The new method for the discrete Fourier transform computation over a finite field is introduced. This method is a nontrivial generalization of the Duhamel-Hollmann algorithm with replacement of the Toeplitz convolution calculation by the normalized cyclic convolution calculation. Both algorithms have the smallest multiplicative complexity.
In this paper, we consider Tyler's robust covariance M-estimator under group symmetry constraints. We assume that the covariance matrix is invariant to the conjugation action of a unitary matrix group, referred to as group symmetry. Examples of group symmetric structures include circulant, perHermitian, and proper quaternion matrices. We introduce a group symmetric version of Tyler's estimator (STyler) and provide an iterative fixed point algorithm to compute it. The classical results claim that at least n=p+1 sample points in general position are necessary to ensure the existence and uniqueness of Tyler's estimator, where p is the ambient dimension. We show that the STyler requires significantly less samples. In some groups, even two samples are enough to guarantee its existence and uniqueness. In addition, in the case of elliptical populations, we provide high probability bounds on the error of the STyler. These, too, quantify the advantage of exploiting the symmetry structure. Finally, these theoretical results are supported by numerical simulations.
A normalized cyclic convolution is a cyclic convolution when one of its factors is a fixed polynomial. Herein, a novel method for constructing a normalized cyclic convolution over a finite field is introduced. This novel method is the first constructive and best known method for even lengths. This method can be applied for computing discrete Fourier transforms over finite fields.