信号处理的数学方法与算法详解

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"《数学方法与信号处理算法》是一本深度探讨信号分析理论的经典著作，由Todd K. Moon和Wyne C. Stirling两位作者共同编著，两位作者分别来自犹他州立大学和杨百翰大学。本书专为对信号处理领域有深入兴趣的专业人士设计，旨在提供一套全面且深入的数学工具和算法体系。该书的核心内容涵盖了信号处理中的关键数学方法，包括但不限于傅里叶变换、滤波器设计、谱分析、小波分析、随机过程等基础概念和理论。这些方法对于理解无线通信、图像处理、音频信号分析、控制系统等领域至关重要。书中不仅阐述了理论原理，还提供了实用的计算技术和应用实例，使得读者能够将理论知识转化为实际操作能力。在编写过程中，作者注重数学严谨性和实用性相结合，确保每个概念都配以清晰的解释和详尽的推导，便于读者逐步掌握复杂的技术细节。此外，书中还包含了丰富的参考资料和索引，便于进一步深入研究和扩展知识。《数学方法与信号处理算法》不仅适合在学术界进行深入研究的研究生和教师，也适用于工程技术人员和从事信号处理实践的工程师，他们在解决实际问题时可以从中找到适用的解决方案和策略。由于其内容深度和广泛性，这本书是信号处理专业学习者不可或缺的参考书籍，也是专业发展和知识更新的重要资源。"

xiv

Conten&

....................

18.9 Inequality constraints: Kuhn-Tucker conditions

.........

18.9.1

Second-order conditions for inequality constraints

......................

18.9.2

An extension: Fritz John conditions

.......................................................

8.10 Exercises

18.11 References

.....................................................

Shortest-Path Algorithms and Dynamic Programming

...........................................

19.1 Definitions for graphs

..........................................

19.2 Dynamic programming

...........................................

19.3 The Viterbi algorithm

....................................

19.4

Code for the Viterbi algorithm

..............

19.4.1

Related algorithms: Dijkstra's and Warshall's

...........

19.4.2

Complexity comparisons of Viterbi and Dijkstra

Applications of path search algorithms

.........................

19.5 Maximum-likelihood sequence estimation

................

19.5.1

The intersymbol interference

(ISI)

channel

............................

19.5.2 Code-division multiple access

.................................

19.5.3 Convolutional decoding

......................

19.6

HMM likelihood analysis and HMM training

......................................

19.6.1 Dynamic warping

...........................

19.7 Alternatives to shortest-path algorithms

19.8 Exercises

.......................................................

.....................................................

19.9 References

Linear Programming

...............................

20.1 Introduction to linear programming

..............................

20.2

Putting a problem into standard form

.................

20.2.1

Inequality constraints and slack variables

..........................................

20.2.2 Free variables

..............................

20.2.3 Variable-bound constraints

...........................

20.2.4

Absolute value in the objective

...........................

20.3

Simple examples of linear programming

...................

20.4

Computation of the linear programming solution

.........................................

20.4.1 Basic variables

...............................................

20.4.2 Pivoting

.....................

20.4.3 Selecting variables on which to pivot

..........

20.4.4 The effect of pivoting on the value of the problem

.......................

20.4.5 Summary of the simplex algorithm

..................

20.4.6 Finding the initial basic feasible solution

...................

20.4.7

MATLAB@

code for linear programming

.................

20.4.8 Matrix notation for the silnplex algorithm

20.5 Dual problems

..................................................

....................................

20.6 Karmarker's algorithm for LP

...................

20.6.1 Conversion to Karmarker standard form

............................

20.6.2 Convergence of the algorithm

................................

20.6.3 Su~nrnary and extensions

Examples and applications of linear programming

....................................

20.7 Linear-phase

FIR

filter design

.......................

20.7.1 Least-absolute-error approximation

...........................................

20.8 Linear optimal control

Contents

..................................................

20.9 Exercises

................................................

20.10 References

Basic Concepts and Definitions

..........................................

A.! Set theory and notation

.........................................

A.2 Mappings and functions

...............................................

A.3 Convex functions

...............................................

A.4

and

Notation

. .

A.5 Continulty

.....................................................

. .

.................................................

A.6 Differentiation

..................

A.6.1 Differentiation with a single real variable

....................

A.6.2 Partial derivatives and gradients on

...................

A.6.3 Linear approximation using the gradient

............................................

A.6.4 Taylor series

...................................

A.7 Basic constrained optimization

...........................

A.8

The Holder and Minkowski inequalities

A.9 Exercises

......................................................

.....................................................

References

Completing the Square

.................................................

The scalar case

................................................

B.2 The matrix case

......................................................

B.3 Exercises

C Basic Matrix Concepts

..........................................

Notational conventions

......................................

C.2 Matrix Identity and Inverse

.............................................

C.3 Transpose and trace

.....................................

