Editing Linear time-invariant system (section)

=== Z and discrete-time Fourier transforms ===
The eigenfunction property of exponentials is very useful for both analysis and insight into LTI systems.  The [[Z transform]]
<math display="block">H(z) = \mathcal{Z}\{h[n]\} = \sum_{n=-\infty}^\infty h[n] z^{-n}</math>

is exactly the way to get the eigenvalues from the impulse response.{{clarify|date=September 2020}}  Of particular interest are pure sinusoids; i.e. exponentials of the form <math>e^{j \omega n}</math>, where <math>\omega \in \mathbb{R}</math>.  These can also be written as <math>z^n</math> with <math>z = e^{j \omega}</math>{{clarify|date=September 2020}}. The [[discrete-time Fourier transform]] (DTFT) <math>H(e^{j \omega}) = \mathcal{F}\{h[n]\}</math> gives the eigenvalues of pure sinusoids{{clarify|date=September 2020}}.  Both of <math>H(z)</math> and <math>H(e^{j\omega})</math> are called the ''system function'', ''system response'', or ''transfer function''.

Like the one-sided Laplace transform, the Z transform is usually used in the context of one-sided signals, i.e. signals that are zero for t<0. The discrete-time Fourier transform [[Fourier series]] may be used for analyzing periodic signals.

Due to the convolution property of both of these transforms, the convolution that gives the output of the system can be transformed to a multiplication in the transform domain. That is,
<math display="block">y[n] = (h*x)[n] = \sum_{m=-\infty}^\infty h[n-m] x[m] = \mathcal{Z}^{-1}\{H(z)X(z)\}.</math>

Just as with the Laplace transform transfer function in continuous-time system analysis, the Z transform makes it easier to analyze systems and gain insight into their behavior.