Mathematical Methods and Algorithms for Signal Processing

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• Presents mathematics with applications drawn from signal processing literature and practice to motivate learning in a wide variety of areas
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• Application topics include both traditional areas? such as spectral estimation, adaptive filtering, detection, and estimation? and also more recent topics such as blind source separation and the EM algorithm
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• A foundation in vector spaces is presented as a unifying framework for a variety of signal processing concepts, including least-squares, minimum mean-squares, wavelet transforms, digital communications, subspace methods, and more
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• Many exercises are provided to reinforce understanding, extend the material, and apply the concepts
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  1  Introduction and Foundations                  3
    1.1  What is signal processing?                3
    1.2  Mathematical topics embraced by signal    5
    processing
    1.3  Mathematical models                       6
    1.4  Models for linear systems and signals     7
      1.4.1  Linear discrete-time models           7
      1.4.2  Stochastic MA and AR models           12
      1.4.3  Continuous-time notation              20
      1.4.4  Issues and applications               21
      1.4.5  Identification of the modes           26
      1.4.6  Control of the modes                  28
    1.5  Adaptive filtering                        28
      1.5.1  System identification                 29
      1.5.2  Inverse system identification         29
      1.5.3  Adaptive predictors                   29
      1.5.4 Interference cancellation              30
    1.6  Gaussian random variables and random      31
    processes
      1.6.1  Conditional Gaussian densities        36
    1.7  Markov and Hidden Markov Models           37
      1.7.1  Markov models                         37
      1.7.2  Hidden Markov models                  39
    1.8  Some aspects of proofs                    41
      1.8.1  Proof by computation: direct proof    43
      1.8.2  Proof by contradiction                45
      1.8.3  Proof by induction                    46
    1.9  An application: LFSRs and Massey's        48
    algorithm
      1.9.1  Issues and applications of LFSRs      50
      1.9.2  Massey's algorithm                    52
      1.9.3  Characterization of LFSR length in    53
      Massey's algorithm
    1.10  Exercises                                58
    1.11  References                               67
II  Vector Spaces and Linear Algebra               69
  2  Signal Spaces                                 71
    2.1  Metric spaces                             72
      2.1.1  Some topological terms                76
      2.1.2  Sequences, Cauchy sequences, and      78
      completeness
      2.1.3  Technicalities associated with the    82
      L(p) and L(Infinity) spaces
    2.2  Vector spaces                             84
      2.2.1  Linear combinations of vectors        87
      2.2.2  Linear independence                   88
      2.2.3  Basis and dimension                   90
      2.2.4  Finite-dimensional vector spaces      93
      and matrix notation
    2.3  Norms and normed vector spaces            93
      2.3.1  Finite-dimensional normed linear      97
      spaces
    2.4  Inner products and inner-product spaces   97
      2.4.1  Weak convergence                      99
    2.5  Induced norms                             99
    2.6  The Cauchy-Schwarz inequality             100
    2.7  Direction of vectors: Orthogonality       101
    2.8  Weighted inner products                   103
      2.8.1  Expectation as an inner product       105
    2.9  Hilbert and Banach spaces                 106
    2.10  Orthogonal subspaces                     107
    2.11  Linear transformations: Range and        108
    nullspace
    2.12  Inner-sum and direct-sum spaces          110
    2.13  Projections and orthogonal projections   113
      2.13.1  Projection matrices                  115
    2.14  The projection theorem                   116
    2.15  Orthogonalization of vectors             118
    2.16  Some final technicalities for            121
    infinite dimensional spaces
    2.17  Exercises                                121
    2.18  References                               129
  3  Representation and Approximation in Vector    130
  Spaces
    3.1  The Approximation problem in Hilbert      130
    space
      3.1.1  The Grammian matrix                   133
    3.2  The Orthogonality principle               135
      3.2.1  Representations in                    136
      infinite-dimensional space
    3.3  Error minimization via gradients          137
    3.4  Matrix Representations of                 138
    least-squares problems
      3.4.1  Weighted least-squares                140
      3.4.2  Statistical properties of the         140
      least-squares estimate
    3.5  Minimum error in Hilbert-space            141
    approximations
Applications of the orthogonality theorem
    3.6  Approximation by continuous polynomials   143
