Convolution and I/O in the Time domain

Cuthbert Nyack
The system of interest is illustrated above. An input fi(t) is applied to a system with impulse response I(t) and the output fo(t) is to be determined.
The meaning of the impulse response I(t) is illustrated above. If the input is a unit impulse(area = 1), then the output I(t) is the impulse response.
If the input consists of several impulses with magnitude fi(tau) then the output must be obtained by summing the outputs produced by all the input impulses.
If the input is continuous, then it can be approximated by a set of impulses of magnitude fi(tau)Delta tau and the output obtained by a summation as above. As delta tau tends to zero, then the summation can be represented by an integration which is the familiar convolution integral.

In the time domain, the output can be obtained by taking the convolution of the input with the impulse response of the system. Time domain here means that no reference is made to the frequency spectrum of the input or the frequency response of the system. All calculations are made with functions which have time as their dependent variable. In the frequency domain, the output spectrum is determined by simple multiplication of the input spectrum with the transfer function.

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Copyright © 1995 Cuthbert A. Nyack.