Amazon no longer offers textbook rentals. We do!

Mathematical Methods and Algorithms for Signal Processing

by: Moon, Todd K.; Stirling, Wynn C.

Mathematical Methods and Algorithms for Signal Processing

by: Moon, Todd K.; Stirling, Wynn C.

ISBN 13:
9780201361865
ISBN 10:
0201361868
Edition: 1st
Format: Paperback
Copyright: 08/04/1999
Publisher: Pearson

Rent From $175.55

New From $231.97

List Price $233.31 Save

Rent $175.55

TERM	PRICE	DUE
SEMESTER	$184.78	Dec 13
QUARTER	$175.55	Aug 16
SHORT TERM	$166.31	Jul 02

Added Benefits of Renting

Free Shipping Both Ways
Highlight/Take Notes Like You Own It
Purchase/Extend Before Due Date

Add to Cart

List Price $233.31 Save $1.34

New $231.97

Usually Ships in 2-3 Business Days

We Buy This Book Back!

Included with your book

Free Shipping On Every Order

Add to Cart

Note: Supplemental materials are not guaranteed with Rental or Used book purchases.

Extend or Purchase Your Rental at Any Time

Need to keep your rental past your due date? At any time before your due date you can extend or purchase your rental through your account.

Summary

This textbook bridges the gap between introductory signal processing classes and the mathematics prevalent in contemporary signal processing research and practice. Moon (Utah State University) and Stirling (Brigham Young) treat linear algebra, statistical signal processing, iterative algorithms, and optimization. The CD-ROM contains algorithms and exercises written in MATLAB.

I Introduction and Foundations

(68)

1 Introduction and Foundations

(66)

1.1 What is signal processing?

(2)

1.2 Mathematical topics embraced by signal processing

(1)

1.3 Mathematical models

(1)

1.4 Models for linear systems and signals

(21)

1.4.1 Linear discrete-time models

(5)

1.4.2 Stochastic MA and AR models

(8)

1.4.3 Continuous-time notation

(1)

1.4.4 Issues and applications

(5)

1.4.5 Identification of the modes

(2)

1.4.6 Control of the modes

(1)

1.5 Adaptive filtering

(3)

1.5.1 System identification

(1)

1.5.2 Inverse system identification

(1)

1.5.3 Adaptive predictors

(1)

1.5.4 Interference cancellation

(1)

1.6 Gaussian random variables and random processes

(6)

1.6.1 Conditional Gaussian densities

(1)

1.7 Markov and Hidden Markov Models

(4)

1.7.1 Markov models

(2)

1.7.2 Hidden Markov models

(2)

1.8 Some aspects of proofs

(7)

1.8.1 Proof by computation: direct proof

(2)

1.8.2 Proof by contradiction

(1)

1.8.3 Proof by induction

(2)

1.9 An application: LFSRs and Massey's algorithm

(10)

1.9.1 Issues and applications of LFSRs

(2)

1.9.2 Massey's algorithm

(1)

1.9.3 Characterization of LFSR length in Massey's algorithm

(5)

1.10 Exercises

(9)

1.11 References

(2)

II Vector Spaces and Linear Algebra

(366)

2 Signal Spaces

(59)

2.1 Metric spaces

(12)

2.1.1 Some topological terms

(2)

2.1.2 Sequences, Cauchy sequences, and completeness

(4)

2.1.3 Technicalities associated with the L(p) and L(Infinity) spaces

(2)

2.2 Vector spaces

(9)

2.2.1 Linear combinations of vectors

(1)

2.2.2 Linear independence

(2)

2.2.3 Basis and dimension

(3)

2.2.4 Finite-dimensional vector spaces and matrix notation

(1)

2.3 Norms and normed vector spaces

(4)

2.3.1 Finite-dimensional normed linear spaces

(1)

2.4 Inner products and inner-product spaces

(2)

2.4.1 Weak convergence

(1)

2.5 Induced norms

(1)

2.6 The Cauchy-Schwarz inequality

100

(1)

2.7 Direction of vectors: Orthogonality

101

(2)

2.8 Weighted inner products

103

(3)

2.8.1 Expectation as an inner product

105

(1)

2.9 Hilbert and Banach spaces

106

(1)

