Chapter-02-Defining-Signals-and-Systems

Chapter-02-Defining-Signals-and-Systems - Chapter 2...

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Unformatted text preview: Chapter 2 Defining Signals and Systems The previous chapter describes the representation of signals and systems as functions, concentrating on how to select the domain and range. This chapter is concerned with how to give more complete definitions of these functions. In particular, we need an assignment rule , which specifies how to assign an element in the range to each element in the domain. There are many ways to give an assignment rule. A theme of this chapter is that these different ways have complementary uses. Procedural descriptions of the assignment rule, for example, are more convenient for synthesizing signals or constructing implementations of a system in software or hardware. Mathematical descriptions are more convenient for analyzing signals and systems and determining their properties. In practice it is often necessary to use several descriptions of assignment rules in combination, be- cause of their complementary uses. In designing systems, a practicing engineer is often reconciling these diverse views to ensure, for instance, that a particular hardware device or piece of software in- deed implements a system that is specified mathematically. We begin with a discussion of functions in general, and then specialize to signals and systems. 2.1 Defining functions A function f : X Y assigns to each element in X a single element in Y , as illustrated in figure 2.1 . This assignment can be defined by declaring the mathematical relationship between the value in X and the value in Y , by graphing or enumerating the possible assignments, by giving a procedure for determining the value in Y given a value in X , or by composing simpler functions. We go over each of these in more detail in this section. Example 2.1: In section 1.1.5 we mentioned that sequences are a special kind of function. An infinite sequence s is a function that maps the natural numbers into some set Y , as illustrated in figure 2.2 . This function fully defines any infinite sequence of elements in Y . 41 42 CHAPTER 2. DEFINING SIGNALS AND SYSTEMS X x 1 x 2 x 3 x 4 Y y 1 y 2 y 3 y 4 f : X Y Figure 2.1: A function f : X Y assigns to each element in X a single ele- ment in Y . Naturals 1 2 3 4 Y y 1 y 2 y 3 y 4 s : Naturals Y ... Figure 2.2: An infinite sequence s is a function s : Naturals Y that assigns to each element in Naturals a single element in Y . 2.1. DEFINING FUNCTIONS 43 2.1.1 Declarative assignment Consider the function Square : Reals Reals given by 2200 x Reals , Square ( x ) = x 2 . (2.1) In ( 2.1 ), we have used the universal quantifier symbol 2200 , which means for all or for every to declare the relationship between values in the domain of the function and values in the range....
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This note was uploaded on 09/29/2010 for the course EE 20N taught by Professor Ayazifar during the Spring '08 term at University of California, Berkeley.

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Chapter-02-Defining-Signals-and-Systems - Chapter 2...

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