Finite field wavelet transforms

For graduate students, research engineers, practicing professionals.

The primary focus of this work is to explore wavelets and filter banks over finite fields and their intersection with error control coding and data encryption. It develops the theory of wavelet transforms over finite fields which provides a general wavelet decomposition of sequences defined over finite fields - an approach that has a rich history in signal processing for the representation of real-valued signals, but which has been lacking in the finite field case. Along with wavelet theory on finite fields, this work introduces the first application of this theory to error correcting codes and data security. It also introduces a finite-field wavelet based public key encryption technique.

1. Introduction.

2. Background Review.
Introduction. Brief Review of Number Theory and Finite Fields. Discrete Fourier Transform over Finite Fields. Basefield Transforms over Finite Fields. Wavelets for Discrete-Time Signals. Previous Work on Finite-Field Wavelets. Some Basic Concepts of Error-Control Coding. Summary.

3. Finite-Field Wavelet Basis Functions.
Introduction. Finite-Field Discrete-Time Basis. Construction of Mother Wavelet and Scaling Function. Summary.

4. Double Circulant Wavelet Block Codes.
Introduction. Structure of Double Circulant Wavelet Coding. Maximum-Distance Separable Codes. Double Circulant Self-Dual Codes. Decoding Wavelet Codes. Summary.

5. Theory of Paraunitary Filter Banks Over Fields of Characteristic Two.
Introduction. Background Review. Unitary Matrices Over GF( r). PU Matrices in Fields of Characteristic Two. Factorization of PU Matrices in GF( r). Summary.

6. Arbitrary-Rate Wavelet Block Codes.
Introduction. Structure of Wavelet Coding. Rate 1/L Maximum-Distance Separable Codes. Arbitrary-Rate Wavelet Block Codes. Arbitrary-Rate Maximum-Distance Separable Codes. Decoding Arbitrary-Rate Wavelet Block Codes. Summary.

7. Wavelet Convolutional Codes.
Introduction. Structure of Wavelet Convolutional Codes. Algebraic Properties of Wavelet Convolutional Encoders. Syndrome Generators and Dual Encoders. Self-Dual and Self-Orthogonal Convolutional Codes. Time-Varying Wavelet Convolutional Codes and Bipartite Trellises. Summary.

8. Concluding Remarks.
Contributions. Suggestions for Future Research.

Appendix A. Proofs for Chapter 5.
Appendix B. Brief Review of Trellis Structures.
Bibliography.
Vita.

• New mathematical tool (wavelet transforms over finite fields), can be used in signal processing and communications.
• Novel error control coding methods based on wavelets, introduces the first application of the finite-field wavelets to error control coding.
• Novel security coding methods based on wavelets, introduces the first application of the finite-field wavelets to data encryption.
• Impact on basic science, brings together researchers from the areas of applied discrete mathematics, signal processing and communications research and provides new connections and new interpretations.
• Impact on technology and products, provides students with orientations toward practical applications, reducing decoding complexity, and satisfying low computing requirements.
• Unique t