Adaptive Generalized Eigen-decomposition Algorithms And Its Application In Communication

Generalized eigen-decomposition(GED) problems arise in a large number of disciplines of engineering and scientific, and GED is an extremely useful statistical tool in such applications. For instance, Linear Discriminant Analysis(LDA) can be formulated as generalized eigen-decomposition problem; GED can be used for spectral estimation and blind source separation in signal processing. In wireless communication, GED can be used for design of a 2D rake receiver for DS-CDMA systems, and to estimate the optimal weight of antenna array.Mathematically speaking, the eigenvalues of a matrix pair (A,B) are the roots of its characteristic polynomial det(A -λB). In our case, A and B are Hermitian, andB is positive definite. Hence, all the generalized eigenvalues are real and positive. Many analytical techniques have been developed in the linear algebra literature to compute the generalized eigenvectors by now, but most of them assume A,B areboth known. These numerical techniques are computationally prohibitive or theyrequire blocks of data. For many applications, adaptive online algorithms are desired. A,B are both unknown in environments where conditions change slowly over time,and need to be estimated on-line directly from their respective stochastic data, thereby tracking becomes a key issue. Only fast on-line algorithms can adapt quickly to the changing environment. Compared to numerical techniques, the adaptive online algorithms need more research.In thesis, the generalized symmetric eigenvalue problem can be recasted into an unconstrained optimization problem. Two novel algorithms are derived, each based on our proposed unconstrained cost functions. A novel quasi-Newton algorithm for adaptively estimating the generalized eigenvectors of matrix pair by making use of an approximation of its Hessian matrix is derived. By minimizing the cost function with project approximation, an adaptive algorithm based on recursive least-squares technique is proposed for finding the generalized eigen-vectors. We first consider the case of a single eigenvector corresponding to one eigenvector and then extend it to the case of more than one eigenvector based on deflation technique. A rigorous analysis of the properties of the cost functions is presented. And a proof to show the convergence behavior of the algorithms is presented. Simulations results show the good performance of the algorithms. We will demonstrate the application of GED in the design of a MC-DS-CDMA receiver for direct-sequence spread spectrum signals and beamforming technique for the antenna array in the CDMA systems, which based on the Maximum Signal to Interference and Noise Ratio (MSINR) criterion. It shown that the proposed GED algorithms can apply, and the results show that the proposed algorithms have fast convergence and excellent tracking capability which are important for practical time-varying communication environment.In this paper, we systematic study the adaptive generalized decomposition problem and establish a complete set of algorithms theoretical framework.