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