7. Input-Output Maps

For the state linear system as introduced in Chap. 6, two input-output maps are introduced. The first is in time domain, and writes the output y as a convolution of the input with the impulse response. The second one is the transfer function, which is in frequency domain. Instead of introducing the transfer function via the Laplace transform, we do it via exponential solutions, i.e., for the input given by \(u(t) =u_0 e^{st}\) we search for an output of a similar form, \(y(t) = y_0 e^{st}\). The mapping \(u_0 \rightarrow y_0\) is our transfer function. The relation between this transfer function and the Laplace transform of the impulse response is studied. Furthermore, it is shown that for systems which are input-output stable the Laplace transform of the output equals the transfer function times the Laplace transform of the input. The chapter ends with a set of 27 exercises and a notes and references section.