Editing Linear time-invariant system (section)

=== Fourier and Laplace transforms ===
The eigenfunction property of exponentials is very useful for both analysis and insight into LTI systems.  The one-sided [[Laplace transform]]
<math display="block">H(s) \mathrel{\stackrel{\text{def}}{=}} \mathcal{L}\{h(t)\} \mathrel{\stackrel{\text{def}}{=}} \int_0^\infty h(t) e^{-s t} \, \mathrm{d} t</math>
is exactly the way to get the eigenvalues from the impulse response.  Of particular interest are pure sinusoids (i.e., exponential functions of the form <math>e^{j \omega t}</math> where <math>\omega \in \mathbb{R}</math> and <math>j \mathrel{\stackrel{\text{def}}{=}} \sqrt{-1}</math>). The [[Fourier transform]] <math>H(j \omega) = \mathcal{F}\{h(t)\}</math> gives the eigenvalues for pure complex sinusoids. Both of <math>H(s)</math> and <math>H(j\omega)</math> are called the ''system function'', ''system response'', or ''transfer function''.

The Laplace transform is usually used in the context of one-sided signals, i.e. signals that are zero for all values of ''t'' less than some value. Usually, this "start time" is set to zero, for convenience and without loss of generality, with the transform integral being taken from zero to infinity (the transform shown above with lower limit of integration of negative infinity is formally known as the [[bilateral Laplace transform]]).

The Fourier transform is used for analyzing systems that process signals that are infinite in extent, such as modulated sinusoids, even though it cannot be directly applied to input and output signals that are not [[square integrable]].  The Laplace transform actually works directly for these signals if they are zero before a start time, even if they are not square integrable, for stable systems.  The Fourier transform is often applied to spectra of infinite signals via the  [[Wiener–Khinchin theorem]] even when Fourier transforms of the signals do not exist.

Due to the convolution property of both of these transforms, the convolution that gives the output of the system can be transformed to a multiplication in the transform domain, given signals for which the transforms exist
<math display="block">y(t) = (h*x)(t) \mathrel{\stackrel{\text{def}}{=}} \int_{-\infty}^\infty h(t - \tau) x(\tau) \, \mathrm{d} \tau \mathrel{\stackrel{\text{def}}{=}} \mathcal{L}^{-1}\{H(s)X(s)\}.</math>

One can use the system response directly to determine how any particular frequency component is handled by a system with that Laplace transform. If we evaluate the system response (Laplace transform of the impulse response) at complex frequency {{nowrap|''s'' {{=}} ''jω''}}, where {{nowrap|''ω'' {{=}} 2''πf''}}, we obtain |''H''(''s'')| which is the system gain for frequency ''f''. The relative phase shift between the output and input for that frequency component is likewise given by arg(''H''(''s'')).