Joint Diagonalization Algorithm Research Articles

Co‐channel interference cancellation based on BSS for multi‐satellite MIMO communication systems with FFR

SummaryMultiple‐input multiple‐output (MIMO) technology can boost the capacity of conventional satellite communications, and full frequency reuse (FFR) can further improve the capacity by making full use of frequency resources. However, FFR will cause severe co‐channel interference and lead to a degradation of channel capacity. Blind source separation (BSS) algorithm can be used to eliminate co‐channel interference and does not require prior information, which avoids the extra occupation of channel frequency resources. While the multi‐satellite MIMO communication system have delay mixing characteristic, the separation performance will be degraded if the delay is ignored. In this paper, DMBSS algorithm for delay mixing is proposed. Firstly, the DMBSS algorithm utilizes Taylor expansion to achieve dimension expansion of observed signals and so as to transform the delay mixing into instantaneous mixing with extended dimension. Secondly, complex non‐orthogonal joint diagonalization algorithm is used to solve the separation matrix equation of extended observation signals and extended source signals and thus to eliminate the co‐channel interference. Simulation results show that for multi‐satellite MIMO communication scenarios with FFR, the proposed algorithm can effectively improve the performance of co‐channel interference cancellation and reduce the value of BER. Furthermore, the influence of order of Taylor expansion on the separation performance is also analyzed and simulated in this paper to show the cost performance of the proposed algorithm.

Generalized Unitary Joint Diagonalization Algorithm Based on Approximate Givens Rotations

Generalized Unitary Joint Diagonalization Algorithm Based on Approximate Givens Rotations

An alternating least-squares algorithm for approximate joint diagonalization of Hermitian matrices

Approximate joint diagonalization (AJD) of a set of target matrices is one of the approaches for blind identification, blind source separation (BSS), and blind source extraction. This approach is useful for applications with measurement noise and/or estimation errors. This paper presents an alternating least-squares (ALS) algorithm for solving AJD problem. The ALS algorithm consists of two processing phases. The first processing phase works for finding diagonal matrices by minimizing the indirect least-squares criterion. A mixing matrix is estimated by minimizing the constrained direct least-squares criterion in the second processing phase. Our ALS algorithm can circumvent numerically unstable behavior by improving the condition number of the estimated mixing matrix in the second processing phase. The proposed algorithm is shown through experimental results to have better estimation performance and faster convergence than existing ALS algorithms with lower computational requirement in the rectangular mixing case.

Joint diagonalization for a pair of Hermitian quaternion matrices and applications to color face recognition

A new joint diagonalization algorithm for a pair of Hermitian quaternion matrices is derived incorporating real structure-preserving strategy. The structure-preserving joint diagonalization algorithm leads to a novel two-dimensional quaternion linear discriminant analysis (2D-QLDA) method for color face recognition and image reconstruction. 2D-QLDA is mathematically characterized by Hermitian quaternion generalized eigenvalue problem. A weighted norm is obtained as a new measurement to determine the distances among Fisher feature matrices, which helps us avoid generating projected images explicitly. Numerical results based on the real face databases indicate that 2D-QLDA performs better than other 2D-LDA-like methods in color face recognition and is effective in image reconstruction.

A two-step algorithm for joint EigenValue decomposition - Application to canonical polyadic decomposition of fluorescence spectra

In this paper, we propose a new Joint EigenValue Decomposition (JEVD) algorithm. JEVD problem belongs to the family of joint diagonalization problems. Hence, JEVD algorithms aim at estimating the common basis of eigenvectors of a matrix set. This problem occurs in many signal processing applications. It has notably allowed to develop efficient algorithms for the Canonical Polyadic Decomposition (CPD) of multiway arrays. The proposed JEVD algorithm is based on an original two-step approach. The first step consists in transforming the considered matrix set into a set of positive definite matrices. In this purpose, we introduce an ad hoc joint symmetrization algorithm. This first step allows us to transform the JEVD problem into a simpler orthogonal joint diagonalization problem. The second step is then performed using an efficient orthogonal joint diagonalization algorithm of the literature. Eventually, the performance of the proposed approach is deeply investigated in the CPD context of multidimensional fluorescence data. More particularly, we consider difficult scenarios such as the cases of an overestimated rank and highly correlated factors.

