Difference between revisions of "Z Transform table" - Rhea

Latest revision as of 07:55, 6 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\$ (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)\$

Back to Collective Table

@@ Line 1: / Line 1: @@
+[[Category:Formulas]]
+[[Category:z-transform]]
+[[Category:inverse z-transform]]
+[[Category:ECE301]]
+[[Category:ECE438]]
+[[Category:ECE]]
+<center><font size= 4>
+'''[[Collective_Table_of_Formulas|Collective Table of Formulas]]'''
+</font size>
+Table of (double-sided)  [[info_z-transform|Z Transform]] Pairs and Properties
+(Used in [[ECE301]], [[ECE438]], [[ECE538]])
+</center>
+----
 {|
 |-
-! style="background: none repeat scroll 0% 0% rgb(228, 188, 126); font-size: 110%;" colspan="2" | Z Transform Pairs and Properties
+! style="background: none repeat scroll 0% 0% rgb(238, 238, 238);" colspan="3" | (double-sided) [[info_z-transform|Z Transform]] and its Inverse
 |-
-! style="background: none repeat scroll 0% 0% rgb(238, 238, 238);" colspan="2" | Definition Z Transform and its Inverse
+| 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;" |  Single-side Z Transform
+| align="right" style="padding-right: 1em;" |  [[info_inverse_z-transform| Inverse Z Transform]]
-| <math>X(z)=\mathcal{L}(x[n])=\sum^{\infty}_{n=0}x[n]z^{-n}</math>
+| <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)]]
 |-
-| align="right" style="padding-right: 1em;" |  Double-side Z Transform
-| <math>X(z)=\mathcal{L}(x[n])=\sum_{n=-\infty}^{\infty}x[n]z^{-n}</math>
 |-
-| align="right" style="padding-right: 1em;" | Inverse Z Transform
-| <math>x[n]=\mathcal{L}^{-1}(X(z))=\frac{1}{2\pi j}\oint_{c}X(z)z^{n-1}dz</math>
 |}
 {|
 |-
-! style="background: none repeat scroll 0% 0% rgb(238, 238, 238);" colspan="4" | 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;" |
-| Signal <math> x[n] </math>
+| Signal
 |
-| Transform <math> X(z) </math>
+| Transform
 |
-| ROC
+| Region of convergence (ROC)
 |-
 | align="right" style="padding-right: 1em;" | Unit impulse signal
@@ Line 31: / Line 50: @@
 | <math> 1\ </math>
 |
-| <math> All\ z\ </math>
+| All complex <math>  z\ </math> including <math>\infty</math>
 |-
 | align="right" style="padding-right: 1em;" | Unit step signal
 | <math> u[n]\ </math>
 |
-| <math> \frac{1}{1-z^{-1}} </math>
+| <math> \frac{1}{1-z^{-1}} \ </math>
 |
-| <math> |z| > 1\ </math>
+| <math> |z| > 1\ </math> [[Compute_z-transform_u_n_ECE301S11|(computation)]]
 |-
 | align="right" style="padding-right: 1em;" |
@@ Line 52: / Line 71: @@
 | <math> z^{-m}\ </math>
 |
-| <math> All\ z, except\ 0\ (if\ m>0)\ or\\ \infty \ (if\ m<0)\ </math>
+| <math> All\ z,\ except\ </math>
+|-
+|-
+| align="right" style="padding-right: 1em;" |
+|
+|
+|
+|
+| <math>  0\ (if\ m>0)\ or\ </math>
+|-
+| align="right" style="padding-right: 1em;" |
+|
+|
+|
+|
+| <math>  \infty \ (if\ m<0)\ </math>
 |-
 | align="right" style="padding-right: 1em;" |
-| <math>te^{-at}u(t)\ </math>, where <math>a\in {\mathbb R}, a>0 </math>
+| <math> \alpha^{n}u[n]\ </math>
 |
-| <math>\left( \frac{1}{a+i2\pi f}\right)^2</math>
+| <math> \frac{1}{1-\alpha z^{-1}} </math>
+|
+| <math> |z| > | \alpha |\ </math>
+|-
+| align="right" style="padding-right: 1em;" |
+| <math> -\alpha^{n}u[-n-1]\ </math>
+|
+| <math> \frac{1}{1-\alpha z^{-1}} </math>
 |
+| <math> |z| < | \alpha |\ </math>
 |-
-| align="right" style="padding-right: 1em;" | CTFT of a cosine
