Signals and Systems - Electrical Engineering

3.2 The Two-Sided Laplace Transform 169

The two-sided Laplace transform of a continuous-time functionf(t)is

F(s)=L[f(t)]=

∫∞

−∞

f(t)e−stdt s∈ROC (3.2)

where the variables=σ+j, withas the frequency in rad/sec andσas a damping factor. ROC stands for

the region of convergence—that is, where the integral exists.

The inverse Laplace transform is given by

f(t)=L−^1 [F(s)]=

1

2 πj

σ∫+j∞

σ−j∞

F(s)estds σ∈ROC (3.3)

Remarks

n The Laplace transform F(s)provides a representation of f(t)in the s-domain, which in turn can be con-
verted back into the original time-domain functon in a one-to-one manner using the region of convergence.
Thus,

F(s) ROC ⇔ f(t)

n If f(t)=h(t), the impulse response of an LTI system, then H(s)is called thesystemortransfer function
of the system and it characterizes the system in the s-domain just like h(t)does in the time-domain. If f(t)
is a signal, then F(s)is its Laplace transform.
n The inverse Laplace transform in Equation (3.3) can be understood as the representation of f(t)(whether
it is a signal or an impulse response) by an infinite summation of complex exponentials with weights
F(s)at each. The computation of the inverse Laplace transform using Equation (3.3) requires complex
integration. Algebraic methods will be used later to find the inverse Laplace transform, thus avoiding the
complex integration.

Laplace and Heaviside

The Marquis Pierre-Simon de Laplace (1749–1827) [2, 7] was a French mathematician and astronomer. Although from hum-

ble beginnings he became royalty by his political abilities. As an astronomer, he dedicated his life to the work of applying

the Newtonian law of gravitation to the entire solar system. He was considered an applied mathematician and, as a member

of the Academy of Sciences, knew other great mathematicians of the time such as Legendre, Lagrange, and Fourier. Besides

his work on celestial mechanics, Laplace did significant work in the theory of probability from which the Laplace transform

probably comes. He felt that “the theory of probabilities is only common sense expressed in number.” Early transformations

similar to Laplace’s had been used by Euler and Lagrange. It was, however, Oliver Heaviside (1850–1925) who used the

Laplace transform in the solution of differential equations. Heaviside, an Englishman, was a self-taught electrical engineer,

mathematician, and physicist [76].