Collective Table of Formulas

Table of (double-sided) Z Transform Pairs and Properties

(Used in ECE301, ECE438, ECE538)



(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)\ $

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