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:

$$f(t) = \frac{1}{2 \pi j} \int_{\sigma - j \infty}^{\sigma + j... ... t} \ ds, \qquad F(s) = \int_{0}^{\infty} f(t) \ e^{-s t} \ dt$$ (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 $\sigma$ 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 ($\sigma<0)$), stationary ($\sigma=0$), or growing ($\sigma>0$) sinusoidals (remember that e^st under the integral represents a growing or decaying sinewave, and with imaginary s a pure sinewave).

The time derivative $\dot f(t)$ has the Laplace transform sF(s), the second derivative $\ddot f(t)$ has s² F(s), etc. Suppose now that an analog data-acquisition or data-processing system is characterized by the linear differential equation

$$c_2 \ddot f(t)+c_1 \dot f(t)+c_0 f(t)= d_2 \ddot g(t)+d_1 \dot g(t)+d_0 g(t)$$ (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

 

c₂ s² F(s)+c₁ s F(s)+c₀ F(s)= d₂ s² G(s)+d₁ s G(s)+d₀ G(s) (8)

from which we get

$$G(s)=\frac{c_2 s^2 + c_1 s + c_0}{d_2 s^2 + d_1 s + d_0} F(s)$$ (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. (Time-invariant means that the properties of the system, and hence the coefficients of the differential equation, do not depend on time.)

The rational function

$$H(s)=\frac{c_2 s^2 + c_1 s + c_0}{d_2 s^2 + d_1 s + d_0}$$ (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

$$G(s)=H(s) \cdot F(s)$$ (11)

with F(s) and G(s) representing the Laplace transforms of the input and output signals.