Engineering Handbook/Mathematics/Z Transformation

Z Transformation Properties

	Time domain	Z-domain	ROC
Notation	$x[n]={\mathcal {Z}}^{-1}\{X(z)\}$	$X(z)={\mathcal {Z}}\{x[n]\}$	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 $z=\infty$ 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	$\operatorname {Re} \{x[n]\}\$	${\frac {1}{2}}\left[X(z)+X^{*}(z^{*})\right]$	ROC
Imaginary part	$\operatorname {Im} \{x[n]\}\$	${\frac {1}{2j}}\left[X(z)-X^{*}(z^{*})\right]$	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 ^{\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]=\lim _{z\rightarrow \infty }X(z)\$, If $x[n]\,$ causal

• Final value theorem

$x[\infty ]=\lim _{z\rightarrow 1}(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	$\delta [n]\,$	$1\,$	${\mbox{all }}z\,$
2	$\delta [n-n_{0}]\,$	$z^{-n_{0}}\,$	$z\neq 0\,$
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	$a^{n}u[n]\,$	${\frac {1}{1-az^{-1}}}$	$|z|>|a|\,$
12	$-a^{n}u[-n-1]\,$	${\frac {1}{1-az^{-1}}}$	$|z|<|a|\,$
13	$na^{n}u[n]\,$	${\frac {az^{-1}}{(1-az^{-1})^{2}}}$	$|z|>|a|\,$
14	$-na^{n}u[-n-1]\,$	${\frac {az^{-1}}{(1-az^{-1})^{2}}}$	$|z|<|a|\,$
15	$n^{2}a^{n}u[n]\,$	${\frac {az^{-1}(1+az^{-1})}{(1-az^{-1})^{3}}}$	$|z|>|a|\,$
16	$-n^{2}a^{n}u[-n-1]\,$	${\frac {az^{-1}(1+az^{-1})}{(1-az^{-1})^{3}}}$	$|z|<|a|\,$
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	$a^{n}\cos(\omega _{0}n)u[n]\,$	${\frac {1-az^{-1}\cos(\omega _{0})}{1-2az^{-1}\cos(\omega _{0})+a^{2}z^{-2}}}$	$|z|>|a|\,$
20	$a^{n}\sin(\omega _{0}n)u[n]\,$	${\frac {az^{-1}\sin(\omega _{0})}{1-2az^{-1}\cos(\omega _{0})+a^{2}z^{-2}}}$	$|z|>|a|\,$