Editing Z-transform (section)

==Table of common Z-transform pairs==
Here:
:<math>u : n \mapsto u[n] = \begin{cases} 1, & n \ge 0 \\ 0, & n < 0 \end{cases}</math>
is the [[Heaviside step function|unit (or Heaviside) step function]] and
:<math>\delta : n \mapsto \delta[n] = \begin{cases} 1, & n = 0 \\ 0, & n \ne 0 \end{cases}</math>
is the [[Kronecker delta#Digital signal processing|discrete-time unit impulse function]] (cf [[Dirac delta function]] which is a continuous-time version). The two functions are chosen together so that the unit step function is the accumulation (running total) of the unit impulse function.

{| class="wikitable"
|-
! !! Signal, <math>x[n]</math> !! Z-transform, <math>X(z)</math> !! ROC
|-
|  1 || <math>\delta[n]</math> || 1 || all ''z''
|-
|  2 || <math>\delta[n-n_0]</math> || <math> z^{-n_0}</math> || <math> z \neq 0</math>
|-
|  3 || <math>u[n] \,</math> || <math> \frac{1}{1-z^{-1} }</math> || <math>|z| > 1</math>
|-
|  4 ||<math>  -u[-n-1]</math> ||  <math> \frac{1}{1 - z^{-1}}</math> ||<math>|z| < 1</math>
|-
|  5 ||<math> n u[n]</math> || <math> \frac{z^{-1}}{( 1-z^{-1} )^2}</math> || <math>|z| > 1</math>
|-
|  6 ||<math> - n u[-n-1] \,</math> || <math> \frac{z^{-1} }{ (1 - z^{-1})^2 }</math> ||<math> |z| < 1</math>
|-
|  7 ||<math>n^2 u[n]</math> || <math>  \frac{ z^{-1} (1 + z^{-1} )}{(1 - z^{-1})^3} </math> || <math>|z| > 1\,</math>
|-
|  8 ||<math> - n^2 u[-n - 1] \,</math> || <math>  \frac{ z^{-1} (1 + z^{-1} )}{(1 - z^{-1})^3} </math> || <math>|z| < 1\,</math>
|-
|  9 ||<math>n^3 u[n]</math> || <math> \frac{z^{-1} (1 + 4 z^{-1} + z^{-2} )}{(1-z^{-1})^4} </math> || <math>|z| > 1\,</math>
|-
| 10 ||<math>- n^3 u[-n -1]</math> || <math> \frac{z^{-1} (1 + 4 z^{-1} + z^{-2} )}{(1-z^{-1})^4} </math> || <math>|z| < 1\,</math>
|-
| 11 ||<math>a^n u[n]</math> ||  <math> \frac{1}{1-a z^{-1}}</math> ||<math> |z| > |a|</math>
|-
| 12 ||<math>-a^n u[-n-1]</math> ||  <math> \frac{1}{1-a z^{-1}}</math> ||<math>|z| < |a|</math>
|-
| 13 ||<math>n a^n u[n]</math> ||  <math> \frac{az^{-1} }{ (1-a z^{-1})^2 }</math> || <math>|z| > |a|</math>
|-
| 14 ||<math>-n a^n u[-n-1]</math> || <math> \frac{az^{-1} }{ (1-a z^{-1})^2 }</math> ||<math> |z| < |a|</math>
|-
| 15 ||<math>n^2 a^n u[n]</math> || <math> \frac{a z^{-1} (1 + a z^{-1}) }{(1-a z^{-1})^3} </math> || <math>|z| > |a|</math>
|-
| 16 ||<math>- n^2 a^n u[-n -1]</math> || <math> \frac{a z^{-1} (1 + a z^{-1}) }{(1-a z^{-1})^3} </math> || <math>|z| < |a|</math>
|-
| 17 ||<math> \left(\begin{array}{c} n + m - 1 \\ m - 1 \end{array} \right) a^n u[n]</math> [[Permutation#k-permutations of n|<sup>[14]</sup>]]|| <math> \frac{1}{(1-a z^{-1})^m} </math>, for positive integer <math>m</math><ref name=forouzan/> || <math>|z| > |a|</math>
|-
| 18 ||<math> (-1)^m \left(\begin{array}{c} -n - 1 \\ m - 1 \end{array} \right) a^n u[-n -m]</math> || <math> \frac{1}{(1-a z^{-1})^m} </math>, for positive integer <math>m</math><ref name=forouzan/> || <math>|z| < |a|</math>
|-
| 19 ||<math>\cos(\omega_0 n) u[n]</math> || <math> \frac{ 1-z^{-1} \cos(\omega_0)}{ 1-2z^{-1}\cos(\omega_0)+ z^{-2}}</math> ||<math> |z| >1</math>
|-
| 20 ||<math>\sin(\omega_0 n) u[n]</math> || <math> \frac{ z^{-1} \sin(\omega_0)}{ 1-2z^{-1}\cos(\omega_0)+ z^{-2} }</math> ||<math> |z| >1</math>
|-
| 21 ||<math>a^n \cos(\omega_0 n) u[n]</math>||<math>\frac{1-a z^{-1} \cos( \omega_0)}{1-2az^{-1}\cos(\omega_0)+ a^2 z^{-2}}</math>||<math>|z|>|a|</math>
|-
| 22 ||<math>a^n \sin(\omega_0 n) u[n]</math>||<math> \frac{ az^{-1} \sin(\omega_0) }{ 1-2az^{-1}\cos(\omega_0)+ a^2 z^{-2} }</math> ||<math>|z|>|a|</math>
|}