Research On Joint Diagonalization In Blind Source Separation Of Non-circular Signals

Blind source separation(BSS)refers to the problem of recovering mutually independent(or uncorrelated)but unknown source signals from their mixtures captured by a number of sensors without a prior information about both the sources and the mixing coefficients.In the past three decades,BSS has received much attention in the research fields of signal processing and neural networks,which has various applications in speech and audio processing,wireless communication,image recognition and enhancement,data analysis and biomedical signal processing,etc.This dissertation emphasizes on the theory and algorithm of instantaneous blind source separation of non-circular signals based on joint diagonalization(JD).The main contributions of this dissertation are summarized as follows.1.The model of the instantaneous BSS problem is formulated.And different possible assumptions about the source signal,noise and mixing channel are introduced.Based on the non-circularity of the independent or uncorrelated source signals,the target matrices which possess the jointly diagonalizable structures are constructed from the statistics of the observations.The detailed analyses on the merits and drawbacks of the commonly-used cost functions for joint diagonalization are done,which provide the theoretical basis for JD-based BSS algorithms.2.We investigated algorithms for orthogonal joint diagonalization(OJD)by one-dimensional global analytical search.The diagonalizer is constructed as the product of a set of modified Givens rotations that dependent on a single parameter s,instead of rotation angle,or sine and cosine functions of a Jacobi angle,or two related parameters c and s.By using the modified Givens rotation matrix,the cost function can be expressed as a simple function with respect to s.This parameter can be obtained in closed-form by maximizing a local cost function of s for each current index pair.It is shown with a strict algebraic proof that the proposed algorithm is globally convergent according to Lyapunov function and LaSalle's invariance principle.To deal with the complex-valued joint diagonalization problem,we present two extended algorithms.The first complex-valued algorithm can indirectly deal with the complex target matrices since the proposed real-valued algorithm can be slightly modified and generalized to complex case after the target matrices are transformed into real symmetric ones.By taking advantage of the special structures of the transformed target matrices and the rotation matrix,the computational cost can be significantly reduced.The second algorithm can directly handle the complex-valued joint diagonalization problem by a bi-Givens strategy and one-dimensional global analytical search,since the elements of the modified rotation matrices are set to be real or pure imaginary.The simulations are performed to illustrate the fast convergence,low complexity and good performance of the proposed algorithms.3.We investigated the non-unitary joint diagonalization algorithms using complex rotations.For non-circular sources,both the conventional and conjugate correlation matrices can be used to improve the performance of the proposed algorithm.The un-mixing matrix can be decomposed into a product of complex rotation matrices which is the combination of complex Givens and complex hyperbolic rotations.By ingeniously using the structure of the complex rotation matrix and the adequate expression of the concerned variables,the minimization of the simplified Frobenius norm criterion function can be converted into the problem of generalized eigenvalue decomposition of two 3×3 real symmetric matrices.The optimal parameters can be obtained from a normalized eigenvector.The proposed algorithm has fast convergence rate.The dependence of some existing algorithms on the structure of the target matrices is released,which expands its scope of application and relaxes some restrictions applied to it.4.We investigated the non-unitary joint diagonalization based on Gaussian Newton method.In order to fully explore the statistical information of the non-circular sources,a fast parametric structures based joint diagonalization algorithm is presented for the non-unitary diagonalization of two sets of complex target matrices.Firstly,based on the assumption that the norms of the off-diagonal parts of the target matrices and updating matrix are small,the second-order approximation of the contract function is obtained.The problem of simultaneous diagonalization can be converted into a serious of linear least-squares problems.By considering the real and imaginary parts of the elements in the updating matrix separately,we get the estimation of the elements of the updating matrix.Secondly,by ingenious utilization of the structure information of the matrices,the computational complexity for estimating the diagonalizer is significantly reduced.The proposed algorithm relaxes several constraints on the target matrices,e.g.unitarity,positive-definiteness or Hermitian assumptions,and thus has more general utilizations.Numerical simulations are provided to illustrate the good performance and highly efficient computation of the proposed algorithm.5.We investigated the non-unitary joint diagonalization based on a modified stationary point method for the blind separation of the non-circular sources.To fully utilize the statistical information available in the non-circular sources,both the correlation and conjugate correlation matrices are used to improve the performance of separation.The proposed algorithm employs the least-squares approximate joint diagonalization criterion,which is minimized with respect to mixing matrix and diagonal matrices,alternatively.In the first stage,fixed the diagonal matrices,then the criterion becomes a quartic polynomial form with respect to the mixing matrix.We firstly search a linear approximation to the gradient function near the previous value of the mixing matrix.A set of linear equations is secondly constructed by letting this approximation be equal to zero.And the current estimation is thirdly obtained by solving such a set of linear equations.In the second stage,fixed the mixing matrix,the diagonal matrices are determined through a least square optimization procedure.Numerical examples are presented to illustrate that the proposed algorithm possesses potential advantages both in terms of convergent performance and convergence speed comparing with the state-of-the-art algorithms.