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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)\$

## Alumni Liaison

Ph.D. on Applied Mathematics in Aug 2007. Involved on applications of image super-resolution to electron microscopy

Francisco Blanco-Silva