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Question: are the pairs listed below for the single-side transform, or the double-side transform? It would be good to have a separate table for each. -pm
 
  
 
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Revision as of 06:24, 5 April 2012

This Collective table of formulas is proudly sponsored
by the Nice Guys of Eta Kappa Nu.

Visit us at the HKN Lounge in EE24 for hot coffee and fresh bagels only $1 each!

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Z Transform Pairs and Properties
Definition Z Transform and its Inverse
(Double-side) Z Transform $ X(z)=\mathcal{L}(x[n])=\sum_{n=-\infty}^{\infty}x[n]z^{-n} $ (info)
Inverse Z Transform $ x[n]=\mathcal{L}^{-1}(X(z))=\frac{1}{2\pi j}\oint_{c}X(z)z^{n-1}dz $
Z Transform Pairs
Signal $ x[n] $ Transform $ X(z) $ ROC
Unit impulse signal $ \delta[n]\ $ $ 1\ $ $ All\ z\ $
Unit step signal $ u[n]\ $ $ \frac{1}{1-z^{-1}} $ $ |z| > 1\ $
$ -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\ $
Z Transform Properties
Signal $ x[n] $ Transform $ X(f) $ ROC
$ x[n]\ $ $ X(z)\ $ $ R\ $
$ x_{1}[n]\ $ $ X_{1}(z)\ $ $ R_{1}\ $
$ x_{2}[n]\ $ $ X_{2}(z)\ $ $ R_{2}\ $
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\ $
$ z_{0}^{n}x[n]\ $ $ X(\frac{z}{z_{0}})\ $ $ z_{0}R\ $
$ a^{n}x[n]\ $ $ X(a^{-1}z)\ $ $ 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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Correspondence Chess Grandmaster and Purdue Alumni

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