A unitary joint diagonalization algorithm for nonsymmetric higher‐order tensors based on Givens‐like rotations

SummaryBased on Givens‐like rotations, we present a unitary joint diagonalization algorithm for a set of nonsymmetric higher‐order tensors. Each unitary rotation matrix only depends on one unknown parameter which can be analytically obtained in an independent way following a reasonable assumption and a complex derivative technique. It can serve for the canonical polyadic decomposition of a higher‐order tensor with orthogonal factors. Furthermore, based on cross‐high‐order cumulants of observed signals, we show that the proposed algorithm can be applied to solve the joint blind source separation problem. The simulation results reveal that the proposed algorithm has a competitive performance compared with those of several existing related methods.

Non-orthogonal approximate joint diagonalization of non-Hermitian matrices in the least-squares sense

A non-orthogonal approximate joint diagonalization (AJD) algorithm of a set of non-Hermitian matrices is presented. Specifically, the proposed algorithm aims to find two distinct general (not necessarily orthogonal nor square) diagonalizing matrices which minimize the least-squares (LS) criterion based on the gradient and an optimal rank-1 approximation approach. It can be used to compute the canonical polyadic decomposition (CPD) of the third-order tensor. Simulation results demonstrate that the proposed algorithm has good convergence, robustness and accuracy properties. The joint blind source separation (JBSS) problem of two datasets can be effectively solved based on the proposed algorithm.

Parameter Estimation of Multi Frequency Hopping Signals Based on Space-Time-Frequency Distribution

Frequency hopping spread spectrum (FHSS) communication is widely used in military and civil communication, and the parameter estimation of frequency hopping (HF) signals is of great significance. In order to estimate the parameters of multiple frequency hopping signals effectively, a blind parameter estimation algorithm based on space-time frequency analysis (STFA) and matrix joint diagonalization (JDM) is proposed. Firstly, the time domain signal received by the linear array is converted to the space-time frequency domain through the space-time frequency transformation, and the space-time frequency distribution (STFD) of the signal is obtained. Then the time-frequency point is extracted from the space-time frequency distribution map, the extraction of the hop is completed by the method of finding an “island”, and the space-time frequency matrix of each hop is constructed, and then the preliminary estimation of each jump frequency, jump time and jump period is completed. Finally, the space-time-frequency matrix of the same hop received by different array elements is jointly diagonalized by the matrix joint diagonalization algorithm, and the diagonalization matrix is obtained. On the basis of the diagonalization matrix, the root-MUSIC algorithm is used to complete the direction of arrival (DOA) estimation of the frequency hopping signal and the separation of the frequency hopping radio. The simulation results show that the proposed algorithm is effective in parameter estimation of multi-hopping signals. It can estimate the parameters of −4 dB signal-to-noise ratio (SNR). The accuracy rate of parameter (hop period, DOA, hop start time, hop end time, frequency hopping frequency set) estimation reaches 73.26%, and the sparse linear regression (SLR) algorithm reaches 70.15%. When the signal-to-noise ratio reaches 5 dB, the accuracy of estimation can reach 94.74%, and the SLR reach 85.64%. It has a good effect on parameter estimation of multi-hopping signals.

Blind separation of temporally correlated noncircular sources using complex matrix joint diagonalization

In this paper, we introduce a new blind source separation (BSS) method for temporal correlated noncircular sources that uses widely linear filter (WLF) model to efficiently describe the temporal structure. The algorithm consists of a WLF coefficients estimator followed by complex matrix joint diagonalization. In the derivation of the new BSS algorithm, an emerging matrix joint diagonalization issue needs to be addressed. Subsequently, a complex matrix joint diagonalization algorithm based on successive Shear and Givens rotations (SGR) is proposed to deal with the new joint diagonalization problem. Simulations of SGR show that it converges fast and converges to excellent value, and it shows a satisfactory anti-perturbation performance of matrix for the new BSS algorithm. We compare the performance of the new BSS algorithm with several competing algorithms, and the simulation results demonstrate the superior performance of the proposed BSS algorithm for both noncircular complex Gaussian and non-Gaussian sources.

