An Exact FFT Recovery Theory: A Nonsubtractive Dither Quantization Approach with Applications

Abstract

Fourier transform is undoubtedly one of the cornerstones of digital signal processing (DSP). The introduction of the now famous FFT algorithm has not only breathed a new lease of life into an otherwise latent classical DFT algorithm, but also led to an explosion in applications that have now far transcended the confines of the DSP field. For a good accuracy, the digital implementation of the FFT requires that the input and/or the 2 basis functions be finely quantized. This paper exploits the use of coarse quantization of the FFT signals with a view to further improving the FFT computational efficiency while preserving its computational accuracy and simplifying its architecture. In order to resolve this apparent conflict between preserving an excellent computational accuracy while using a quantization scheme as coarse as can be desired, this paper advances new theoretical results which form the basis for two new and practically attractive FFT estimators that rely on the principle of 1 bit nonsubtractive dithered quantization (NSDQ). The proposed theory is very well substantiated by the extensive simulation work carried out in both noise-free and noisy environments. This makes the prospect of implementing the 2 proposed 1 bit FFT estimators on a chip both practically attractive and rewarding and certainly worthy of a further pursuit.