There seems to be no generally accepted notation for the two-sided transform; the
In science and engineering applications, the argument t often represents time (in seconds), and the function f(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 on an output which is a higher value of t. In population ecology, the argument t often represents spatial displacement in a dispersal kernel.
When working with functions of time, f(t) is called the time domain representation of the signal, while F(s) is called the s-domain (or Laplace 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.
In terms of the Fourier transform, we may also obtain the two-sided Laplace transform, as 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.
Since there are no superexponentially growing linear feedback networks, 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 general integral transforms, or even general harmonic analysis.
In that framework and nomenclature, Laplace transforms are simply another form of Fourier analysis, even if more general in hindsight.
may be defined in terms of the two-sided Laplace transform by On the other hand, we also have where
), so either version of the Laplace transform can be defined in terms of the other.
as above, and conversely we can get the two-sided transform from the Mellin transform by The moment-generating function of a continuous probability density function ƒ(x) can be expressed as
The following properties can be found in Bracewell (2000) and Oppenheim & Willsky (1997) Most properties of the bilateral Laplace transform are very similar to properties of the unilateral Laplace transform, but there are some important differences: Let
be functions with bilateral Laplace transforms
be a function with bilateral Laplace transform
If f is a locally integrable function (or more generally a Borel measure locally of bounded variation), then the Laplace transform F(s) of f converges provided that the limit exists.
The set of values for which F(s) converges absolutely is either of the form Re(s) > a or else Re(s) ≥ a, where a is an extended real constant, −∞ ≤ a ≤ ∞.
The constant a is known as the abscissa of absolute convergence, and depends on the growth behavior of f(t).
[3] Analogously, the two-sided transform converges absolutely in a strip of the form a < Re(s) < b, and possibly including the lines Re(s) = a or Re(s) = b.
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.
Therefore, the region of convergence is a half-plane of the form Re(s) > a, possibly including some points of the boundary line Re(s) = a.
In the region of convergence Re(s) > Re(s0), the Laplace transform of f can be expressed by integrating by parts as the integral That is, in the region of convergence F(s) can effectively be expressed as the absolutely convergent Laplace transform of some other function.
In engineering applications, a function corresponding to a linear time-invariant (LTI) system is stable if every bounded input produces a bounded output.
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 also Bracewell (2000)):