MATLAB Code Implementation of Least Squares Method for Localization Networks

In this technical article, we explore the implementation of least squares method using MATLAB code for constructing localization networks and configuring node parameters. The least squares method serves as an optimization technique for data fitting and identifying optimal curves in the presence of measurement errors. Our implementation focuses on calculating inter-node distances through matrix operations and utilizing these distance measurements to determine precise node positions through linear algebra solutions. The MATLAB code employs the backslash operator (\) for solving linear systems Ax=b, which efficiently computes the least squares solution using QR decomposition. For node configuration, we demonstrate how to initialize anchor positions and mobile nodes using coordinate arrays, while implementing distance measurement functions that handle Gaussian noise through random number generation. The algorithm continuously refines position estimates by minimizing the sum of squared residuals between measured and calculated distances. We also share optimal node placement strategies using geometric optimization techniques to achieve superior localization accuracy, including grid-based and circular deployment patterns. The complete MATLAB implementation includes functions for data preprocessing, coefficient matrix construction, and iterative refinement using gradient descent approaches when dealing with nonlinear problems. This comprehensive guide provides practical insights into building robust localization networks through mathematical optimization and proper node configuration techniques.