信号与系统第二版第一章知识点 - 图文

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SIGNALS AND SYSTEMS

1.0 INTRODUCTION

As described in the Foreword, the intuitive notions of signals and systems arise in a rich variety of contexts. Moreover, as we will see in this book, there is an analytical framework—that is, a language for describing signals and systems and an extremely powerful set of tools for analyzing them—that applies equally well to problems in many field. In this chapter, we begin our development of the analytical framework for signal and system by introducing their mathematical description and representation. In the chapters that follow, we build on this foundation in order to develop and describe additional concepts and methods that add considerably both to our understanding of signals and systems and to our ability to analyze and solve problems involving signals and systems that arise in a broad array of applications.

1.1 CONTINUOUS-TIME AND DISCRETE-TIME SIGNALS 1.1.1Examples and Mathematical Representation

Signals may describe a wide variety of physical phenomena. Although signals can be represented in many ways, in all case the information in a signal is contained in a pattern of variations of some form. For example, consider the simple circuit in Figure 1.1. In this case, the patterns of variation over time in the source and capacitor voltages, vs and vc, are examples of signals. Similarly, as depicted in Figure 1.2, the variations over time of the applied force f and the resulting automobile velocity v are signals. As another example, consider the human vocal mechanism, which produces speech by creating fluctuations in acoustic pressure. Figure 1.3 is an illustration of a recording of such a speech signal, obtained by using a microphone to sense variations in acoustic pressure, which

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Figure 1.1 A simple RC circuit with Figure 1.2 An automobile responding to Source voltage vs and capacitor an applied force f from the engine and to Voltage vc. a retarding frictional force ?v Proportional to the automobile’s velocity v .

are then converted in to an electrical signal. As can be seen in the figure, different sounds

correspond to different patterns in the variations of acoustic pressure, and the human vocal system produces intelligible speech by generating particular sequences of these patterns. Alternatively, for the monochromatic picture, shown in figure 1.4, it is the pattern of variations in brightness across the image that is important.

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Figure 1.4 A monochromatic picture

Signals are represented mathematically as functions of one or more independent variables. For example, a speech signal can be represented mathematically by acoustic pressure as a function of time, and a picture can be represented by brightness as a function of two spatial variables. In this book, we focus our attention on signals involving a single independent variable. For convenience, we will generally refer to the independent variable as time, although it may not in fact represent time in specific applications. For example in geophysics, signals representing variation with depth of physical quantities such as density, porosity, and electrical resistivity are used to study the structure of the earth. Also, knowledge of the variations of air pressure, temperature, and wind speed with altitude are extremely important in meteorological investigations. Figure1.5 depicts a typical examples of annual average vertical wind profile as a function of height. The measured variations of wind speed with height are used in examining weather patterns, as well as wind conditions that may affect an aircraft during final approach and landing.

Throughout this book we will be considering two basic types of signals; continuous-time signals and discrete-time signals. In the case of continuous-time signals the independent variable is continuous, and thus these signals are defined for a continuum of values of the independent variable.

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Figure 1.6 An example of a discrete time signal; The weekly Dow Jones stock market Index from January 5 1929, to January 4, 1930.

On the other hand, discrete-time signals are defined only a discrete-times and consequently, for these signals the independent variable takes on only a discrete set of values. A speech signal as a function of time and atmospheric pressure as a function of altitude are examples of continuous-time signals. The weekly Dow Jones stock market index, as illustrated in figure 1.6, is an example of a discrete time signal. Other examples of discrete-time signals can be found in demographic studies in which various attributes, such as average budget, crime rate or pounds of fish caught, are tabulated against such discrete variables as family size, total population, or type of fishing vessel, respectively.

To distinguish between continuous-time and discrete-time signals, we will use the symbol t to denote the continuous time independent variable and n to denote the discrete tine independent variable. In addition, for continuous-time signals we will enclose the independent variable in parentheses (.), whereas for discrete-time signals we will use brackets [.] to enclose the independent variable. We will also have frequent occasions when it will be useful to represent signals graphically. Illustrations of a continuous-time signal x(t) and a discrete time signal x[n] are shown in figure 1.7. it is important to note that the discrete-time signal x[n] is defined only for integer value of the independent variable. Our choice of graphical representation for x[n] as discrete time sequence.

A discrete-time signal x[n] may represent a phenomenon for which the independent variable is inherently discrete. Signals such as demographic data are examples of this. On the other hand, a very important class of discrete time signals arises from the sampling of continuous time signals. In this case, the discrete-time signal x[n] represents successive samples of an underlying phenomenon for which the independent variable is continuous. Because of their speed, computational power, and flexibility, modern digital processors are used to implement many practical systems, ranging from digital autopilots to digital audio systems. Such systems require the use of discrete-time sequences representing sampled versions of continuous-time signals—e.g.,

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Figure1.7 Graphical representations of (a) continuous time and (b) discrete time signals. aircraft position, velocity, and heading for an autopilot or speech and music for an audio system. Also, pictures in newspapers or in this book, for that matter actually consist of a very fine grid of point. And each of this points represents a sample of the brightness of the corresponding point in the original image. No matter what the source of the data, however , the signal x[n] is defined only for integer values of n. it makes no more sense to refer to the 3.5th sample of a digital speech signal than it does to refer to the average budget for a family with 2.5 family members.

Throughout most of this book we will treat discrete time signals and continuous time signals separately but in parallel, so that we can draw on insights developed in one setting to aid our understanding of another. In chapter 7 we will return to the question of sampling, and in that context we will bring continuous time and discrete time concepts together in order to examine the relationship between a continuous time signal and a discrete time signal obtained from it by sampling.

1.1.2 Signal Energy and Power

From the range of examples provided so far, we see that signals may represent a broad variety of phenomena. In many, but not all, applications, the signals we consider are directly related to physical quantities capturing power and energy in a physical system. For example, if v(t) and i(t) are, respectively, the voltage and current across a resistor with resistance R, then the instantaneous power is

p(t) = v(t)i(t) =

12v(t) (1.1) R5

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