    3.7  Approximation by discrete polynomials     145
    3.8  Linear regression                         147
    3.9  Least-squares filtering                   149
      3.9.1  Least-squares prediction and AR       154
      spectrum estimation
    3.10  Minimum mean-square estimation           156
    3.11  Minimum mean-squared error (MMSE)        157
    filtering
    3.12  Comparison of least squares and          161
    minimum mean squares
    3.13  Frequency-domain optimal filtering       162
      3.13.1  Brief review of stochastic           162
      processes and Laplace transforms
      3.13.2  Two-sided Laplace transforms and     165
      their decompositions
      3.13.3  The Wiener-Hopf equation             169
      3.13.4  Solution to the Wiener-Hopf          171
      equation
      3.13.5  Examples of Wiener filtering         174
      3.13.6  Mean-square error                    176
      3.13.7  Discrete-time Wiener filters         176
    3.14  A dual approximation problem             179
    3.l5  Minimum-norm solution of                 182
    underdetermined equations
    3.16  Iterative Reweighted LS (IRLS) for       183
    L(p) optimization
    3.17  Signal transformation and generalized    186
    Fourier series
    3.18  Sets of complete orthogonal functions    190
      3.18.1  Trigonometric functions              190
      3.18.2  Orthogonal polynomials               190
      3.18.3  Sinc functions                       193
      3.18.4  Orthogonal wavelets                  194
    3.19  Signals as points: Digital               208
    communications
      3.19.1  The detection problem                210
      3.19.2  Examples of basis functions used     212
      in digital communications
      3.19.3  Detection in nonwhite noise          213
    3.20  Exercises                                215
    3.21  References                               228
  4  Linear Operators and Matrix Inverses          229
    4.1  Linear operators                          230
      4.1.1  Linear functionals                    231
    4.2  Operator norms                            232
      4.2.1  Bounded operators                     233
      4.2.2  The Neumann expansion                 235
      4.2.3  Matrix norms                          235
    4.3  Adjoint operators and transposes          237
      4.3.1  A dual optimization problem           239
    4.4  Geometry of linear equations              239
    4.5  Four fundamental subspaces of a linear    242
    operator
      4.5.1  The four fundamental subspaces        246
      with non-closed range
    4.6  Some properties of matrix inverses        247
      4.6.1  Tests for invertibility of matrices   248
    4.7  Some results on matrix rank               249
      4.7.1  Numeric rank                          250
    4.8  Another look at least squares             251
    4.9  Pseudoinverses                            251
    4.10  Matrix condition number                  253
    4.11  Inverse of a small-rank adjustment       258
      4.11.1  An application: the RLS filter       259
      4.11.2  Two RLS applications                 261
    4.12  Inverse of a block (partitioned)         264
    matrix
      4.12.1  Application: Linear models           267
    4.13  Exercises                                268
    4.14  References                               274
  5  Some Important Matrix Factorizations          275
    5.1  The LU factorization                      275
      5.1.1 Computing the determinant using the    277
      LU factorization
      5.1.2  Computing the LU factorization        278
    5.2  The Cholesky factorization                283
      5.2.1  Algorithms for computing the          284
      Cholesky factorization
    5.3  Unitary matrices and the QR               285
    factorization
      5.3.1  Unitary matrices                      285
      5.3.2  The QR factorization                  286
      5.3.3  QR factorization and least-squares    286
      filters
      5.3.4  Computing the QR factorization        287
      5.3.5  Householder transformations           287
      5.3.6  Algorithms for Householder            291
      transformations
      5.3.7  QR factorization using Givens         293
      rotations
      5.3.8  Algorithms for QR factorization       295
      using Givens rotations
      5.3.9  Solving least-squares problems        296
      using Givens rotations
      5.3.10  Givens rotations via CORDIC          297
      rotations
      5.3.11  Recursive updates to the QR          299
      factorization
    5.4  Exercises                                 300
    5.5  References                                304
  6  Eigenvalues and Eigenvectors                  305
    6.1  Eigenvalues and linear systems            305
    6.2  Linear dependence of eigenvectors         308
    6.3  Diagonalization of a matrix               309
      6.3.1  The Jordan form                       311
      6.3.2  Diagonalization of self-adjoint       312
      matrices
    6.4  Geometry of invariant subspaces           316
    6.5  Geometry of quadratic forms and the       318
    minimax principle
    6.6  Extremal quadratic forms subject to       324
    linear constraints
    6.7  The Gershgorin circle theorem             324