Hybrid Joint Diagonalization Algorithms

This paper deals with a hybrid joint diagonalization (JD) problem considering both Hermitian and transpose congruences. Such problem can be encountered in certain non-circular signal analysis applications including blind source separation. We introduce new Jacobi-like algorithms using Givens or a combination of Givens and hyperbolic rotations. These algorithms are compared with state-of-the-art methods and their performance gain, especially in the high dimensional case, is assessed through simulation experiments including examples related to blind separation of non-circular sources.

LU-based Jacobi-like algorithms for non-orthogonal joint diagonalization

In this paper, based on the LU decomposition, we propose three non-orthogonal Jacobi-like alternating iterative algorithms with two strategies for solving the joint diagonalization problem of a set of Hermitian matrices. In this kind of algorithm, each transformation includes one upper triangular iterative step and one lower triangular iterative step, and each step involves one parameter. The optimal parameter of each step is derived analytically. The convergence of our proposed algorithms is proven. According to this convergence analysis, the existing GNJD algorithm is revisited. Finally, numerical simulations are presented to illustrate the effectiveness of the proposed algorithms in comparison with existing ones.

Approximate Joint Singular Value Decomposition Algorithm Based on Givens-Like Rotation

An approximate joint singular value decomposition algorithm is proposed for a set of $K (K\geq 2)$ complex matrices. It can be seen as an orthogonal non-Hermitian approximate joint diagonalization algorithm. We exploit a Givens-like rotation method based on a special parameterization of the updating matrices and a reasonable approximation. The main points consist of the presentation of the new parameter structure, analytical derivation of the elementary updating matrices. High accuracy and fast convergence rate are obtained. Numerical simulations illustrate the overall good performance of the proposed algorithm.

Measurement-induced nonlocality in arbitrary dimensions in terms of the inverse approximate joint diagonalization

Here we focus on the measurement induced nonlocality and present a redefinition in terms of the skew information subject to a broken observable. It is shown that the obtained quantity possesses an obvious operational meaning, can tackle the noncontractivity of the measurement induced nonlocality and has analytic expressions for many quantum states. Most importantly, an inverse approximate joint diagonalization algorithm, due to its simplicity, high efficiency, stability, and state independence, is presented to provide almost analytic expressions for any quantum state, which can also shed light on other aspects in physics.

Multi-Target Angle Tracking Algorithm for Bistatic Multiple-Input Multiple-Output (MIMO) Radar Based on the Elements of the Covariance Matrix.

In this paper, we consider the problem of tracking the direction of arrivals (DOA) and the direction of departure (DOD) of multiple targets for bistatic multiple-input multiple-output (MIMO) radar. A high-precision tracking algorithm for target angle is proposed. First, the linear relationship between the covariance matrix difference and the angle difference of the adjacent moment was obtained through three approximate relations. Then, the proposed algorithm obtained the relationship between the elements in the covariance matrix difference. On this basis, the performance of the algorithm was improved by averaging the covariance matrix element. Finally, the least square method was used to estimate the DOD and DOA. The algorithm realized the automatic correlation of the angle and provided better performance when compared with the adaptive asymmetric joint diagonalization (AAJD) algorithm. The simulation results demonstrated the effectiveness of the proposed algorithm. The algorithm provides the technical support for the practical application of MIMO radar.

A Jacobi Generalized Orthogonal Joint Diagonalization Algorithm for Joint Blind Source Separation

Joint blind source separation (J-BSS) has emerged as a data-driven technique for multi-set data fusion applications. In this paper, we propose a Jacobi generalized orthogonal joint diagonalization (GOJD) algorithm for J-BSS of multiset signals. By the use of second-order statistics, we can obtain multiple sets of auto-covariance and cross-covariance matrices from the multi-set signals, which together admit a GOJD formulation. For computing the GOJD, we propose a computationally efficient Jacobi algorithm, which uses a sequence of Givens rotations to simultaneously diagonalize the covariance matrices. In comparison with other GOJD algorithms, the proposed algorithm is shown to have fast convergence. Moreover, as the optimal Givens rotation matrix in each update is calculated in closed-form, this algorithm is computationally very efficient. In the application aspect, we have considered the scenario where different data sets in J-BSS may have different number of sources, among which there exist both similar components that are consistently present in multiple data sets, and diverse components that are uniquely present in each data set. We have shown how J-BSS based on the proposed GOJD algorithm can effectively extract both similar and diverse source components. Simulation results are given to show the nice performance of the proposed algorithm, with regards to both speed and accuracy, in comparison with other algorithms of similar type.

