## 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,

$X(z) = \sum^{\infty}_{n = -\infty} x[n]z^{-n}\!$

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

$X(z) = \sum^{\infty}_{n = -\infty} x[n]z^{-n}\!$

$= \sum^{\infty}_{n = -\infty} x[n](re^{j\omega})^{-n}\!$
$= \sum^{\infty}_{n = -\infty} x[n]r^{-n}e^{-j\omega n}\!$
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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 $|z| > \frac{1}{2}\!$.

## Alumni Liaison

Ph.D. 2007, working on developing cool imaging technologies for digital cameras, camera phones, and video surveillance cameras.

Buyue Zhang