Jacobi-like Joint Diagonalization Algorithm Based on QR Decomposition

doi:10.12068/j.issn.1005-3026.2024.03.001

Abstract

Abstract:

In order to improve the blind separation performance of approximate joint diagonalization of real matrix sets and to avoid trivial solutions， a Jacobi?like joint diagonalization algorithm based on QR decomposition is proposed. Using the numerical stability of QR decomposition， the Jacobi rotation matrix is used to decompose the separation matrix into the product of several elementary triangular matrices and orthogonal matrices. The structure of Jacobi rotation matrix and the related elements of the target matrix transformation are used to obtain the optimal parameters. The high-dimensional minimization problem is iteratively transformed into a series of low-dimensional sub-problems， which enhances the recovery accuracy of the source signal. The algorithm complexity is reduced by solving the simplified Frobenius-norm objective function. The simulation results of mixed electrocardiogram （ECG） signals show that compared with QRJ2D， LUCJD and EGJLUD， the proposed algorithm has certain advantages in separation accuracy and convergence speed.

Key words: blind source separation（BSS）, non-orthogonal joint diagonalization, QR decomposition, Jacobi-like algorithm, ECG signal model

CLC Number:

TN 911.7

Ce JI, Ye LI, Bo-qun LI. Jacobi-like Joint Diagonalization Algorithm Based on QR Decomposition[J]. Journal of Northeastern University(Natural Science), 2024, 45(3): 305-313.

Figures/Tables 18

Fig.1 Flowchart of the proposed algorithm

Fig.2 Simulated ECG signal waveform

Fig.3 Simulated observed ECG signals

Fig.4 Separation results of simulated observed signals with the proposed algorithm

Fig.5 Separation results of simulated observed signals with QRJ2D algorithm

Fig.6 Separation results of simulated observed signals with LUCJD algorithm

Fig.7 Separation results of simulated observed signals with EGJLUD algorithm

Table 1 Correlation coefficient of four algorithms

算法	恢复信号1	恢复信号2
本文算法	0.993 0	0.989 5
QRJ2D	0.985 2	0.973 1
LUCJD	0.990 8	0.986 3
EGJLUD	0.991 5	0.982 3

Table 2 Crosstalk error and convergence time

算法	串音误差/dB	收敛时间/s
本文算法	-23.259	1.695
QRJ2D	-9.986	1.697
LUCJD	-21.951	1.850
EGJLUD	-17.999	1.863

Table 3 SIR of four algorithms

算法	恢复信号1	恢复信号2
本文算法	35.749	21.615
QRJ2D	23.433	9.166
LUCJD	26.339	20.747
EGJLUD	30.052	17.826

Fig.8 Crosstalk error of four algorithms under different SNR

Fig.9 1~4 lead ECG signals from DaISy database

Fig.10 Clinical ECG signal separation results with the proposed algorithm

Fig.11 Clinical ECG signal separation results with QRJ2D algorithm

Fig.12 Clinical ECG signal separation results with LUCJD algorithm

Fig.13 Clinical ECG signal separation results with EGJLUD algorithm

Table 4 SNR of four algorithms

算法	信噪比/dB
本文算法	8.17
QRJ2D	7.69
LUCJD	7.72
EGJLUD	7.84

Table 5 Time complexity of four algorithms

算法	时间复杂度
本文算法	$O K N 3$
QRJ2D	$O K N 3$
LUCJD	$O 3 K N 3 + 4 N 3$
EGJLUD	$O 4 K N 3 + N 3$

Table 5 Time complexity of four algorithms

算法	时间复杂度
本文算法	$O K N 3$
QRJ2D	$O K N 3$
LUCJD	$O 3 K N 3 + 4 N 3$
EGJLUD	$O 4 K N 3 + N 3$

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