In this paper, two different methods are used to study the cyclic structure solution and the optimal approximation of the quaternion Stein equation AXB - X = F . Firstly, the matrix equation equivalent to the ta...In this paper, two different methods are used to study the cyclic structure solution and the optimal approximation of the quaternion Stein equation AXB - X = F . Firstly, the matrix equation equivalent to the target structure matrix is constructed by using the complex decomposition of the quaternion matrix, to obtain the necessary and sufficient conditions for the existence of the cyclic solution of the equation and the expression of the general solution. Secondly, the Stein equation is converted into the Sylvester equation by adding the necessary parameters, and the condition for the existence of a cyclic solution and the expression of the equation’s solution are then obtained by using the real decomposition of the quaternion matrix and the Kronecker product of the matrix. At the same time, under the condition that the solution set is non-empty, the optimal approximation solution to the given quaternion circulant matrix is obtained by using the property of Frobenius norm property. Numerical examples are given to verify the correctness of the theoretical results and the feasibility of the proposed method. .展开更多
In order to calculate the cross-correlation of two color images treated as vector in a holistic manner,a rapid vertical/parallel decomposition algorithm for quaternion is presented.The calculation for decomposition is...In order to calculate the cross-correlation of two color images treated as vector in a holistic manner,a rapid vertical/parallel decomposition algorithm for quaternion is presented.The calculation for decomposition is reduced from 21 times to 4 times real number multiplications with the same results.An algorithm for cross-correlation of color images based on decomposition in time domain is put forward,in which some properties pointed out in this paper can be utilized to reduce the computational complexity.Simulation results show the effectiveness and superiority of the proposed method.展开更多
In this paper we derive a practical method of solving simultaneously the problem of Schmidt decomposition of quaternion matrix and the orthonormalization of vectors in a generalized unitary space by using elementary c...In this paper we derive a practical method of solving simultaneously the problem of Schmidt decomposition of quaternion matrix and the orthonormalization of vectors in a generalized unitary space by using elementary column operations on matrices over the quaternion field.展开更多
本文证明了长方四元数矩阵奇异值的一些不等式:设H为四元数体,A∈H^(n×m),B∈H^(m×k),S=min{n,k},1≤l≤s,则 sum from i=1 to l σ_i(AB)≤sum from i=1 to l σ_i(A)σ_i(B) (ⅰ) sum from i=1 to l σ_s _(i+1)(AB)≥sum f...本文证明了长方四元数矩阵奇异值的一些不等式:设H为四元数体,A∈H^(n×m),B∈H^(m×k),S=min{n,k},1≤l≤s,则 sum from i=1 to l σ_i(AB)≤sum from i=1 to l σ_i(A)σ_i(B) (ⅰ) sum from i=1 to l σ_s _(i+1)(AB)≥sum from i+j=m+s-l+1 σ_i(A)σ_j(B) (ⅱ) multiply from i=1 to l σ_i(A)σ_(m-i+1) (B)≤multiply from i=1 to l σ_i(AB)≤multiply from i=1 to l σ_i(A)σ_i(B) (ⅲ) 其中,σ_1(A)≥σ_2(A)≥…≥σ_m(A)≥0是A的从大到小的奇异值,当i>m时,σ_1(A)(?)0。不等式(ⅰ),(ⅱ),(ⅲ)包含或加强了文[3]、[4]、[5]的一些基本结果。展开更多
在"Joint Estimation of DOA and Polarization with CLD Pair Cylindrical Array Based on Quaternion Model"一文中,其阵列流形是通过特征值分解反推得到的,该文的作者认为信号导向矢量可以通过阵列流形的每一列除以该列首...在"Joint Estimation of DOA and Polarization with CLD Pair Cylindrical Array Based on Quaternion Model"一文中,其阵列流形是通过特征值分解反推得到的,该文的作者认为信号导向矢量可以通过阵列流形的每一列除以该列首元素得到。在本文中我们指出,通过特征值分解反推得到的阵列流形,其每列与真实值存在一个模糊系数,由于四元数乘法不满足交换律,从而使得导向矢量无法通过阵列流形的每一列除以该列首元素得到。最后还给出一种求解导向矢量的方法,并通过计算机仿真进行验证。展开更多
