Research On Joint (Block-) Diagonalization Based Blind Source Separation Algorithms

Blind source separation (BSS) aims to extract independent but unobservered source signals from their mixtures captured by a number of sensors without knowing the mixing coefficients. In the past two decades, BSS has received much attention in various fields, such as speech and audio processing, image enhancement and biomedical signal processing. This dissertation emphasizes on the theory and algorithm of BSS in both instantaneous case and convolutive case. The joint diagonalization (JD) and joint block-diagonalization (JBD) are the essential tools to resolve the instantaneous BSS and convolutive BSS, respectively. Accordingly, the main contributions of this dissertation are summarized as follows:1. An identifiability theorem of non-orthogonal (non-unitary) JD was given with wide applicability, which provides a theoretical basis for non-orthogonal (non-unitary) JD based BSS algorithms. It was shown with a strict algebraic proof that the extracted mixing matrix from a group of objective matrices constructed from a common mixing matric, is essentially equal to the real mixing matrix, i.e., there is only a difference of one generalized permutation matrix between the extracted and the real mixing matrices.2. Based on the permutation and scaling indeterminacies for the BSS problems, a novel tri-quadratic fitting function was proposed by improving the classical least squares cost function. The new fitting function is quadratic if any two of the three unknown parameter sets are fixed. Meanwhile, the corresponding non-orthogonal JD algorithm, tri-quadratic iterative algorithm (TIA), was given to optimize the new quadratic fitting function. Neither the (conjugate) symmetry of the objective matrix nor the orthogonality of the mixing function is required in the TIA algorithm. Convergence of TIA algorithm was analyzed by utilizing the definition of the Lyapunov function and the LaSalle'Invariance Principle. The analytical results indicate that the TIA converges to the invariant set of the tri-quadratic fitting function that contains the acceptable set of the BSS.3. Taking the sum of the F-norms of all off-diagonal sub-matrices as a criterion, a novel orthogonal JBD algorithm, QJBD (quadratic JBD), was proposed to estimate the whole mixing matrix through minimizing a sequence of restrictive quadratic subfunctions corresponding to mixing submatrices. Both theoretical analysis and simulation results show that QJBD has much lower complexity and faster convergence speed than that of the classical Jacobi-like methods without performance loss. Furthermore, different from Jacobi-like methods, QJBD is not sensitive to the channel order and initialization values.4. Analysis shows that there is only a difference of one permutation matrix between the estimated and the real separation matrices, if the J-Di algorithm that is used traditionally for non-orthogonal JD, is directly applied to JBD. Accordingly, a non-orthogonal JBD algorithm, GH-NOJBD, was proposed by improving the J-Di algorithm. GH-NOJBD solves the convolutive-mixture BSS directly in time domain and does not require pre-whitening of the observed signals. Then, obviously there is no extra error introduced by pre-whitening and as a result, there is no need to require at least one objective matrix to be positive definite. What is the most important is that the separation performance of the proposed GH-NOJBD algorithm is superior to that of the Jacobi-like JBD counterparts.5. To overcome the disadvantages of the existing ZJBD algorithm (easily generating singular solutions) and the proposed GH-NOJBD algorithm (sensitive to noise), an alternating and iterative approach (ALS-NOJBD) for non-orthogonal JBD was proposed to estimate all the parameters of the undetermined matrix jointly. Because the correlation matrix group of the received signals has a three-factor-product JBD structure, standard least-squares expressions could be deduced with respect to the three factorable matrices by cutting pieces and then reprogramming of the 3-D matrix composed of the correlation matrix group. This is just the motivation of the ALS-NOJBD algorithm. Simulation results show that compared with traditional JBD algorithms, ALS-NOJBD has better and more stable separation performance without limitation in initial parameter choosing, and effectively overcome the disadvantages of both ZJBD and GH-NOJBD algorithms.6. Inspired by the TIA for JD, its counterpart for JBD, TIA-NOJBD algorithm was proposed. In each sub-step, a closed solution is derived by minimizing the cost function associated with one parameter-group while fixing the others. Compared with ALS-NOJBD, TIA-NOJBD also has a stable separation performance but at least one order of magnitude lower computation complexity. Besides, TIA-NOJBD is superior to all other JBD algorithms (including those proposed in this thesis, QJBD,GH-NOJBD,ALS-NOJBD) in three aspects, i.e., application conditions, computation complexity and convergency performance.7. A multistage decomposition based on single component tracking (MTD) was proposed to resolve the convolutive BSS in time-frequency domain. Unlike the previous frequency-domain methods, MTD estimates one set of inverse FIR parameters corresponding to only one of the separated signals at a time. This feature of MTD makes there are no internal-permutation and internal-scaling ambiguities which are the inherent disadvantages of the frequency-domain convolutive BSS algorithms. From a certain point of view, MTD can be considered as an extential form of JD in time-frequency domain.