Two-sided Laplace Transform

In mathematics, the two-sided Laplace transform or bilateral Laplace transform is an integral transform closely related to the Fourier transform, the Mellin transform, and the ordinary or one-sided Laplace transform. If ƒ(t) is a real or complex valued function of the real variable t defined for all real numbers, then the two-sided Laplace transform is defined by the integral

$\mathcal{B} \left\{f(t)\right\} = F(s) = \int_{-\infty}^\infty e^{-st} f(t) \,dt.$

The integral is most commonly understood as an improper integral, which converges if and only if each of the integrals

exists. There seems to be no generally accepted notation for the two-sided transform; the used here recalls "bilateral". The two-sided transform used by some authors is

$\mathcal{T}\left\{f(t)\right\} = s\mathcal{B}\left\{f\right\} = sF(s) = s \int_{-\infty}^\infty e^{-st} f(t) \, dt.$

In pure mathematics the argument t can be any variable, and Laplace transforms are used to study how differential operators transform the function.

In science and engineering applications, the argument t often represents time (in seconds), and the function ƒ(t) often represents a signal or waveform that varies with time. In these cases, the signals are transformed by filters, that work like a mathematical operator, but with a restriction. They have to be causal, which means that the output in a given time t cannot depend of input values in higher values of t.

When working with functions of time, ƒ(t) is called the time domain representation of the signal, while F(s) is called the s-domain representation. The inverse transformation then represents a synthesis of the signal as the sum of its frequency components taken over all frequencies, whereas the forward transformation represents the analysis of the signal into its frequency components.

Read more about Two-sided Laplace Transform: Relationship To Other Integral Transforms, Properties, Causality

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