Difference between revisions of "Z Transform table" - Rhea

Revision as of 17:39, 3 March 2015

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\$
	$-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\$
	$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)\$

Go to Relevant Course Page: ECE 438

Go to Relevant Course Page: ECE 538

Back to Collective Table

@@ Line 10: / Line 10: @@
 </font size>
-Table of (double-sided) Z Transform Pairs and Properties
+Table of (double-sided)  [[info_z-transform|Z Transform]] Pairs and Properties
 click [[Collective_Table_of_Formulas|here]] for [[Collective_Table_of_Formulas|more formulas]]
@@ Line 21: / Line 21: @@
 {|
 |-
-! style="background: none repeat scroll 0% 0% rgb(238, 238, 238);" colspan="3" | (double-sided) Z Transform and its Inverse
+! style="background: none repeat scroll 0% 0% rgb(238, 238, 238);" colspan="3" | (double-sided) [[info_z-transform|Z Transform]] and its Inverse
 |-
-| align="right" style="padding-right: 1em;" |  (Double-side) Z Transform
+| align="right" style="padding-right: 1em;" |   [[info_z-transform|(Double-side)  Z Transform]]
 | <math>X(z)=\mathcal{Z}(x[n])=\sum_{n=-\infty}^{\infty}x[n]z^{-n}</math>
 |[[info_z-transform|(info)]]
 |-
-| align="right" style="padding-right: 1em;" | Inverse Z Transform
+| align="right" style="padding-right: 1em;" |  [[info_inverse_z-transform| Inverse Z Transform]]
 | <math>x[n]=\mathcal{Z}^{-1}(X(z))=\frac{1}{2\pi j}\oint_{c}X(z)z^{n-1}dz</math>
 | [[info_inverse_z-transform|(info)]]
-|-}
+|-
+|}
 {|
 |-
-! style="background: none repeat scroll 0% 0% rgb(238, 238, 238);" colspan="6" | Z Transform Pairs
+! style="background: none repeat scroll 0% 0% rgb(238, 238, 238);" colspan="6" | (double-sided)  [[info_z-transform|Z Transform]] Pairs
 |-
 | align="right" style="padding-right: 1em;" |
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 {|
 |-
-! style="background: none repeat scroll 0% 0% rgb(238, 238, 238);" colspan="6" | Z Transform Properties
+! style="background: none repeat scroll 0% 0% rgb(238, 238, 238);" colspan="6" | (double-sided)  [[info_z-transform|Z Transform]] Properties
 |-
 ! colspan="6" | Below <math>x[n]</math>, <math>x_1[n]</math> and <math>x_2[n]</math> are DT signals with z-transforms <math>X(z)</math>, <math>X_1(Z)</math>, <math>X_2(z)</math>, and region of convergence (ROC) <math>R</math>, <math>R_1</math>, <math>R_2</math>, respectively.
@@ Line 274: / Line 276: @@
 {|
 |-
-! style="background: none repeat scroll 0% 0% rgb(238, 238, 238);" colspan="2" | Other Z Transform Properties
+! style="background: none repeat scroll 0% 0% rgb(238, 238, 238);" colspan="2" | Other  [[info_z-transform|Z Transform]] Properties
 |-
 | align="right" style="padding-right: 1em;" | Initial Value Theorem

Difference between revisions of "Z Transform table" - Rhea

Revision as of 17:39, 3 March 2015

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