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::
   :label: eq_FT
	   
   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.

Let's redo this analysis for a real valued function:
$x(t)=\cos(\w t)$. We know that

.. math::
   x(t) = \cos(\w t) = \half (e^{j\w t} + e^{-j\w t})

Because we are considering a linear system we have:

.. math::
   y(t) &= (\op Lx)(t) = \half( \op L(e^{j\w t}) + \op L(e^{-j\w t}) )\\
   &= \half\left( H(\w)e^{j\w t} + H(-\w) e^{-j\w t} \right)

Note that from the definition of the Fourier transform (Eq. :eq:`eq_FT`)we can see that
if $h$ is real valued function (which in practice it always is) that
$H(-\w) = H^\star(\w)$. Writing $H(\w)$ in polar notation:

.. math::
   H(\w) &= |H(\w)| e^{j\angle H(\w)}\\
   H(-\w) &= H^\star(\w) = |H(\w)| e^{-j\angle H(\w)}

So

.. math::
   y(t) &= \half |H(\w)| \left( e^{j(\w t+\angle H(\w)} + e^{-j(\w t +\angle H(\w)} \right)\\
   &= |H(\w)| \cos( \w t + \angle H(\w))


So for a linear system with impulse response $h$ we have

.. math::
   x(t) = e^{j\w t} \quad\xrightarrow{\quad L\quad}\quad y(t) = H(\w) e^{j\w t}

and

.. math::
   x(t) = \cos(\w t) \quad\xrightarrow{\quad L\quad}\quad y(t) = |H(w)| \cos( \w t + \angle H(\w))

Evidently the use of complex exponential functions allows for a much
more convenient (and short) notation. Complex numbers are really
indispensible in signal processing theory.