Linear time-invariant system

In electrical engineering, specifically in signal processing and control theory, LTI system theory investigates the effects of a linear, time-invariant system on an arbitrary input.

For example, suppose the input signal is $x(t)$ where its index set is the real line, i.e., $t\in \mathbb {R}$. The linear operator $\mathbb {H}$ on this index set is a two-dimensional function

$h(t_{1},t_{2}){\mbox{ where }}t_{1},t_{2}\in \mathbb {R}$

The linear transformation of $x(t)$ is the superposition integral

$y(t_{1})=\int _{-\infty }^{\infty }h(t_{1},t_{2})\,x(t_{2})\,dt_{2}$

If the linear operator $\mathbb {H}$ is also time-invariant, the following property holds

$h(t_{1},t_{2})=h(t_{1}+\tau ,t_{2}+\tau )=h(t_{1}-t_{2},0)\,\,\forall \,\tau \in \mathbb {R}$

We usually drop the zero second argument to $h(t_{1},t_{2})$ for brevity of notation so that the superposition integral now becomes the familiar convolution integral used in filtering

$y(t_{1})=\int _{-\infty }^{\infty }h(t_{1}-t_{2})\,x(t_{2})\,dt_{2}$

Thus, the convolution integral represents the effect of a linear, time-invariant system on any input function. For a finite-dimensional analog, see the article on a circulant matrix.

Complex exponentials as eigenfunctions

The complex exponential functions $e^{j\omega t}{\mbox{ where }}\omega \in \mathbb {R}$ are eigenfunctions of a linear, time-invariant operator. A simple proof illustrates this concept.

Suppose the input is $x(t)=\,\!e^{j\omega t}$. The transformation of this function is then

$\int _{-\infty }^{\infty }h(t-\tau )\,e^{j\omega \tau }\,d\tau$

which is equivalent to the following by the commutative property of convolution

$\int _{-\infty }^{\infty }h(\tau )\,e^{j\omega (t-\tau )}\,d\tau =e^{j\omega t}\int _{-\infty }^{\infty }h(\tau )\,e^{-j\omega \tau }\,d\tau =e^{j\omega t}{\mathcal {F}}\{h(t)\}=e^{j\omega t}H(\omega )$

where ${\mathcal {F}}$ is the Fourier transform.

So, $\,\!e^{j\omega t}$ is an eigenfunction of an LTI system because the system response is itself scaled by an amount $H(\omega )$. Therefore, the eigenvalue spectrum is the Fourier transform of the operator $\mathbb {H}$.