+| align="right" style="padding-right: 1em;" |
-| <math>\cos(\omega_0 t) \ </math>
+| <math> n\alpha^{n}u[n]\ </math>
 |
-| <math> \frac{1}{2} \left[\delta (f - \frac{\omega_0}{2\pi}) + \delta (f + \frac{\omega_0}{2\pi})\right] \ </math>
+| <math> \frac{\alpha z^{-1}}{(1-\alpha z^{-1})^{2}} </math>
 |
+| <math> |z| > | \alpha |\ </math>
 |-
-| align="right" style="padding-right: 1em;" | CTFT of a sine
+| align="right" style="padding-right: 1em;" |
-| <math>sin(\omega_0 t)  \ </math>
+| <math> -n\alpha^{n}u[-n-1]\ </math>
 |
-| <math>\frac{1}{2i} \left[\delta (f - \frac{\omega_0}{2\pi}) - \delta (f + \frac{\omega_0}{2\pi})\right]</math>
+| <math> \frac{\alpha z^{-1}}{(1-\alpha z^{-1})^{2}} </math>
 |
+| <math> |z| < | \alpha |\ </math>
 |-
-| align="right" style="padding-right: 1em;" | CTFT of a rect
+| align="right" style="padding-right: 1em;" | Single-side cosine signal
-| <math>\left\{\begin{array}{ll}1, &  \text{ if }|t|<T,\\ 0, & \text{else.}\end{array} \right. \ </math>
+| <math> [\cos{\omega_{0}n}]u[n]\ </math>
 |
-| <math> \frac{\sin \left(2\pi Tf \right)}{\pi f}  \ </math>
+| <math> \frac{1-[\cos{\omega_{0}}]z^{-1}}{1-[2\cos{\omega_{0}}]z^{-1}+z^{-2}} </math>
 |
+| <math> |z| > 1\ </math>
 |-
-| align="right" style="padding-right: 1em;" | CTFT of a sinc
+| align="right" style="padding-right: 1em;" | Single-side sine signal
-| <math>\frac{2 \sin \left( W t  \right)}{\pi t }  \ </math>
+| <math> [\sin{\omega_{0}n}]u[n]\ </math>
 |
-| <math>\left\{\begin{array}{ll}1, &  \text{ if }|f| <\frac{W}{2\pi},\\ 0, & \text{else.}\end{array} \right.  \ </math>
+| <math> \frac{1-[\sin{\omega_{0}}]z^{-1}}{1-[2\cos{\omega_{0}}]z^{-1}+z^{-2}} </math>
 |
+| <math> |z| > 1\ </math>
 |-
-| align="right" style="padding-right: 1em;" | CTFT of a periodic function
+| align="right" style="padding-right: 1em;" |
-| <math>\sum^{\infty}_{k=-\infty} a_{k}e^{ikw_{0}t}</math>
+| <math> [r^{n}\cos{\omega_{0}n}]u[n]\ </math>
 |
-| <math>\sum^{\infty}_{k=-\infty}a_{k}\delta(f-\frac{kw_{0}}{2\pi}) \ </math>
+| <math> \frac{1-[r\cos{\omega_{0}}]z^{-1}}{1-[2r\cos{\omega_{0}}]z^{-1}+r^{2}z^{-2}} </math>
 |
+| <math> |z| > r\ </math>
 |-
-| align="right" style="padding-right: 1em;" | CTFT of an impulse train
+| align="right" style="padding-right: 1em;" |
-| <math>\sum^{\infty}_{n=-\infty} \delta(t-nT)  \ </math>
+| <math> [r^{n}\sin{\omega_{0}n}]u[n]\ </math>
 |
-| <math>\frac{1}{T}\sum^{\infty}_{k=-\infty}\delta(f-\frac{k}{T}) \ </math>
+| <math> \frac{1-[r\sin{\omega_{0}}]z^{-1}}{1-[2r\cos{\omega_{0}}]z^{-1}+r^{2}z^{-2}} </math>
 |
+| <math> |z| > r\ </math>
+|-
 |}
 {|
 |-
-! style="background: none repeat scroll 0% 0% rgb(238, 238, 238);" colspan="4" | CT Fourier 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.
 |-
 | align="right" style="padding-right: 1em;" |
-| <span class="texhtml">''x''(''t'')</span>
+| Signal
-| <math>\longrightarrow</math>
+|
-| <math> X(f) </math>
+| Z-Transform
+|
+| ROC
 |-
-| align="right" style="padding-right: 1em;" | multiplication property
+| align="right" style="padding-right: 1em;" | Linearity
-| <math>x(t)y(t) \ </math>
+| <math> ax_{1}[n]+bx_{2}[n]\ </math>
 |
-| <math> X(f)*Y(f) =\int_{-\infty}^{\infty} X(\theta)Y(f-\theta)d\theta</math>
+| <math> aX_{1}(z)+bX_{2}[z]\ </math>
+|
+| <math> At\ least\ the\ intersection\ of\ R_{1}\ and\ R_{2}\ </math>
 |-
-| align="right" style="padding-right: 1em;" | convolution property