2.10 Orthogonal subspaces

107

(1)

2.11 Linear transformations: Range and nullspace

108

(2)

2.12 Inner-sum and direct-sum spaces

110

(3)

2.13 Projections and orthogonal projections

113

(3)

2.13.1 Projection matrices

115

(1)

2.14 The projection theorem

116

(2)

2.15 Orthogonalization of vectors

118

(3)

2.16 Some final technicalities for infinite dimensional spaces

121

(1)

2.17 Exercises

121

(8)

2.18 References

129

(1)

3 Representation and Approximation in Vector Spaces

130

(99)

3.1 The Approximation problem in Hilbert space

130

(5)

3.1.1 The Grammian matrix

133

(2)

3.2 The Orthogonality principle

135

(2)

3.2.1 Representations in infinite-dimensional space

136

(1)

3.3 Error minimization via gradients

137

(1)

3.4 Matrix Representations of least-squares problems

138

(3)

3.4.1 Weighted least-squares

140

(1)

3.4.2 Statistical properties of the least-squares estimate

140

(1)

3.5 Minimum error in Hilbert-space approximations

141

(2)

Applications of the orthogonality theorem

3.6 Approximation by continuous polynomials

143

(2)

3.7 Approximation by discrete polynomials

145

(2)

3.8 Linear regression

147

(2)

3.9 Least-squares filtering

149

(7)

3.9.1 Least-squares prediction and AR spectrum estimation

154

(2)

3.10 Minimum mean-square estimation

156

(1)

3.11 Minimum mean-squared error (MMSE) filtering

157

(4)

3.12 Comparison of least squares and minimum mean squares

161

(1)

3.13 Frequency-domain optimal filtering

162

(17)

3.13.1 Brief review of stochastic processes and Laplace transforms

162

(3)

3.13.2 Two-sided Laplace transforms and their decompositions

165

(4)

3.13.3 The Wiener-Hopf equation

169

(2)

3.13.4 Solution to the Wiener-Hopf equation

171

(3)

3.13.5 Examples of Wiener filtering

174

(2)

3.13.6 Mean-square error

176

(1)

3.13.7 Discrete-time Wiener filters

176

(3)

3.14 A dual approximation problem

179

(3)

3.l5 Minimum-norm solution of underdetermined equations

182

(1)

3.16 Iterative Reweighted LS (IRLS) for L(p) optimization

183

(3)

3.17 Signal transformation and generalized Fourier series

186

(4)

3.18 Sets of complete orthogonal functions

190

(18)

3.18.1 Trigonometric functions

190

(1)

3.18.2 Orthogonal polynomials

190

(3)

3.18.3 Sinc functions

193

(1)

3.18.4 Orthogonal wavelets

194

(14)

3.19 Signals as points: Digital communications

208

(7)

3.19.1 The detection problem

210

(2)

3.19.2 Examples of basis functions used in digital communications

212

(1)

3.19.3 Detection in nonwhite noise

213

(2)

3.20 Exercises

215

(13)

3.21 References

228

(1)

4 Linear Operators and Matrix Inverses

229

(46)

4.1 Linear operators

230

(2)

4.1.1 Linear functionals

231

(1)

4.2 Operator norms

232

(5)

4.2.1 Bounded operators

233

(2)

4.2.2 The Neumann expansion

235

(1)

4.2.3 Matrix norms

235

(2)

4.3 Adjoint operators and transposes

237

(2)

4.3.1 A dual optimization problem

239

(1)

4.4 Geometry of linear equations

239

(3)

4.5 Four fundamental subspaces of a linear operator

242

(5)

4.5.1 The four fundamental subspaces with non-closed range

246

(1)

4.6 Some properties of matrix inverses

247

(2)

4.6.1 Tests for invertibility of matrices

248

(1)

4.7 Some results on matrix rank

249

(2)

4.7.1 Numeric rank

250

(1)

4.8 Another look at least squares

251

(1)

4.9 Pseudoinverses

251

(2)

4.10 Matrix condition number

253

(5)

4.11 Inverse of a small-rank adjustment

258

(6)

4.11.1 An application: the RLS filter

259

(2)

4.11.2 Two RLS applications

261

(3)

4.12 Inverse of a block (partitioned) matrix

264

(4)

