[8.8 SUMMARY](#page-13-0)

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8.8 SUMMARY

A signal bandlimited to B Hz can be reconstructed exactly from its samples if the sampling rate fs > 2B Hz (the sampling theorem). Such a reconstruction, although possible theoretically, poses practical problems such as the need for ideal filters, which are unrealizable or are realizable only with infinite delay. Therefore, in practice, there is always an error in reconstructing a signal from its samples. Moreover, practical signals are not bandlimited, which causes an additional error (aliasing error) in signal reconstruction from its samples. When a signal is sampled at a frequency fs Hz, samples of a sinusoid of frequency (fs/2) + x Hz appear as samples of a lower frequency (fs/2) − x Hz. This phenomenon, in which higher frequencies appear as lower frequencies, is known as aliasing. Aliasing error can be reduced by bandlimiting a signal to fs/2 Hz (half the sampling frequency). Such bandlimiting, done prior to sampling, is accomplished by an anti-aliasing filter that is an ideal lowpass filter of cutoff frequency fs/2 Hz.

The sampling theorem is very important in signal analysis, processing, and transmission because it allows us to replace a continuous-time signal with a discrete sequence of numbers. Processing a continuous-time signal is therefore equivalent to processing a discrete sequence of numbers. This leads us directly into the area of digital filtering (discrete-time systems). In the field of communication, the transmission of a continuous-time message reduces to the transmission of a sequence of numbers. This opens doors to many new techniques of communicating continuous-time signals by pulse trains.

The dual of the sampling theorem states that for a signal timelimited to τ seconds, its spectrum X(ω) can be reconstructed from the samples of X(ω) taken at uniform intervals not greater than 1/τ Hz. In other words, the spectrum should be sampled at a rate not less than τ samples/Hz.

To compute the direct or the inverse Fourier transform numerically, we need a relationship between the samples of x(t) and X(ω). The sampling theorem and its dual provide such a quantitative relationship in the form of a discrete Fourier transform (DFT). The DFT computations are greatly facilitated by a fast Fourier transform (FFT) algorithm, which reduces the number of computations from something on the order of N^{2 0} to N⁰ logN0.