2025/04/17 by Saulo Queiroz, Queiroz, Saulo, João P. Vilela +7
Computer Science · Engineering · #Advanced Electrical Measurement Techniques #Computational Complexity (cs.CC) #Data Structures and Algorithms (cs.DS) #Digital Filter Design and Implementation #FOS: Computer and information sciences #FOS: Electrical engineering #Signal Processing (eess.SP) #Sparse and Compressive Sensing Techniques #electronic engineering #information engineering
paper · pdf · doi:10.48550/arxiv.2504.12551
openalex publication_date 2025/04/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In~\citesic-magazine-2025, the authors show that the square index coefficients (SICs) of the N-point discrete Fourier transform (DFT) -- that is, the coefficients Xk√(N) for k = 0, 1, …, √(N) - 1 -- can be losslessly compressed from N to √(N) points, thereby accelerating the computation of these specific DFT coefficients accordingly. Following up on that, in this article we generalize SICs into what we refer to as rectangular index coefficients (RICs) of the DFT, formalized as XkL, k=0,1,⋯,C-1, in which the integers C and L are generic roots of N such that N=LC. We present an algorithm to compress the N-point input signal x into a C-point signal \mathbfx at the expense of O(N) complex sums and no complex multiplication. We show that a DFT on \mathbfx is equivalent to a DFT on the RICs of x. In cases where specific frequencies of x are of interest -- as in harmonic analysis -- one can conveniently adjust the signal parameters (e.g., frequency resolution) to align the RICs with those frequencies, and use the proposed algorithm to compute them significantly faster. If N is a power of two -- as required by the fast Fourier transform (FFT) algorithm -- then C can be any power of two in the range [2, N/2] and one can use our algorithm along with FFT to compute all RICs in O(Clog C) time complexity.