Table of (double-sided) Z Transform Pairs and Properties
| (double-sided) Z Transform and its Inverse | ||
|---|---|---|
| (Double-side) Z Transform | $ X(z)=\mathcal{Z}(x[n])=\sum_{n=-\infty}^{\infty}x[n]z^{-n} \ $ | (info) |
| Inverse Z Transform | $ x[n]=\mathcal{Z}^{-1}(X(z))=\frac{1}{2\pi j}\oint_{c}X(z)z^{n-1}dz \ $ | (info) |
| (double-sided) Z Transform Pairs | |||||
|---|---|---|---|---|---|
| Signal | Transform | Region of convergence (ROC) | |||
| Unit impulse signal | $ \delta[n]\ $ | $ 1\ $ | All complex $ z\ $ including $ \infty $ | ||
| Unit step signal | $ u[n]\ $ | $ \frac{1}{1-z^{-1}} \ $ | $ |z| > 1\ $ (computation) | ||
| $ -u[-n-1]\ $ | $ \frac{1}{1-z^{-1}} $ | $ |z| < 1\ $ | |||
| Shifted unit impulse signal | $ \delta[n-m]\ $ | $ z^{-m}\ $ | $ All\ z,\ except\ $ | ||
| $ 0\ (if\ m>0)\ or\ $ | |||||
| $ \infty \ (if\ m<0)\ $ | |||||
| $ \alpha^{n}u[n]\ $ | $ \frac{1}{1-\alpha z^{-1}} $ | $ |z| > | \alpha |\ $ | |||
| $ -\alpha^{n}u[-n-1]\ $ | $ \frac{1}{1-\alpha z^{-1}} $ | $ |z| < | \alpha |\ $ | |||
| $ n\alpha^{n}u[n]\ $ | $ \frac{\alpha z^{-1}}{(1-\alpha z^{-1})^{2}} $ | $ |z| > | \alpha |\ $ | |||
| $ -n\alpha^{n}u[-n-1]\ $ | $ \frac{\alpha z^{-1}}{(1-\alpha z^{-1})^{2}} $ | $ |z| < | \alpha |\ $ | |||
| Single-side cosine signal | $ [\cos{\omega_{0}n}]u[n]\ $ | $ \frac{1-[\cos{\omega_{0}}]z^{-1}}{1-[2\cos{\omega_{0}}]z^{-1}+z^{-2}} $ | $ |z| > 1\ $ | ||
| Single-side sine signal | $ [\sin{\omega_{0}n}]u[n]\ $ | $ \frac{1-[\sin{\omega_{0}}]z^{-1}}{1-[2\cos{\omega_{0}}]z^{-1}+z^{-2}} $ | $ |z| > 1\ $ | ||
| $ [r^{n}\cos{\omega_{0}n}]u[n]\ $ | $ \frac{1-[r\cos{\omega_{0}}]z^{-1}}{1-[2r\cos{\omega_{0}}]z^{-1}+r^{2}z^{-2}} $ | $ |z| > r\ $ | |||
| $ [r^{n}\sin{\omega_{0}n}]u[n]\ $ | $ \frac{1-[r\sin{\omega_{0}}]z^{-1}}{1-[2r\cos{\omega_{0}}]z^{-1}+r^{2}z^{-2}} $ | $ |z| > r\ $ | |||
| (double-sided) Z Transform Properties | |||||
|---|---|---|---|---|---|
| Below $ x[n] $, $ x_1[n] $ and $ x_2[n] $ are DT signals with z-transforms $ X(z) $, $ X_1(Z) $, $ X_2(z) $, and region of convergence (ROC) $ R $, $ R_1 $, $ R_2 $, respectively. | |||||
| Signal | Z-Transform | ROC | |||
| Linearity | $ ax_{1}[n]+bx_{2}[n]\ $ | $ aX_{1}(z)+bX_{2}[z]\ $ | $ At\ least\ the\ intersection\ of\ R_{1}\ and\ R_{2}\ $ | ||
| Time shifting | $ x[n-n_{0}]\ $ | $ z^{-n_{0}}X(z)\ $ | $ R,\ except\ for\ the\ possible\ addition\ $ | ||
| $ or\ deletion\ of\ the\ origin\ $ | |||||
| Scaling in the z-domain | $ e^{j\omega_{0}n}x[n]\ $ | $ X(e^{j\omega_{0}}z)\ $ | $ R\ $ | ||
| Modulation (proof) | $ a^{n}x[n]\ $ | $ X(a^{-1}z)\ $ | $ |a_0| R $ (Scaled version of) $ R\ $ | ||
| $ (i.e.,\ |a|R=\ the\ set\ of\ points\ {|a|z}\ for\ z\ in\ R)\ $ | |||||
| Time reversal | $ x[-n]\ $ | $ X(z^{-1})\ $ | $ R^{1/k}\ (i.e.,\ the\ set\ of\ points\ z^{1/k},\ $ | ||
| $ where\ z\ is\ in\ R)\ $ | |||||
| Time expansion | $ x^{(k)}= \begin{cases} x[r], &n=rk \\ 0, &n\neq rk \end{cases} $ | $ X(z^{k})\ $ | $ R^{1/k}\ (i.e.,\ the\ set\ of\ points\ z^{1/k},\ $ | ||
| $ where\ z\ is\ in\ R)\ $ | |||||
| Conjugation | $ x^{*}[n]\ $ | $ X^{*}(z^{*})\ $ | $ R\ $ | ||
| Convolution | $ x_{1}[n]*x_{2}[n]\ $ | $ X_{1}(z)X_{2}(z)\ $ | $ At\ least\ the\ intersection\ of\ R_{1}\ and\ R_{2}\ $ | ||
| First difference | $ x[n]-x[n-1]\ $ | $ (1-z^{-1})X(z)\ $ | $ At\ least\ the\ intersection\ of\ R\ and\ |z|>0\ $ | ||
| Accumulation | $ \sum_{k=-\infty}^{n}x[k]\ $ | $ \frac{1}{(1-z^{-1})}X(z)\ $ | $ At\ least\ the\ intersection\ of\ R\ and\ |z|>1\ $ | ||
| Differentiation in the z-domain | $ nx[n]\ $ | $ -z\frac{dX(z)}{dz}\ $ | $ R\ $ | ||
| Other Z Transform Properties | |
|---|---|
| Initial Value Theorem | $ If\ x[n]=0\ for\ n<0,\ then\ x[0]=\lim_{z\rightarrow \infty}X(z)\ $ |
