In this topic, you study the Table of Z-Transform.
Definition: The inverse Z-transform of $X[z]$ is
\[{x_n} = \frac{1}{{2\pi j}}\oint\limits_C {X[z]{z^{n – 1}}dz} \]
Using above property, the inverse Z-transform of Basic Functions are
| X[z] | x(n) | Region of convergence (ROC) |
| \[1\] | \[\delta [n] \] | \[|z| > 1\] |
| \[\frac{{a{z^{ – 1}}}}{{{{(1 – {z^{ – 1}})}^2}}}\] | \[{a^n}u[n] \] | \[|z| > |a|\] |
| \[\frac{{a{z^{ – 1}}}}{{{{(1 – {z^{ – 1}})}^2}}}\] | \[ – n{a^n}u[ – n – 1] \] | \[|z| < |a|\] |
| \[\frac{1}{{1 – {z^{ – 1}}}}\] | \[u[n] \] | \[|z| > 1\] |
| \[\frac{1}{{1 – {z^{ – 1}}}}\] | \[-u[-n-1] \] | \[|z| < 1\] |
| \[\frac{{z.\sin {\omega _o}}}{{{z^2} – 2z\cos {\omega _o} + 1}}\] | \[\sin ({\omega _o}n)u[n]\] | \[|z| > 1\] |
| \[\frac{{z(z – \cos {\omega _o})}}{{{z^2} – 2z\cos {\omega _o} + 1}}\] | \[\cos ({\omega _o}n)u[n]\] | \[|z| > 1\] |