An Approximation for the 32-point Discrete Fourier Transform

TL;DR

This work presents a 32-point approximate DFT constructed via Frobenius-norm minimization of the exact $32$-point DFT matrix against a low-complexity candidate, using a rounding strategy that confines multipliers to a small set and thus aims at null multiplicative complexity. A normalization matrix $S$ ensures the energy of the transformed basis stays near one, with the optimization constrained by $\hat{f}_{k,n} \in \{0,\pm1\} \times \{0,\pm1\}$. The resulting transform is implemented efficiently through a fast algorithm obtained by factorizing the approximate matrix as $\hat{F}_{32}^* = W_8 W_7 W_6 W_5 W_4 W_3 W_2 W_1$, where the $W_i$ are sparse; this factorization is noted as unique in the literature. Arithmetic-count results show the base approximate DFT requires $0$ real multiplications and $128$ real additions for real inputs, while the fast algorithm requires $0$ real multiplications and $144$ real additions, with $W_8$ contributing only data swaps. The explicit approximate transform matrix is provided in the Appendix, enabling practical deployment in low-power contexts such as beam-forming.

Abstract

This brief note aims at condensing some results on the 32-point approximate DFT and discussing its arithmetic complexity.