## Some common pairs – z – transform formulas

### Some common pairs

 $$x(t)$$ $$X(Z)$$ $$\delta$$ $$1$$ $$u(n)$$ $$\frac {Z}{Z-1}$$ $$u(-n-1)$$ $$-\frac {Z}{Z-1}$$ $$\delta(n-m)$$ $$z^{-m}$$ $$a^nu[n]$$ $$\frac {Z}{Z-a}$$ $$a^nu[-n-1]$$ $$-\frac {Z}{Z-a}$$ $$na^nu[n]$$ $$\frac {aZ}{|Z-a|^2}$$ $$na^nu[-n-1]$$ $$-\frac {aZ}{|Z-a|^2}$$ $$a^n\cos \omega nu[n]$$ $$\frac {Z^2-aZ\cos \omega}{Z^2-2aZ\cos \omega+a^2}$$ $$a^n\sin \omega nu[n]$$ $$\frac {aZ\sin \omega}{Z^2-2aZ\cos \omega+a^2}$$ $$nu(n)$$ $$\frac {Z^{-1}}{(1-Z^{-1})^3}$$ $$u(n-1)$$ $$Z^{-1}\frac {1}{(1-Z^{-1})}$$ $$\delta(n+m)$$ $$z^m$$

### Example:

Find z-transform of the following sequence x(n)=10 sin (0.25πn)u(n)

### Solution:

$$X(z)=10Z (\sin (0.25\pi n)u(n))$$
$$X(z)= \frac {10 \sin (0.25\pi)z}{z^2-2z\cos (0.25\pi)+1}$$
$$X(z)= \frac {7.07z}{z^2-1.414z+1}$$