自适应滤波器算法原理与应用

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自适应滤波器自适应滤波器是一种基于算法的信号处理技术，用于实时调整滤波器的系数，以适应信号的变化。自适应滤波器的应用非常广泛，包括音频信号处理、图像处理、通信系统、雷达系统等。自适应滤波器的核心思想是使用算法来调整滤波器的系数，以使其能够适应信号的变化。常用的自适应滤波器算法包括Least Mean Square（LMS）算法、Normalized Least Mean Square（NLMS）算法、Recursive Least Square（RLS）算法等。 LMS算法是最常用的自适应滤波器算法之一。该算法的基本思想是使用梯度下降法来调整滤波器的系数，以使其能够最小化均方误差。LMS算法的优点是计算复杂度低、实时性好，但其收敛速度较慢。 NLMS算法是LMS算法的改进版本。该算法引入了归一化因子，以提高算法的收敛速度。NLMS算法的优点是收敛速度快、计算复杂度低，但其需要选择合适的步长因子。 RLS算法是基于递归最小二乘法的自适应滤波器算法。该算法的优点是收敛速度快、计算复杂度低，但其需要选择合适的遗忘因子。自适应滤波器的应用非常广泛，包括： 1. 音频信号处理：自适应滤波器可以用于音频信号的降噪、echo消除等。 2. 图像处理：自适应滤波器可以用于图像的去噪、edge检测等。 3. 通信系统：自适应滤波器可以用于通信系统中的channel均衡、信道估计等。 4. 雷达系统：自适应滤波器可以用于雷达系统中的目标检测、跟踪等。 Sayed的大作《Adaptive Filters》是自适应滤波器领域的权威著作，详细介绍了自适应滤波器的各种算法原理和应用。该书使用大量的实例来诠释自适应滤波器的算法原理，是不可多得的好书。自适应滤波器是一种非常重要的信号处理技术，广泛应用于各个领域。了解自适应滤波器的算法原理和应用是非常重要的。

xiv

Problems and Computer Projects

CONTENTS

595

PART

IX:

FAST RLS ALGORITHMS

36 Hyperbolic Rotations

36.1 Hyperbolic Givens Rotations

36.2 Hyperbolic Householder Transformations

36.3 Hyperbolic Basis Rotations

37 Fast Array Algorithm

37.1

37.2

37.3

37.4

37.A

Time-Update of the Gain Vector

Time-Update of the Conversion Factor

Initial Conditions

Array Algorithm

Chandrasekhar Filter

38 Regularized Prediction Problems

38.1 Regularized Backward Prediction

38.2 Regularized Forward Prediction

38.3 Low-Rank Factorization

39 Fast Fixed-Order Filters

39.1 Fast Transversal Filter

39.2

FAEST

Filter

39.3 Fast Kalman Filter

39.4 Stability Issues

Summary and Notes

Problems and Computer Projects

602

605

608

612

613

614

618

621

622

624

627

628

630

63 1

633

639

642

PART

LAlllCE FILTERS

40 Three Basic Estlmation Problems

40.1 Motivation for Lattice Filters

40.2 Joint Process Estimation

40.3 Backward Estimation Problem

40.4

Forward Estimation Problem

40.5

Time and Order-Update Relations

41 Lattice Filter Algorithms

41.1 Significance

Data Structure

41.2 A Posteriori-Based Lattice Filter

653

654

656

659

662

664

669

672

Preface

am indebted to the many readers and colleagues who have written to me on several occasions with

encouraging feedback

my earlier textbook

Fundamentals ofAdaptive Filtering

(Wiley,

NJ,

2003).

Their enthusiastic comments encouraged me to pursue this second project in an effort to create a

revised version for teaching purposes. During this exercise,

decided to remove some advanced

material and move select topics to the problems.

also opted to fundamentally restructure the entire

text into eleven consecutive

parts

with each part consisting of a series of focused lectures and ending

with bibliographic comments, problems, and computer projects. I believe this restructuring into

a sequence of lectures will provide readers and instructors with more flexibility in designing and

managing their courses. I also collected most background material on random variables and linear

algebra into three chapters at the beginning of the book. Students and readers have found

this

material

of independent interest in its own right. At the same time,

decided to maintain the same general style

and features of the earlier publication in terms of presentation and exposition, motivation, problems,

computer projects, summary, and bibliographic notes. These features have been well received by our

readers.

AREA

STUDY

Adaptive filtering is a topic of immense practical relevance and deep theoretical challenges that

persist even to this date. There are several notable texts on the subject that describe many of the

features that have marveled students and researchers over the years. In this textbook, we choose

to step back and to take a broad look at the field. In

doing, we feel that we are able to bring

forth, to the benefit of the reader, extensive commonalities that exist among different classes of

adaptive algorithms and even among different filtering theories. We are also able to provide a uniform

treatment of the subject in a manner that addresses some existing limitations, provides additional

insights, and allows for extensions of current theory.

