = \frac{1}{2\pi i} \oint X(z)\,z^{n-1}\,dz\]Applications
The Z-transform is fundamental to digital filter design, difference equation analysis, and digital control systems. IIR and FIR filters are specified by their pole-zero patterns, and the discrete Fourier transform (DFT) evaluates $X(z)$ on the unit circle at $N$ equally spaced points.