Eigenfunctions
==============

A remarkable fact of linear systems is that the complex exponentials
are *eigenfunctions* of a linear system. I.e. if we take a complex
exponential $x(t)=\exp(j\omega t)$ as input, the output is a complex
exponential, *with the same frequency* as the input but multiplied
with a complex constant (dependent on the frequency).

   .. tikz:: Complex Exponentials are the Eigenfunctions of a CT LTI
             Linear System

      \bXInput{A}
      \bXBloc[6]{B}{$h(t)$}{A}
      \bXLink[\$e^{-j\omega t}\$]{A}{B}
      \bXOutput[12]{C}{B}
      \bXLink[\$y(t)=H(\omega) e^{-j\omega t}\$]{B}{C}





Consider the system with impulse response $h$ then the output is given
by:

.. math::
   y(t) = \int_{-\infty}^{\infty} e^{j\omega(t-u)} h(u) du

We can simplify this as

.. math::
   y(t) = e^{j \omega t} \int_{-\infty}^{\infty} e^{-j\omega u} h(u) du

Observe that the integral only depends on $\omega$ and we denote it as
$H(\omega)$, then:

.. math::
   y(t) =  e^{j \omega t} H(\omega)

i.e. in case the input of a linear system is a sinusoidal signal
(complex exponential) the output is the exponential function
(sinusoidal function *with the same frequency*) multiplied with a
complex factor $H(\omega)$ that is completely characterized by the
linear system (its impulse response).

The function $H$

.. math::
   H(\omega) = \int_{-\infty}^{\infty} e^{-j\omega u} h(u) du

is called the Fourier transform of $h$. The Fourier transform will
play a major role in this lecture series.