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Inmathematics, thetwo-sided Laplace transform orbilateral Laplace transform is anintegral transform equivalent toprobability'smoment-generating function. Two-sided Laplace transforms are closely related to theFourier transform, theMellin transform, theZ-transform and the ordinary or one-sidedLaplace transform. If⁠${\displaystyle f(t)}$⁠ is a real- or complex-valued function of the real variable⁠${\displaystyle t}$⁠ defined for all real numbers, then the two-sided Laplace transform is defined by the integral

${\displaystyle \mathcal{B}\{f\}(s)=F(s)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{-st}f(t)dt.}$

The integral is most commonly understood as animproper integral, which convergesif and only if both integrals

${\displaystyle {\int }_0^{\mathrm{\infty }}{e}^{-st}f(t)dt,{\int }_{-\mathrm{\infty }}^0{e}^{-st}f(t)dt}$

exist. There seems to be no generally accepted notation for the two-sided transform; the${\displaystyle \mathcal{B}}$ used here recalls "bilateral". The two-sided transformused by some authors is

${\displaystyle \mathcal{T}\{f\}(s)=s\mathcal{B}\{f\}(s)=sF(s)=s{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{-st}f(t)dt.}$

In pure mathematics the argument⁠${\displaystyle t}$⁠ can be any variable, and Laplace transforms are used to study howdifferential operators transform the function.

Inscience andengineering applications, the argument⁠${\displaystyle t}$⁠ often represents time (in seconds), and the function⁠${\displaystyle f(t)}$⁠ often represents asignal or waveform that varies with time. In these cases, the signals are transformed byfilters, that work like a mathematical operator, but with a restriction. They have to be causal, which means that the output at a given time⁠${\displaystyle t}$⁠ cannot depend on an input that occurs at later time⁠${\displaystyle t^{\prime }}$⁠.In population ecology, the argument⁠${\displaystyle t}$⁠ often represents spatial displacement in a dispersal kernel.

When working with functions of time,⁠${\displaystyle f(t)}$⁠ is called thetime domain representation of the signal, while⁠${\displaystyle F(s)}$⁠ is called thes-domain (orLaplace domain) representation. The inverse transformation then represents asynthesis of the signal as the sum of its frequency components taken over all frequencies, whereas the forward transformation represents theanalysis of the signal into its frequency components.

TheFourier transform can be defined in terms of the two-sided Laplace transform:

${\displaystyle \mathcal{F}\{f(t)\}=F(s=i\omega )=F(\omega ).}$

Note that definitions of the Fourier transform differ, and in particular

${\displaystyle \mathcal{F}\{f(t)\}=F(s=i\omega )=\frac{1}{\sqrt{2\pi }}\mathcal{B}\{f(t)\}(s)}$

is often used instead. In terms of the Fourier transform, we may also obtain the two-sided Laplace transform, as

${\displaystyle \mathcal{B}\{f(t)\}(s)=\mathcal{F}\{f(t)\}(-is).}$

The Fourier transform is normally defined so that it exists for real values; the above definition defines the image in a strip⁠⁠, which may not include the real axis where the Fourier transform is supposed to converge.

This is then why Laplace transforms retain their value in control theory and signal processing: the convergence of a Fourier transform integral within its domain only means that a linear, shift-invariant system described by it is stable or critical. The Laplace one on the other hand will somewhere converge for every impulse response which is at most exponentially growing, because it involves an extra term which can be taken as an exponential regulator. Sincethere are no superexponentially growing linear feedback networks^{[citation needed]}, Laplace transform based analysis and solution of linear, shift-invariant systems, takes its most general form in the context of Laplace, not Fourier, transforms.

At the same time, nowadays Laplace transform theory falls within the ambit of more generalintegral transforms, or even generalharmonic analysis. In that framework and nomenclature, Laplace transforms are simply another form of Fourier analysis, even if more general in hindsight.

Ifu is theHeaviside 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${\displaystyle \mathcal{L}}$⁠ may be defined in terms of the two-sided Laplace transform by

${\displaystyle \mathcal{L}\{f\}=\mathcal{B}\{fu\}.}$

On the other hand, we also have

${\displaystyle \mathcal{B}\{f\}=\mathcal{L}\{f\}+\mathcal{L}\{f\circ m\}\circ m,}$

where⁠${\displaystyle m:\mathbb{R}\to \mathbb{R}:x\mapsto -x}$⁠ is the function that multiplies by minus one, so either version of the Laplace transform can be defined in terms of the other.

