Time Domain Representation of Continuous and Discrete Signals

Deterministic such as current, voltage, power, or energy represented as vectors or matrices by MATLAB®. In this book, following the widely accepted industrial standards, signals are classifi ed in two broad categories Analog Discrete Analog signals are signals capable of changing at any time. This type of signals is also referred as continuous time signals, meaning that continuous amplitude imply that the amplitude of the signal can take any value.

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Discrete time signals, however, are signals defi ned at some instances of time, over a time interval t ∈ [t0, t1]. Therefore discrete signals are given as a sequence of points, also called samples over time such as t = nT, for n = 0, ±1, ±2, …, ±N, whereas all other points are undefi ned. An analog or continuous signal is denoted by f(t), whereas a discrete signal is represented by f(nT) or in short without any loss of generality by f(n), as indicated in Figure 1.1 by dots.

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An analog signal f(t) can be converted into a discrete signal f(nT) by sampling f(t) with a constant sampling rate T (a time also referred as Ts ), where n is an integer over the range −∞ < n < +∞ large but fi nite. Therefore a large, but fi nite number of samples also referred to as a sequence can be generated. Since the sampling rate is constant (T), a discrete signal can simply be represented by f(nT) or f(n), without any loss of information (just a scaling factor of T).

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Continuous time systems or signals usually model physical systems and are best described by a set of differential equations. The analogous model for discrete models is described by a set of difference equations. Signals that occur in nature are usually analog.

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But if a signal is processed by a computer or any digital device the continuous signal must be converted to a discrete sequence (using an analog to digital converter, denoted by A/D), or mathematically by a fi nite sequence of numbers that represent its amplitude at the sampling instances. Discrete signals take the value of the continuous signals at equally spaced time intervals (nT).

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Those values can be considered an ordered sequence, meaning that the discrete signal represents mathematically the sequence: f(0), f(1), f(2), f(3), …, f(n). The spacing T between consecutive samples of f(t) is called the sampling interval or the sampling period (also referred to as Ts ).

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