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D.3 Trigonometric Identities D.4 Power Series Expansions D.5 Exponential and Logarithmic Functions D.6 Some Definite Integrals Index
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1
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Signals and Systems
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1.1 INTRODUCTION
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The concept and theory of signals and systems are needed in almost all electrical engineering fields and in many other engineering and scientific disciplines as well. In this chapter we introduce the mathematical description and representation of signals and systems and their classifications. We also define several important basic signals essential to our studies.
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SIGNALS AND CLASSIFICATION OF SIGNALS
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A signal is a function representing a physical quantity or variable, and typically it contains information about the behavior or nature of the phenomenon. For instance, in a RC circuit the signal may represent the voltage across the capacitor or the current flowing in the resistor. Mathematically, a signal is represented as a function of an independent variable t. Usually t represents time. Thus, a signal is denoted by x ( t ) .
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A. Continuous-Time and Discrete-Time Signals: A signal x(t) is a continuous-time signal if t is a continuous variable. If t is a discrete variable, that is, x ( t ) is defined at discrete times, then x ( t ) is a discrete-time signal. Since a discrete-time signal is defined at discrete times, a discrete-time signal is often identified as a sequence of numbers, denoted by {x,) o r x[n], where n = integer. Illustrations of a continuous-time signal x ( t ) and of a discrete-time signal x[n] are shown in Fig. 1-1.
Fig. 1-1 Graphical representation of (a) continuous-time and ( 6 )discrete-time signals.
A discrete-time signal x[n] may represent a phenomenon for which the independent variable is inherently discrete. For instance, the daily closing stock market average is by its nature a signal that evolves at discrete points in time (that is, at the close of each day). On the other hand a discrete-time signal x[n] may be obtained by sampling a continuous-time
SIGNALS AND SYSTEMS
[CHAP. 1
signal x(t) such as x(to), +,)' or in a shorter form as or where we understand that x, = x [ n ] =x(t,) and x,'s are called samples and the time interval between them is called the sampling interval. When the sampling intervals are equal (uniform sampling), then x,, = x [ n ] =x(nT,) where the constant T, is the sampling interval. A discrete-time signal x[n] can be defined in two ways: 1. We can specify a rule for calculating the nth value of the sequence. For example, x[O], x [ l ] , ..., x [ n ] , . .. xo, x ~ ,. . ,x,, . . .
~ ( t , ) .. * ,
2. We can also explicitly list the values of the sequence. For example, the sequence shown in Fig. l-l(b) can be written as (x,)
( . . . , 0,0,1,2,2,1,0,1,0,2,0,0,... )
We use the arrow to denote the n = 0 term. We shall use the convention that if no arrow is indicated, then the first term corresponds to n = 0 and all the values of the sequence are zero for n < 0.
(c,)
= a(a,)
+C, = aa,
= constant
B. Analog and Digital Signals:
If a continuous-time signal x(l) can take on any value in the continuous interval (a, b), where a may be - 03 and b may be + m, then the continuous-time signal x(t) is called an analog signal. If a discrete-time signal x[n] can take on only a finite number of distinct values, then we call this signal a digital signal. C. Real and Complex Signals: A signal x(t) is a real signal if its value is a real number, and a signal x(t) is a complex signal if its value is a complex number. A general complex signal ~ ( t is a function of the )
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