Anderson & Moore - Optimal Filtering in Information Science

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"Anderson-1979-Optimal filtering copy.pdf" 是一本由Brian D. O. Anderson和John B. Moore合著的书籍，属于"Optimal Filtering"（最优滤波）这一主题，是Prentice-Hall, Inc.在1979年出版的信息与系统科学系列书籍之一，由Thomas Kailath担任编辑。本书详细探讨了最优滤波理论，这是信号处理和控制工程中的一个关键领域。最优滤波涉及如何从噪声中提取有用信息，尤其是在系统存在不确定性的情况下。它广泛应用于雷达探测、通信系统、导航系统以及许多其他需要实时数据处理的应用。 Anderson和Moore的著作涵盖了滤波理论的核心概念，包括卡尔曼滤波器（Kalman Filter），这是一种用于估计动态系统的状态的数学方法，特别适用于线性高斯系统。他们可能讨论了滤波器的设计、性能分析以及滤波器在实际问题中的应用。书中可能包含以下内容： 1. 滤波理论基础：介绍了滤波的基本原理，如线性系统理论和随机过程的概念。 2. 最优滤波算法：详细阐述了如何构建和实现最优滤波器，包括预测滤波和更新滤波步骤。 3. 卡尔曼滤波器：全面讲解了卡尔曼滤波的数学框架，包括状态空间模型、预测和更新方程，以及协方差矩阵的更新。 4. 非线性滤波：可能探讨了扩展卡尔曼滤波等处理非线性问题的方法。 5. 应用实例：书中可能包含了雷达探测、通信系统、航空导航等领域的应用案例，展示最优滤波的实际效果。 6. 系统建模：讨论了如何建立动态系统的数学模型，这对于滤波器设计至关重要。 7. 噪声处理：涉及噪声的来源、特性、测量以及如何在滤波过程中考虑噪声的影响。这本书对于电气工程、控制工程和信号处理专业的学生和从业者来说是一本宝贵的参考资料，它提供了深入理解滤波理论和技术的途径，并为解决实际工程问题提供了理论基础。通过阅读此书，读者可以学习到如何利用最优滤波技术在复杂环境中提高数据处理的精度和效率。

6 INTRODUCTION

Ch.

linear systems with associated gaussian random processes; it transpires that

the optimal filter in a minimum mean-square-error sense

then linear.

course, many applications involve nonlinear systems and/or nongaussian

random processes, and for these situations, the optimal filter

nonlinear.

However, the plain fact

the matter

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

used as a substitute for an

optimal nonlinear filter, or some form of nonlinear filter may be derived which

is in some way a modification

a linear filter or, sometimes, a collection

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

discussion

linear filtering, since one can hardly begin to study nonlinear

filtering with any effectiveness without a knowledge

linear filtering.

Finite dimensionality.

turns 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

generally what happens in the modeling

process. The finite-dimensional modeling

the physical system then leads to

an associated finite-dimensional filter. This filter

will be suboptimal to the

extent that the model

the physical system

in some measure an inaccurate

reflection

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

do in places in this book,

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

are easier

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

filters.

1.4

OUTLINE

THE

BOOK

The book falls naturally into three parts.

The first

part

the book

devoted to the formulation and solution

the basic Kalman filtering problem.

the end

the first section

Chapter

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

the end

Chapter

have seen it in use.

The second

part

the book

concerned with a deeper examination

the operational

and

computational properties

the filter.

For

example, there

discussion

time-invariant filters, including special techniques for com-

puting these filters, and filter stability;

the, Kalman filter

shown to have a

signal-to-noise ratio enhancement property.

In the third

part

the book, there are a number

developments taking

Ch. 1

REFERENCES 7

off from the basic theory.

For

example, the topics

smoothers, nonlinear

and adaptive filters, and spectral factorization are all covered.

There

also a collection

appendices to which the reader will probably

refer on a number

occasions. These deal with probability theory and

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

theory.

and large,

expect a reader to know some,

but

not all,

the

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

allow learning

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

a new idea

encountered, they will permit the reader to learn much, simply

by using the ideas.

Last,

make the point

that

there are many ideas developed in the

problems. Many are

not

routine.

REFERENCES

[1]

SKILLING,

H.,

Electrical Engineering Circuits,

John

Wiley & Sons,

Inc.,

New

York,

1957.