Non-orthogonal joint diagonalization algorithm preventable ill conditioned solutions for blind source separation

The general joint diagonalization problem involves estimating the separating matrix only given mixing signals interrelated matrix set. For nonorthogonal joint diagonalization based on the weighted least-squares criterion, the algorithms may converge to trivial (zero) solution. Certainly, the trivial solution can simply be avoided by adopting some constraint on the diagonalizing matrix or penalty terms. However, free of zero solution is not enough, especially for the blind signal separation (BSS). Actually, ill-conditioned diagonalizer even though nonzero makes the objective function unstable or even divergence in the process optimization. Therefore, it is necessary to prevent the iterative solutions from degenerating ill-conditioned forms. To solve this problem, a novel nonleast-squares criterion for non-orthogonal joint diagonalization is proposed. It is imposed constrainted terms on diagonalizers, which are induced form the mathematic define of the ill condition matrix. Finally, Computer simulations indicate that the new algorithm yields diagonalizers which not only minimize the diagonalization error but also have as small condition numbers as possible, meanwhile, degenerate solutions are avoided strictly.

Blind Separation of Cyclostationary Sources Sharing Common Cyclic Frequencies Using Joint Diagonalization Algorithm

We propose a new method for blind source separation of cyclostationary sources, whose cyclic frequencies are unknown and may share one or more common cyclic frequencies. The suggested method exploits the cyclic correlation function of observation signals to compose a set of matrices which has a particular algebraic structure. The aforesaid matrices are automatically selected by proposing two new criteria. Then, they are jointly diagonalized so as to estimate the mixing matrix and retrieve the source signals as a consequence. The nonunitary joint diagonalization (NU-JD) is ensured by Broyden-Fletcher-Goldfarb-Shanno (BFGS) method which is the most commonly used update strategy for implementing a quasi-Newton technique. The efficiency of the method is illustrated by numerical simulations in digital communications context, which show good performances comparing to other state-of-the-art methods.

New Jacobi-like algorithms for non-orthogonal joint diagonalization of Hermitian matrices

In this paper, two new algorithms are proposed for non-orthogonal joint matrix diagonalization under Hermitian congruence. The idea of these two algorithms is based on the so-called Jacobi algorithm for solving the eigenvalues problem of Hermitian matrix. The algorithms are then called ‘general Jabobi-like diagonalization’ algorithms (GERALD). They are based on the search of two complex parameters by the minimization of a quadratic criterion corresponding to a measure of diagonality. Lastly, numerical simulations are conducted to illustrate the effective performances of the GERALD algorithms.

Convolutive Blind Source Separation Using an Iterative Least-Squares Algorithm for Non-Orthogonal Approximate Joint Diagonalization

In this paper, we present an approach of recovering signal waveforms of speech sources from observed signals in noisy and reverberant environments. The approach is based on approximate joint diagonalization estimate to provide interference suppression of source signals and reduce echoes and distortions of separated signals. In the proposed approach, the mixing matrix is estimated by minimizing the constrained direct least-squares (LS) criterion in direct model. Exclusively under the condition where the estimated mixing matrix is not of full rank, it is replaced by a full-rank matrix. The unmixing matrix from the estimated mixing matrix is obtained by setting the frequency response of the composite mixing-unmixing filter to identity matrix. The cross-spectral density diagonal matrices of the source signals are precisely estimated by minimizing the indirect LS criterion in indirect model. These operations are fulfilled by using alternating least-squares algorithm. The correlation between the interfrequency power ratios is used to prevent a misalignment permutation of the unmixing matrix. Finally, we compare the proposed BSS with a number of conventional BSS methods in noisy and reverberant environments under both artificial and actual conditions.

Relative gradient based algorithms for general joint diagonalization of complex matrices

This article deals with the problem of joint diagonalization of hermitian and/or complex symmetric matrices. Within the framework of gradient algorithms, we develop various algorithms which are based on different levels of approximation of the classical diagonalization criterion. The algorithms are based on a multiplicative update and on the derivation of an optimal step-size. One of the algorithms is a generalization of DOMUNG to the complex case. Finally, in the blind source separation context, computer simulations illustrate the relative performances of some proposed algorithms in comparison to the true gradient one.