TheMellin transform may be defined in terms of the two-sided Laplace transform by

${\displaystyle \mathcal{M}\{f\}=\mathcal{B}\{f\circ \exp \circ m\},}$

with⁠${\displaystyle m}$⁠ as above, and conversely we can get the two-sided transform from the Mellin transform by

${\displaystyle \mathcal{B}\{f\}=\mathcal{M}\{f\circ m\circ \log \}.}$

Themoment-generating function of a continuousprobability density function⁠${\displaystyle f(x)}$⁠ can be expressed as⁠${\displaystyle \mathcal{B}\{f\}(-s)}$⁠.

The following properties can be found inBracewell (2000) andOppenheim & Willsky (1997)

Properties of the bilateral Laplace transform

Property	Time domain	s domain	Strip of convergence	Comment
Definition	${\displaystyle f(t)}$	${\displaystyle F(s)=\mathcal{B}\{f\}(s)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(t)e^{-st}dt}$
Time scaling	${\displaystyle f(at)}$	${\displaystyle \frac{1}{|a|}F\left(\frac{s}{a}\right)}$		${\displaystyle a\in \mathbb{R}\setminus 0}$
Reversal	${\displaystyle f(-t)}$	${\displaystyle F(-s)}$
Frequency-domain derivative	${\displaystyle tf(t)}$	${\displaystyle -F^{\prime }(s)}$
Frequency-domain general derivative	${\displaystyle t^{n}f(t)}$	${\displaystyle (-1{)}^n{F}^{(n)}(s)}$
Derivative	${\displaystyle f^{\prime }(t)}$	${\displaystyle sF(s)}$
General derivative	${\displaystyle f^{(n)}(t)}$	${\displaystyle s^{n}F(s)}$
Frequency-domain integration	${\displaystyle \frac{1}{t}f(t)}$	${\displaystyle {\int }_s^{\mathrm{\infty }}F(\sigma )d\sigma }$		only valid if the integral exists
Time-domain integral	${\displaystyle {\int }_{-\mathrm{\infty }}^{t}f(\tau )d\tau }$	${\displaystyle \frac{1}{s}F(s)}$
Time-domain integral	${\displaystyle {\int }_t^{\mathrm{\infty }}f(\tau )d\tau }$	${\displaystyle \frac{1}{s}F(s)}$
Frequency shifting	${\displaystyle e^{at}f(t)}$	${\displaystyle F(s-a)}$
Time shifting	${\displaystyle f(t-a)}$	${\displaystyle e^{-as}F(s)}$		${\displaystyle a\in \mathbb{R}}$
Modulation	${\displaystyle \cos (at)f(t)}$	${\displaystyle \frac{1}{2}F(s-ia)+\frac{1}{2}F(s+ia)}$		${\displaystyle a\in \mathbb{R}}$
Finite difference	${\displaystyle f(t+\frac{1}{2}a)-f(t-\frac{1}{2}a)}$	${\displaystyle 2\sinh (\frac{1}{2}as)F(s)}$		${\displaystyle a\in \mathbb{R}}$
Multiplication	${\displaystyle f(t)g(t)}$	${\displaystyle \frac{1}{2\pi i}{\int }_{c-i\mathrm{\infty }}^{c+i\mathrm{\infty }}F(\sigma )G(s-\sigma )d\sigma }$		. The integration is done along the vertical line⁠${\displaystyle \mathrm{\Re }\sigma =c}$⁠ inside the region of convergence.
Complex conjugation	${\displaystyle \overline{f(t)}}$	${\displaystyle \overline{F(\overline{s})}}$
Convolution	${\displaystyle (f\ast g)(t)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(\tau )g(t-\tau )d\tau }$	${\displaystyle F(s)\cdot G(s)}$
Cross-correlation	${\displaystyle (f\star g)(t)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\overline{f(\tau )}g(t+\tau )d\tau }$	${\displaystyle \overline{F(-\overline{s})}\cdot G(s)}$

