1. The z-Transform The z-Transform is the more general case of the discrete-time Fourier transform. For the DT Fourier transform $ z = e^{j\omega } $ with $ \omega $ real $ \Rightarrow |z| = 1 $. When z is not restricted to 1, it has the form $ re^{j\omega} $. This can be developed into the more general case of transform called z-Transform. The development of the z-Transform is outlined in Chapter 10.1 of the Oppenheim and Wilsky text.
- $ X(z) = \sum_{n = -\infty}^{+\infty}x[n]z^{-n} $
2. Region of Convergence for the z-Transform
3. The Inverse z-Transform
The derivation of the inverse z-Transform equation is outlined in chapter 10.3 of the text (pg 757-8).
- $ x[n] = \frac{1}{2\pi j} \oint X(z)z^{n-1}\,dz $
This is a closed loop integral around a CCW rotation centered at the origin with radius r. r can be any value for which X(z) converges.
| 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) $ | |||
z-Transform Pairs
| z-Transform Pairs | |||
|---|---|---|---|
| # | Signal | Transform | ROC |
| 1 | $ \,\!\delta[n] $ | $ \,\! 1 $ | All $ \,\! z $ |
| 2 | $ \,\!u[n] $ | $ \,\!\frac{1}{1-z^{-1}} $ | $ \,\! |z| > 1 $ |
| 3 | $ \,\!-u[-n-1] $ | $ \,\!\frac{1}{1-z^{-1}} $ | $ \,\! |z| < 1 $ |
| 4 | $ \,\!\delta [n-m] $ | $ \,\! z^{-m} $ | All $ \,\!z $ except 0 (if $ \,\! m > 0 $) or $ \,\!\infty\mbox{(if } m < 0 \mbox{)} $ |
| 5 | $ \,\!\alpha^{n}u[n] $ | $ \,\! \frac{1}{1-\alpha z^{-1}} $ | $ \,\! |z| > |\alpha| $ |
| 6 | $ \,\! -\alpha^{n}u[-n-1] $ | $ \,\!\frac{1}{1-\alpha z^{-1}} $ | $ \,\! |z| < |\alpha| $ |
| 7 | $ \,\! n\alpha^{n}u[n] $ | $ \,\! \frac{\alpha z^{-1}}{(1-\alpha z^{-1})^{2}} $ | $ \,\! |z| > |\alpha| $ |
| 8 | $ \,\! -n\alpha^{n}u[-n-1] $ | $ \,\! \frac{\alpha z^{-1}}{(1-\alpha z^{-1})^{2}} $ | $ \,\! |z| < |\alpha| $ |
| 9 | $ \,\! [cos(\omega_0 n)]u[n] $ | $ \,\! \frac{1-[cos(\omega_0)]z^{-1}}{1-[2cos(\omega_0)]z^{-1}+z^{-2}} $ | $ \,\! |z| > 1 $ |
| 10 | $ \,\! [sin(\omega_0 n)]u[n] $ | $ \,\! \frac{1-[cos(\omega_0)]z^{-1}}{1-[2cos(\omega_0)]z^{-1}+z^{-2}} $ | $ \,\! |z| > 1 $ |
| 11 | $ \,\! [r^{n}cos(\omega_0 n)]u[n] $ | $ \,\! \frac{1-[rcos(\omega_0)]z^{-1}}{1-[2rcos(\omega_0)]z^{-1}+r^{2}z^{-2}} $ | $ \,\! |z| > r $ |
| 12 | $ \,\! [r^{n}sin(\omega_0 n)]u[n] $ | $ \,\! \frac{1-[rcos(\omega_0)]z^{-1}}{1-[2rcos(\omega_0)]z^{-1}+r^{2}z^{-2}} $ | $ \,\! |z| > r $ |
Recommended Exercises: 10.1, 10.2, 10.3, 10.4, 10.6, 10.7, 10.8, 10.9, 10.10, 10.11, 10.13, 10.15, 10.21, 10.22, 10.23, 10.24, 10.25, 10.26, 10.27, 10.30, 10.31, 10.32, 10.33, 10.43, 10.44.