4.12.1 Application: Linear models

267

(1)

4.13 Exercises

268

(6)

4.14 References

274

(1)

5 Some Important Matrix Factorizations

275

(30)

5.1 The LU factorization

275

(8)

5.1.1 Computing the determinant using the LU factorization

277

(1)

5.1.2 Computing the LU factorization

278

(5)

5.2 The Cholesky factorization

283

(2)

5.2.1 Algorithms for computing the Cholesky factorization

284

(1)

5.3 Unitary matrices and the QR factorization

285

(15)

5.3.1 Unitary matrices

285

(1)

5.3.2 The QR factorization

286

(1)

5.3.3 QR factorization and least-squares filters

286

(1)

5.3.4 Computing the QR factorization

287

(1)

5.3.5 Householder transformations

287

(4)

5.3.6 Algorithms for Householder transformations

291

(2)

5.3.7 QR factorization using Givens rotations

293

(2)

5.3.8 Algorithms for QR factorization using Givens rotations

295

(1)

5.3.9 Solving least-squares problems using Givens rotations

296

(1)

5.3.10 Givens rotations via CORDIC rotations

297

(2)

5.3.11 Recursive updates to the QR factorization

299

(1)

5.4 Exercises

300

(4)

5.5 References

304

(1)

6 Eigenvalues and Eigenvectors

305

(64)

6.1 Eigenvalues and linear systems

305

(3)

6.2 Linear dependence of eigenvectors

308

(1)

6.3 Diagonalization of a matrix

309

(7)

6.3.1 The Jordan form

311

(1)

6.3.2 Diagonalization of self-adjoint matrices

312

(4)

6.4 Geometry of invariant subspaces

316

(2)

6.5 Geometry of quadratic forms and the minimax principle

318

(6)

6.6 Extremal quadratic forms subject to linear constraints

324

(1)

6.7 The Gershgorin circle theorem

324

(3)

Application of Eigendecomposition methods

6.8 Karhunen-Loeve low-rank approximations and principal methods

327

(3)

6.8.1 Principal component methods

329

(1)

6.9 Eigenfilters

330

(6)

6.9.1 Eigenfilters for random signals

330

(2)

6.9.2 Eigenfilter for designed spectral response

332

(2)

6.9.3 Constrained eigenfilters

334

(2)

6.10 Signal subspace techniques

336

(4)

6.10.1 The signal model

336

(1)

6.10.2 The noise model

337

(1)

6.10.3 Pisarenko harmonic decomposition

338

(1)

6.10.4 MUSIC

339

(1)

6.11 Generalized eigenvalues

340

(2)

6.11.1 An application: ESPRIT

341

(1)

6.12 Characteristic and minimal polynomials

342

(2)

6.12.1 Matrix polynomials

342

(2)

6.12.2 Minimal polynomials

344

(1)

6.13 Moving the eigenvalues around: Introduction to linear control

344

(3)

6.14 Noiseless constrained channel capacity

347

(3)

6.15 Computation of eigenvalues and eigenvectors

350

(5)

6.15.1 Computing the largest and smallest eigenvalues

350

(1)

6.15.2 Computing the eigenvalues of a symmetric matrix

351

(1)

6.15.3 The QR iteration

352

(3)

6.16 Exercises

355

(13)

6.17 References

368

(1)

7 The Singular Value Decomposition

369

(27)

7.1 Theory of the SVD

369

(3)

7.2 Matrix structure from the SVD

372

(1)

7.3 Pseudoinverses and the SVD

373

(2)

7.4 Numerically sensitive problems

375

(2)

7.5 Rank-reducing approximations: Effective rank

377

(1)

Applications of the SVD

7.6 System identification using the SVD

378

(3)

7.7 Total least-squares problems

381

(5)

7.7.1 Geometric interpretation of the TLS solution

385

(1)

7.8 Partial total least squares

386

(3)

7.9 Rotation of subspaces

389

(1)

7.10 Computation of the SVD

390

(2)

7.11 Exercises

392

(3)

7.12 References

395

(1)

8 Some Special Matrices and Their Applications

396

(26)

8.1 Modal matrices and parameter estimation

396

(3)

8.2 Permutation matrices

399

(1)

8.3 Toeplitz matrices and some applications

400

(9)

