Two-sided Laplace transform

Relationship to the Fourier transform

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

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

Note that definitions of the Fourier transform differ, and in particular :\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 :\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 a 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 , 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.

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 \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 m:\mathbb{R}\to\mathbb{R} is the function that multiplies by minus one (m(x) = -x), 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 m 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 ƒ(x) can be expressed as \mathcal{B}{f}(-s).

Properties

The following properties can be found in and

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 f_1(t) and f_2(t) be functions with bilateral Laplace transforms F_1(s) and F_2(s) in the strips of convergence \alpha_{1,2}. Let c\in\mathbb{R} with \max(-\beta_1,\alpha_2). Then Parseval's theorem holds: : \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 f(t) be a function with bilateral Laplace transform F(s) in the strip of convergence \alpha. Let c\in\mathbb{R} with \alpha. 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 f,g for which the two-sided Laplace transforms \mathcal{T} {f}, \mathcal{T} {g} exist, if \mathcal{T}{f} = \mathcal{T} {g}, i.e. \mathcal{T}{f}(s) = \mathcal{T}{g}(s) for every value of s\in\mathbb R, then f=g 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 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 : \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 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 ≤ ∞. (This follows from the dominated convergence theorem.) The constant a is known as the abscissa of absolute convergence, and depends on the growth behavior of f(t). Analogously, the two-sided transform converges absolutely in a strip of the form a

Similarly, the set of values for which F(s) 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 s = s0, then it automatically converges for all s with Re(s) Re(s0). 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 :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 F(s) 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 f 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 ):

References

• Van der Pol, Balthasar, and Bremmer, H., Operational Calculus Based on the Two-Sided Laplace Integral, Chelsea Pub. Co., 3rd ed., 1987.
• .

References

1. {{harvnb. LePage. 1980
2. {{harvnb. Widder. 1941
3. {{harvnb. Widder. 1941
4. {{harvnb. Widder. 1941