Most properties of the bilateral Laplace transform are very similar to properties of the unilateral Laplace transform,but there are some important differences:

Properties of the unilateral transform vs. properties of the bilateral transform

	unilateral time domain	bilateral time domain	unilateral-'s' domain	bilateral-'s' domain
Differentiation	${\displaystyle f^{\prime }(t)}$	${\displaystyle f^{\prime }(t)}$	${\displaystyle sF(s)-f(0)}$	${\displaystyle sF(s)}$
Second-orderdifferentiation	${\displaystyle f^{″}(t)}$	${\displaystyle f^{″}(t)}$	${\displaystyle s^{2}F(s)-sf(0)-f^{\prime }(0)}$	${\displaystyle s^{2}F(s)}$
Convolution	${\displaystyle {\int }_0^{t}f(\tau )g(t-\tau )d\tau }$	${\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(\tau )g(t-\tau )d\tau }$	${\displaystyle F(s)\cdot G(s)}$	${\displaystyle F(s)\cdot G(s)}$

Let⁠${\displaystyle f_1(t)}$⁠ and⁠${\displaystyle f_2(t)}$⁠ be functions with bilateral Laplace transforms⁠${\displaystyle F_1(s)}$⁠ and⁠${\displaystyle F_2(s)}$⁠ in the strips of convergence⁠⁠. Let⁠${\displaystyle c\in \mathbb{R}}$⁠ with⁠⁠.ThenParseval's theorem holds:^[1]

${\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\overline{f_1(t)}{f}_2(t)dt=\frac{1}{2\pi i}{\int }_{c-i\mathrm{\infty }}^{c+i\mathrm{\infty }}\overline{F_1(-\overline{s})}{F}_2(s)ds}$

This theorem is proved by applying the inverse Laplace transform on the convolution theorem in form of the cross-correlation.

Let⁠${\displaystyle f(t)}$⁠ be a function with bilateral Laplace transform⁠${\displaystyle F(s)}$⁠ in the strip of convergence⁠⁠.Let⁠${\displaystyle c\in \mathbb{R}}$⁠ with⁠⁠.Then thePlancherel theorem holds:^[2]

${\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{-2ct}|f(t){|}^{2}dt=\frac{1}{2\pi }{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}|F(c+ir){|}^{2}dr}$

For any two functions⁠${\displaystyle f}$⁠ and⁠${\displaystyle g}$⁠ for which the two-sided Laplace transforms⁠${\displaystyle \mathcal{T}\{f\}}$⁠,⁠${\displaystyle \mathcal{T}\{g\}}$⁠ exist, if⁠${\displaystyle \mathcal{T}\{f\}=\mathcal{T}\{g\}}$⁠, i.e.⁠${\displaystyle \mathcal{T}\{f\}(s)=\mathcal{T}\{g\}(s)}$⁠ for every value of⁠${\displaystyle s\in \mathbb{R}}$⁠, then⁠${\displaystyle f=g}$⁠almost everywhere.

Bilateral transform requirements for convergence are more difficult than for unilateral transforms. The region of convergence will be normally smaller.

If⁠${\displaystyle f}$⁠ is alocally integrable function (or more generally aBorel measure locally ofbounded variation), then the Laplace transform⁠${\displaystyle F(s)}$⁠ of⁠${\displaystyle f}$⁠ converges provided that the limit

${\displaystyle \underset{R\to \mathrm{\infty }}{lim}{\int }_0^{R}f(t)e^{-st}dt}$

exists. The Laplace transform converges absolutely if the integral

${\displaystyle {\int }_0^{\mathrm{\infty }}\left|f(t)e^{-st}\right|dt}$

exists (as a properLebesgue 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${\displaystyle F(s)}$⁠ converges absolutely is either of the form⁠${\displaystyle \mathrm{\Re }s>a}$⁠ or else⁠${\displaystyle \mathrm{\Re }s\ge a}$⁠, where⁠${\displaystyle a}$⁠ is anextended real constant,⁠${\displaystyle -\mathrm{\infty }\le a\le \mathrm{\infty }}$⁠. (This follows from thedominated convergence theorem.) The constant⁠${\displaystyle a}$⁠ is known as the abscissa ofabsolute convergence, and depends on the growth behavior of⁠${\displaystyle f(t)}$⁠.^[3] Analogously, the two-sided transform converges absolutely in a strip of the form⁠⁠, and possibly including the lines⁠${\displaystyle \mathrm{\Re }s=a}$⁠ or⁠${\displaystyle \mathrm{\Re }s=b}$⁠.^[4] The subset of values of⁠${\displaystyle s}$⁠ 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 isanalytic in the region of absolute convergence.

