Fast Computation of the Discrete Fourier Transform Rectangular Index Coefficients

2025-04-17

AI Summary

To address high computational redundancy in the discrete Fourier transform (DFT) when only a subset of frequency coefficients is required, this paper introduces Rectangular Index Coefficients (RICs), enabling arbitrary factorization (N = L imes C) and compressing an (N)-point input into a (C)-point signal. A multiplication-free signal compression algorithm enables exact, low-complexity extraction of RICs—requiring only (O(N)) complex additions and zero complex multiplications—to equivalently obtain the desired DFT coefficients. Integrating FFT optimization further reduces RIC computation to (O(C log C)) when (N) is a power of two, achieving an acceleration ratio of (N/C). This work generalizes prior square-index-based approaches to a universal rectangular indexing scheme, supporting on-demand configuration of frequency resolution to align with critical harmonics. As a result, it significantly improves real-time performance and energy efficiency in sparse-spectrum applications such as harmonic analysis.

Technology Category

• Data Mining & Knowledge Management: Data Compression
• Search and Optimization: Mixed Discrete/Continuous Search
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines

Abstract

In~cite{sic-magazine-2025}, the authors show that the square index coefficients (SICs) of the (N)-point discrete Fourier transform (DFT) -- that is, the coefficients (X_{ksqrt{N}}) for (k = 0, 1, ldots, sqrt{N} - 1) -- can be losslessly compressed from (N) to (sqrt{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 $X_{kL}, k=0,1,cdots,C-1$, 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 $mathbf{x}$ into a $C$-point signal $mathbf{hat{x}}$ at the expense of $mathcal{O}(N)$ complex sums and no complex multiplication. We show that a DFT on $mathbf{hat{x}}$ is equivalent to a DFT on the RICs of $mathbf{x}$. In cases where specific frequencies of (mathbf{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 $mathcal{O}(Clog C)$ time complexity.

Problem

• Generalize square index coefficients to rectangular index coefficients (RICs).
• Develop algorithm to compress N-point signal to C-point.
• Accelerate DFT computation for specific frequencies using RICs.

Innovation

• Generalizes square index coefficients to rectangular index coefficients
• Compresses N-point signal to C-point with O(N) sums
• Combines with FFT for O(C log C) time complexity