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The Z-Transform
Similar to the Laplace Transform, the Z-Transform is an extension of the Fourier Transform, in this case the DT Fourier Transform. As previously defined, the response, $ y[n]\! $, of a DT LTI system is $ y[n] = H(z)z^n\! $, where $ H(z) = \sum^{\infty}_{n = -\infty} h[n]z^{-n}\! $. When $ z = e^{j\omega}\! $ with $ \omega\! $ real, this summation equals the Fourier Transform of $ h[n]\! $. When $ z\! $ is not restricted to this value, the summation is know as the Z-Transform of $ h[n]\! $. To be exact,
where $ z\! $ is a complex variable. This is sometimes denoted as $ X(z) = Z(x[n])\! $.
Relationship between Z-Transform and Fourier Transform
The Fourier Transform at $ \omega\! $ is equal to the Z-Transform at $ e^{j\omega}\! $, as shown below.
$ X(\omega) = X(e^{j\omega})\! $
If we look at the unit circle with radius $ r\! $ and $ X(z) = X(re^{j\omega})\! $, then
$ X(z) = \!F(x[n]r^{-n})\! $ because
$ X(z) = \sum^{\infty}_{n = -\infty} x[n]z^{-n}\! $
$ = \sum^{\infty}_{n = -\infty} x[n](re^{j\omega})^{-n}\! $
$ = \sum^{\infty}_{n = -\infty} x[n]r^{-n}e^{-j\omega n}\! $ |_____________________| | F.T. of $ x[n]r^{-n}\! $
Region of Convergence
Similar to the Laplace Transform, the Z-Transform sum does not always converge and a region of convergence is required for each problem asking for a Z-Transform. The set of complex numbers, $ z\! $, such that the Z-Transform of $ x[n]\! $ converges is called the "Region of Convergence" (ROC) of $ X(z)\! $. To find the properties of the ROC, please see some of my classmates' pages.
General Example of Z-Transform
Find the Z-Transform of $ x[n] = \frac{u[n]}{a^n}\! $.
$ X(z) = \sum^{\infty}_{n = -\infty} x[n]z^{-n}\! $
$ = \sum^{\infty}_{n = -\infty} \frac{u[n]}{a^n}z^{-n}\! $
$ = \sum^{\infty}_{n = 0} \frac{z^{-n}}{a^n}\! $
$ = \sum^{\infty}_{n = 0} (\frac{1}{az})^n\! $
if $ |\frac{1}{az}| \geq 1\! $, then $ X(z)\! $ diverges. else,
$ X(z) = \frac{1}{1-\frac{1}{az}}\! $, by the geometric series formula.
$ = \frac{az}{az-1}\! $
Therefore, the Z-Transform is $ \frac{az}{az-1}\! $, ROC is $ |\frac{1}{az}| < 1\! $ or $ |z| > \frac{1}{2}\! $.
