<p>Spherical harmonic analysis (SHA) and synthesis (SHS) are widely used by researchers in various fields. Both numerical integration and least-squares methods can be employed for analysis and synthesis. However, these approaches, when calculated via summation, are computationally intensive. Although the Fast Fourier Transform (FFT) algorithm is efficient, it is traditionally limited to processing global grid points starting from zero longitude. In this paper, we derive an improved FFT algorithm for spherical harmonic analysis and synthesis. The proposed algorithm eliminates the need for grid points to start at zero longitude, thereby expanding the applicability of FFT-based methods. Numerical experiments demonstrate that the new algorithm retains the computational efficiency of conventional FFT while achieving accuracy comparable to the summation method. Consequently, it enables direct harmonic coefficient calculation from global grid data without requiring interpolation to align with zero longitude. Additionally, the algrithm can generate grid points with equi-angular spacing using the improved FFT algorithm, starting from non-zero longitudes. To address the loss of orthogonality in latitude due to discrete spherical grids, a quadrature weight factor—dependent on grid type (e.g., regular or Gauss grid)—is incorporated, as summarized in this study.</p>