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The Laplace transformation
A signal that has a definite beginning in
time (such as the seismic waves from an earthquake)
can be decomposed into exponentially growing,
stationary, or exponentially decaying
sinusoidal signals with the Laplace integral transformation:

(6) 
The first integral defines the inverse transformation
(the synthesis of the given signal) and the second
integral the forward transformation (the analysis).
It is assumed here that the signal begins at or
after the time origin. s is a
complex variable that may assume any value
for which the second integral converges
(depending on f(t), it may not converge when shas a negative real part). The Laplace
transform is then said to "exist" for this
value of s. The real parameter
which defines
the path of integration for the inverse
transformation (the first integral) can be arbitrarily
chosen as long as it remains on the right side of all
singularities of F(s) in the complex splane. This
parameter decides whether f(t) is synthesized
from decaying (), stationary (), or
growing ()
sinusoidals (remember that
e^{st} under the integral
represents a growing or decaying sinewave,
and with imaginary s a pure sinewave).
The time
derivative
has the Laplace transform sF(s),
the second derivative
has s^{2} F(s),
etc. Suppose now that an analog dataacquisition
or dataprocessing system is characterized
by the linear differential equation

(7) 
where f(t) is the input signal, g(t) is the
output signal, and the c_{i} and d_{i} are constants.
We may then subject each term in
the equation to a Laplace transformation and obtain
c_{2} s^{2} F(s)+c_{1} s F(s)+c_{0} F(s)=
d_{2} s^{2} G(s)+d_{1} s G(s)+d_{0} G(s)

(8) 
from which we get

(9) 
We have thus expressed the Laplace transform
of the output signal by the Laplace transform
of the input signal, multiplied by a known
rational function of s. From this we can obtain the
output signal itself by an inverse Laplace
transformation. This means, we can solve the
differential equation by transforming it into
an algebraic equation for the Laplace transforms.
Of course, this is only practical when we are able
to evaluate the integrals analytically, which is
the case for a wide range of "mathematical"
signals. Real signals would have to be approximated by
mathematical functions for a transformation. The method
can obviously be applied to linear and time
invariant differential equations of any order.
(Timeinvariant means that the properties of the
system, and hence the coefficients of the differential
equation, do not depend on time.)
The rational function

(10) 
is the (Laplace) transfer function of the
system described by the differential equation (7). It
contains the same information on the system
as the differential equation itself. Generally, the transfer function
H(s) of an LTI system is the complex function for which

(11) 
with F(s) and G(s) representing the Laplace
transforms of the input and output signals.
Next: The Fourier transformation
Up: Basic Theory
Previous: The complex notation
Erhard Wielandt
20021108