[2)

STORER,

J. E., Passive Network Synthesis,

McGraw-Hill

Book

Company,

New

York,

1957.

(3)

TERMAN,

E., Electronic and Radio Engineering, McGraw-Hill Book

Com-

pany,

New

York,

1955.

(4)

VITERBI,

J.,

Principles

Coherent Communication, McGraw-Hill

Book

Com-

pany,

New

York,

1966.

[5]

Gow,

B.,

and

RADER,

Digital Processing

Signals, McGraw-Hill

Book.

Company,

New

York,

1969.

[6]

RABINER,

R.,

and

GOLD,

Theory and Application

Digital Signal Pro-

cessing, Prentice-Hall, Inc., Englewood Cliffs,

N.J.,

1975.

[7]

OPPENHEIM,

A. V.,

and

SCHAFER,

Digital Signal Processing, Prentice-

Hall,

Inc.,

Englewood

Cliffs,

N.J.,

1975.

[8]

WIENER,

N.,

Extrapolation, Interpolation & Smoothing

Stationary Time Series,

The

M.l.T.

Press,

Cambridge,

Mass., 1949.

[9]

KoLMOGOROY,

N.,

"Interpolation

and

Extrapolation,"

Bull.

/'academie

des sciences

U.S.S.R., Ser.

Math.

1941, pp. 3-14.

(10)

WAINSTEIN,

L.A.,

and

ZUBAKOY,

Extraction

Signals from Noise,

Prentice-Hall, Inc., Englewood Cliffs,

N.J.,

1962.

[11)

KALMAN,

E.,

and

R. S. BucY,

"New

Results

Linear Filtering

and

Predic-

tion

Theory,"

Basic Eng., Trans.

ASME,

Series

Vol. 83,

No.

3, 1961,

pp. 95-108.

[12]

KALMAN,

E.,

New

Approach

Linear

Filtering

and

Prediction

Prob-

lems," J. Basic Eng., Trans.

ASME,

Series

Vol. 82,

No.

1, 1960, pp. 35-45.

FILTERING.

LINEAR

SYSTEMS.

AND

ESTIMATION

Ch.

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

and

even formally identify the system with the model.

discussing filtering and related problems, it

implicit

that

the systems

under

consideration are noisy. The noise may arise in a number

ways.

For

example, inputs

the

system may be unknown

and

unpredictable except for

their

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

the aid

a noisy sensor, i.e., one

that

contributes on a generally

random

basis some inaccuracy

the measurement

the system

output.

Again,

outputs

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

channel.

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

that

the

output

measurement process

noisy. On most occasions,

the

inputs also

will be assumed

be noisy.

Now

let us consider exactly

what

we mean by filtering. Suppose there

some

quantity

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

tion whose value we would like to know

each instant

time.

For

the sake

argument, assume the system in question

a continuous time system, and

the quantity in question

denoted by

• ). *

may be

that

this quantity

not

directly measurable,

that

it can only be measured with error.

any

case, we shall suppose

that

noisy measurements

·)

are available, with z(

·)

not

the same as

s(.

The

term

filtering

used in two senses. First, it

used as a generic term:

filtering

the recovery from z(

·)

s( · ), or

approximation to

· ),

even some information

abouts(.).

other words, noisy measurements

system are used

obtain information

about

some quantity

that

is essentially

internal

the system. Second, it

used to distinguish a certain kind

information processing from two related kinds, smoothing

and

prediction.

this sense, filtering means the recovery

time t

some information

about

s(t) using measurements

till time t. The

important

thing

note

the

triple occurrence

the time argument

First,

are concerned with obtain-

ing information

about

·)

time t, i.e., s(t). Second, the information

available

time t,

not

some later time. Third, measurements right

to,

but

not

after, time t are used. [If information

about

s(t)

be available

time t,

then

causality rules

out

the use

measurements taken later

than

time

tin

producing

this

information.]

example

the application

filtering in everyday life

in radio

reception. Here the signal

interest

the voice signal. This signal

used

modulate a high frequency carrier

that

transmitted

a radio receiver. The

received signal is inevitably corrupted by noise,

and

so, when demodulated,

filtered

recover as well as possible the original signal.

•Almost without exception throughout the book,

x(t)

will denote the value taken by

a function

time t, and x(

·)

will

denote that function. Therefore,

x(t)

is a number, and

x(.)

infinite set

pairs, {t, x(t)},

fort

ranging over all possible values.

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