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Chapter 10

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.

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