C.4 Block (partitioned) matrices

...................................................

C.5 Determinants

..........................

C.5.1 Basic properties of determinants

.............................

C.5.2 Formulas for the determinant

.........................

C.5.3 Determinants and matrix inverses

......................................................

C.6 Exercises

.....................................................

C.7 References

D Random Processes

.............................

D.l Definitions of means and correlations

....................................................

D.2 Stationarity

.................................

D.3 Power spectral-density functions

.............................

D.4

Linear systems with stochastic inputs

.....................

D.4.1 Continuous-time signals and systems

........................

D.4.2 Discrete-time signals and systems

....................................................

D.5 References

Derivatives and Gradients

........

Derivatives of vectors and scalars with respect to a real vector

.................................

1.1 Some important gradients

.................

E.2

Derivatives of real-valued functions of real matrices

.........

E.3

Derivatives of matrices with respect to scalars, and vice versa

......................................

E.4 The transformation principle

................................

E.5 Derivatives of products of matrices

List

Figures

.......................

Input loutput relation for a transfer function

...................

Realization of the AR part of a transfer function

.................................

Realization of a transfer function

Realization of a transfer function with state-variable labels

..........

Prediction error

................................................

Linear predictor as an inverse system

.............................

...........................................

PSD input and output

...............................

Representation of an adaptive filter

Identification of an unknown plant

...............................

......................

Adapting to the inverse of an unknown plant

An adaptive predictor

...........................................

Configuration for interference cancellation

........................

The Gaussian density

...........................................

Demonstration of the central limit theorem

........................

.....................

Plot of two-dimensional Gaussian distribution

A simple Markov model

........................................

A hidden Markov model

........................................

HMM

with four states

.......................................

.................................

Binary symmetric channel model

LFSR realization

...............................................

Alternative LFSR realization

....................................

A binary LFSR and its output

...................................

Simple feedback configuration

Illustration of the triangle inequality

..............................

Quantization of the vector

.....................................

................................

Comparison of

and

metrics

is interior.

is exterior. and

is neither interior nor exterior

....

Illustration of open and closed sets

...............................

..............................................

The function

(t)

Illustration of Gibbs phenomenon

................................

...............................................

subspace of

IR3

A triangle inequality interpretation

...............................

Unit spheres in

IR2

under various

norm

.........................

Chebyshev polynomials

To(t)

through

Ts(t)

for

]

........

..........................

A space and its orthogonal complement

............................................

Disjoint lines in

.....................

Decomposition of

into disjoint components

Orthogonal projection finds the closest point in

............

Orthogonal projection onto the space spanned by several vectors

...

xviii

List

Figures

........................................

The projection theorem

.....................

The first steps of the Gram-Schmidt process

.........................

Third step of the Gram-Schmidt process

.........................................

The parallelogram law

.....................................

Functions to orthogonalize

....................................

The approximation problem

........................

Approximation with one and two vectors

...............................

An error surface for two variables

............................................

Projection solution

..............................

Statistician's Pythagorean theorem

Comparison of LS. WLS. and Taylor series approximations to

...

.............

discrete function and the error in its approximation

............................................

Data for regression

Illustration of least-squares and weighted least-squares lines

.......

................................

Least-squares equalizer example

An equalizer problem

..........................................

.................................

Contour plot of an error surface

Pole-zero plot of rational

(s)

poles.

zeros)

............

as the output of a linear system driven by white noise

...........

as the output of a linear system driven by

....................

The optimal filter as the cascade of a whitening filter and a Wiener

filter with white-noise inputs

...................................

...............................

Minimum norm to a linear variety

...............

Magnitude response for filters designed using IRLS

Legendre polynomials

po(t)

through

ps(t)

for

.........

A function

(t)

and its projection onto

and

V-l

...............

The simplest scaling and wavelet functions

.......................

Illustration of scaling and wavelet functions

......................

...............................

Illustration of a wavelet transform

Multirate interpretation of wavelet transform

.....................

Illustration of the inverse wavelet transform

......................

Filtering interpretation of an inverse wavelet transform

............

................................

Perfect reconstruction filter bank

Two basis functions. and some functions represented by using them

Implementations of digital receiver processing

...................

Digital receiver

processing

.....................................

In~plementation of a matched filter rece~ver

......................

PSK signal constellation and detection example

..................

Illustration of concepts of various signal constellations

............

Block diagram for detection processing

..........................

Geometry of the operator norm

.................................

Intersections of lines form solutions of systems of linear equations

Intersecting planes:

(a)

no solution (b) infinite number

solutions

The four

fundamental

subspaces of a matrix operator

..............

.................................

Operation of the pseudoinverse

Demonstration

an ill-conditioned linear system

................

Condition

the Hilbert matrix

.................................

Condition number for a bad idea

................................

........................................

RLS

adaptive

ualiztr

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信号处理的数学方法与算法详解

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