Doppler Spectrum Estimation by Ramanujan Fourier Transforms

TL;DR

This work tackles Doppler spectrum estimation for weather radar signals, highlighting limitations of standard DFT/FFT approaches for aperiodic content. It introduces Ramanujan sums $c_q(n)$ and the Ramanujan-Fourier Transform (RFT) as a new framework to analyze arithmetic-like sequences and estimate the Doppler PSD, complementing traditional time- and frequency-domain methods. The authors derive the RFT formalism, including inversion, orthogonality $\sum_{n=1}^{q} c_q^2(n)=q\phi(q)$, and multiplicativity, and validate the approach on real WSR-88D radar data, showing PSD estimates comparable to FFT with potential data reduction and improved sensitivity to low-magnitude spectrum components. The results suggest that RFT-based spectrum estimation can offer faster or more robust performance for certain weather radar signals and could enable real-time implementations using coprime resonances and number-theoretic coefficients $x_q$, paving the way for broader adoption of Ramanujan-based signal processing in meteorology.

Abstract

The Doppler spectrum estimation of a weather radar signal in a classic way can be made by two methods, temporal one based in the autocorrelation of the successful signals, whereas the other one uses the estimation of the power spectral density PSD by using Fourier transforms. We introduces a new tool of signal processing based on Ramanujan sums cq(n), adapted to the analysis of arithmetical sequences with several resonances p/q. These sums are almost periodic according to time n of resonances and aperiodic according to the order q of resonances. New results will be supplied by the use of Ramanujan Fourier Transform (RFT) for the estimation of the Doppler spectrum for the weather radar signal.

Figures (6)

• Figure 1: The received Doppler spectrum
• Figure 2: Time series I and Q
• Figure 3: Doppler spectrum
• Figure 4: Ramanujan Spectrum for the range cell of N${}^\circ$1
• Figure 5: Mean wind velocity estimation