When Ramanujan sums meet affine Fourier transform

Ramanujan Fourier transform is one of the efficient multiresolution analysis tools, but it only works for stationary or periodically characterized signals. Affine Fourier transform, a general form of Fourier transform, is suitable for nonstationary signal analysis but with fixed resolution. In this study, the Ramanujan sums are combined with the affine Fourier transform to provide affine multiresolution analysis tools for nonstationary signals. First, affine Ramanujan sums are proposed and their main properties are introduced, based on which the affine Ramanujan Fourier transform (ARFT) for infinite sequences is proposed. Then, the first type ARFT (ARFT-I) for finite sequences is obtained by truncating the ARFT. Meanwhile, the transform matrix of ARFT-I is not orthogonal in general. Thus, the affine Ramanujan subspaces are constructed by affine Ramanujan sums and their canonical circular time shifts, among which different subspaces are orthogonal. Subsequently, the second type ARFT (ARFT-II) for finite sequences is developed, where the transform matrix is block orthogonal. Independent variables of ARFTs are chirp periods, which provides novel and direct methods to estimate (hidden) chirp periods. Different simulations, such as radar moving target detection and velocity estimation, are designed to show superior performance of ARFTs than affine Fourier transform based algorithms.