[1.3 CLASSIFICATION OF](#page-7-0) SIGNALS

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1.3 CLASSIFICATION OF SIGNALS

Classification helps us better understand and utilize the items around us. Cars, for example, are classified as sports, offroad, family, and so forth. Knowing you have a sports car is useful in deciding whether to drive on a highway or on a dirt road. Knowing you want to drive up a mountain, you would probably choose an offroad vehicle over a family sedan. Similarly, there are several classes of signals. Some signal classes are more suitable for certain applications than others. Further, different signal classes often require different mathematical tools. Here we shall consider only the following classes of signals, which are suitable for the scope of this book:

• 1. Continuous-time and discrete-time signals
• 2. Analog and digital signals
• 3. Periodic and aperiodic signals
• 4. Energy and power signals
• 5. Deterministic and probabilistic signals

1.3-1 Continuous-Time and Discrete-Time Signals

A signal that is specified for a continuum of values of time t (Fig. 1.10a) is a continuous-time signal, and a signal that is specified only at discrete values of t (Fig. 1.10b) is a discrete-time signal. Telephone and video camera outputs are continuous-time signals, whereas the quarterly gross national product (GNP), monthly sales of a corporation, and stock market daily averages are discrete-time signals.

1.3-2 Analog and Digital Signals

The concept of continuous time is often confused with that of analog. The two are not the same. The same is true of the concepts of discrete time and digital. A signal whose amplitude can take on any value in a continuous range is an analog signal. This means that an analog signal amplitude can take on an infinite number of values. A digital signal, on the other hand, is one whose amplitude can take on only a finite number of values. Signals associated with a digital computer are digital because they take on only two values (binary signals). A digital signal whose amplitudes can take on M values is an M-ary signal of which binary (M = 2) is a special case. The terms continuous time and discrete time qualify the nature of a signal along the time (horizontal) axis. The terms analog and digital, on the other hand, qualify the nature of the signal amplitude (vertical axis). Figure 1.11 shows examples of signals of various types. It is clear that analog is not necessarily continuous-time and digital need not be discrete-time. Figure 1.11c shows an example of an analog discrete-time signal. An analog signal can be converted into a digital signal [analog-to-digital (A/D) conversion] through quantization (rounding off ), as explained in Sec. 8.3.

Figure 1.10 (a) Continuous-time and (b) discrete-time signals.

1.3-3 Periodic and Aperiodic Signals

A signal x(t) is said to be periodic if for some positive constant T⁰

$$x(t) = x(t + T_0) \qquad \text{for all } t \tag{1.7}$$

The smallest value of T⁰ that satisfies the periodicity condition of Eq. (1.7) is the fundamental period of x(t). The signals in Figs. 1.2b and 1.3e are periodic signals with periods 2 and 1, respectively. A signal is aperiodic if it is not periodic. Signals in Figs. 1.2a, 1.3a, 1.3b, 1.3c, and 1.3d are all aperiodic.

By definition, a periodic signal x(t) remains unchanged when time-shifted by one period. For this reason, a periodic signal must start at t = −∞: if it started at some finite instant, say, t = 0, the time-shifted signal x(t + T0) would start at t = −T⁰ and x(t + T0) would not be the same as 80 CHAPTER 1 SIGNALS AND SYSTEMS

Figure 1.11 Examples of signals: (a) analog, continuous time; (b) digital, continuous time; (c) analog, discrete time; and (d) digital, discrete time.

Figure 1.12 A periodic signal of period T0.

x(t). Therefore, a periodic signal, by definition, must start at t = −∞ and continue forever, as illustrated in Fig. 1.12.