8.3.1 Durbin's algorithm

402

(1)

8.3.2 Predictors and lattice filters

403

(4)

8.3.3 Optimal predictors and Toeplitz inverses

407

(1)

8.3.4 Toeplitz equations with a general right-hand side

408

(1)

8.4 Vandermonde matrices

409

(1)

8.5 Circulant matrices

410

(6)

8.5.1 Relations among Vandermonde, circulant, and companion matrices

412

(1)

8.5.2 Asymptotic equivalence of the eigenvalues of Toeplitz and circulant matrices

413

(3)

8.6 Triangular matrices

416

(1)

8.7 Properties preserved in matrix products

417

(1)

8.8 Exercises

418

(3)

8.9 References

421

(1)

9 Kronecker Products and the Vec Operator

422

(13)

9.1 The Kronecker product and Kronecker sum

422

(3)

9.2 Some applications of Kronecker products

425

(3)

9.2.1 Fast Hadamard transforms

425

(1)

9.2.2 DFT computation using Kronecker products

426

(2)

9.3 The vec operator

428

(3)

9.4 Exercises

431

(2)

9.5 References

433

(2)

III Detection, Estimation, and Optimal Filtering

435

(186)

10 Introduction to Detection and Estimation, and Mathematical Notation

437

(23)

10.1 Detection and estimation theory

437

(5)

10.1.1 Game theory and decision theory

438

(2)

10.1.2 Randomization

440

(1)

10.1.3 Special cases

441

(1)

10.2 Some notational conventions

442

(2)

10.2.1 Populations and statistics

443

(1)

10.3 Conditional expectation

444

(1)

10.4 Transformations of random variables

445

(1)

10.5 Sufficient statistics

446

(7)

10.5.1 Examples of sufficient statistics

450

(1)

10.5.2 Complete sufficient statistics

451

(2)

10.6 Exponential families

453

(3)

10.7 Exercises

456

(3)

10.8 References

459

(1)

11 Detection Theory

460

(82)

11.1 Introduction to hypothesis testing

460

(2)

11.2 Neyman-Pearson theory

462

(21)

11.2.1 Simple binary hypothesis testing

462

(1)

11.2.2 The Neyman-Pearson lemma

463

(3)

11.2.3 Application of the Neyman-Pearson lemma

466

(1)

11.2.4 The likelihood ratio and the receiver operating characteristic (ROC)

467

(1)

11.2.5 A Poisson example

468

(1)

11.2.6 Some Gaussian examples

469

(11)

11.2.7 Properties of the ROC

480

(3)

11.3 Neyman-Pearson testing with composite binary hypotheses

483

(2)

11.4 Bayes decision theory

485

(14)

11.4.1 The Bayes principle

486

(1)

11.4.2 The risk function

487

(2)

11.4.3 Bayes risk

489

(1)

11.4.4 Bayes tests of simple binary hypotheses

490

(4)

11.4.5 Posterior distributions

494

(4)

11.4.6 Detection and sufficiency

498

(1)

11.4.7 Summary of binary decision problems

498

(1)

11.5 Some M-ary problems

499

(4)

11.6 Maximum-likelihood detection

503

(1)

11.7 Approximations to detection performance: The union bound

503

(1)

11.8 Invariant Tests

504

(8)

11.8.1 Detection with random (nuisance) parameters

507

(5)

11.9 Detection in continuous time

512

(8)

11.9.1 Some extensions and precautions

516

(4)

11.10 Minimax Bayes decisions

520

(12)

11.10.1 Bayes envelope function

520

(3)

11.10.2 Minimax rules

523

(1)

11.10.3 Minimax Bayes in multiple-decision problems

524

(4)

11.10.4 Determining the least favorable prior

528

(1)

11.10.5 A minimax example and the minimax theorem

529

(3)

11.11 Exercises

532

(9)

11.12 References

541

(1)

12 Estimation Theory

542

(49)

12.1 The maximum-likelihood principle

542

(5)

12.2 ML estimates and sufficiency

547

(1)

12.3 Estimation quality

548

(13)

12.3.1 The score function

548

(2)

12.3.2 The Cramer-Rao lower bound

550

(2)

12.3.3 Efficiency

552

(1)