Application of Eigendecomposition methods
    6.8  Karhunen-Loeve low-rank approximations    327
    and principal methods
      6.8.1  Principal component methods           329
    6.9  Eigenfilters                              330
      6.9.1  Eigenfilters for random signals       330
      6.9.2  Eigenfilter for designed spectral     332
      response
      6.9.3  Constrained eigenfilters              334
    6.10  Signal subspace techniques               336
      6.10.1  The signal model                     336
      6.10.2  The noise model                      337
      6.10.3  Pisarenko harmonic decomposition     338
      6.10.4  MUSIC                                339
    6.11  Generalized eigenvalues                  340
      6.11.1  An application: ESPRIT               341
    6.12  Characteristic and minimal polynomials   342
      6.12.1  Matrix polynomials                   342
      6.12.2  Minimal polynomials                  344
    6.13  Moving the eigenvalues around:           344
    Introduction to linear control
    6.14  Noiseless constrained channel capacity   347
    6.15  Computation of eigenvalues and           350
    eigenvectors
      6.15.1  Computing the largest and            350
      smallest eigenvalues
      6.15.2  Computing the eigenvalues of a       351
      symmetric matrix
      6.15.3  The QR iteration                     352
    6.16  Exercises                                355
    6.17  References                               368
  7  The Singular Value Decomposition              369
    7.1  Theory of the SVD                         369
    7.2  Matrix structure from the SVD             372
    7.3  Pseudoinverses and the SVD                373
    7.4  Numerically sensitive problems            375
    7.5  Rank-reducing approximations:             377
    Effective rank
Applications of the SVD
    7.6  System identification using the SVD       378
    7.7  Total least-squares problems              381
      7.7.1  Geometric interpretation of the       385
      TLS solution
    7.8  Partial total least squares               386
    7.9  Rotation of subspaces                     389
    7.10  Computation of the SVD                   390
    7.11  Exercises                                392
    7.12  References                               395
  8  Some Special Matrices and Their               396
  Applications
    8.1  Modal matrices and parameter estimation   396
    8.2  Permutation matrices                      399
    8.3  Toeplitz matrices and some applications   400
      8.3.1  Durbin's algorithm                    402
      8.3.2  Predictors and lattice filters        403
      8.3.3  Optimal predictors and Toeplitz       407
      inverses
      8.3.4  Toeplitz equations with a general     408
      right-hand side
    8.4  Vandermonde matrices                      409
    8.5  Circulant matrices                        410
      8.5.1  Relations among Vandermonde,          412
      circulant, and companion matrices
      8.5.2  Asymptotic equivalence of the         413
      eigenvalues of Toeplitz and circulant
      matrices
    8.6  Triangular matrices                       416
    8.7  Properties preserved in matrix products   417
    8.8  Exercises                                 418
    8.9  References                                421
  9  Kronecker Products and the Vec Operator       422
    9.1  The Kronecker product and Kronecker sum   422
    9.2  Some applications of Kronecker products   425
      9.2.1  Fast Hadamard transforms              425
      9.2.2  DFT computation using Kronecker       426
      products
    9.3  The vec operator                          428
    9.4  Exercises                                 431
    9.5  References                                433
III  Detection, Estimation, and Optimal            435
Filtering
  10  Introduction to Detection and Estimation,    437
  and Mathematical Notation
    10.1  Detection and estimation theory          437
      10.1.1  Game theory and decision theory      438
      10.1.2  Randomization                        440
      10.1.3  Special cases                        441
    10.2  Some notational conventions              442
      10.2.1  Populations and statistics           443
    10.3  Conditional expectation                  444
    10.4  Transformations of random variables      445
    10.5  Sufficient statistics                    446
      10.5.1  Examples of sufficient statistics    450
      10.5.2  Complete sufficient statistics       451
    10.6  Exponential families                     453
    10.7  Exercises                                456
    10.8  References                               459
  11  Detection Theory                             460
    11.1  Introduction to hypothesis testing       460
    11.2  Neyman-Pearson theory                    462
      11.2.1  Simple binary hypothesis testing     462
      11.2.2  The Neyman-Pearson lemma             463
      11.2.3  Application of the Neyman-Pearson    466
      lemma
      11.2.4  The likelihood ratio and the         467
      receiver operating characteristic
      11.2.5  A Poisson example                    468
      11.2.6  Some Gaussian examples               469
      11.2.7  Properties of the ROC                480