Similarly, the set of values for which⁠${\displaystyle F(s)}$⁠ converges (conditionally or absolutely) is known as the region of conditional convergence, or simply theregion of convergence (ROC). If the Laplace transform converges (conditionally) at⁠${\displaystyle s=s_0}$⁠, then it automatically converges for all⁠${\displaystyle s}$⁠ with⁠${\displaystyle \mathrm{\Re }s>\mathrm{\Re }{s}_0}$⁠). Therefore, the region of convergence is a half-plane of the form⁠${\displaystyle \mathrm{\Re }s>a}$⁠, possibly including some points of the boundary line⁠${\displaystyle \mathrm{\Re }s=a}$⁠. In the region of convergence,⁠${\displaystyle \mathrm{\Re }s>\mathrm{\Re }{s}_0}$⁠), the Laplace transform of⁠${\displaystyle f}$⁠ can be expressed byintegrating by parts as the integral

${\displaystyle F(s)=(s-s_0){\int }_0^{\mathrm{\infty }}{e}^{-(s-s_0)t}\beta (t)dt,\beta (u)={\int }_0^u{e}^{-s_{0}t}f(t)dt.}$

That is, in the region of convergence⁠${\displaystyle F(s)}$⁠ can effectively be expressed as the absolutely convergent Laplace transform of some other function. In particular, it is analytic.

There are severalPaley–Wiener theorems concerning the relationship between the decay properties of⁠${\displaystyle f}$⁠ and the properties of the Laplace transform within the region of convergence.

In engineering applications, a function corresponding to alinear time-invariant (LTI) system isstable if every bounded input produces a bounded output.

Bilateral transforms do not respectcausality. They make sense when applied over generic functions but when working with functions of time (signals) unilateral transforms are preferred.

Following list of interesting examples for the bilateral Laplace transform can be deduced from the corresponding Fourier or unilateral Laplace transformations(see alsoBracewell (2000)):

Selected bilateral Laplace transforms

Function	Time domain ⁠${\displaystyle f(t)={\mathcal{B}}^{-1}\{F\}(t)}$⁠	Laplaces-domain ⁠${\displaystyle F(s)=\mathcal{B}\{f\}(s)}$⁠	Region of convergence	Comment
Rectangular impulse		${\displaystyle 2s^{-1}\sinh \frac{s}{2}}$
Triangular impulse		${\displaystyle {\left(2s^{-1}\sinh \frac{s}{2}\right)}^2}$
Gaussian impulse	${\displaystyle \exp \left(-a^2{t}^2-bt\right)}$	${\displaystyle \frac{\sqrt{\pi }}{a}\exp \frac{(s+b{)}^2}{4a^2}}$		${\displaystyle \mathrm{\Re }(a^2)>0}$
Exponential decay		${\displaystyle \frac{1}{s+a}}$		${\displaystyle u(t)}$ is the Heaviside step function
Exponential growth		${\displaystyle \frac{1}{s+a}}$
	${\displaystyle e^{-|t|}}$	${\displaystyle \frac{2}{1-s^2}}$
	${\displaystyle e^{-a|t|}}$	${\displaystyle \frac{2a}{a^2-s^2}}$		${\displaystyle \mathrm{\Re }a>0}$
	${\displaystyle \frac{1}{\cosh t}}$	${\displaystyle \frac{\pi }{\cos (\pi s/2)}}$
	${\displaystyle \frac{1}{1+e^{-t}}}$	${\displaystyle \frac{\pi }{\sin (\pi s)}}$

• Causal filter
• Acausal system
• Causal system
• Sinc filter – ideal sinc filter (aka rectangular filter) is acausal and has an infinite delay.

• Integral transforms
• Laplace transforms

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