+| align="right" style="padding-right: 1em;" | Time shifting
-| <math>x(t)*y(t) \!</math>
+| <math> x[n-n_{0}]\ </math>
 |
-| <math> X(f)Y(f) \!</math>
+| <math> z^{-n_{0}}X(z)\ </math>
+|
+| <math> R,\ except\ for\ the\ possible\ addition\ </math>
 |-
-| align="right" style="padding-right: 1em;" | time reversal
+| align="right" style="padding-right: 1em;" |
-| <math>\ x(-t) </math>
+|
+|
+|
+|
+| <math> or\ deletion\ of\ the\ origin\ </math>
+|-
+| align="right" style="padding-right: 1em;" | Scaling in the z-domain
+| <math> e^{j\omega_{0}n}x[n]\ </math>
 |
-| <math>\ X(-f)</math>
+| <math> X(e^{j\omega_{0}}z)\ </math>
+|
+| <math> R\ </math>
+|-
+| align="right" style="padding-right: 1em;" | Modulation [[Practice_prove_modulation_property_z_transform| (proof)]]
+| <math> a^{n}x[n]\ </math>
+|
+| <math> X(a^{-1}z)\ </math>
+|
+| <math> |a_0| R </math> (Scaled version of) <math> R\ </math>
+|-
+| align="right" style="padding-right: 1em;" |
+|
+|
+|
+|
+| <math> (i.e.,\ |a|R=\ the\ set\ of\ points\ {|a|z}\ for\ z\ in\ R)\ </math>
+|-
+| align="right" style="padding-right: 1em;" | Time reversal
+| <math> x[-n]\ </math>
+|
+| <math> X(z^{-1})\ </math>
+|
+| <math> R^{1/k}\ (i.e.,\ the\ set\ of\ points\ z^{1/k},\ </math>
+|-
+| align="right" style="padding-right: 1em;" |
+|
+|
+|
+|
+| <math> where\ z\ is\ in\ R)\ </math>
+|-
+| align="right" style="padding-right: 1em;" | Time expansion
+| <math> x^{(k)}=
+\begin{cases}
+x[r],  &n=rk \\
+, &n\neq rk
+\end{cases}  </math>
+|
+| <math> X(z^{k})\ </math>
+|
+| <math> R^{1/k}\ (i.e.,\ the\ set\ of\ points\ z^{1/k},\ </math>
+|-
+| align="right" style="padding-right: 1em;" |
+|
+|
+|
+|
+| <math> where\ z\ is\ in\ R)\ </math>
+|-
+| align="right" style="padding-right: 1em;" | Conjugation
+| <math> x^{*}[n]\ </math>
+|
+| <math> X^{*}(z^{*})\ </math>
+|
+| <math> R\ </math>
+|-
+| align="right" style="padding-right: 1em;" | Convolution
+| <math> x_{1}[n]*x_{2}[n]\ </math>
+|
+| <math> X_{1}(z)X_{2}(z)\ </math>
+|
+| <math> At\ least\ the\ intersection\ of\ R_{1}\ and\ R_{2}\ </math>
+|-
+| align="right" style="padding-right: 1em;" | First difference
+| <math> x[n]-x[n-1]\ </math>
+|
+| <math> (1-z^{-1})X(z)\ </math>
+|
+| <math> At\ least\ the\ intersection\ of\ R\ and\ |z|>0\  </math>
+|-
+| align="right" style="padding-right: 1em;" | Accumulation
+| <math> \sum_{k=-\infty}^{n}x[k]\ </math>
+|
+| <math> \frac{1}{(1-z^{-1})}X(z)\ </math>
+|
+| <math> At\ least\ the\ intersection\ of\ R\ and\ |z|>1\  </math>
+|-
+| align="right" style="padding-right: 1em;" | Differentiation in the z-domain
+| <math> nx[n]\ </math>
+|
+| <math> -z\frac{dX(z)}{dz}\ </math>
+|
+| <math> R\  </math>
+|-
 |}
 {|
 |-
-! style="background: none repeat scroll 0% 0% rgb(238, 238, 238);" colspan="2" | Other CT Fourier 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;" | Parseval's relation
+|-
-| <math>\int_{-\infty}^{\infty} |x(t)|^2 dt = \int_{-\infty}^{\infty} |X(f)|^2 df</math>
+| align="right" style="padding-right: 1em;" | Initial Value Theorem
+| <math> If\ x[n]=0\ for\ n<0,\ then\ x[0]=\lim_{z\rightarrow \infty}X(z)\ </math>
 |}
 ----
 [[Collective_Table_of_Formulas|Back to Collective Table]]
-[[Category:Formulas]]

Difference between revisions of "Z Transform table" - Rhea

Latest revision as of 07:55, 6 March 2015

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