Implementation of GS Algorithm Using Fourier Transform in Two-Dimensional Space

This document presents the implementation of the Gram-Schmidt (GS) algorithm in MATLAB, utilizing Fourier transform for two-dimensional spatial applications. The GS algorithm is a fundamental signal processing technique that operates in the frequency domain to achieve functions such as noise suppression and feature extraction. Fourier transform serves as a mathematical tool for converting signals from the time domain to the frequency domain, decomposing signals into different frequency components to facilitate spectral analysis. By integrating the GS algorithm with Fourier transform, we can effectively process and analyze signals in two-dimensional space. This combined approach involves key MATLAB functions including fft2 for 2D Fast Fourier Transform, ifft2 for inverse transformation, and orthogonalization procedures for vector processing. The implementation typically follows these steps: performing 2D Fourier transform on input data, applying GS orthogonalization in the frequency domain, and reconstructing the processed signal through inverse Fourier transform. This methodology finds extensive applications in image processing, audio signal analysis, communication systems, and other domains requiring spatial frequency manipulation. The code implementation emphasizes efficient matrix operations and proper handling of complex frequency domain data to ensure accurate signal reconstruction.