# Engineering Tables/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 $n u[n] \,$ $\frac{z^{-1}}{( 1-z^{-1} )^2}$ $|z| > 1\,$
6 $- n u[-n-1] \,$ $\frac{z^{-1} }{ (1 - z^{-1})^2 }$ $|z| < 1 \,$
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 $n^3 u[n] \,$ $\frac{z^{-1} (1 + 4 z^{-1} + z^{-2} )}{(1-z^{-1})^4}$ $|z| > 1\,$
10 $- n^3 u[-n -1] \,$ $\frac{z^{-1} (1 + 4 z^{-1} + z^{-2} )}{(1-z^{-1})^4}$ $|z| < 1\,$
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 $n a^n u[n] \,$ $\frac{az^{-1} }{ (1-a z^{-1})^2 }$ $|z| > |a|\,$
14 $-n a^n u[-n-1] \,$ $\frac{az^{-1} }{ (1-a z^{-1})^2 }$ $|z| < |a|\,$
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 $a^n \cos(\omega_0 n) u[n] \,$ $\frac{ 1-a z^{-1} \cos( \omega_0) }{ 1-2az^{-1}\cos(\omega_0)+ a^2 z^{-2} }$ $|z| > |a|\,$
20 $a^n \sin(\omega_0 n) u[n] \,$ $\frac{ az^{-1} \sin(\omega_0) }{ 1-2az^{-1}\cos(\omega_0)+ a^2 z^{-2} }$ $|z| > |a|\,$