Study On A Recursive Eigenvector Estimation Algorithm And Its Application In Spectral Estimation

In many applications of communication and engineering fields, eigenvectors of a data matrix need to be calculated. Eigenvector estimation plays a critical role in a variety of modern signal processing applications like communications, array signal processing, modern spectral estimation, etc. For example subspace-based high-resolution methods have been applied successfully for both temporal and spatial domain spectral analysis. For the spectral estimation of sinusoidal signals mixing with white noise, eigenvector can be used to estimate power and frequency spectrums directly and precisely. Validity and superiority of the algorithm proposed in this paper can be seen when it is compared with MUSIC spectral estimation. Some difficulties and technical requirements in the traditional eigenvector estimation are as follows: (1) An algorithm should have reasonable construction and lower computational complexity. (2) Its convergence should be fast with favorable error property and can estimate eigenvector correctly. (3) It should have high robustness and can raise signal noise ratio. (4) It can directly estimate eigenvector. Eigenvector can be calculated from subspace spanned by eigenvector or directly estimated by complicated computation. There are very few recursive eigenvector estimation algorithms with lower computational complexity. To resolve the above problem, two new recursive eigenvector estimation algorithms are proposed in this paper. One is based on the relation between KLT and sliding DCT when considering the equivalence of weight coefficient in an adaptive filter and the estimated eigenvector. The eigenvector can be estimated by choosing a reasonable filter desired response. The other is based on the improvement of subspace iteration method and utilization of deflation technique. Both algorithms can effectively reduce amount of calculation and complexity with high estimation accuracy, and it is easily implemented on time. 1. An Adaptive Eigenvector Estimation Algorithm Karhunen-Loeve transform (KLT) is an orthogonal transformation. A signal vector x after KLT, it is a same dimensional vector y in which correlation in each component is eliminated and y approaches to x under least-mean-square criteria. Therefore KLT is also called an optimal transformation. Discrete cosine transform (DCT) is also a kind of orthogonal transformation. The DCT is widely applied in speech and image processing. KLT is a signal-dependent transformation, and its implementation requires the estimation of correlation matrix of input vector, the diagonalization of the matrix, and the construction of the required basis vectors. These computations make the KLT impractical for real-time applications. Fortunately, DCT provides a predetermined set of basis vectors which are a good approximation to KLT. Moreover for a stationary zero-mean and first-order Markov process that is deemed to be sufficiently general in signal-processing, DCT is asymptotically equivalent to the KLT. Even if the above hypothesis can not be satisfied, DCT can still maintain favorable performance. Based on the above approximation between KLT and DCT, a new adaptive recursive eigenvector estimation algorithm is proposed in the forth chapter in this paper. The equivalence of the eigenvector estimation and weight coefficient vector computation in an adaptive filter is the theoretical base of the algorithm. Every eigenvector is calculated as an adaptive filter weight coefficient vector. Recursive least squares (RLS) algorithm is applied in this new algorithm, which can raise better property than that of least mean square (LMS) algorithm. The output of KLT and the output of RLS adaptive filter have the same form and the error between KLT and sliding DCT also has the same form with which eigenvector can be estimated as RLS weight vector. The output of KLT is equivalent to the output of RLS filter and the output of sliding DCT is equivalent to expectation response of RLS filter. The output of the above filter is the estimation of the desired response. The estimation error controls and adjusts the weight vector of the filter, which makes the estimation of output to be same with the desired response. The proposed algorithm is an iterative method so that it is more suitable for the real time implementation and hardware realization. The superiority of the proposed algorithm will be shown when the order of the adaptive filter increases. 2. Subspace Iteration Eigenvector Estimation Algorithm To generalize the power method, subspace iteration algorithm can be used for the resolution of some maximum eigenvalues and corresponding eigenvectors. Rayleigh-Ritz approximation principle is applied to subspace iteration to accelerate...