Two-sided Laplace transform

The Fourier transform can be defined in terms of the two-sided Laplace transform of a function : : \mathcal{F}\{f(t)\} = \left. F(s) \right|_{s = i\omega} = F(i\omega). Note that definitions of the Fourier transform differ, and in particular : \mathcal{F}\{f(t)\} = \left. F(s) \right|_{s = i\omega} = \frac{1}{\sqrt{2\pi}} \mathcal{B}\{f(t)\}(s) is often used instead. In terms of the Fourier transform, we may also obtain the two-sided Laplace transform where , as : \mathcal{B}\{f(t)\}(s) = \mathcal{F}\{f(t)\}(-is). The Fourier transform of a function is defined over real frequencies, i.e. for pure imaginary (where ); for functions that are upper-bounded in an absolute sense by an exponential function, the above definition converges in a strip for some and , which may not include the imaginary -axis. Laplace transform is often used in control theory and signal processing, as it generally deals with linear causal system responses that may grow exponentially. The Fourier transform integral fails to converge for exponentially growing signals, which means that a linear, shift-invariant system described by it is stable or critical. The Laplace transform, on the other hand, will somewhere converge for every (causal) impulse response that grows at most exponentially. Since , Laplace transform based analysis and solution of linear, shift-invariant systems, takes its most general form in the context of Laplace, rather than Fourier, transforms. Laplace transform theory falls within the ambit of more general integral transforms, or even general harmonic analysis. In that framework and nomenclature, Laplace transforms are simply a form of Fourier analysis, though more general. == Relationship to other integral transforms ==

If u is the Heaviside step function, equal to zero when its argument is less than zero, to one-half when its argument equals zero, and to one when its argument is greater than zero, then the Laplace transform {{tmath| \mathcal{L} }} may be defined in terms of the two-sided Laplace transform by : \mathcal{L}\{f\} = \mathcal{B}\{f u\}. On the other hand, we also have : \mathcal{B}\{f\} = \mathcal{L}\{f\} + \mathcal{L}\{f\circ m\}\circ m, where {{tmath| m : \mathbb{R}\to\mathbb{R} : x \mapsto -x }} is the function that multiplies by minus one, so either version of the Laplace transform can be defined in terms of the other. The Mellin transform may be defined in terms of the two-sided Laplace transform by : \mathcal{M}\{f\} = \mathcal{B}\{f \circ {\exp} \circ m\}, with as above, and conversely we can get the two-sided transform from the Mellin transform by : \mathcal{B}\{f\} = \mathcal{M}\{f \circ m \circ \log \}. The moment-generating function of a continuous probability density function can be expressed as {{tmath| \mathcal{B}\{f\}(-s) }}. == Properties ==

The following properties can be found in and F \left ({s \over a} \right) Most properties of the bilateral Laplace transform are very similar to properties of the unilateral Laplace transform, but there are some important differences: Parseval's theorem and Plancherel's theorem Let and be functions with bilateral Laplace transforms and in the strips of convergence {{tmath| \alpha_{1,2} \int_{-\infty}^{\infty} \overline{f_1(t)}\,f_2(t)\,dt = \frac{1}{2\pi i} \int_{c-i\infty}^{c+i\infty} \overline{F_1(-\overline{s})}\,F_2(s)\,ds This theorem is proved by applying the inverse Laplace transform on the convolution theorem in form of the cross-correlation. Let be a function with bilateral Laplace transform in the strip of convergence . Let {{tmath| c\in\mathbb{R} }} with . Then the Plancherel theorem holds: : \int_{-\infty}^{\infty} e^{-2c\,t} \, |f(t)|^2 \,dt = \frac{1}{2\pi} \int_{-\infty}^{\infty} |F(c+ir)|^2 \, dr Uniqueness For any two functions and for which the two-sided Laplace transforms {{tmath| \mathcal{T} \{f\} }}, {{tmath| \mathcal{T} \{g\} }} exist, if {{tmath|1= \mathcal{T}\{f\} = \mathcal{T} \{g\} }}, i.e. {{tmath|1= \mathcal{T}\{f\}(s) = \mathcal{T}\{g\}(s) }} for every value of , then almost everywhere. == Region of convergence ==

Bilateral transform requirements for convergence are more difficult than for unilateral transforms. The region of convergence will be normally smaller. If is a locally integrable function (or more generally a Borel measure locally of bounded variation), then the Laplace transform of converges provided that the limit : \lim_{R\to\infty}\int_0^R f(t)e^{-st}\, dt exists. The Laplace transform converges absolutely if the integral : \int_0^\infty \left|f(t)e^{-st}\right|\, dt exists (as a proper Lebesgue integral). The Laplace transform is usually understood as conditionally convergent, meaning that it converges in the former instead of the latter sense. The set of values for which converges absolutely is either of the form or else , where is an extended real constant, . (This follows from the dominated convergence theorem.) The constant is known as the abscissa of absolute convergence, and depends on the growth behavior of . Analogously, the two-sided transform converges absolutely in a strip of the form , and possibly including the lines or . The subset of values of for which the Laplace transform converges absolutely is called the region of absolute convergence or the domain of absolute convergence. In the two-sided case, it is sometimes called the strip of absolute convergence. The Laplace transform is analytic in the region of absolute convergence. Similarly, the set of values for which converges (conditionally or absolutely) is known as the region of conditional convergence, or simply the region of convergence (ROC). If the Laplace transform converges (conditionally) at , then it automatically converges for all with ). Therefore, the region of convergence is a half-plane of the form , possibly including some points of the boundary line . In the region of convergence, ), the Laplace transform of can be expressed by integrating by parts as the integral : F(s) = (s-s_0)\int_0^\infty e^{-(s-s_0)t}\beta(t)\, dt,\quad \beta(u) = \int_0^u e^{-s_0t}f(t)\, dt. That is, in the region of convergence can effectively be expressed as the absolutely convergent Laplace transform of some other function. In particular, it is analytic. There are several Paley–Wiener theorems concerning the relationship between the decay properties of and the properties of the Laplace transform within the region of convergence. In engineering applications, a function corresponding to a linear time-invariant (LTI) system is stable if every bounded input produces a bounded output. == Causality ==

Bilateral transforms do not respect causality. They make sense when applied over generic functions but when working with functions of time (signals) unilateral transforms are preferred. == Table of selected bilateral Laplace transforms ==

Following list of interesting examples for the bilateral Laplace transform can be deduced from the corresponding Fourier or unilateral Laplace transformations (see also ): == See also ==