该文将四元数理论应用到双基地集中式多输入多输出(MIMO)雷达的角度估计中。文中通过传统数据模型构造四元数矩阵,提出了基于四元数的求根-多重信号分类(Root MUltiple SIgnal Classification,Root-MUSIC)的MIMO雷达中角度估计算法,该...该文将四元数理论应用到双基地集中式多输入多输出(MIMO)雷达的角度估计中。文中通过传统数据模型构造四元数矩阵,提出了基于四元数的求根-多重信号分类(Root MUltiple SIgnal Classification,Root-MUSIC)的MIMO雷达中角度估计算法,该算法通过奇异值分解和Root-MUSIC来估计出发射角(Direction Of Departure,DOD)和接收角(Direction Of Arrival,DOA)。该算法的角度估计性能远优于现有文献的方法,并且无需谱峰搜索,复杂度大大降低。仿真结果验证了算法的有效性。展开更多
文摘In this paper, two different methods are used to study the cyclic structure solution and the optimal approximation of the quaternion Stein equation AXB - X = F . Firstly, the matrix equation equivalent to the target structure matrix is constructed by using the complex decomposition of the quaternion matrix, to obtain the necessary and sufficient conditions for the existence of the cyclic solution of the equation and the expression of the general solution. Secondly, the Stein equation is converted into the Sylvester equation by adding the necessary parameters, and the condition for the existence of a cyclic solution and the expression of the equation’s solution are then obtained by using the real decomposition of the quaternion matrix and the Kronecker product of the matrix. At the same time, under the condition that the solution set is non-empty, the optimal approximation solution to the given quaternion circulant matrix is obtained by using the property of Frobenius norm property. Numerical examples are given to verify the correctness of the theoretical results and the feasibility of the proposed method. .
基金supported by the National Natural Science Foundation of China (6060402160874054)
文摘In order to calculate the cross-correlation of two color images treated as vector in a holistic manner,a rapid vertical/parallel decomposition algorithm for quaternion is presented.The calculation for decomposition is reduced from 21 times to 4 times real number multiplications with the same results.An algorithm for cross-correlation of color images based on decomposition in time domain is put forward,in which some properties pointed out in this paper can be utilized to reduce the computational complexity.Simulation results show the effectiveness and superiority of the proposed method.
文摘In this paper we derive a practical method of solving simultaneously the problem of Schmidt decomposition of quaternion matrix and the orthonormalization of vectors in a generalized unitary space by using elementary column operations on matrices over the quaternion field.
文摘本文证明了长方四元数矩阵奇异值的一些不等式:设H为四元数体,A∈H^(n×m),B∈H^(m×k),S=min{n,k},1≤l≤s,则 sum from i=1 to l σ_i(AB)≤sum from i=1 to l σ_i(A)σ_i(B) (ⅰ) sum from i=1 to l σ_s _(i+1)(AB)≥sum from i+j=m+s-l+1 σ_i(A)σ_j(B) (ⅱ) multiply from i=1 to l σ_i(A)σ_(m-i+1) (B)≤multiply from i=1 to l σ_i(AB)≤multiply from i=1 to l σ_i(A)σ_i(B) (ⅲ) 其中,σ_1(A)≥σ_2(A)≥…≥σ_m(A)≥0是A的从大到小的奇异值,当i>m时,σ_1(A)(?)0。不等式(ⅰ),(ⅱ),(ⅲ)包含或加强了文[3]、[4]、[5]的一些基本结果。
文摘在"Joint Estimation of DOA and Polarization with CLD Pair Cylindrical Array Based on Quaternion Model"一文中,其阵列流形是通过特征值分解反推得到的,该文的作者认为信号导向矢量可以通过阵列流形的每一列除以该列首元素得到。在本文中我们指出,通过特征值分解反推得到的阵列流形,其每列与真实值存在一个模糊系数,由于四元数乘法不满足交换律,从而使得导向矢量无法通过阵列流形的每一列除以该列首元素得到。最后还给出一种求解导向矢量的方法,并通过计算机仿真进行验证。
文摘该文将四元数理论应用到双基地集中式多输入多输出(MIMO)雷达的角度估计中。文中通过传统数据模型构造四元数矩阵,提出了基于四元数的求根-多重信号分类(Root MUltiple SIgnal Classification,Root-MUSIC)的MIMO雷达中角度估计算法,该算法通过奇异值分解和Root-MUSIC来估计出发射角(Direction Of Departure,DOD)和接收角(Direction Of Arrival,DOA)。该算法的角度估计性能远优于现有文献的方法,并且无需谱峰搜索,复杂度大大降低。仿真结果验证了算法的有效性。
