2_Function_Def

2_Function_Def - Chapter 2 Real Functions Chapter Outline...

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19 Chapter 2 Real Functions Chapter Outline 2.1 CONTINUOUS-TIME FUNCTIONS. ........................................................ 19 2.2 COMMON FUNCTIONS . ........................................................................ 22 2.3 DISCRETE-TIME FUNCTIONS. ............................................................. 28 2.4 HOMEWORK FOR CHAPTER 2. ............................................................ 32 This chapter has two primary goals. First, we review basic mathematical background required for the rest of the text. Second, we establish the mathematical notation, particularly with respect to functions, which we will use throughout the text. In Sections 2.1 and 2.2 we review the concept of a function, and we introduce the notation of a number of basic functions including singularity functions. Because these functions are used extensively, it is suggested that readers familiarize themselves with the notation. Section 2.3 introduces the notation of a discrete function. This material provides the starting point for the discussion of discrete signals and systems. Summary of Sections Section 2.1: We introduce some basic concepts of real functions. Section 2.2: We define several common functions including singularity functions and some special functions. Section 2.3: We introduce discrete functions. Coverage of the Text This chapter is self-contained. 2.1 CONTINUOUS-TIME FUNCTIONS In this text we assume that the reader is familiar with elementary set theory. Notation : Let R be the set of real numbers.
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20 Chapter 2 Real Functions Definition 2.1.1 : An interval is a set with one of the forms () ( , ) ( , ] [, ) [ , ] iIt t t t t t ii I t t t t t t iii I t t t t t t iv I t t t t t t 1 =∈ << {} = <≤ = ≤< = ≤≤ = -+ - + - + - + - + R R R R 2 3 4 We say that the interval (i) is an open interval. We say that the interval (iv) is a closed interval if -∞ < < < < ∞ tt t . ▲▲ Intervals (ii) and (iii) are neither open nor closed. Notation : If t - =-∞ or t + =∞ , then the strict inequality applies. When we write It t t 1 - R (2.1.1) we mean that t can take on arbitrarily large positive values, but not infinity. Definition 2.1.2 : Let I 1 and I 2 be two intervals. Let f be a rule which assigns a member of I 2 , t 2 I 2 , to each member of I 1 , t 1 I 1 ; write f ( t 1 ) = t 2 . Then we say f is a (real) function ; write f ( t ). The interval on which a function f ( t ) is defined, I 1 , is called the domain of the function. The interval from which the function takes values, I 2 , is called the range . A function can take on only one value. Obviously, functions can be defined for any set, not just intervals. A function is not defined outside its domain. The domain of definition plays an important role in the definitions of the Laplace and Fourier transforms below. Functions are distinguished, in part by their domains of definition. Let us consider the function sin t . Suppose we define the interval t 1 0 R (2.1.2) Now define the function fI ft t 1 : 11 →= R , ( ) sin . (2.1.3) This function is shown in Figure 2.1.1. In particular, the negative real axis is not in Figure 2.1.1, because it is not in the domain of definition. Consider next the interval t 2 =∈ - = RR . (2.1.4) Define a second function
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Section 2.1 Continuous-Time Functions 21 t Figure 2.1.1 Function ft t t 1 0 ( ) sin , =≥ t Figure 2.1.2 Function tt t 2 0 00 () sin , , = < t Figure 2.1.3 Function t t 3 ( ) sin , =- < < ff t t 2 : RR →= < ,( ) sin , , 2 0 (2.1.5)
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This note was uploaded on 09/10/2009 for the course ECE 60367 taught by Professor Meehan during the Spring '09 term at Virginia Tech.

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2_Function_Def - Chapter 2 Real Functions Chapter Outline...

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