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*).

Since:

and:

we get:

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 ...

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