Lecture 16 - Notes

February 19, 2016

The Bilateral Z-Transform

definition: We define the Bilateral Z-Transform, for was given signal $x[n]$, as

$$\begin{aligned} X(z) &= \sum_{n=-\infty}^{\infty} x[n]z^{-n} \end{aligned}$$

where $z \in \mathbb C$.

Fourier Transform Relationship

For values of $z$ in the region $|z|=1$, known as the unit circle, we can express the transform as a function of a single, real variable, $\omega$, by defining $z = e^{j\omega}$. And the bi-lateral transform reduces to a Fourier series:

$$\begin{aligned} \sum_{n=-\infty}^{\infty} x[n]z^{-n} = \sum_{n=-\infty}^{\infty} x[n]e^{-j \omega n} \end{aligned}$$