Z Transform
Table of Contents
Given a time domain discrete signal, i.e., a sequence, it can be represented in the complex frequency domain by Z transform. It is essentially an extension of discrete time Fourier transform (DTFT).
Definition
Bilateral transform
\begin{align*} X(z) = \sum_{n=-\infty}^{\infty} x(n) z^{-n} \end{align*}Unilateral transform
\begin{align*} X(z) = \sum_{n=0}^{\infty} x(n) z^{-n} \end{align*}Inverse transform
\begin{align*} x(n) &= \frac{1}{2\pi j}\oint_C X(z)z^{n-1} dz \\ &= \sum_{k} \text{Res} \left[ X(z)z^{n-1}, z_k \right]. \end{align*}- Region of convergence (ROC): Considering the fact that \(X(z)\) is an infinite summation, the convergence must be ensured. To this end, ROC is defined, which is comprised of the candidate values of \(z\) keeping the convergence. \(z\) is termed complex frequency.
Properties
Denote the ROC as \(\alpha < |z| < \beta\), \(0 \le\alpha < \beta\).
Common to both bilateral and unilateral transforms
\begin{align*} a^n x(n), \quad a \neq 0. &\leftrightarrow X \left( \frac{z}{a} \right), \quad \alpha |a| < |z| < \beta |a|. \\ x_1(n) * x_2(n) &\leftrightarrow X_1(z) X_2(z), \quad \max(\alpha_1, \alpha_2) < |z| < \min(\beta_1, \beta_2). \\ n^mx(n), \quad m > 0. &\leftrightarrow \left(-z\frac{d}{dz}\right)^m X(z), \quad \alpha < |z| < \beta.\\ \frac{x(n)}{n+m}, \quad n+m > 0. &\leftrightarrow z^m \int_z^{\infty}X(\eta)\eta^{-(m+1)}d\eta, \quad \alpha < |z| < \beta. \\ x^*(n) &\leftrightarrow X^*(z^*) \\ x_1(n) * x_2(n) &\leftrightarrow X_1(z) X_2(z) \\ x(-n) &\leftrightarrow X\left(\frac{1}{z}\right), \quad \frac{1}{\beta} < |z| < \frac{1}{\alpha}.\\ \sum_{i=-\infty}^nx(i) &\leftrightarrow \frac{z}{z-1}X(z), \quad \max(\alpha, 1) < |z| < \beta. \end{align*}Dedicated to bilateral transform
\begin{align*} x(n + m) \leftrightarrow z^m X(z), \quad \alpha < |z| < \beta \\ \end{align*}Dedicated to unilateral transform
\begin{align*} x(n-m), \quad m > 0. &\leftrightarrow z^{-m}X(z) + \sum_{n=0}^{m-1}x(n-m)z^{-n}, \quad |z| > \alpha. \\ x(n+m), \quad m > 0. &\leftrightarrow z^mX(z) - \sum_{n=0}^{m-1}x(n)z^{m-n}, \quad |z| > \alpha. \end{align*}
Initial Value Theorem
\begin{align*}
x(0) = \lim_{z\to\infty}X(z)
\end{align*}
Final Value Theorem
\begin{align*}
\lim_{n\to\infty}x(n) = \lim_{z\to 1}(z-1)X(z)
\end{align*}
Transform Pairs
\begin{align*}
\delta(n) &\leftrightarrow 1 \\
\varepsilon(n) &\leftrightarrow \frac{z}{z-1}, \quad |z| > 1. \\
a^n\varepsilon(n) &\leftrightarrow \frac{z}{z-a}, \quad |z| > 1. \\
n\varepsilon(n) &\leftrightarrow \frac{z}{(z-1)^2}, \quad |z| > 1. \\
\delta(n-m), \quad m > 0. &\leftrightarrow z^{-m}, \quad |z| > 0. \\
-\varepsilon(-n-1) &\leftrightarrow \frac{z}{z-1}, \quad |z| < 1. \\
-a^n\varepsilon(-n-1) &\leftrightarrow \frac{z}{z-a}, \quad |z| < a. \\
-n\varepsilon(-n-1) &\leftrightarrow \frac{z}{(z-1)^2}, \quad |z| < 1. \\
\delta(n+m), \quad m > 0. &\leftrightarrow z^m
\end{align*}