12.3.4 Asymptotic properties of maximum-likelihood estimators

553

(3)

12.3.5 The multivariate normal case

556

(3)

12.3.6 Minimum-variance unbiased estimators

559

(2)

12.3.7 The linear statistical model

561

(1)

12.4 Applications of ML estimation

561

(7)

12.4.1 ARMA parameter estimation

561

(4)

12.4.2 Signal subspace identification

565

(1)

12.4.3 Phase estimation

566

(2)

12.5 Bayes estimation theory

568

(1)

12.6 Bayes risk

569

(11)

12.6.1 MAP estimates

573

(1)

12.6.2 Summary

574

(1)

12.6.3 Conjugate prior distributions

574

(3)

12.6.4 Connections with minimum mean-squared estimation

577

(1)

12.6.5 Bayes estimation with the Gaussian distribution

578

(2)

12.7 Recursive estimation

580

(4)

12.7.1 An example of non-Gaussian recursive Bayes

582

(2)

12.8 Exercises

584

(6)

12.9 References

590

(1)

13 The Kalman Filter

591

(30)

13.1 The state-space signal model

591

(1)

13.2 Kalman filter I: The Bayes approach

592

(3)

13.3 Kalman filter I: The innovations approach

595

(9)

13.3.1 Innovations for processes with linear observation models

596

(1)

13.3.2 Estimation using the innovations process

597

(1)

13.3.3 Innovations for processes with state-space models

598

(1)

13.3.4 A recursion for P(t|t-1)

599

(2)

13.3.5 The discrete-time Kalman filter

601

(1)

13.3.6 Perspective

602

(1)

13.3.7 Comparison with the RLS adaptive filter algorithm

603

(1)

13.4 Numerical considerations: Square-root filters

604

(2)

13.5 Application in continuous-time systems

606

(1)

13.5.1 Conversion from continuous time to discrete time

606

(1)

13.5.2 A simple kinematic example

606

(1)

13.6 Extensions of Kalman filtering to nonlinear systems

607

(6)

13.7 Smoothing

613

(3)

13.7.1 The Rauch-Tung-Streibel fixed-interval smoother

613

(3)

13.8 Another approach: H(Infinity) smoothing

616

(1)

13.9 Exercises

617

(3)

13.10 References

620

(1)

IV Iterative and Recursive Methods in Signal Processing

621

(128)

14 Basic Concepts and Methods of Iterative Algorithms

623

(47)

14.1 Definitions and qualitative properties of iterated functions

624

(5)

14.1.1 Basic theorems of iterated functions

626

(1)

14.1.2 Illustration of the basic theorems

627

(2)

14.2 Contraction mappings

629

(2)

14.3 Rates of convergence for iterative algorithms

631

(1)

14.4 Newton's method

632

(5)

14.5 Steepest descent

637

(6)

14.5.1 Comparison and discussion: Other techniques

642

(1)

Some Applications of Basic Iterative Methods

14.6 LMS adaptive Filtering

643

(5)

14.6.1 An example LMS application

645

(1)

14.6.2 Convergence of the LMS algorithm

646

(2)

14.7 Neural networks

648

(12)

14.7.1 The backpropagation training algorithm

650

(3)

14.7.2 The nonlinearity function

653

(1)

14.7.3 The forward-backward training algorithm

654

(1)

14.7.4 Adding a momentum term

654

(1)

14.7.5 Neural network code

655

(3)

14.7.6 How many neurons?

658

(1)

14.7.7 Pattern recognition: ML or NN?

659

(1)

14.8 Blind source separation

660

(5)

14.8.1 A bit of information theory

660

(2)

14.8.2 Applications to source separation

662

(2)

14.8.3 Implementation aspects

664

(1)

14.9 Exercises

665

(3)

14.10 References

668

(2)

15 Iteration by Composition of Mappings

670

(25)

15.1 Introduction

670

(1)

15.2 Alternating projections

671

(5)

15.2.1 An applications: bandlimited reconstruction

675

(1)

15.3 Composite mappings

676

(1)

15.4 Closed mappings and the global convergence theorem

677

(3)

15.5 The composite mapping algorithm

680

(9)

15.5.1 Bandlimited reconstruction, revisited

681

(1)

