2.3. 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).

Complex Exponentials are the Eigenfunctions of a CT LTI Linear System

Consider the system with impulse response \(h\) then the output is given by:

\[y(t) = \int_{-\infty}^{\infty} e^{j\omega(t-u)} h(u) du\]

We can simplify this as

\[y(t) = e^{j \omega t} \int_{-\infty}^{\infty} e^{-j\omega u} h(u) du\]

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

\[y(t) = e^{j \omega t} H(\omega)\]

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 \(H\)

\[H(\omega) = \int_{-\infty}^{\infty} e^{-j\omega u} h(u) du\]

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