Fast Computation of the Discrete Fourier Transform Rectangular Index Coefficients

TL;DR

This work extends the sparse-DFT idea by introducing Rectangular Index Coefficients (RICs) that compress an $N$-point signal to a $C$-point vector so that the $C$-point DFT of the compressed signal yields the $X_{kL}$ coefficients, with no multiplier operations in the compression step. It shows how $N=LC$ and $L$ and $C$ are chosen from a power-of-two decomposition, enabling $X_{kL} = \hat{X}_k$ where $\hat{x}_c=\sum_{l=0}^{L-1} x_{lC+c}$, and provides a detailed normalization framework for DFT/IDFT mappings. The proposed algorithm achieves multiplierless compression with $\mathcal{O}(N)$ additions and, when combined with FFT, $\mathcal{O}(C\log C)$ multiplications, yielding an overall complexity of $\mathcal{O}(N) + \mathcal{O}(C\log C)$; SIC is recovered as a special case when $L=C$. This enables fast computation of chosen frequency indices with adjustable resolution, and the paper provides concrete DFT/IDFT examples and discusses normalization. Future work includes error analysis, spectral leakage assessment, and extending the approach to odd-indexed coefficients.

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 $X_{k\sqrt{N}}$ 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 $\mathcal{O}(C\log C)$ time complexity.