In practice, the expression of the signal is not available, so the signal has to be digitized (truncated and sampled) and then analyzed numerically by a digital computer.
As computer can only process finite amount of data, a signal of
must be truncated, i.e., it is assumed
to be periodic with a finite period . (We cannot assume
the signal is 0 outside the time range T, as that would still be an infinite
signal. Repeated signals carry no information while zero signals still do.)
The Fourier transform of this periodic signal now becomes
Fourier series expansion (Eq. 3.38):
Sometimes it is desirable to study the frequency characteristics of a signal over a short duration . However, the time window and the frequency resolution (the gap between two consecutive frequency components available) are inversely related, i.e., it is impossible to increase both the temporal resolution (small ) and frequency resolution (small ). When one of the resolutions is improved, the other must suffer. This relationship is similar to Heisenberg's Uncertainty Principle in quantum physics which states that it is impossible to exactly measure both the position and the momentum of a particle at the same time. The more precisely one of the quantities is determined, the less precisely the other is known. To better address this issue of time and frequency resolution, Wavelet transform can be used.
As computer can only process discrete values, the continuous signal
must be sampled:
The nth Fourier coefficient is:
Alternatively, the truncation and sampling of the discretization process can be carried out in a different order, sampling first followed by truncation.
The signal becomes discrete after sampling with rate :
The sampled signal is then truncated by a time window and becomes periodic
with time samples in a period . The spectrum become discrete (with
gap ) as well as periodic (with period ). There are also
coefficients in a period :