    11.3  Neyman-Pearson testing with composite    483
    binary hypotheses
    11.4  Bayes decision theory                    485
      11.4.1  The Bayes principle                  486
      11.4.2  The risk function                    487
      11.4.3  Bayes risk                           489
      11.4.4  Bayes tests of simple binary         490
      hypotheses
      11.4.5  Posterior distributions              494
      11.4.6  Detection and sufficiency            498
      11.4.7  Summary of binary decision           498
      problems
    11.5  Some M-ary problems                      499
    11.6  Maximum-likelihood detection             503
    11.7  Approximations to detection              503
    performance: The union bound
    11.8  Invariant Tests                          504
      11.8.1  Detection with random (nuisance)     507
      parameters
    11.9  Detection in continuous time             512
      11.9.1  Some extensions and precautions      516
    11.10  Minimax Bayes decisions                 520
      11.10.1  Bayes envelope function             520
      11.10.2  Minimax rules                       523
      11.10.3  Minimax Bayes in                    524
      multiple-decision problems
      11.10.4  Determining the least favorable     528
      prior
      11.10.5  A minimax example and the           529
      minimax theorem
    11.11  Exercises                               532
    11.12  References                              541
  12  Estimation Theory                            542
    12.1  The maximum-likelihood principle         542
    12.2  ML estimates and sufficiency             547
    12.3  Estimation quality                       548
      12.3.1  The score function                   548
      12.3.2 The Cramer-Rao lower bound            550
      12.3.3  Efficiency                           552
      12.3.4  Asymptotic properties of             553
      maximum-likelihood estimators
      12.3.5  The multivariate normal case         556
      12.3.6  Minimum-variance unbiased            559
      estimators
      12.3.7  The linear statistical model         561
    12.4  Applications of ML estimation            561
      12.4.1  ARMA parameter estimation            561
      12.4.2  Signal subspace identification       565
      12.4.3  Phase estimation                     566
    12.5  Bayes estimation theory                  568
    12.6  Bayes risk                               569
      12.6.1  MAP estimates                        573
      12.6.2  Summary                              574
      12.6.3  Conjugate prior distributions        574
      12.6.4  Connections with minimum             577
      mean-squared estimation
      12.6.5  Bayes estimation with the            578
      Gaussian distribution
    12.7  Recursive estimation                     580
      12.7.1  An example of non-Gaussian           582
      recursive Bayes
    12.8  Exercises                                584
    12.9  References                               590
  13  The Kalman Filter                            591
    13.1  The state-space signal model             591
    13.2  Kalman filter I: The Bayes approach      592
    13.3  Kalman filter I: The innovations         595
    approach
      13.3.1  Innovations for processes with       596
      linear observation models
      13.3.2  Estimation using the innovations     597
      process
      13.3.3  Innovations for processes with       598
      state-space models
      13.3.4  A recursion for P(t|t-1)             599
      13.3.5  The discrete-time Kalman filter      601
      13.3.6  Perspective                          602
      13.3.7  Comparison with the RLS adaptive     603
      filter algorithm
    13.4  Numerical considerations: Square-root    604
    filters
    13.5  Application in continuous-time systems   606
      13.5.1  Conversion from continuous time      606
      to discrete time
      13.5.2  A simple kinematic example           606
    13.6  Extensions of Kalman filtering to        607
    nonlinear systems
    13.7  Smoothing                                613
      13.7.1  The Rauch-Tung-Streibel              613
      fixed-interval smoother
    13.8  Another approach: H(Infinity)            616
    smoothing
    13.9  Exercises                                617
    13.10  References                              620
IV  Iterative and Recursive Methods in Signal      621
Processing
  14  Basic Concepts and Methods of Iterative      623
  Algorithms
    14.1  Definitions and qualitative              624
    properties of iterated functions
      14.1.1  Basic theorems of iterated           626
      functions
      14.1.2  Illustration of the basic theorems   627
    14.2  Contraction mappings                     629
    14.3  Rates of convergence for iterative       631
    algorithms
    14.4  Newton's method                          632
    14.5  Steepest descent                         637
      14.5.1  Comparison and discussion: Other     642
      techniques
Some Applications of Basic Iterative Methods
    14.6  LMS adaptive Filtering                   643
      14.6.1  An example LMS application           645
      14.6.2  Convergence of the LMS algorithm     646
    14.7  Neural networks                          648
