# Electronics Communication/Signal Processing/Signal Transformation/Z Transform
## Z Transform 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 Transform Table

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 | $a^{n}\sin(\omega _{0}n)u[n]\,$ | ${\frac {az^{-1}\sin(\omega _{0})}{1-2az^{-1}\cos(\omega _{0})+a^{2}z^{-2}}}$ | $|z|>|a|\,$ |