The z-transform

Generalizing the dtft

The backbone of the DTFT is the complex harmonic sinusoid: $e^{j\omega n}$. This type of signal is actually a specific example of a more broader class of signals $z^n$, where $z$ is a complex number. Signals in this class are called complex exponentials. As $z^n=|z|^n e^{j(\angle z)n}$, we see that complex exponentials are simply complex harmonic sinusoids, with exponential function envelopes. When $|z|\lt 1$, they decay in time:

Since complex harmonic sinusoids have the special property that they are eigenvectors of LTI systems, we may ask if the more general signal class of complex exponentials does, as well. To test, we will find the an LTI system's output (i.e., the result of the convolution sum) to some generic complex exponential $z^n$: $\begin{align*}y[n]&=z^n\ast h[n]\\&=\sum\limits_{m=-\infty}^\infty x[n-m]h[n]\\&=\sum\limits_{m=-\infty}^\infty z^{n-m}h[n]\\&=\sum\limits_{m=-\infty}^\infty z^n z^{-m} h[n]\\&=z^n \sum\limits_{m=-\infty}^\infty z^{-m} h[n]\\&=\left(\sum\limits_{m=-\infty}^\infty z^{-m} h[n]\right)z^n\\&=H(z)z^n~,~H(z)=\sum\limits_{m=-\infty}^\infty z^{-m} h[n] \end{align*}$If we input a complex exponential into some LTI system, then indeed the output is simply a scaled version of that same exponential. So complex exponentials are also eigenvectors of LTI systems. Now, the amount a given LTI system scales a complex exponential $z^n$ is very important, so much so that it has a special name--the system's transfer function (it tells us how the system "transfers" the input into the output)--and a special label, $H(z)$.

This transfer function of LTI systems, $H(z)$, is clearly a generalization of the frequency response $H(\omega)$. The frequency response indicated how an LTI system $H$ scales input complex sinusoids $e^{j\oemga}$, and likewise the transfer function $H(z)$ tells us how the system scales input complex exponentials $z^n$. As $e^{j\omega n}=z^n|_{z=e^{j\omega}}$ it follows that $H(\omega)=H(z)|_{z=e^{j\omega}}$. So it would seem that the DTFT of a function (for $H(\omega)$ is the DTFT of $h[n]$) can be generalized to be a broader kind of transform, which is represented by $H(z)$. This broader class of transform is called the z-transform.

The z-transform

There are a few things we can note immediately about the z-transform. First, unlike either transform we have considered (the DFT and DTFT), it is a complex function of a COMPLEX variable. Second, and like the DTFT (but not the DFT), the z-transform may not necessarily exist for all $z$ and/or all $x[n]$. And third, like both other transforms, the z-transform diagonalizes LTI systems. By this, we mean that if a signal $x[n]$ is expressed in terms of its z-transform, then the z-transform of the output at various $z$ is simply the pointwise multiplication of the input's transform with the transform of the impulse response (i.e., the transfer function: $Y(z)=H(z)X(z)$