Applied and Numerical Harmonic Analysis

1 Review of algebra.- 1.1 Introduction.- 1.2 Definitions and examples of groups.- 1.3 Subgroups, cosets, and quotients.- 1.4 Ideals.- 1.5 Mappings.- 1.6 Finitely generated abelian groups.- 1.6.1 Cyclic groups.- 1.6.2 Free abelian groups.- References.- 2 Linear algebra and abelian groups.- 2.1 Introduction.- 2.2 Vector space L(A).- 3 Fourier transform over A.- 3.1 Introduction.- 3.2 Character groups.- 3.3 Character formulas.- 3.4 Duality theory.- 3.5 Character group basis.- 3.6 Fourier transform.- 3.7 Shift and multiplication operators.- References.- Problems.- 4 Poisson summation formula.- 4.1 Introduction.- 4.2 Statement and proof.- 4.3 Fourier transform of periodic functions.- 4.4 Periodization-decimation operators.- References.- Problems.- 5 Zak transform.- 5.1 Introduction.- 5.2 Fourier analysis on A x A*.- 5.3 Zak transform.- 5.4 Functional equation.- 5.5 Fourier and Zak transform.- 5.6 Isometry.- 5.7 Algorithm for computing Zak transform.- References.- Problems.- 6 Weyl-Heisenberg systems.- 6.1 Introduction.- 6.2 Translates.- 6.3 W-H systems.- 6.4 Sampling rates.- 6.5 Divide-and-conquer.- References.- Problems.- 7 Zak transform and W-H systems.- 7.1 Introduction.- 7.2 Basic results.- 7.3 Fundamental formulas.- 7.4 Zak space characterization of W-H systems.- 7.4.1 Critical sampling subgroup.- 7.4.2 Integer over-sampling subgroup.- 7.4.3 General sampling subgroup.- 7.5 Zero set characterization.- 7.5.1 Critical sampling subgroup.- 7.5.2 Integer over-sampling subgroup.- Problems.- 8 Algorithms for W-H systems.- 8.1 Introduction.- 8.2 Critical sampling algorithm.- 8.3 Integer over-sampling algorithm.- 8.3.1 Reducing the problem.- 8.4 General over-sampling algorithm.- 8.4.1 Reducing the problem.- References.- Problems.- 9 Orthogonal projection theorem.- 9.1 Introduction.- 9.2 Orthogonal projection algorithm.- 9.3 Iterative W-H coefficient set algorithm.- 10 Cross-ambiguity function.- 10.1 Introduction.- 10.2 Basic results.- 10.3 Direct algorithm.- 10.4 Critical sampling algorithm.- 10.5 Integer over-sampling algorithm.- 10.6 General divide-and-conquer algorithm.- References.- Problems.- 11 Ambiguity surfaces.- 11.1 Introduction.- 11.2 Fourier transform of ambiguity surfaces.- 11.3 Formulas D1 and D2.- References.- 12 Orthonormal W-H systems.- 12.1 Introduction.- 12.2 Orthonormal W-H systems.- 12.2.1 Critical sampling subgroup.- 12.2.2 Integer over-sampling subgroup.- 12.2.3 Over-sampling subgroup ?.- References.- Problems.- 13 Duality.- 13.1 Introduction.- 13.2 Biorthogonal.- 13.3 Algorithms for computing biorthogonals.- 13.3.1 ?-periodization.- 13.3.2 Critical sampling subgroup.- 13.3.3 Integer over-sampling subgroup.- 13.3.4 General over-sampling subgroup.- References.- Problems.- 14 Frames.- 14.1 Introduction.- 14.2 Frame Operator.- 14.2.1 Critical sampling subgroup.- 14.2.2 Integer over-sampling subgroup.- 14.2.3 General over-sampling subgroup.- 14.3 Frames.- 14.4 Frame biorthogonals.- 14.5 Operator interpretation.- 14.6 Tight frames.- References.- 15 Implementation.- 15.1 Introduction.- 15.2 Tensor product.- 15.3 Multidimensional arrays.- 15.3.1 Two-dimensional arrays.- 15.3.2 Multidimensional arrays.- 15.4 Computing the Zak transform.- 15.4.1 1-dimensional Zak transform.- 15.4.2 Two-dimensional Zak transform.- References.- Problems.- 16 Algebra of multirate structures.- 16.1 Introduction.- 16.2 Algebra.- 16.3 Exact sequences.- 16.4 Main theorem.- 16.5 Representatives mapping theorem.- 16.6 Integer matrices.- References.- Problems.- 17 Multirate structures.- 17.1 Introduction.- 17.2 Decimation operator.- 17.2.1 Elementary formulas.- 17.2.2 Shift-invariant operators.- 17.3 Polyphase representation.- 17.3.1 Analysis bands.- 17.3.2 Synthesis bands.- 17.4 Integer rate bands.- 17.5 Reduction theorems.- 17.5.1 Subgroup of a band.- 17.5.2 Nonuniform integer sampling rate filter banks.- 17.5.3 Rational sampling rate filter banks.- 17.6 Decimators and expanders.- References.- 18 A Time-frequency search for stock market anomalies.- 18.1 Introduction.- 18.2 Time-frequency trees.- 18.2.1 Adaptive segmentation via time-frequency trees.- 18.2.2 Recombination to a nondyadic split.- 18.3 Analysis of the log-differenced DJIA and S&P 500 data.- References.