CAJ | 학술논문

基于多维尺度分析的到达时间差定位算法是一种新型而高效的到达时间差定位算法，它通过多维尺度分析，将到达时间差定位问题建模为矩阵范数的最优化问题，然后通过子空间分析将该最优化问题转化为线性方程求解问题。指出该算法在推导过程中的一处疏漏，即在转化为线性方程求解问题过程中，并不能通过标量积矩阵的正定性得到目标线性方程，因为标量积矩阵并非正定。针对二维定位的情形给出一种严格的证明。该证明针对位置坐标矩阵列向量的线性相关性进行分类，当该矩阵的列向量线性无关时，目标线性方程成立；而当该矩阵的列向量线性相关时，通过分析该矩阵的列秩和行秩，可以得出参考点和目标点所必须满足的几何排布条件，并验证在该条件下目标线性方程仍然成立。
기우다유척도분석적도체시간차정위산법시일충신형이고효적도체시간차정위산법，타통과다유척도분석，장도체시간차정위문제건모위구진범수적최우화문제，연후통과자공간분석장해최우화문제전화위선성방정구해문제。지출해산법재추도과정중적일처소루，즉재전화위선성방정구해문제과정중，병불능통과표량적구진적정정성득도목표선성방정，인위표량적구진병비정정。침대이유정위적정형급출일충엄격적증명。해증명침대위치좌표구진렬향량적선성상관성진행분류，당해구진적렬향량선성무관시，목표선성방정성립；이당해구진적렬향량선성상관시，통과분석해구진적렬질화행질，가이득출삼고점화목표점소필수만족적궤하배포조건，병험증재해조건하목표선성방정잉연성립。
The multi-dimensional scaling (MDS )-based TDOA algorithm is a new and efficient TDOA localization algorithm. The basic idea is to model the TDOA localization problem into a problem of optimizing the norm of a special matrix, which is then transformed into a set of linear equations by subspace analysis. In this paper,an omission in the derivation process of the original algorithm is pointed out, i. e. , during the process of transforming the matrix norm optimization problem into the linear equations,the objective linear equations should not be derived by the positive definiteness of the scalar product matrix,because the scalar product matrix is not positive definite. This paper presents a rigorous proof in the two-dimensional. In the case when the column vectors of the position coordinates matrix are linearly independent,the objective linear equations hold. In the case when the above column vectors are linearly dependent,by analyzing the column rank and the row rank of the matrix,the necessary condition on the geometric arrangement of the reference points and the target point is obtained,under which the objective linear equations still hold. By this way, the correctness of the MDS algorithm is confirmed.