7.7 ALIASING IN DISCRETE-TIME SAMPLING

The problem of choosing the sampling rate for a given signal was discussed in Section 3.8. The Fourier transform now provides us with a more analytical way to look this problem, and the related problem of aliasing when we sample a signal.

In Section 3.5, sampling was introduced as a multiplication of the continuous-time analog signal with the sampling pulses (the “railing” function). The Fourier series expansion of the sampling impulses of frequency O_s can be shown to be:

(7.45)

We know that sampling is equivalent to multiplying the signal x(t) by the railing function r(t) to obtain the sampled signal x_s(t). To simplify notation, we replace the cosine function with its complex exponential equivalent: