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The total energy expended over the time interval t1?t?t2 is
?1t2?t1
t2t1p(t)dt??t2t112v(t)dt (1.2) Rand the average power over this time interval is
?t2t1p(t)dt =
1t2?t1
?t2t112v(t)dt (1.3) RSimilarly, for the automobile depicted in Figure 1.2, the instantaneous power dissipated through friction is p(t) = bv 2(t), and we can then define the total energy and average power over a time interval in the same way as in eqs.(1.2) and (1.3).
With simple physical examples such as these as motivation, it is a common and worthwhile convention to use similar terminology for power and energy for any continuous-time signal x(t) or any discrete-time signal x[n]. Moreover, as we will see shortly, we will frequently find it convenient to consider signals that take on complex values. In this case, the total energy over the time interval t1≤ t ≤ t2 in a continuous-time signal x(t) is defined as
?t2t1|x(t)|2dt， （1.4）
where |x| denotes the magnitude of the (possibly complex) number x. The time-averaged power is obtained by dividing eq.(1.4) by the length, t1 - t2，of the time interval. Similarly, the total energy in a discrete-time signal x[n] over the time interval n1 ≤ n ≤ n2 is defined as
n?n1?|x[n]|n22, (1.5)
and dividing by the number of points in the interval, n1 - n2+1, yields the average power over the interval. Ii is important to remember that the terms ―power‖ and ―energy‖ are used here indepen- dently of whether the quantities in eqs.(1.4) and (1.5) actually are related to physical energy.1 Never- theless, we will find it convenient to use these terms in a general fashion.
Furthermore, in many systems we will be interested in examining power and energy in signals over an infinite time interval, i.e., for ???t??? or for ???n???. In these cases, we define the total energy as limits of eqs. (1.4) and (1.5) as the time interval increases without bound. That is, in continuous time,
E?ΔlimT???T?T|x(t)|dt?2?????|x(t)|dt, (1.6)
2and in discrete time,
E??limN??N??N?|x(t)|?N2?n????|x[n]|??2. (1.7)
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Even if such a relationship does exist, eqs. (1.4) and (1.5) may have the wrong dimensions and scalings. For
example, comparing eqs.(1.2) and (1.4), we see that if x(t) represents the voltage across a resistor, then eqs.(1.4) must be divided by the resistance (measured, for example, in ohms) to obtain units of physical energy.
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Note that some signals the integral in eq.(1.6)and (1.7) might not converge—e.g., if x(t) or x[n] equal a nonzero constant value for all time. Such signals have infinite energy, while signals with E∞ < ∞ have finite energy.
In an analogous fashion, we can define the time-averaged power over an infinite interval as
P??limand
1T??2T?T?T|x(t)|2dt (1.8)
?N1P??|x[n]|2 (1.9) ?2N?1n??Nin continuous time and discrete time, respectively. With these definitions, we can identify three important classes of signals. The first of these is the class of signals with finite total energy, i.e.,
those signals for which E∞ < ∞. Such a signal must have zero average power, since in the continuous-time case, for example, we see from eq. (1.8) that
P??limE??0. (1.10)
T??2TAn example of a finite-energy signal is a signal that takes on the value 1 for 0?t?1 and 0 otherwise. In this case, E∞ =1 and P∞ = 0.
A second class of signals are those with finite average power P∞. From what we have just seen, if P∞>0, then, of necessity, E∞ = ∞. This, of course, makes sense, since if there is a nonzero average energy per unit time (i.e., nonzero power), then integrating or summing this over an infinite time interval yields an infinite amount of energy. For example, the constant signal x[n] = 4 has infinite energy, but average power P∞ =16. There are also signals for which neither P∞ nor E∞ are finite. A simple example is the signal x(t) = t. We will encounter other examples of signals in each of these classes in the remainder of this and the following chapters.
1.2 TRANSFORMATIONS OF THE INDEPENDENT VARIABLE
A central concept in signal and system analysis is that of the transformation of a signal. For example, in an aircraft control system, signals corresponding to the actions of the pilot are transformed by electrical and mechanical systems into changes in aircraft thrust or the positions of aircraft control surfaces such as the rudder or ailerons, which in turn are transformation through the dynamics and kinematics of the vehicle into changes in aircraft velocity and heading. Also, in a high-fidelity system, an input signal representing music as recorded on a cassette or compact disc is modified in order to enhance desirable characteristics, to remove recording noise, or to balance the several components of the signal (e.g., treble and bass). In this section, we focus on a very limited but important class of elementary signal transformations that involve simple modification of the independent variable, i.e., the time axis. As we will see in this and subsequent sections of this chapter, these elementary transformations allow us to introduce several basic properties of signals and systems. In later chapters, we will find that they also play an important role in defining and characterizing far richer and important classes of systems.
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1.2.1 Examples of Transformations of the Independent Variable
A simple and very important example of transforming the independent variable of a signal is a time shift. A time shift in discrete time is illustrated in Figure 1.8, in which we have two signals x[n] and x[n-n0] that are identical in shape, but that are displaced or shifted relative to each other. We will also encounter tine shifts in continuous time, as illustrated in Figure 1.9, in which x(t-t0) represents a delayed (if t0 is positive) or advanced (if t0 is negative) version of x(t). Signals that are related in this fashion arise in applications such as radar, sonar and seismic signal processing, in which several receivers at different locations observe a signal being transmitted through a medium (water, rock, air, etc.). In this case, the difference in propagation time from the point of origin of the transmitted signal to any two receivers results in a time shift between the signals at the two receivers.
A second basic transformation of the time axis is that of time reversal. For example, as illustrated in Figure 1.10, the signal x[-n] is obtained from the signal x[n] by a reflection about n = 0 (i.e., by reversing the signal). Similarly, as depicted in Figure 1.11, the signal x(-t) is obtained from the signal x(t) by a reflection t = 0. Thus, if x(t) represents an audio tape recording, then x(-t) is the same tape recording played backward. Another transformation is that of time scaling. In Figure 1.12 we have illustrated three signals, x(t), x(2t), and x(t/2), that are related by linear scale changes in the independent variable. If we again think of the example of x(t) as a tape recording, then x(2t) is that recording played at twice the speed, and x(t/2) is the recording played at half-speed.
It is often of interest to determine the effect of transforming the independent variable of a given signal of the form x(??t + ?), where ? and ? are given numbers. Such a transformation of the independent variable preserve the shape of x(t), except that the resulting signal may be linearly stretched if |?|<1, linearly compressed if |?|<1, reversed in time if |?|<0, and shifted in time if ? is nonzero. This is illustrated in the following set of examples.
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Example 1.1
Given the signal x(t) shown in Figure 1.13(a), the signal x(t+1) corresponds to an advance (shift to the left) by one unit along the t axis as illustrated in Figure 1.13(b). Specifically, we note that the value of x(t) at t =t0 occurs in x(t+1) at t = t0-1. For example, the value of x(t) at t =1 is found in x(t+1) at t =1-1= 0. Also, since x(t) is zero for t < 0, we have x(t+1) zero for t < -1. Similarly, since x(t) is zero for t >2, x(t+1) is zero for t > 1.
Let us also consider the signal x(-t+1), which may be obtained by replacing t with –t in x(t+1). That is, x(-t+1) is the time reversed version of x(t+1). Thus, x(-t+1) may be obtained graphically by reflecting x(t+1) about the t axis as shown in Figure 1.13(c).
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