Evaluation of Fourier transform estimation schemes of multidimensional signals using random sampling

Equidistributed sampling sequences for spectral analysis

“…We must remark that there are variations of random sampling sequences which can provide Fourier transform estimates with faster asymptotic convergence rates [5], [6]. Nonetheless, these advanced estimates also converge at the rate of 0.5 1 N and do not show the fast convergence unless the number of collected samples is considerably large, which however defeats the whole object of random sampling of employing low sampling rates [7], [8].…”

“…Also, the decay rate of the error of the calculated Fourier transform of signals with bounded variation is ( ) log N N which is faster than that for standard random sampling, i.e. [4], although there are random sampling schemes [5], [6] which can show faster decay rate but they require further restrictions on the processed signal and relatively high number of samples [7], [8]. These sequences are also known as quasi-random sequences [9] and used for numerical calculation of definite integrals.…”

Efficient multidimensional sampling scheme for Fourier transform estimation

“…Several such methods have been presented and thoroughly analysed for onedimensional (1-D) signals in [2]- [5] and the K-dimensional (K-D) case in [6]. The term random sampling refers to the fact that the sampling instants are randomly distributed according to a pre-defined probability density function(s).…”

“…Fourier transform estimation based on total random sampling has been studied in [2], [3], for the 1-D case and in [6] for the K-D case. These methods approximate the Fourier transform using N samples with estimation quality independent of the positioning of the spectral components of the signal in the frequency domain.…”

“…The mean-square error in the estimated Fourier transform decays at the rate of 1/ N . This rate has been significantly improved by using stratified sampling [4], [6] and its enhanced version antithetical stratified sampling [5], [6]. Although these methods also converge at the rate of 1/ N for small N, it can be shown that they accelerate with increasing N to 1 2/ 1/ K N + in the case of stratified sampling and even 1 4/ 1/ K N + for antithetical stratified sampling, provided that the processed signal is adequately smooth [6].…”