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Hopefully you are familiar with the notion of the eigenvectors of a "matrix system," if not they do a quick review of eigen-stuff . We can develop the same ideas for LTI systems acting on signals. A linear time invariant (LTI) system $\mathscr{H}$ operating on a continuous input $f(t)$ to produce continuous time output $y(t)$
is mathematically analogous to an $N$ x $N$ matrix $A$ operating on a vector $x\in {\u2102}^{N}$ to produce another vector $b\in {\u2102}^{N}$ (seeMatrices and LTI Systemsfor an overview).
Just as an eigenvector of $A$ is a $v\in {\u2102}^{N}$ such that $Av=\lambda v$ , $\lambda \in \mathbb{C}$ ,
we can define an eigenfunction (or eigensignal ) of an LTI system $\mathscr{H}$ to be a signal $f(t)$ such thatEigenfunctions are the simplest possible signals for $\mathscr{H}$ to operate on: to calculate the output, we simply multiply the input by a complex number $\lambda $ .
The class of LTI systems has a set of eigenfunctions in common: the complex exponentials $e^{st}$ , $s\in \mathbb{C}$ are eigenfunctions for all LTI systems.
We can prove [link] by expressing the output as a convolution of the input $e^{st}$ and the impulse response $h(t)$ of $\mathscr{H}$ :
Since the action of an LTI operator on its eigenfunctions $e^{st}$ is easy to calculate and interpret, it is convenient to represent an arbitrary signal $f(t)$ as a linear combination of complex exponentials. The Fourier series gives us this representation for periodic continuous timesignals, while the (slightly more complicated) Fourier transform lets us expand arbitrary continuous time signals.
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