Sampling and the Sampling Theorem
When we do the Fourier or Laplace transform, or when we do their inverse transform we inevitably sample the signal—either in the time domain (former) or frequency domain (latter). The moment we migrate from equations to numbers sampling happens. Question then arises—how fine of a resolution does sampling needs to happen at? As a demonstrate vehicle we assume a time signal of finite bandwidth ω b . We show in the chapter that in order for us to be able to reconstruct the signal back from the Fourier transform the signal must have been sampled at a frequency at least twice the bandwidth frequency ω b . Sampling in the time domain is tantamount of multiplying by the train delta function. The time train function has a Fourier transform which is also a train function, but now in the frequency domain. By making the spacing of the delta functions close enough in the time domain, the spacing in the frequency domain becomes larger. If this spacing is large enough then the spectrum of the sampled signal is repeated (via convolution in the frequency domain) in such a way that each copy of the spectrum which is centered at one of the frequency delta functions does not interfere with the other. As such, doing a low-frequency filtering should enable us to extract the original transform and use to figure the inverse transform (i.e., reconstruct the signal). We show real examples that demonstrate—numerically and graphically—how under-sampling results in problems and over-sampling works perfectly fine.