Discrete-Time Systems
Sampling
When we were looking at the frequency-shifting and modulation properties, I promised you that I'd tell you what the example was for.
Well, it all has to do with discrete-time sampling. Computers and digital systems typically don't work in continuous time; they operate on a clock that causes them to look at a real-world input only once every so often. This precludes ordinary integration techniques, and so everything has to be expressed as sums and we need to do some tricks to emulate what we see in the real continuous-time world.
Another problem with this approach is that our maths is all geared up to work with continuous signals... and not very well with discrete time. Thus, we need some way of modelling the sampling discipline in continuous time. The way this is done is to multiply the function by a periodic sampling function, that models the input behaviour of the sampler.
The obvious sampling function is an infinite series of Dirac impulse functions - and you should now be able to see where that example comes in useful. It demonstrates that by sampling, we have introduced additional frequencies into the function, at intervals of the sampling frequency. Therefore, if we want to obtain the original frequency distribution, we must use a low-pass filter that removes the sampling noise. This filter is placed at half the sampling frequency. It should be obvious, therefore, that the sampling rate must be high enough that this filter does not remove any of the original signal:
where ω0 is the maximum frequency in the signal, and ωs is the sampling frequency. The frequency 2ω0 is called the Nyquist frequency of the signal, and the above statement the Nyquist sampling theorem.
In practice, of course, filters are not ideal. In fact, filters with excellent frequency response often have appalling phase response, and vice-versa. In order to avoid phase distortion, therefore, filters which have worse frequency response are used - so the sampling rate must be higher to avoid roll-off of the audible frequency spectrum. This is termed oversampling.
However, we can't really do a lot with sampling unless we have a discrete, computable algorithm for Fourier transformation. This is where the discrete Fourier transform (DFT) comes in, and its more efficient counterpart, the fast Fourier transform (FFT).