ABSTRACT
Discrete-time signals are seldom found in nature. Numerous discrete signals are used to verify results or theories, for example, the daily change of the stock market in economics, the daily maximum temperature at a particular place for the climate change, and so on. However, most of the signals used by engineers and scientists are the outputs of transducers that, by nature, are continuous signals. Therefore, we are forced to digitize these signals and proceed to manipulate them at their digitized form. Hence, a continuous-time signal f(t) is digitized (sampled by a constant amount of time T) and becomes a set of numbers {f(nT)}. The sampling theorem tells us that for those signals with finite spectrum (band-limited signals) and maximum frequency ωN(known as the Nyquist frequency), we must use a sampling frequency ω s = 2 π/T or T = 2 π/ω s https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315215136/85154af4-39d8-4b43-ae90-9b7d84af8606/content/eq199.tif"/> , which should be twice as large as the Nyquist frequency, or, equivalently, the sampling time T must be less than onehalf of the Nyquist time, T N = 2 π/ω N . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315215136/85154af4-39d8-4b43-ae90-9b7d84af8606/content/eq200.tif"/> In this chapter, we shall assume that all the detected signals are band limited. This is a reasonable assumption, because almost all of the signals have been transferred through a finite-frequency filter (low-pass filter, prefiltering), transducer.
