Optimal Filtering in Information and System Sciences

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"OptimalFiltering.CourierCorporation2012AndersonBrianDOandJohnBMoore.pdf" 是一本由Brian D.O. Anderson和John B. Moore合著的书籍，属于Prentice-Hall的"INFORMATION AND SYSTEM SCIENCES SERIES"，由Thomas Kailath编辑。该书详细探讨了最优滤波理论，是信息和系统科学领域的重要著作。在书中，作者深入浅出地阐述了最优滤波的基本概念和方法。最优滤波是信号处理和控制工程中的核心概念，它涉及到如何从噪声中提取有用信息，尤其是在存在不确定性的情况下，如何设计滤波器来最大化信息的提取效率或最小化误差。这一理论在雷达探测、通信系统、控制系统等多个领域有着广泛的应用。 "Optimal Filtering"可能涵盖以下几个关键知识点： 1. **滤波理论基础**：介绍了滤波的基本原理，包括线性滤波器、非线性滤波器，以及卡尔曼滤波等经典滤波方法。这些滤波器用于消除信号中的噪声，改善信号质量。 2. **最优性标准**：讨论了衡量滤波器性能的各种指标，如均方误差（MSE）、最小化协方差等，这些都是确定滤波器最优性的关键。 3. **随机过程与统计推断**：书中可能涉及随机过程的基础知识，如马尔可夫过程、高斯过程等，以及如何利用这些工具进行统计推断和预测。 4. **kalman滤波**：作为最优滤波的一个典型例子，Kalman滤波器在动态系统状态估计中扮演重要角色，其数学模型和算法是理解最优滤波的关键。 5. **应用案例**：书中可能会提供实际应用案例，如雷达目标跟踪、遥感数据处理、经济预测等，以帮助读者理解和掌握最优滤波的实际应用。 6. **噪声处理**：噪声的来源、特性、测量方法也是书中的一部分，因为滤波的主要目的是在噪声环境中提取信号。 7. **系统模型与控制**：最优滤波常常与系统模型紧密相关，书中可能涵盖了如何建立动态系统的数学模型，以及如何设计滤波器以优化系统性能。 8. **滤波器设计与实现**：讨论了滤波器的设计方法，包括离散时间与连续时间滤波器的设计，以及如何在实际系统中实现这些滤波算法。这本书对电气工程、信号处理、控制科学、通信工程等领域的研究人员和工程师来说是一本宝贵的参考书，通过深入学习，读者能够掌握如何在复杂环境下设计和应用最优滤波技术。

~ INTRODUCTION

Ch. 1

linear systems with associated gaussian random processes; it transpires that

the optimal filter in a minimum mean-square-error sense is then linear. Of

course, man y applications involve nonlinear systems and/or nongaussian

random processes, and for these situations, the optimal filter is nonlinear.

However, the plain fact of the matter is that optimal nonlinear filter design

and implementation are very hard, if not impossible, in many instances. For

this reason, a suboptimal linear filter may often be used as a substitute for an

optimal nonlinear filter, or some form of nonlinear filter maybe derived which

is in some way a modification of a linear filter or, sometimes, a collection of

linear filters. These approaches are developed in this book and follow our

discussion of linear filtering, since one can hardly begin to study nonlinear

filtering with any effectiveness without a knowledge of linear filtering.

Finite u?mensionufity. Itturns out that finite-dimensional filters should

be used when the processes being filtered are associated with finite-dimen-

sional systems. Now most physical systems are not finite dimensional;

however, almost all infinite-dimensional systems can be approximated by

finite-dimensional systems, and this is generally what happens in the modeling

process. The finite-dimensional modeling of the physical system then leads to

an associated finite-dimensional filter. This filter will be suboptimal to the

extent that the model of the physical system is in some measure an inaccurate

reflection of physical reality. Why should one use a suboptimal filter?

Though one can without too much difficulty discuss infinite-dimensional

filtering problems in discrete time, and this we do in places in this book,

finite-dimensional filters are very much to be preferred on two grounds: they

are easier to design, and far easier to implement, than infinite-dimensional

filters.

1.4 OUTLINE OF THE BOOK

The book falls naturally into three parts.

The first part of the book is devoted to the formulation and solution of

the basic Kalman filtering problem. By the end of the first section of Chapter

3, the reader should know the fundamental Kalman filtering result, and by

the end of Chapter 3, have seen it in use.

The second part of the book is concerned with a deeper examination of

the operational and computational properties of the filter. For example, there

is discussion of time-invariant filters, including special techniques for com-

puting these filters, and filter stability; the: Kalman filter is shown to have a

signal-to-noise ratio enhancement property.

In the third part of the book, there area number of developments taking

Ch. 1

REFERENCES 7

off

from the basic theory. For example, the topics of smoothers, nonlinear

and adaptive filters, and spectral factorization are all covered.

