Sampling and Frequency Representation of Discrete-Time Signals

In this chapter, we present the theory of signal sampling, the operation permitting to transform a continuous-time signal into a discrete-time signal in the form of an infinite set of signal samples. We will introduce the notion of the sampling frequency and formulate the Nyquist-Shannon sampling theorem. Focused on a periodic spectrum of the sampled signal, we will show the consequences of the aliasing effect, which occurs if the sampling frequency is too small, that is, smaller than the Nyquist frequency, defined as twice the maximal frequency of an analog signal. Normalization of the angular frequency in a spectrum of the sampled signal will allow us to define the Discrete-time Fourier Transform. On the other hand, we will show that the Discrete Fourier Transform is a result of discretization of the frequency domain.