# 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]\$ $a_{1}X_{1}(z)+a_{2}X_{2}(z)\$ At least the intersection of ROC$_1$ and ROC$_2$ 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]\$ $X_{1}(z)X_{2}(z)\$ At least the intersection of ROC$_1$ and ROC$_2$ Correlation $r_{x_{1},x_{2}}(l)=x_{1}[l]*x_{2}[-l]\$ $R_{x_{1},x_{2}}(z)=X_{1}(z)X_{2}(z^{-1})\$ At least the intersection of ROC of X$_1$(z) and X$_2$($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|\,$