There is also a collection of appendices to which the reader will probably

refer on a number of occasions. These deal with probability theory and

random processes, matrix theory, linear systems, and Lyapunov stability

theory. By and large, we expect a reader to know some, but not all, of the

material in these appendices. They are too concentrated in presentation to

allow learning of the ideas from scratch. However, if they are consulted when

a new idea is encountered, they will permit the reader to learn much, simply

by using the ideas.

Last, we make the point that there are many ideas developed in the

problems. Many are not routine.

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

[11]

[12]

REFERENCES

SKILLING,H. H.,

Electrical Engineering Circuits, John Wiley & Sons, Inc.,

New York, 1957.

STORER,J. E.,

Pamive Network Synrhesis, McGraw-Hill Book Company, New

York, 1957.

TERMAN,F. EL,

Electronic and Radio Engineering, McGraw-Hill Book Com-

pany, New York, 1955.

VITERBI,A. J.,

Principles of Coherent Communication, McGraw-Hill Book Com-

pany, New York, 1966.

GOLD, B., and C. M. RADER,

Digital Processing of Signals, McGraw-Hill Book’

Company, New York, 1969.

RABINER,L. R., and B.

GOLD, Theory and Application of Digital Signal Pro-

cessing,

Prentice-Hall, Inc., Englewood Cliffs, N. J., 1975.

OPPENHEIM,A. V., and R. W. SCHAFER,

Digital Signal Processing, Prentice-

Hall, Inc., Englewood Cliffs, N.J., 1975.

WIENER,N.,

Extrapolation, Interpolation & Smoothing of Stationary Time Series,

The M.I.T. Press, Cambridge, Mass., 1949.

KOLMOGOROV,A. N.,

“Interpolation and Extrapolation,”

Bull. de l’acadkmie

des sciences de U.S. S. R.,

Ser. Math. 5, 1941, pp. 3-14.

WAINSTEIN,L. A., and V. D. ZUBAKOV,

Extraction of Signals from Noise,

Prentice-Hall, Inc., Englewood Cliffs, N. J., 1962.

KALMAN,R. E., and R. S. BUCY,“New Results in Linear Filtering and Predic-

tion Theory,”

J. of Basic Eng., Trans. ASME, Series D, Vol. 83, No. 3, 1961,

pp. 95-108.

KALMAN, R. E., “A New Approach to Linear Filtering and Prediction Prob-

lems,”

J. Basic Eng., Trans. ASME, Series D, Vol. 82, No. 1, 1960, pp. 35-45.

FILTERING, LINEAR SYSTEMS, AND ESTIMATION

Ch. 2

Later, we shall pose specific mathematical models for some systems, and

even formally identify the system with the model.

In discussing filtering and related problems, it is implicit that the systems

under consideration are noisy. The noise may arise in a number of ways. For

example, inputs to the system may be unknown and unpredictable except for

their statistical properties, or outputs from the system may be derived with

the aid of a noisy sensor, i.e., one that contributes on a generally random

basis some inaccuracy to the measurement of the system output. Again,

outputs may only be observed via a sensor after transmission over a noisy

channel.

In virtually all the problems we shall discuss here, it will be assumed that

the output measurement process is noisy. On most occasions, the inputs also

will be assumed to be noisy.

Now let us consider exactly what we mean by

jiltering. Suppose there is

some quantity (possibly a vector quantity) associated with the system opera-

tion whose value we would like to know at each instant of time. For the sake

of argument, assume the system in question is a continuous time system, and

the quantity in question is denoted by s(. ).* It may be that this quantity is

not directly measurable, or that it can only be measured with error. In any

case, we shall suppose that noisy measurements Z(.) are available, with Z(. )

not the same as S(.).

The term filtering is used in two senses. First, it is used as a generic term:

filtering is the recovery from Z(.) of s(.), or an approximation to s(.), or

even some information about S(.). In other words, noisy measurements of a

system are used to obtain information about some quantity that is essentially

internal to the system. Second, it is used to distinguish a certain kind of

information processing from two related kinds, smoothing and prediction.

In this

sense, filtering means the recovery at time tof some information about

s(t)using measurements up till time t.The important thing to note is the

triple occurrence of the time argument t.First, we are concerned with obtain-

ing information about S(. ) at time t,i.e., s(t).Second, the information is

available at time t,not at some later time. Third, measurements right up to,

but not after, time t are used. [If information about s(t) is to be available at

time t, then causality rules out the use of measurements taken later than time

t in producing this information.]

An example of the application of filtering in everyday life is in radio

reception. Here the signal of interest is the voice signal. This signal is used to

modulate a high frequency carrier that is transmitted to a radio receiver. The

received signal is inevitably corrupted by noise, and so, when demodulated,

it is filtered to recover as well as possible the original signal.

●Almost without exeeption throughout the book,

x(t) will denote the value taken by

a function at time I, and x(. ) will denote that function. Therefore, x(t) is a number, and

x(. ) an infinite set of pairs, {t, x(t)}, for / ranging over all possible values.

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