The diffusion MRI community lacks a spectral perspective to investigate the intricate relationship between the diffusion MRI signals and cellular structures. Our study proposes a new simulation method that can provide a Fourier representation of the signal. Common spectral methods suffer from the singularity of the heat kernel, which requires infinite Fourier modes. We overcome the singularity by formulating the solution in a convolutional form and splitting the heat kernel into a singular part and a smooth part. Numerical experiments are performed to demonstrate the convergence of our method.