[3.1 INTRODUCTION](#page-9-0)

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3.1 INTRODUCTION

A discrete-time signal is basically a sequence of numbers. Such signals arise naturally in inherently discrete-time situations such as population studies, amortization problems, national income models, and radar tracking. They may also arise as a result of sampling continuous-time signals in sampled data systems and digital filtering. Such signals can be denoted by x[n], y[n], and so on, where the variable n takes integer values, and x[n] denotes the nth number in the sequence labeled x. In this notation, the discrete-time variable n is enclosed in square brackets instead of parentheses, which we have reserved for enclosing continuous-time variables, such as t.

Systems whose inputs and outputs are discrete-time signals are called discrete-time systems. A digital computer is a familiar example of this type of system. A discrete-time signal is a sequence of numbers, and a discrete-time system processes a sequence of numbers x[n] to yield another sequence y[n] as the output.†

A discrete-time signal, when obtained by uniform sampling of a continuous-time signal x(t), can also be expressed as x(nT), where T is the sampling interval and n, the discrete variable taking on integer values. Thus, x(nT) denotes the value of the signal x(t) at t = nT. The signal x(nT) is a sequence of numbers (sample values), and hence, by definition, is a discrete-time signal. Such a signal can also be denoted by the customary discrete-time notation x[n], where x[n] = x(nT). A typical discrete-time signal is depicted in Fig. 3.1, which shows both forms of notation. By way of an example, a continuous-time exponential x(t) = e−^t , when sampled every T = 0.1 seconds, results in a discrete-time signal x(nT) given by

$$x(nT) = e^{-nT} = e^{-0.1n}$$

^† There may be more than one input and more than one output.

Figure 3.2 Processing a continuous-time signal by means of a discrete-time system.

Clearly, this signal is a function of n and may be expressed as x[n]. Such representation is more convenient and will be followed throughout this book, even for signals resulting from sampling continuous-time signals.

Digital filters can process continuous-time signals by discrete-time systems, using appropriate interfaces at the input and the output, as illustrated in Fig. 3.2. A continuous-time signal x(t) is first sampled to convert it into a discrete-time signal x[n], which is then processed by a discrete-time system to yield the output y[n]. A continuous-time signal y(t) is finally constructed from y[n]. We shall use the notations C/D and D/C for conversion from continuous to discrete time and from discrete to continuous time. By using the interfaces in this manner, we can use an appropriate discrete-time system to process a continuous-time signal. As we shall see later in our discussion, discrete-time systems have several advantages over continuous-time systems. For this reason, there is an accelerating trend toward processing continuous-time signals with discrete-time systems.

3.1-1 Size of a Discrete-Time Signal

Arguing along the lines similar to those used for continuous-time signals, the size of a discrete-time signal x[n] will be measured by its energy Ex, defined by

$$E_x = \sum_{n=-\infty}^{\infty} |x[n]|^2$$

\n(3.1)

This definition is valid for real or complex x[n]. For this measure to be meaningful, the energy of a signal must be finite. A necessary condition for the energy to be finite is that the signal amplitude must → 0 as |n|→∞. Otherwise the sum in Eq. (3.1) will not converge. If Ex is finite, the signal is called an energy signal.

In some cases, for instance, when the amplitude of x[n] does not → 0 as |n|→∞, then the signal energy is infinite, and a more meaningful measure of the signal in such a case would be the time average of the energy (if it exists), which is the signal power Px, defined by

$$P_{x} = \lim_{N \to \infty} \frac{1}{2N + 1} \sum_{-N}^{N} |x[n]|^{2}$$

In this equation, the sum is divided by 2N + 1 because there are 2N + 1 samples in the interval from −N to N. For periodic signals, the time averaging need be performed over only one period in view of the periodic repetition of the signal. If Px is finite and nonzero, the signal is called a power signal. As in the continuous-time case, a discrete-time signal can either be an energy signal or a power signal, but cannot be both at the same time. Some signals are neither energy nor power signals.

EXAMPLE 3.1 Computing DT Energy and Power

Find the energy of the signal x[n] = n(u[n] − u[n − 6]), shown in Fig. 3.3a and the power for the periodic signal y[n] in Fig. 3.3b.

By definition,

$$E_x = \sum_{n=0}^{5} n^2 = 55$$

A periodic signal x[n] with period N⁰ is characterized by the fact that

$$x[n] = x[n+N_0]$$

The smallest value of N⁰ for which the preceding equation holds is the fundamental period. Such a signal is called N⁰ periodic. Figure 3.3b shows an example of a periodic signal y[n] of period N⁰ = 6 because each period contains 6 samples. Note that if the first sample is taken at n = 0, the last sample is at n = N⁰ − 1 = 5, not at n = N⁰ = 6. Because the signal y[n] is periodic, its power Py can be found by averaging its energy over one period. Averaging the energy over one period, we obtain

$$P_{y} = \frac{1}{6} \sum_{n=0}^{5} n^{2} = \frac{55}{6}$$

Figure 3.3 (a) Energy and (b) power computations for a signal.

DR ILL 3.1 DT Signal Classification: Energy, Power, and Neither

Show that the signal x[n] = anu[n] is an energy signal of energy Ex = 1/(1 − |a| ²) if |a| < 1, that it is a power signal of power Px = 0.5 if |a| = 1, and that it is neither an energy signal nor a power signal if |a| > 1.