Editing Linear time-invariant system (section)

=== Exponentials as eigenfunctions ===
An [[eigenfunction]] is a function for which the output of the operator is a scaled version of the same function. That is,
<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>A e^{s t}</math>, where <math>A, s \in \mathbb{C}</math>, are [[eigenfunction]]s of a [[linear]], [[time-invariant]] operator. A simple proof illustrates this concept. Suppose the input is <math>x(t) = A e^{s t}</math>. The output of the system with impulse response <math>h(t)</math> is then
<math display="block">\int_{-\infty}^\infty h(t - \tau) A e^{s \tau}\, \mathrm{d} \tau</math>
which, by the commutative property of [[convolution]], is equivalent to
<math display="block">\begin{align}
    \overbrace{\int_{-\infty}^\infty h(\tau) \, A e^{s (t - \tau)} \, \mathrm{d} \tau}^{\mathcal{H} f}
  &= \int_{-\infty}^\infty h(\tau) \, A e^{s t} e^{-s \tau} \, \mathrm{d} \tau \\[4pt]
  &= A e^{s t} \int_{-\infty}^{\infty} h(\tau) \, e^{-s \tau} \, \mathrm{d} \tau \\[4pt]
  &= \overbrace{\underbrace{A e^{s t}}_{\text{Input}}}^{f} \, \overbrace{\underbrace{H(s)}_{\text{Scalar}}}^{\lambda} \, , \\
\end{align}</math>

where the scalar
<math display="block">H(s) \mathrel{\stackrel{\text{def}}{=}} \int_{-\infty}^\infty h(t) e^{-s t} \, \mathrm{d} t</math>
is dependent only on the parameter ''s''.

So the system's response is a scaled version of the input. In particular, for any <math>A, s \in \mathbb{C}</math>, the system output is the product of the input <math>A e^{st}</math> and the constant <math>H(s)</math>. Hence, <math>A e^{s t}</math> is an [[eigenfunction]] of an LTI system, and the corresponding [[eigenvalue]] is <math>H(s)</math>.

==== Direct proof ====
It is also possible to directly derive complex exponentials as eigenfunctions of LTI systems.

Let's set <math>v(t) = e^{i \omega t}</math> some complex exponential and <math>v_a(t) = e^{i \omega (t+a)}</math> a time-shifted version of it.

<math>H[v_a](t) = e^{i\omega a} H[v](t)</math> by linearity with respect to the constant <math>e^{i \omega a}</math>.

<math>H[v_a](t) = H[v](t+a)</math> by time invariance of <math>H</math>.

So <math>H[v](t+a) = e^{i \omega a} H[v](t)</math>. Setting <math>t = 0</math> and renaming we get:
<math display="block">H[v](\tau) = e^{i\omega \tau} H[v](0)</math>
i.e. that a complex exponential <math>e^{i \omega \tau}</math> as input will give a complex exponential of same frequency as output.