A continuous linear system whose input is x(t) produces a response y(t). This system is regulated by a linear differential equation with constant coefficients that links x(t) and y(t). The most general expression of this differential equation is in the form:
By assuming that x(t) = y(t) = 0 for t < 0, we will show that if we apply the Laplace transform to the differential equation (2.14), we will obtain an explicit relation between the Laplace transforms of x(t) and y(t).
The relation of the Laplace transforms of the input and output of the system gives the system transmittance, or even what we can term the transfer function. It equals:
This means that whatever the nature of the input (unit sample sequence, unit step signal, unit ramp signal), we can easily obtain the Laplace transform of the output:
The frequency transform of the output generated by the system can ...