Fast transform and frequency estimation algorithm of finite Ramanujan Fourier transformation

A new Ramanujan transformation (RFT) is an arithmetic transformation based on Ramanujan sums, well adapted to the analysis of signals with fractional frequency. First, spectrum characteristic for the finite Ramanujan transform and the distribution model of Ramanujan base vectors were presented. Second, the fast algorithm for RFT was derived and the multiplication computation amount of the Ramanujan transformation with that of the fast Fourier transformation was compared. Furthermore, a recursive frequency estimation algorithm for RFT and the frequency resolution analysis had been presented. Finally, over the non-Gaussian noise, the frequency estimation performance comparison of RFT and Fourier transformation has shown that the normalized mean square error (MSE) of RFT can reach at 10^-3 for the non-Gaussian noise with the SNR equal to -20 dB.