      14.7.1  The backpropagation training         650
      algorithm
      14.7.2  The nonlinearity function            653
      14.7.3  The forward-backward training        654
      algorithm
      14.7.4  Adding a momentum term               654
      14.7.5  Neural network code                  655
      14.7.6  How many neurons?                    658
      14.7.7  Pattern recognition: ML or NN?       659
    14.8  Blind source separation                  660
      14.8.1  A bit of information theory          660
      14.8.2  Applications to source separation    662
      14.8.3  Implementation aspects               664
    14.9  Exercises                                665
    14.10  References                              668
  15  Iteration by Composition of Mappings         670
    15.1  Introduction                             670
    15.2  Alternating projections                  671
      15.2.1  An applications: bandlimited         675
      reconstruction
    15.3  Composite mappings                       676
    15.4  Closed mappings and the global           677
    convergence theorem
    15.5  The composite mapping algorithm          680
      15.5.1  Bandlimited reconstruction,          681
      revisited
      15.5.2  An example: Positive sequence        681
      determination
      15.5.3  Matrix property mappings             683
    15.6  Projection on convex sets                689
    15.7  Exercises                                693
    15.8  References                               694
  16  Other Iterative Algorithms                   695
    16.1  Clustering                               695
      16.1.1  An example application: Vector       695
      quantization
      16.1.2  An example application: Pattern      697
      recognition
      16.1.3  k -means Clustering                  698
      16.1.4  Clustering using fuzzy k-means       700
    16.2  Iterative methods for computing          701
    inverses of matrices
      16.2.1  The Jacobi method                    702
      16.2.2  Gauss-Seidel iteration               703
      16.2.3  Successive over-relaxation (SOR)     705
    16.3  Algebraic reconstruction techniques      706

    16.4  Conjugate-direction methods              708
    16.5  Conjugate-gradient method                710
    16.6  Nonquadratic problems                    713
    16.7  Exercises                                713
    16.8  References                               715
  17  The EM Algorithm in Signal Processing        717
    17.1  An introductory example                  718
    17.2  General statement of the EM algorithm    721
    17.3  Convergence of the EM algorithm          723
      17.3.1  Convergence rate: Some               724
      generalizations
Example applications of the EM algorithm
    17.4  Introductory example, revisited          725
    17.5  Emission computed tomography (ECT)       725
    image reconstruction
    17.6  Active noise cancellation (ANC)          729
    17.7  Hidden Markov models                     732
      17.7.1  The E- and M-steps                   734
      17.7.2  The forward and backward             735
      probabilities
      17.7.3  Discrete output densities            736
      17.7.4  Gaussian output densities            736
      17.7.5  Normalization                        737
      17.7.6  Algorithms for HMMs                  738
    17.8  Spread-spectrum, multiuser               740
    communication
    17.9  Summary                                  743
    17.10  Exercises                               744
    17.11  References                              747
V  Methods of Optimization                         749
  18  Theory of Constrained Optimization           751
    18.1  Basic definitions                        751
    18.2  Generalization of the chain rule to      755
    composite functions
    18.3  Definitions for constrained              757
    optimization
    18.4  Equality constraints: Lagrange           758
    multipliers
      18.4.1  Examples of equality-constrained     764
      optimization
    18.5  Second-order conditions                  767
    18.6  Interpretation of the Lagrange           770
    multipliers
    18.7  Complex constraints                      773
    18.8  Duality in optimization                  773
    18.9  Inequality constraints: Kuhn-Tucker      777
    conditions
      18.9.1  Second-order conditions for          783
      inequality constraints
      18.9.2  An extension: Fritz John             783
      conditions
    18.10  Exercises                               784
    18.11  References                              786
  19  Shortest-Path Algorithms and Dynamic         787
  Programming
    19.1  Definitions for graphs                   787
    19.2  Dynamic programming                      789
    19.3  The Viterbi algorithm                    791
    19.4  Code for the Viterbi algorithm           795
      19.4.1  Related algorithms: Dijkstra's       798
      and Warshall's
      19.4.2  Complexity comparisons of Viterbi    799
      and Dijkstra
Applications of path search algorithms
    19.5  Maximum-likelihood sequence estimation   800
      19.5.1  The intersymbol interference         800
      (ISI) channel
      19.5.2  Code-division multiple access        804
