Difference between revisions of "Z Transform table" - Rhea

Revision as of 12:13, 27 November 2010

Z Transform Pairs and Properties
Definition Z Transform and its Inverse
Single-side Z Transform	$X(z)=\mathcal{L}(x[n])=\sum^{\infty}_{n=0}x[n]z^{-n}$
Double-side Z Transform	$X(z)=\mathcal{L}(x[n])=\sum_{n=-\infty}^{\infty}x[n]z^{-n}$
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\$
Single-side cosine signal	$[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\$

CT Fourier Transform Properties
	$x (t)$	$\longrightarrow$	$$ X(f) $$
multiplication property	$x(t)y(t) \$		$X(f)*Y(f) =\int_{-\infty}^{\infty} X(\theta)Y(f-\theta)d\theta$
convolution property	$x(t)*y(t) \!$		$X(f)Y(f) \!$
time reversal	$\ x(-t)$		$\ X(-f)$

Other CT Fourier Transform Properties
Parseval's relation	$\int_{-\infty}^{\infty} \|x(t)\|^2 dt = \int_{-\infty}^{\infty} \|X(f)\|^2 df$

@@ Line 83: / Line 83: @@
 |-
 | align="right" style="padding-right: 1em;" | Single-side cosine signal
-| <math> [\cos{\omega_{0}n}]u[n] </math>
+| <math> [\cos{\omega_{0}n}]u[n]\ </math>
 |
 | <math> \frac{1-[\cos{\omega_{0}}]z^{-1}}{1-[2\cos{\omega_{0}}]z^{-1}+z^{-2}} </math>
@@ Line 89: / Line 89: @@
 | <math> |z| > 1\ </math>
 |-
-| align="right" style="padding-right: 1em;" | CTFT of a periodic function
+| align="right" style="padding-right: 1em;" | Single-side sine signal
-| <math>\sum^{\infty}_{k=-\infty} a_{k}e^{ikw_{0}t}</math>
+| <math> [\sin{\omega_{0}n}]u[n]\ </math>
 |
-| <math>\sum^{\infty}_{k=-\infty}a_{k}\delta(f-\frac{kw_{0}}{2\pi}) \ </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;" |
+| <math> [r^{n}\cos{\omega_{0}n}]u[n]\ </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;" | Single-side cosine signal
-| <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>
 |}