We do not have any illusions about the difficulties that arise in any attempt at understanding

adaptive filters more fully. This is because adaptive filters are, by design, time-variant, nonlinear,

and stochastic systems. Any one of these qualifications alone would have resulted in a formidable

system to study. Put them together and you face an almost impossible task. It is

wonder then that

current practice tends to study different adaptive schemes separately, with techniques and assump-

tions that are usually more suitable for one adaptation form over another. It is also no surprise that

most treatments of adaptive filters, including the one adopted in this textbook, need to rely on some

simplifying assumptions in order to make filter analysis and design a more tractable objective.

Still, in our view, three desirable features of any study of adaptive filters would be

(1)

to attempt

to keep the number of simplifying assumptions to a minimum,

(2)

to delay their use until necessary,

and

(3)

to apply similar assumptions uniformly across different classes of adaptive algorithms. This

last feature enables

to evaluate and compare the performance of adaptive schemes under similar

assumptions on the data, while delaying the use of assumptions enables

to extract the most infor-

mation possible about actual filter performance. In our discussions in this book we pay particular

attention to these three features throughout the presentation.

addition, we share the conviction that a thorough understanding of the performance and limi-

tations of adaptive filters requires a solid grasp of the fundamentals of least-mean-squares estimation

xvii

xviii

PREFACE

theory. These fundamentals help the designer understand what it is that an adaptive filter is trying

to accomplish and how well it performs in

this

regard. For

this

reason, Parts

(Optimal

Estimation)

and

(Linear Estimation)

the

book are designed to provide the reader with a self-contained and

easy-to-follow exposition of estimation theory, with a focus on topics that are relevant to the subject

matter of the book. In these initial parts, special emphasis is placed on geometric interpretations of

several fundamental results.

The

reader is advised to pay close attention to these interpretations since

it will become clear, time and again, that cumbersome algebraic manipulations can often be simpli-

fied by recourse to geometric constructions. These constructions not only provide a more lasting

appreciation for the results of the book, but they also expose the reader to powerful tools that can be

useful in other contexts as well, other than adaptive filtering and estimation theory.

The reader is further advised to master the convenience of the vector notation, which is used

extensively throughout

this

book. Besides allowing a compact exposition of ideas and a compact

representation of results, the vector notation also allows

to exploit to great effect several important

results from linear algebra and matrix theory and to capture, in elegant ways, many revealing charac-

teristics of adaptive filters. We cannot emphasize strongly enough the importance

linear algebraic

and matrix tools in our presentation, as well as the elegance that they bring to the subject. The com-

bined power of the geometric point of view and the vector notation is perhaps best exemplified by our

detailed treatment later in

this

book of least-squares theory and its algorithmic variants.

course,

the reader

exposed to geometric and vector formulations in the early chapters of the book.

STRUCTURE

THE

BOOK

The book is divided into eleven core parts, in addition to a leading part on

Background Marerial

and

a trailing part on

References and Indices.

Table P.l lists the various parts. Each

the core parts,

numbered

through

XI,

consists of four distinctive elements in the following order: (i) a series of lec-

tures where the concepts are introduced, (ii) a

summary

of all lectures combined, (iii) bibliographic

commentary, and (iv) problems and computer projects.

Lectures and Concepts.

In the early parts of the book, each concept is motivated from first

principles; starting from the obvious and ending with the more advanced. We follow

this

route of

presentation until the reader develops enough maturity in the field.

the book progresses, we ex-

pect the reader to become more sophisticated and, therefore, we cut back on the “obvious.”

Summaries.

For ease of reference, at the end of each part, we collect a summary of the key con-

cepts and results introduced in the respective lectures.

Bibliographic Commentaries.

In the remarks at the end of each part we provide a wealth of

references

the main contributors to the results discussed in the respective lectures. Rather than

scatter references throughout the lectures, we find it useful to collect all references at the end of the

part in the form of a narrative. We believe that

this

way

presentation gives the reader a more

focused perspective on how

the

references and the contributions relate to each other both in time and

context.

Problems.

The book contains a significant number of problems, some more challenging than others

and some more applied than others. The problems should be viewed as an

integral

part of the text,

especially since additional results appear in them. It

for

this

reason, and also for the benefit of the

reader, that we have chosen to formulate and design most problems in a guided manner. Usually,

and especially in

the

more challenging cases, a problem

starts

by stating its objective followed by

a sequence

guided steps until the final answer

attained. In most cases, the answer to each

step appears stated in the body of the problem. In

this

way, a reader would know what the answer

should be, even if the reader fails to solve the problem. Thus rather than ask the reader to “find an

expression for

x,”,

we would generally ask instead to “show that

is given by

. .

.”

and then

give the expression for

All

instructors can request copies

free

solutions manual from

the

publisher.

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