An FFT Optimal Scaling Vector Search Scheme Based on Dynamic Programming

Abstract

The fast Fourier transform (FFT) is widely used in digital signal processing. However, hardware implementations of the discrete Fourier transform (DFT)/FFT are limited by word length constraints, necessitating truncation, saturation, and scaling operations to balance hardware resources and computational performance. The choice of a suitable scaling vector significantly affects FFT efficiency. This paper introduces a novel estimation model for FFT error power, accounting for both quantization and saturation errors. Based on this model, we propose a dynamic programming (DP)-based scaling vector search scheme to reduce the search space and computational complexity. After 1000 experiments, the model demonstrated a mean and variance of relative error in signal-to-noise ratio (SNR) of 0.063 and 0.35, proving its effectiveness. In a case study of a 1024-point FFT, our model accurately estimated error power. While the exhaustive search yielded an optimal SNR of 65.91 dB, our method reduced the search space by 3600 times, with only a 0.3 dB loss in SNR. In a 256-point FFT hardware implementation, performance improved by over 10 dB. Our scheme achieves SNR performance comparable to other methods when bit width is fixed and superior SNR when bit width is variable. This approach offers guidance for selecting scaling vectors in FFT hardware design.