Polynomial Matrix Eigenvalue Decomposition Research Articles

Enhancement of Noisy Reverberant Speech Using Polynomial Matrix Eigenvalue Decomposition

Speech enhancement is important for applications such as telecommunications, hearing aids, automatic speech recognition and voice-controlled systems. Enhancement algorithms aim to reduce interfering noise and reverberation while minimizing any speech distortion. In this work for speech enhancement, we propose to use polynomial matrices to model the spatial, spectral and temporal correlations between the speech signals received by a microphone array and polynomial matrix eigenvalue decomposition (PEVD) to decorrelate in space, time and frequency simultaneously. We then propose a blind and unsupervised PEVD-based speech enhancement algorithm. Simulations and informal listening examples involving diverse reverberant and noisy environments have shown that our method can jointly suppress noise and reverberation, thereby achieving speech enhancement without introducing processing artefacts into the enhanced signal.

Efficient Implementation of Iterative Polynomial Matrix EVD Algorithms Exploiting Structural Redundancy and Parallelisation

A number of algorithms are capable of iteratively calculating a polynomial matrix eigenvalue decomposition (PEVD), which is a generalisation of the EVD and will diagonalise a parahermitian polynomial matrix via paraunitary operations. While offering promising results in various broadband array processing applications, the PEVD has seen limited deployment in hardware due to the high computational complexity of these algorithms. Akin to low complexity divide-and-conquer (DaC) solutions to eigenproblems, this paper addresses a partially parallelisable DaC approach to the PEVD. A novel algorithm titled parallel-sequential matrix diagonalisation exhibits significantly reduced algorithmic complexity and run-time when compared with existing iterative PEVD methods. The DaC approach, which is shown to be suitable for multi-core implementation, can improve eigenvalue resolution at the expense of decomposition mean squared error, and offers a trade-off between the approximation order and accuracy of the resulting paraunitary matrices.

On the Existence and Uniqueness of the Eigenvalue Decomposition of a Parahermitian Matrix

This paper addresses the extension of the factorization of a Hermitian matrix by an eigenvalue decomposition (EVD) to the case of a parahermitian matrix that is analytic at least on an annulus containing the unit circle. Such parahermitian matrices contain polynomials or rational functions in the complex variable $z$ and arise, e.g., as cross spectral density matrices in broadband array problems. Specifically, conditions for the existence and uniqueness of eigenvalues and eigenvectors of a parahermitian matrix EVD are given, such that these can be represented by a power or Laurent series that is absolutely convergent, at least on the unit circle, permitting a direct realization in the time domain. Based on an analysis of the unit circle, we prove that eigenvalues exist as unique and convergent but likely infinite-length Laurent series. The eigenvectors can have an arbitrary phase response and are shown to exist as convergent Laurent series if eigenvalues are selected as analytic functions on the unit circle, and if the phase response is selected such that the eigenvectors are Holder continuous with $\alpha >\frac{1}{2}$ on the unit circle. In the case of a discontinuous phase response or if spectral majorisation is enforced for intersecting eigenvalues, an absolutely convergent Laurent series solution for the eigenvectors of a parahermitian EVD does not exist. We provide some examples, comment on the approximation of a parahermitian matrix EVD by Laurent polynomial factors, and compare our findings to the solutions provided by polynomial matrix EVD algorithms.

Fetal electrocardiogram estimation using polynomial eigenvalue decomposition

In this paper, we propose the application of polynomial matrix eigenvalue decomposition (PEVD) to the problem of fetal electrocardiogra (ECG) extraction from real ECG recordings obtained from abdominal leads. We model the fetal ECG extraction problem as a broadband sensor array signal processing problem in order to account for the broadband nature of the ECG noise present in the recordings. An algorithm for providing an approximate PEVD is used in order to estimate the broadband noise subspace. Suppression of the broadband noise and maternal ECG is achieved by carrying out an orthonormal projection of the recordings onto the estimated fetal subspace. The proposed scheme was evaluated with multichannel synthetic ECG signals and real ECG data from the PhysioNet Challenge database and is shown to perform favorably as compared to prior-art methods. Results indicate that our method is more robust than the prior-art ones for the task of fetal ECG estimation.

Sequential Matrix Diagonalization Algorithms for Polynomial EVD of Parahermitian Matrices

For parahermitian polynomial matrices, which can be used, for example, to characterize space-time covariance in broadband array processing, the conventional eigenvalue decomposition (EVD) can be generalized to a polynomial matrix EVD (PEVD). In this paper, a new iterative PEVD algorithm based on sequential matrix diagonalization (SMD) is introduced. At every step the SMD algorithm shifts the dominant column or row of the polynomial matrix to the zero lag position and eliminates the resulting instantaneous correlation. A proof of convergence is provided, and it is demonstrated that SMD establishes diagonalization faster and with lower order operations than existing PEVD algorithms.

Novel Field-Programmable Gate Array Architecture for Computing the Eigenvalue Decomposition of Para-Hermitian Polynomial Matrices

In this paper, we introduce a field-programmable gate array (FPGA) hardware architecture for the realization of an algorithm for computing the eigenvalue decomposition (EVD) of para-Hermitian polynomial matrices. Specifically, we develop a parallelized version of the second-order sequential best rotation (SBR2) algorithm for polynomial matrix EVD (PEVD). The proposed algorithm is an extension of the parallel Jacobi method to para-Hermitian polynomial matrices, and as such, it is the first architecture devoted to PEVD. Hardware implementation of the algorithm is achieved via a highly pipelined, nonsystolic FPGA architecture. The efficient hardware solution is accomplished by using the coordinate rotation digital computer algorithm to calculate the trigonometric functions and vector multiplications that are performed by the algorithm. The architecture, which is scalable in terms of the size of the input para-Hermitian matrix, has been designed using the Xilinx system generator tool. We verify the algorithmic convergence of the architecture and demonstrate its decomposition performance through computer simulations and FPGA-in-the-loop hardware co-simulations. Results confirm that the proposed solution gives low execution times while reducing the number of resources required from the FPGA.

Parallel algorithm for computation of second-order sequential best rotations

Algorithms for computing an approximate polynomial matrix eigenvalue decomposition of para-Hermitian systems have emerged as a powerful, generic signal processing tool. A technique that has shown much success in this regard is the sequential best rotation (SBR2) algorithm. Proposed is a scheme for parallelising SBR2 with a view to exploiting the modern architectural features and inherent parallelism of field-programmable gate array (FPGA) technology. Experiments show that the proposed scheme can achieve low execution times while requiring minimal FPGA resources.