| z-Transform Properties | |||
|---|---|---|---|
| Property | Signal | z-Transform | ROC |
| Linearity | $ \,\! ax_1[n] + bx_2[n] $ | $ \,\! aX_1(z)+bX_2(z) $ | At least $ R_1 \cap R_2 $ |
| Time Shifting | $ \,\! x[n-n_0] $ | $ z^{-n_0}X(z) $ | R, except for the possible addition or deletion of the origin |
| Scaling in the z-Domain | $ e^{j\omega_0 n}x[n] $ | $ X(e^{-j\omega_0}z) $ | R |
| $ z_0^nx[n] $ | $ X\Bigg( \frac{z}{z_0} \Bigg) $ | $ z_0 R $ | |
| $ \,\! a^nx[n] $ | $ \,\! X(a^{-1}z) $ | a|R= the set of point {|a|z} for z in R | |
| Time Reversal | $ \,\! x[-n] $ | $ \,\! X(z^{-1}) $ | Inverted R (i.e., R^-1= the set of point z^-1, where z is in R) |
| Time Expansion | $ x_{(k)}[n] = \begin{cases} x[r], & \mbox{if }n=rk \mbox{ for }r\in \mathbb{Z}\\ 0, &\mbox{if }n\neq rk \mbox{ for } r\in \mathbb{Z}\end{cases} $ | $ \,\! X(z^k) $ | $ R^{1/k} $ (i.e., the set of points $ z^{1/k} $, where z is in R) |
| Conjugation | $ \,\! x^{*}[n] $ | $ \,\! X^{*}(z^{*}) $ | R |
| Convolution | $ \,\! x_1[n] * x_2[n] $ | $ \,\! X_1(z)X_2(z) $ | At least the intersection of R_1 and R_2 |
| First Difference | $ \,\! x[n] - x[n-1] $ | $ \,\! (1-z^{-1})X(z) $ | At least the intersection of R and $ |z| > 0 $ |
| Accumulation | $ \sum_{k = -\infty}^{n}x[k] $ | $ \frac{1}{1-z^{-1}}X(z) $ | At least the intersection of R and $ |z| > 1 $ |
| Differentiation in the z-Domain | $ \,\! nx[n] $ | $ -z\frac{dX(z)}{dz} $ | R |
Initial-Value Theorem
If x[n] = 0 for n < 0, then $ x[0] = \lim_{z\rightarrow \infty} X(z) $ | |||
