Wavelet Design And Crosscorrelation Of M-sequences Based On Algebraic And Symbolic Computation

The construction of wavelets and filter banks, and the design of sequences with optimal correlation properties are two important problems in signal processing. Wavelets and Filter banks are useful for image compression, signal estimation, and digital watermark, and sequences with optimal correlation properties are widely used in spreading communication systems, secret communication systems and code- division-multiple-access communication systems. In this thesis, two key problems with respect to the construction of wavelets and filter banks and design of cipher are studied based on the theory of Groebner bases, namely, how the design of wavelets and filter banks can be convert into algebraic problems on the polynomial ring and the Laurent polynomial ring, and accordingly how to improve the existent algorithm to offer an effective way to obtain solutions; And then the crosscorrelation properties of m-sequences are described based on the theory of finite fields.The main works are as follows:Firstly, by analyzing the basic principle of XL-algorithm which is used to solve algebraic equation systems, an improved algorithm is presented based on the theory of Groebner bases. The research results demonstrate that the computation complexity of the improved XL-algorithm is lower than that of XL-algorithm.Secondly, by studying the design principle of general compactly supported orthonormal wavelets and applying the theory of Grobener bases, the algebraic expressions of the corresponding polynomial filter banks are studied in two different approaches. Since it is well-known that exactly symmetric finite-length solutions do not exisit for the orthonormal two-channel filter bank design problem for the exception of Haar solution, a new method to design nearly symmetric orthonormal PR filter banks is presented based on the theory of Grobener bases. In addition, a new method to design multi-channel PRLPFBs is described. Finally, m-sequences are widely used as the spreading sequences in spreading spectrum communication systems and the key sequences in secret communication systems since they have optimal pseudorandom properties and correlation properties. It is well known that m-sequences have optimal autocorrelation function, namely, the autocorrelation of m-sequences have two-value properties, while their crosscorrelation properties have not been described perfectly so far. In this thesis, the crosscorrelation functions of m-sequences and their d = 2 m+ 3 decimation sequences are calculated by using the theory of quadratic forms and exponential sums over finite fields.