15.5.2 An example: Positive sequence determination

681

(2)

15.5.3 Matrix property mappings

683

(6)

15.6 Projection on convex sets

689

(4)

15.7 Exercises

693

(1)

15.8 References

694

(1)

16 Other Iterative Algorithms

695

(22)

16.1 Clustering

695

(6)

16.1.1 An example application: Vector quantization

695

(2)

16.1.2 An example application: Pattern recognition

697

(1)

16.1.3 k -means Clustering

698

(2)

16.1.4 Clustering using fuzzy k-means

700

(1)

16.2 Iterative methods for computing inverses of matrices

701

(5)

16.2.1 The Jacobi method

702

(1)

16.2.2 Gauss-Seidel iteration

703

(2)

16.2.3 Successive over-relaxation (SOR)

705

(1)

16.3 Algebraic reconstruction techniques (ART)

706

(2)

16.4 Conjugate-direction methods

708

(2)

16.5 Conjugate-gradient method

710

(3)

16.6 Nonquadratic problems

713

(1)

16.7 Exercises

713

(2)

16.8 References

715

(2)

17 The EM Algorithm in Signal Processing

717

(32)

17.1 An introductory example

718

(3)

17.2 General statement of the EM algorithm

721

(2)

17.3 Convergence of the EM algorithm

723

(2)

17.3.1 Convergence rate: Some generalizations

724

(1)

Example applications of the EM algorithm

17.4 Introductory example, revisited

725

(1)

17.5 Emission computed tomography (ECT) image reconstruction

725

(4)

17.6 Active noise cancellation (ANC)

729

(3)

17.7 Hidden Markov models

732

(8)

17.7.1 The E- and M-steps

734

(1)

17.7.2 The forward and backward probabilities

735

(1)

17.7.3 Discrete output densities

736

(1)

17.7.4 Gaussian output densities

736

(1)

17.7.5 Normalization

737

(1)

17.7.6 Algorithms for HMMs

738

(2)

17.8 Spread-spectrum, multiuser communication

740

(3)

17.9 Summary

743

(1)

17.10 Exercises

744

(3)

17.11 References

747

(2)

V Methods of Optimization

749

(106)

18 Theory of Constrained Optimization

751

(36)

18.1 Basic definitions

751

(4)

18.2 Generalization of the chain rule to composite functions

755

(2)

18.3 Definitions for constrained optimization

757

(1)

18.4 Equality constraints: Lagrange multipliers

758

(9)

18.4.1 Examples of equality-constrained optimization

764

(3)

18.5 Second-order conditions

767

(3)

18.6 Interpretation of the Lagrange multipliers

770

(3)

18.7 Complex constraints

773

(1)

18.8 Duality in optimization

773

(4)

18.9 Inequality constraints: Kuhn-Tucker conditions

777

(7)

18.9.1 Second-order conditions for inequality constraints

783

(1)

18.9.2 An extension: Fritz John conditions

783

(1)

18.10 Exercises

784

(2)

18.11 References

786

(1)

19 Shortest-Path Algorithms and Dynamic Programming

787

(31)

19.1 Definitions for graphs

787

(2)

19.2 Dynamic programming

789

(2)

19.3 The Viterbi algorithm

791

(4)

19.4 Code for the Viterbi algorithm

795

(5)

19.4.1 Related algorithms: Dijkstra's and Warshall's

798

(1)

19.4.2 Complexity comparisons of Viterbi and Dijkstra

799

(1)

Applications of path search algorithms

19.5 Maximum-likelihood sequence estimation

800

(8)

19.5.1 The intersymbol interference (ISI) channel

800

(4)

19.5.2 Code-division multiple access

804

(2)

19.5.3 Convolutional decoding

806

(2)

19.6 HMM likelihood analysis and HMM training

808

(5)

19.6.1 Dynamic warping

811

(2)

19.7 Alternatives to shortest-path algorithms

813

(2)

19.8 Exercises

815

(2)

19.9 References

817

(1)

20 Linear Programming

818

(37)

20.1 Introduction to linear programming

818

(1)

20.2 Putting a problem into standard form

819

(4)

20.2.1 Inequality constraints and slack variables

819

(1)

20.2.2 Free variables

820

(2)

20.2.3 Variable-bound constraints

822

(1)

