Frequency and Time Domain

Cuthbert Nyack

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.

Frequency Domain

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.