Multiplierless DFT Approximation Based on the Prime Factor Algorithm

TL;DR

This work develops a generic, multiplierless DFT approximation framework based on the Prime Factor Algorithm (PFA) that decomposes large, non-power-of-two transform lengths into coprime ground transforms. By choosing small ground sizes ($N_1\in\{3,11,31\}$) and approximating them with a fixed expansion factor $\alpha=9/8$, the authors construct a fully multiplierless 1023-point DFT approximation via the Good–Thomas mapping, with diagonal scaling $\hat{S}$ that can itself be approximated to preserve multiplierless operation. The approach yields 15 distinct 1023-point approximations and corresponding 3-, 11-, and 31-point ground transforms, achieving substantial reductions in arithmetic complexity while maintaining competitive error metrics and spectral fidelity compared to prior methods. The results suggest strong potential for hardware-efficient large-scale DFT blocks suitable for beamforming, spectrum sensing, and IoT-era signal processing, with scalability to other coprime factorizations. Overall, the paper demonstrates that fully multiplierless, low-complexity DFT approximations are feasible and advantageous for energy- and area-constrained implementations.

Abstract

Matrix approximation methods have successfully produced efficient, low-complexity approximate transforms for the discrete cosine transforms and the discrete Fourier transforms. For the DFT case, literature archives approximations operating at small power-of-two blocklenghts, such as \{8, 16, 32\}, or at large blocklengths, such as 1024, which are obtained by means of the Cooley-Tukey-based approximation relying on the small-blocklength approximate transforms. Cooley-Tukey-based approximations inherit the intermediate multiplications by twiddled factors which are usually not approximated; otherwise the effected error propagation would prevent the overall good performance of the approximation. In this context, the prime factor algorithm can furnish the necessary framework for deriving fully multiplierless DFT approximations. We introduced an approximation method based on small prime-sized DFT approximations which entirely eliminates intermediate multiplication steps and prevents internal error propagation. To demonstrate the proposed method, we design a fully multiplierless 1023-point DFT approximation based on 3-, 11- and 31-point DFT approximations. The performance evaluation according to popular metrics showed that the proposed approximations not only presented a significantly lower arithmetic complexity but also resulted in smaller approximation error measurements when compared to competing methods.

Figures (4)

• Figure 1: Error plots between the filter bank frequency response magnitude for (a) 3-point approximation, (b) 11-point approximation, (c) 31-point approximation, and (d) 1023-point approximation and their exact counterparts.
• Figure 2: Comparison between the magnitude of the filter-bank responses of the approximations and their exact counterparts for the least three performing rows.
• Figure 3: Comparison between the magnitude of the filter-bank responses of the approximation $\hat{\mathbf{F }}^\prime_{1023}$ and the exact DFT for the least three performing rows.
• Figure 4: Comparison between the magnitude responses obtained from madanayake2020fast and $\hat{\mathbf{F}}'_{1023}$ with their exact counterparts, applied to a pure cosine signal.