Editing Linear time-invariant system (section)

=== Exponentials as eigenfunctions ===
An [[eigenfunction]] is a function for which the output of the operator is the same function, scaled by some constant.  In symbols,
<math display="block">\mathcal{H}f = \lambda f ,</math>

where ''f'' is the eigenfunction and <math>\lambda</math> is the [[eigenvalue]], a constant.

The [[exponential function]]s <math>z^n = e^{sT n}</math>, where <math>n \in \mathbb{Z}</math>, are [[eigenfunction]]s of a [[linear]], [[time-invariant]] operator. <math>T \in \mathbb{R}</math> is the sampling interval, and <math>z = e^{sT}, \ z,s \in \mathbb{C}</math>.  A simple proof illustrates this concept.

Suppose the input is <math>x[n] = z^n</math>. The output of the system with impulse response <math>h[n]</math> is then
<math display="block">\sum_{m=-\infty}^{\infty} h[n-m] \, z^m</math>

which is equivalent to the following by the commutative property of [[convolution]]
<math display="block">\sum_{m=-\infty}^{\infty} h[m] \, z^{(n - m)} = z^n \sum_{m=-\infty}^{\infty} h[m] \, z^{-m} = z^n H(z)</math>
where
<math display="block">H(z) \mathrel{\stackrel{\text{def}}{=}} \sum_{m=-\infty}^\infty h[m] z^{-m}</math>
is dependent only on the parameter ''z''.

So <math>z^n</math> is an [[eigenfunction]] of an LTI system because the system response is the same as the input times the constant <math>H(z)</math>.