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Example 1.2
Given the signal x(t), shown in Figure 1.13(a), the signal x(3t/2) corresponds to a linear compression of x(t) by a factor of 2/3 as illustrated in Figure 1.13(d). Specifically we note that the value of x(t) at t = t0 occurs in x(3t/2) at t = 2t0/3. For example, the value of x(t) at t = 1 is found in x(3t/2) at t = (1)2/3 = 2/2. Also, since x(t) is zero for t < 0, we have x(3t/2) zero for t < 0. Similarly, since x(t) is zero for t > 2, x(3t/2) is zero for t >4/3. Example 1.3
Suppose that we would like to determine the effect of transforming the independent variable of a given signal, x(t), to obtain a signal of the form x(?t+?), where ? and ? are given numbers. A systematic approach to doing this is to first delay or advance x(t) in accordance with the value of ?, and then to perform time scaling and/or time reversal on the resulting signal in accordance with the value of ?. The delayed or advanced signal is linearly stretched if |?|<1, linearly compressed if |?| > 1, and reversed in time if?<0.
To illustrate this approach, let us show how x(3t/2+1) may be determined for the signal x(t) shown in Figure 1.13(a). Since ? = 1, we first advance (shift to the left) x(t) by 1 as shown in Figure 1.13(b). Since |?|=3/2, we may linearly compress the shifted signal of Figure 1.13(b) by a factor of 3/2 to obtain the signal shown in Figure 1.13(e).
In addition to their use in representing physical phenomena such the time shift in a sonar signal and the speeding up or reversal of an audiotape, transformations of the independent variable are extremely useful in signal and system analysis. In Section 1.6 and in Chapter 2, we will use transformations of the independent variable to introduce and analyze the properties of systems. These transformations are also important in defining and examining some important properties of signals. 1.2.2
Periodic Signals
An important class of signals that we will encounter frequently throughout this book is the class of periodic signals. A periodic continuous –time signal x (t) has the property that there is a positive value of T for which
x (t) = x ( t + T ) (1.11)
for all values of t. In other words, a periodic signal has the property that it is unchanged by a time shift of T. In this case, we say that x(t) is periodic with period T. Periodic continuous-time signals arise in a variety of contexts. For example, as illustrated in Problem 2.61, the natural response of systems in which energy is conserved, such as ideal LC circuits without resistive energy dissipation and ideal mechanical system without frictional losses, are periodic and, in fact, are composed of some of the basic periodic signals that we will introduce in Section 1.3.
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Figure 1.14 A continuous-time periodic signal.
An example of a periodic continuous-time signal is given in Figure 1.14. From the figure or from eq. (1.11), we can readily deduce that if x (t) is periodic with period T, then x (t) = x (t+ mT) for all t and for any integer m. Thus, x (t) is also periodic with period 2T, 3T, 4T…The fundamental period T0 of x (t) is the smallest positive value of T for which eq. (1.11) holds. This definition of the fundamental period works, except if x (t) is a constant. In this case the fundamental period is undefined, since x (t) is periodic for any choice of T (so there is no smallest positive value). A signal x (t) that is not periodic will be referred to as an aperiodic signal.
Periodic signals are defined analogously in discrete time. Specifically, a discrete-time signal x[n] is periodic with period N, where N is a positive integer, if it is unchanged by a time shift of N, i.e., if
x[n ]= x [n +N ] (1.12)
for all values of n. If eq.(1.12) holds, then x[n] is also periodic with period 2N, 3N,….The fundamental period N0 is the smallest positive value of N for which eq. (1.12) holds. An example of a discrete-time periodic signal with fundamental period N0 = 3 is shown in Figure 1.15.
Figure 1.15 A discrete-time periodic signal with fundamental period N0 = 3.
Example 1.4
Let us illustrate the type of problem solving that may be required in determining whether or not a given signal is periodic. The signal whose periodicity we wish to check is given by
x(t)?{cost()ifsint()ift?0 . (1.13) t?0From trigonometry, we know that cos(t + 2) = cos(t) and sin(t + 2) = sin(t). Thus, considering t >0 and t <0 separately, we see that x (t) does repeat itself over every interval of length 2. However, as illustrate in Figure 1.16, x (t) also has a discontinuity at the time origin that does not recur at any other time. Since every feature in the shape of a periodic signal must recur periodically, we conclude that the signal x (t) is not periodic.
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Figure 1.16 The signal x (t) considered in Example 1.4 1.2.3 Even and Odd Signals
Another set of useful properties of signals relates to their symmetry under time reversal. A signal x (t) or x[n] is referred to as an even signal if it is identical to its time-reversed counterpart, i.e., with its reflection about the origin. In continuous time a signal is even if
x (-t)= x (t), (1.14)
while a discrete-time signal is even if
x [- n] = x [n] (1.15)
A signal is referred to as odd if
x (- t) = - x (t), (1.16) x [- n] = - x [n]. (1.17)
An odd signal must necessarily be 0 at t = 0 or n = 0, since eqs. (1.16) and (1.17) require that x (0) = - x (0) and x [0] = - x [0]. Example of even and odd continuous-time signals are shown in Figure 1.17
Figure 1.17 (a) An even continuous-time signal;
(b) an odd continuous-time signal.
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Figure 1.18 Example of the even-odd decomposition of a discrete-time signal.
An important fact is that any signal can be broken into a sum of two signals, one of which is even and one of which is odd. To see this, consider the signal
(1.18)
which is referred to as the even part of x (t). Similarly, the odd part of x (t) is given by
(1.19)
It is a simple exercise to check that the even part is in fact even, that the odd part is odd, and that x (t) is the sum of the two. Exactly analogous definitions hold in the discrete-time case. An example of the even-odd decomposition of a discrete-time signal is given in Figure 1.18.
1.3 EXPONENTIAL AND SINUSOIDAL SIGNALS
In this section and the next, we introduce several basic continuous-time and discrete-time signals. Not only do these signals occur frequently, but they also serve as basic building block from which we can construct many other signals.
1.3.1 Continuous-Time Complex Exponential and Sinusoidal Signals
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The continuous-time complex exponential signal is of the form
(1.20)
where C and a are in general, complex numbers. Depending upon the values of these parameters, the complex exponential can exhibit several different characteristics. Real Exponential Signals
As illustrated in Figure 1.19, if C and a are real [in which case x(t) is called a real exponential], there are basically two types of-behavior. If a is positive, then as t increases x (t) is a growing exponential, a form that is used in describing many different physical processes, including chain reactions in atomic explosions and complex chemical reactions. If a is negative, then x(t)ix a decaying exponential, a signal that is also used to describe a wide variety of phenomena, including the process of radioactive decay and the responses of RC circuits and damped mechanical systems. In particular, as shown inProblems2.61 and 2.62, the natural responses of the circuit in Figure 1.1 and the automobile in Figure 1.2 are decaying exponentials. Also, we note that for a = 0, x (t) is constant.
Figure 1.19 Continuous-real exponential x (t)=Ceat: (a) a >0; (b) a <0.
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