Frequency and Time Domain
The above diagram summarises the proceedure for obtaining
the output fo(t) from the input fi(t) in the Time domain.
fo(t) is obtained by
taking the convolution of the input fi(t) with the
Unit Impulse Response I(t) of the system. The unit
impulse response is the output when the input is a unit impulse(
an input which is very narrow, high and whose area = 1 ). In
the above calculation, no Transforms are used.
The above diagram and equations summarise the proceedure used
to calculate fo(t) from fi(t) using the Fourier Transform when
the signal is aperiodic. Although
there are now 3 steps instead of 1, this method may often be
simpler because of the convolution operation. If the signal is
periodic, then finding the fourier transform means finding
the coefficients an, bn and the inverse fourier transform means
summing the fourier series.
The above equations summarise the 2 methods of calculation
and also shows that the basis for the equivalence of the 2
methods is the fact that the Fourier Transform of the unit impulse
Response I(t) is the transfer function
A(w). This is because
the Fourier Transform of the unit impulse is constant and
equal to 1.
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Copyright © Cuthbert A. Nyack.