20.2.4 Absolute value in the objective

823

(1)

20.3 Simple examples of linear programming

823

(1)

20.4 Computation of the linear programming solution

824

(12)

20.4.1 Basic variables

824

(2)

20.4.2 Pivoting

826

(2)

20.4.3 Selecting variables on which to pivot

828

(1)

20.4.4 The effect of pivoting on the value of the problem

829

(1)

20.4.5 Summary of the simplex algorithm

830

(1)

20.4.6 Finding the initial basic feasible solution

831

(3)

20.4.7 MATLAB(R) code for linear programming

834

(1)

20.4.8 Matrix notation for the simplex algorithm

835

(1)

20.5 Dual problems

836

(2)

20.6 Karmarker's algorithm for LP

838

(8)

20.6.1 Conversion to Karmarker standard form

842

(2)

20.6.2 Convergence of the algorithm

844

(2)

20.6.3 Summary and extensions

846

(1)

Examples and applications of linear programming

20.7 Linear-phase FIR filter design

846

(3)

20.7.1 Least-absolute-error approximation

847

(2)

20.8 Linear optimal control

849

(1)

20.9 Exercises

850

(3)

20.10 References

853

(2)

A Basic Concepts and Definitions

855

(22)

A.1 Set theory and notation

855

(4)

A.2 Mappings and functions

859

(1)

A.3 Convex functions

860

(1)

A.4 O and o Notation

861

(1)

A.5 Continuity

862

(2)

A.6 Differentiation

864

(5)

A.6.1 Differentiation with a single real variable

864

(1)

A.6.2 Partial derivatives and gradients on R^(m)

865

(2)

A.6.3 Linear approximation using the gradient

867

(1)

A.6.4 Taylor series

868

(1)

A.7 Basic constrained optimization

869

(1)

A.8 The Holder and Minkowski inequalities

870

(1)

A.9 Exercises

871

(5)

A.10 References

876

(1)

B Completing the Square

877

(3)

B.1 The scalar case

877

(2)

B.2 The matrix case

879

(1)

B.3 Exercises

879

(1)

C Basic Matrix Concepts

880

(11)

C.1 Notational conventions

880

(2)

C.2 Matrix Identity and Inverse

882

(1)

C.3 Transpose and trace

883

(2)

C.4 Block (partitioned) matrices

885

(1)

C.5 Determinants

885

(4)

C.5.1 Basic properties of determinants

885

(2)

C.5.2 Formulas for the determinant

887

(2)

C.5.3 Determinants and matrix inverses

889

(1)

C.6 Exercises

889

(1)

C.7 References

890

(1)

D Random Processes

891

(5)

D.1 Definitions of means and correlations

891

(1)

D.2 Stationarity

892

(1)

D.3 Power spectral-density functions

893

(1)

D.4 Linear systems with stochastic inputs

894

(1)

D.4.1 Continuous-time signals and systems

894

(1)

D.4.2 Discrete-time signals and systems

895

(1)

D.5 References

895

(1)

E Derivatives and Gradients

896

(17)

E.1 Derivatives of vectors and scalars with respect to a real vector

896

(3)

E.1.1 Some important gradients

897

(2)

E.2 Derivatives of real-valued functions of real matrices

899

(2)

E.3 Derivatives of matrices with respect to scalars, and vice versa

901

(2)

E.4 The transformation principle

903

(1)

E.5 Derivatives of products of matrices

903

(1)

E.6 Derivatives of powers of a matrix

904

(2)

E.7 Derivatives involving the trace

906

(2)

E.8 Modifications for derivatives of complex vectors and matrices

908

(2)

E.9 Exercises

910

(2)

E.10 References

912

(1)

F Conditional Expectations of Multinomial and Poisson r.v.s

913

(2)

F.1 Multinomial distributions

913

(1)

F.2 Poisson random variables

914

(1)

F.3 Exercises

914

(1)

Bibliography

915

(14)

Index

929

Amazon no longer offers textbook rentals. We do!

Mathematical Methods and Algorithms for Signal Processing

Mathematical Methods and Algorithms for Signal Processing

9780201361865

0201361868

Summary

Author Biography

Table of Contents

Supplemental Materials

Excerpts

Write a Review