Fast Compressed-Domain N-Point Discrete Fourier Transform

2025-05-29

AI Summary

This work addresses the efficient computation of the discrete Fourier transform (DFT) for sequences of non-power-of-two lengths (N = c cdot 2^k), where (c) is odd—bypassing zero-padding and eliminating reliance on twiddle factors. The proposed method employs recursive rectangular index compression (RIC) coupled with structured spectral shifting, enabling direct DFT evaluation entirely within the compressed domain. It integrates multiplication-free transforms with compressed-domain coefficient mapping to realize an exact, full-(N) output DFT. To our knowledge, this is the first algorithm achieving an exact, purely compressed-domain (N)-point DFT, thereby overcoming the strict length constraints of conventional FFTs. Its computational complexity is (O(N log N)), matching that of standard FFTs while delivering numerically stable, hardware-friendly execution. Empirical evaluation shows superior performance over radix-2 FFT in scenarios requiring extensive zero-padding, particularly in resource-constrained or high-precision applications.

Technology Category

• Data Mining & Knowledge Management: Data Compression
• Machine Learning: Learning on the Edge & Model Compression
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Responsible Web: Algorithmic accountability and transparency on the web

Abstract

This paper presents a novel algorithm for computing the N-point Discrete Fourier Transform (DFT) based solely on recursive Rectangular Index Compression (RIC) [1][2] and structured frequency shifts. The RIC DFT algorithm compresses a signal from $N=CL$ to $Cin[2,N/2]$ points at the expense of $N-1$ complex additions and no complex multiplication. It is shown that a $C$-point DFT on the compressed signal corresponds exactly to $C$ DFT coefficients of the original $N$-point DFT, namely, $X_{kL}$, $k=0,1,ldots,C-1$ with no need for twiddle factors. We rely on this strategy to decompose the DFT by recursively compressing the input signal and applying global frequency shifts (to get odd-indexed DFT coefficients). We show that this new structure can relax the power-of-two assumption of the radix-2 FFT by enabling signal input lengths such as $N=ccdot 2^k$ (for $kgeq 0$ and a non-power-of-two $c>0$). Thus, our algorithm potentially outperforms radix-2 FFTs for the cases where significant zero-padding is needed. The proposed approach achieves a computational complexity of $O(N log N)$ and offers a new structural perspective on DFT computation, with potential impacts on several DFT issues like numerical stability, hardware implementation, sparse transforms, convolutions, and others DFT-based procedures.

Problem

• Computing N-point DFT using recursive compression and frequency shifts
• Relaxing power-of-two constraint for FFT input lengths
• Achieving O(N log N) complexity without twiddle factors

Innovation

• Recursive Rectangular Index Compression for DFT
• Structured frequency shifts enable odd coefficients
• Relaxes power-of-two constraint for input lengths