Engineering Handbook/Mathematics/Z Transformation

Z Transformation Properties

	Time domain	Z-domain	ROC
Notation	$x[n]={𝒵}^{-1}\{X(z)\}$	$X(z)=𝒵\{x[n]\}$	ROC: $r_2<|z|<r_1$
Linearity	$a_1{x}_1[n]+a_2{x}_2[n]$	$a_1{X}_1(z)+a_2{X}_2(z)$	At least the intersection of ROC1 and ROC2
Time shifting	$x[n-k]$	$z^{-k}X(z)$	ROC, except $z=0$ if $k>0$ and $z=\mathrm{\infty }$ if $k<0$
Scaling in the z-domain	$a^{n}x[n]$	$X(a^{-1}z)$	$|a|r_2<|z|<|a|r_1$
Time reversal	$x[-n]$	$X(z^{-1})$	$\frac{1}{r_2}<|z|<\frac{1}{r_1}$
Conjugation	$x^{*}[n]$	$X^{*}(z^{*})$	ROC
Real part	$\mathrm{Re}\{x[n]\}$	$\frac{1}{2}[X(z)+X^{*}(z^{*})]$	ROC
Imaginary part	$\mathrm{Im}\{x[n]\}$	$\frac{1}{2j}[X(z)-X^{*}(z^{*})]$	ROC
Differentiation	$nx[n]$	$-z\frac{\mathrm{d}X(z)}{\mathrm{d}z}$	ROC
Convolution	$x_1[n]*x_2[n]$	$X_1(z)X_2(z)$	At least the intersection of ROC1 and ROC2
Correlation	$r_{x_1,x_2}(l)=x_1[l]*x_2[-l]$	$R_{x_1,x_2}(z)=X_1(z)X_2(z^{-1})$	At least the intersection of ROC of X1(z) and X2($z^{-1}$)
Multiplication	$x_1[n]x_2[n]$	$\frac{1}{j2\pi }{\oint }_C{X}_1(v)X_2(\frac{z}{v})v^{-1}\mathrm{d}v$	At least $r_{1l}{r}_{2l}<|z|<r_{1u}{r}_{2u}$
Parseval's relation	${\sum }^{\mathrm{\infty }}{x}_1[n]x_2^{*}[n]$	$\frac{1}{j2\pi }{\oint }_C{X}_1(v)X_2^{*}(\frac{1}{v^{*}})v^{-1}\mathrm{d}v$

• Initial value theorem

$x[0]=\underset{z\to \mathrm{\infty }}{\lim }X(z)$, If $x[n]$ causal

• Final value theorem

$x[\mathrm{\infty }]=\underset{z\to 1}{\lim }(z-1)X(z)$, Only if poles of $(z-1)X(z)$ are inside unit circle

Z Transformation of Functions

Here:

• $u[n]=1$ for $n>=0$, $u[n]=0$ for $n<0$
• $\delta [n]=1$ for $n=0$, $\delta [n]=0$ otherwise

	Signal, $x[n]$	Z-transform, $X(z)$	ROC
1	$\delta [n]$	$1$	$\text{all }z$
2	$\delta [n-n_0]$	$z^{-n_0}$	$z\ne 0$
3	$u[n]$	$\frac{1}{1-z^{-1}}$	$|z|>1$
4	$-u[-n-1]$	$\frac{1}{1-z^{-1}}$	$|z|<1$
5	$nu[n]$	$\frac{z^{-1}}{(1-z^{-1}{)}^2}$	$|z|>1$
6	$-nu[-n-1]$	$\frac{z^{-1}}{(1-z^{-1}{)}^2}$	$|z|<1$
7	$n^{2}u[n]$	$\frac{z^{-1}(1+z^{-1})}{(1-z^{-1}{)}^3}$	$|z|>1$
8	$-n^{2}u[-n-1]$	$\frac{z^{-1}(1+z^{-1})}{(1-z^{-1}{)}^3}$	$|z|<1$
9	$n^{3}u[n]$	$\frac{z^{-1}(1+4z^{-1}+z^{-2})}{(1-z^{-1}{)}^4}$	$|z|>1$
10	$-n^{3}u[-n-1]$	$\frac{z^{-1}(1+4z^{-1}+z^{-2})}{(1-z^{-1}{)}^4}$	$|z|<1$
11	$a^{n}u[n]$	$\frac{1}{1-a{z}^{-1}}$	$|z|>|a|$
12	$-a^{n}u[-n-1]$	$\frac{1}{1-a{z}^{-1}}$	$|z|<|a|$
13	$n{a}^{n}u[n]$	$\frac{a{z}^{-1}}{(1-a{z}^{-1}{)}^2}$	$|z|>|a|$
14	$-n{a}^{n}u[-n-1]$	$\frac{a{z}^{-1}}{(1-a{z}^{-1}{)}^2}$	$|z|<|a|$
15	$n^2{a}^{n}u[n]$	$\frac{a{z}^{-1}(1+a{z}^{-1})}{(1-a{z}^{-1}{)}^3}$	$|z|>|a|$
16	$-n^2{a}^{n}u[-n-1]$	$\frac{a{z}^{-1}(1+a{z}^{-1})}{(1-a{z}^{-1}{)}^3}$	$|z|<|a|$
17	$\cos ({\omega }_{0}n)u[n]$	$\frac{1-z^{-1}\cos ({\omega }_0)}{1-2z^{-1}\cos ({\omega }_0)+z^{-2}}$	$|z|>1$
18	$\sin ({\omega }_{0}n)u[n]$	$\frac{z^{-1}\sin ({\omega }_0)}{1-2z^{-1}\cos ({\omega }_0)+z^{-2}}$	$|z|>1$
19	$a^n\cos ({\omega }_{0}n)u[n]$	$\frac{1-a{z}^{-1}\cos ({\omega }_0)}{1-2a{z}^{-1}\cos ({\omega }_0)+a^2{z}^{-2}}$	$|z|>|a|$
20	$a^n\sin ({\omega }_{0}n)u[n]$	$\frac{a{z}^{-1}\sin ({\omega }_0)}{1-2a{z}^{-1}\cos ({\omega }_0)+a^2{z}^{-2}}$	$|z|>|a|$