Riassunto
This book is intended for those wishing to acquire a working knowledge of orthogonal transforms in the area of digital signal processing. The authors hope that their introduction will enhance the opportunities for interdiscipli- nary work in this field. The book consists of ten chapters. The first seven chapters are devoted to the study of the background, motivation and development of orthogonal transforms, the prerequisites for which are a basic knowledge of Fourier series transform (e.g., via a course in differential equations) and matrix al- gebra. The last three chapters are relatively specialized in that they are di- rected toward certain applications of orthogonal transforms in digital signal processing. As such, a knowlegde of discrete probability theory is an essential additional prerequisite. A basic knowledge of communication theory would be helpful, although not essential. Much of the material presented here has evolved from graduate level courses offered by the Departments of Electrical Engineering at Kansas State University and the University of Texas at Arlington, during the past five years. With advanced graduate students, all the material was covered in one semester. In the case of first year graduate students, the material in the first seven chapters was covered in one semester. This was followed by a prob- lems project-oriented course directed toward specific applications, using the material in the last three chapters as a basis.
Contenuti
One Introduction.- 1.1 General Remarks.- 1.2 Terminology.- 1.3 Signal Representation Using Orthogonal Functions.- 1.4 Book Outline.- References.- Problems.- Two Fourier Representation of Signals.- 2.1 Fourier Representation.- 2.2 Power, Amplitude, and Phase Spectra.- 2.3 Fourier Transform.- 2.4 Relation Between the Fourier Series and the Fourier Transform.- 2.5 Crosscorrelation, Autocorrelation, and Convolution.- 2.6 Sampling Theorem.- 2.7 Summary.- References.- Problems.- Three Fourier Representation of Sequences.- 3.1 Definition of the Discrete Fourier Transform.- 3.2 Properties of the DFT.- 3.3 Matrix Representation of Correlation and Convolution.- 3.4 Relation Between the DFT and the Fourier Transform Series.- 3.5 Power, Amplitude, and Phase Spectra.- 3.6 2-dimensional DFT.- 3.7 Time-varying Fourier Spectra.- 3.8 Summary.- Appendix 3.1.- References.- Problems.- Four Fast Fourier Transform.- 4.1 Statement of the Problem.- 4.2 Motivation to Search for an Algorithm.- 4.3 Key to Developing the Algorithm.- 4.4 Development of the Algorithm.- 4.5 Illustrative Examples.- 4.6 Shuffling.- 4.7 Operations Count and Storage Requirements.- 4.8 Some Applications.- 4.9 Summary.- Appendix 4.1 An FFT Computer Program.- References.- Problems.- Five A Class of Orthogonal Functions.- 5.1 Definition of Sequency.- 5.2 Notation.- 5.3 Rademacher and Haar Functions.- 5.4 Walsh Functions.- 5.5 Summary.- Appendix 5.1 Elements of the Gray Code.- References.- Problems.- Six Walsh-Hadamard Transform.- 6.1 Walsh Series Representation.- 6.2 Hadamard Ordered Walsh-Hadamard Transform (WHT)h.- 6.3 Fast Hadamard Ordered Walsh-Hadamard Transform (FWHT)h.- 6.4 Walsh Ordered Walsh-Hadamard Transform (WHT)W.- 6.5 Fast Walsh Ordered Walsh-Hadamard Transform (FWHT)w.- 6.6 Cyclic and Dyadic Shifts.- 6.7 (WHT)w Spectra.- 6.8 (WHT)h Spectra.- 6.9 Physical Interpretations for the (WHT)h Power Spectrum.- 6.10 Modified Walsh-Hadamard Transform (MWHT).- 6.11 Cyclic and Dyadic Correlation/Convolution.- 6.12 Multidimensional (WHT)h and (WHT)w.- 6.13 Summary.- Appendix 6.1 WHT Computer Program.- References.- Problems.- Seven Miscellaneous Orthogonal Transforms.- 7.1 Matrix Factorization.- 7.2 Generalized Transform.- 7.3 Haar Transform.- 7.4 Algorithms to Compute the HT.- 7.5 Slant Matrices.- 7.6 Definition of the Slant Transform (ST).- 7.7 Discrete Cosine Transform (DCT).- 7.8 2-dimensional Transform Considerations.- 7.9 Summary.- Appendix 7.1 Kronecker Products.- Appendix 7.2 Matrix Factorization.- References.- Problems.- Eight Generalized Wiener Filtering.- 8.1 Some Basic Matrix Operations.- 8.2 Mathematical Model.- 8.3 Filter Design.- 8.4 Suboptimal Wiener Filtering.- 8.5 Optimal Diagonal Filters.- 8.6 Suboptimal Diagonal Filters.- 8.7 2-dimensional Wiener Filtering Considerations.- 8.8 Summary.- Appendix 8.1 Some Terminology and Definitions.- References.- Problems.- Nine Data Compression.- 9.1 Search for the Optimum Transform.- 9.2 Variance Criterion and the Variance Distribution.- 9.3 Electrocardiographic Data Compression.- 9.4 Image Data Compression Considerations.- 9.5 Image Data Compression Examples.- 9.6 Additional Considerations.- 9.7 Summary.- Appendix 9.1 Lagrange Multipliers.- References.- Problems.- Ten Feature Selection in Pattern Recognition.- 10.1 Introduction.- 10.2 The Concept of Training.- 10.3 d-Dimensional Patterns.- 10.4 The 3-Class Problem.- 10.5 Image Classification Experiment.- 10.6 Least-Squares Mapping Technique.- 10.7 Augmented Feature Space.- 10.8 3-Class Least-Squares Minimum Distance Classifier.- 10.9 K-Class Least-Squares Minimum Distance Classifier.- 10.10 Quadratic Classifiers.- 10.11 An ECG Classification Experiment.- 10.12 Summary.- References.- Problems.- Author Index.
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