Another important property of a periodic signal x(t) is that x(t) can be generated by periodic extension of any segment of x(t) of duration T⁰ (the period). As a result, we can generate x(t) from any segment of x(t) having a duration of one period by placing this segment and the reproduction thereof end to end ad infinitum on either side. Figure 1.13 shows a periodic signal x(t) of period T⁰ = 6. The shaded portion of Fig. 1.13a shows a segment of x(t) starting at t = −1 and having a duration of one period (6 seconds). This segment, when repeated forever in either direction, results in the periodic signal x(t). Figure 1.13b shows another shaded segment of x(t) of duration T⁰ starting at t = 0. Again, we see that this segment, when repeated forever on either side, results in x(t). The reader can verify that this construction is possible with any segment of x(t) starting at any instant as long as the segment duration is one period.

Figure 1.13 Generation of a periodic signal by periodic extension of its segment of one-period duration.

An additional useful property of a periodic signal x(t) of period T⁰ is that the area under x(t) over any interval of duration T⁰ is the same; that is, for any real numbers a and b,

$$\int_{a}^{a+T_0} x(t) dt = \int_{b}^{b+T_0} x(t) dt$$

This result follows from the fact that a periodic signal takes the same values at the intervals of T0. Hence, the values over any segment of duration T⁰ are repeated in any other interval of the same duration. For convenience, the area under x(t) over any interval of duration T⁰ will be denoted by

$$\int_{T_0} x(t) \, dt$$

It is helpful to label signals that start at t = −∞ and continue forever as everlasting signals. Thus, an everlasting signal exists over the entire interval −∞ < t < ∞. The signals in Figs. 1.1b and 1.2b are examples of everlasting signals. Clearly, a periodic signal, by definition, is an everlasting signal.

A signal that does not start before t = 0 is a causal signal. In other words, x(t) is a causal signal if

$$x(t) = 0 \qquad t < 0$$

The signals in Figs. 1.3a–1.3c are causal signals. A signal that starts before t = 0 is a noncausal signal. All the signals in Figs. 1.1 and 1.2 are noncausal. Observe that an everlasting signal is always noncausal but a noncausal signal is not necessarily everlasting. The everlasting signal in Fig. 1.2b is noncausal; however, the noncausal signal in Fig. 1.2a is not everlasting. A signal that is zero for all t ≥ 0 is called an anti-causal signal.

Comment. A true everlasting signal cannot be generated in practice for obvious reasons. Why should we bother to postulate such a signal? In later chapters we shall see that certain signals

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(e.g., an impulse and an everlasting sinusoid) that cannot be generated in practice do serve a very useful purpose in the study of signals and systems.

1.3-4 Energy and Power Signals

A signal with finite energy is an energy signal, and a signal with finite and nonzero power is a power signal. The signals in Figs. 1.2a and 1.2b are examples of energy and power signals, respectively. Observe that power is the time average of energy. Since the averaging is over an infinitely large interval, a signal with finite energy has zero power, and a signal with finite power has infinite energy. Therefore, a signal cannot be both an energy signal and a power signal. If it is one, it cannot be the other. On the other hand, there are signals that are neither energy nor power signals. The ramp signal is one such case.

Comments. All practical signals have finite energies and are therefore energy signals. A power signal must necessarily have infinite duration; otherwise, its power, which is its energy averaged over an infinitely large interval, will not approach a (nonzero) limit. Clearly, it is impossible to generate a true power signal in practice because such a signal has infinite duration and infinite energy.

Also, because of periodic repetition, periodic signals for which the area under |x(t)| ² over one period is finite are power signals; however, not all power signals are periodic.

DR ILL 1.6 Neither Energy nor Power

Show that an everlasting exponential e−at is neither an energy nor a power signal for any real value of a. However, if a is imaginary, it is a power signal with power Px = 1 regardless of the value of a.

1.3-5 Deterministic and Random Signals

A signal whose physical description is known completely, in either a mathematical form or a graphical form, is a deterministic signal. A signal whose values cannot be predicted precisely but are known only in terms of probabilistic description, such as mean value or mean-squared value, is a random signal. In this book we shall exclusively deal with deterministic signals. Random signals are beyond the scope of this study.