      19.5.3  Convolutional decoding               806
    19.6  HMM likelihood analysis and HMM          808
    training
      19.6.1  Dynamic warping                      811
    19.7  Alternatives to shortest-path            813
    algorithms
    19.8  Exercises                                815
    19.9  References                               817
  20  Linear Programming                           818
    20.1  Introduction to linear programming       818
    20.2  Putting a problem into standard form     819
      20.2.1  Inequality constraints and slack     819
      variables
      20.2.2  Free variables                       820
      20.2.3  Variable-bound constraints           822
      20.2.4  Absolute value in the objective      823
    20.3  Simple examples of linear programming    823
    20.4  Computation of the linear programming    824
    solution
      20.4.1  Basic variables                      824
      20.4.2  Pivoting                             826
      20.4.3  Selecting variables on which to      828
      pivot
      20.4.4  The effect of pivoting on the        829
      value of the problem
      20.4.5  Summary of the simplex algorithm     830
      20.4.6  Finding the initial basic            831
      feasible solution
      20.4.7  MATLAB(R) code for linear            834
      programming
      20.4.8  Matrix notation for the simplex      835
      algorithm
    20.5  Dual problems                            836
    20.6  Karmarker's algorithm for LP             838
      20.6.1  Conversion to Karmarker standard     842
      form
      20.6.2  Convergence of the algorithm         844
      20.6.3  Summary and extensions               846
Examples and applications of linear programming
    20.7  Linear-phase FIR filter design           846
      20.7.1  Least-absolute-error approximation   847
    20.8  Linear optimal control                   849
    20.9  Exercises                                850
    20.10  References                              853
  A  Basic Concepts and Definitions                855
    A.1  Set theory and notation                   855
    A.2  Mappings and functions                    859
    A.3  Convex functions                          860
    A.4  O and o Notation                          861
    A.5  Continuity                                862
    A.6  Differentiation                           864
      A.6.1  Differentiation with a single real    864
      variable
      A.6.2  Partial derivatives and gradients     865
      on R^
      A.6.3  Linear approximation using the        867
      gradient
      A.6.4  Taylor series                         868
    A.7  Basic constrained optimization            869
    A.8  The Holder and Minkowski inequalities     870
    A.9  Exercises                                 871
    A.10  References                               876
  B  Completing the Square                         877
    B.1  The scalar case                           877
    B.2  The matrix case                           879
    B.3  Exercises                                 879
  C  Basic Matrix Concepts                         880
    C.1  Notational conventions                    880
    C.2  Matrix Identity and Inverse               882
    C.3  Transpose and trace                       883
    C.4  Block (partitioned) matrices              885
    C.5  Determinants                              885
      C.5.1  Basic properties of determinants      885
      C.5.2  Formulas for the determinant          887
      C.5.3  Determinants and matrix inverses      889
    C.6  Exercises                                 889
    C.7  References                                890
  D  Random Processes                              891
    D.1  Definitions of means and correlations     891
    D.2  Stationarity                              892
    D.3  Power spectral-density functions          893
    D.4  Linear systems with stochastic inputs     894
      D.4.1  Continuous-time signals and systems   894
      D.4.2  Discrete-time signals and systems     895
    D.5  References                                895
  E  Derivatives and Gradients                     896
    E.1  Derivatives of vectors and scalars        896
    with respect to a real vector
      E.1.1  Some important gradients              897
    E.2  Derivatives of real-valued functions      899
    of real matrices
    E.3  Derivatives of matrices with respect      901
    to scalars, and vice versa
    E.4  The transformation principle              903
    E.5  Derivatives of products of matrices       903
    E.6  Derivatives of powers of a matrix         904
    E.7  Derivatives involving the trace           906
    E.8  Modifications for derivatives of          908
    complex vectors and matrices
    E.9  Exercises                                 910
    E.10  References                               912
  F  Conditional Expectations of Multinomial       913
  and Poisson r.v.s
    F.1  Multinomial distributions                 913
    F.2  Poisson random variables                  914
    F.3  Exercises                                 914
Bibliography                                       915
Index                                              929</body>
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