Non-Uniform Fourier Transform Method for Dual Objectives

Our primary objective involves employing non-uniform Fourier transform (NUFFT) techniques to develop signal interpolation algorithms. This approach requires sophisticated computational methods, typically implemented through specialized libraries or custom MATLAB/Python scripts utilizing optimization algorithms like conjugate gradient methods. Key implementation considerations include handling non-uniform sampling grids, optimizing computational efficiency through fast Fourier transform (FFT) approximations, and managing numerical accuracy in frequency-domain conversions. The secondary objective focuses on enhancing algorithmic documentation with comprehensive technical details, including mathematical formulations of NUFT operations, window function selections for spectral leakage reduction, and interpolation error analysis. Practical implementation would involve: - Designing NUFFT algorithms using exponential basis functions - Implementing iterative reconstruction methods for signal recovery - Incorporating anti-aliasing filters and oversampling techniques - Validating results through spectral analysis and reconstruction error metrics This dual-pronged approach ensures both theoretical robustness and practical applicability in digital signal processing applications.