An Exact FFT Recovery Theory: A Nonsubtractive Dither Quantization Approach with Applications
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An Exact FFT Recovery Theory: A Nonsubtractive Dither Quantization Approach with Applications L. Cheded and S. Akhtar Systems Engineering Department, King Fahd University of Petroleum and Minerals, KFUPM Box 116, Dhahran 31261, Saudi Arabia Received 27 June 2004; Revised 13 September 2005; Accepted 26 September 2005 Recommended for Publication by Jar-Ferr Kevin Yang 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. Copyright © 2006 Hindawi Publishing Corporation. All rights reserved.
1.
INTRODUCTION
The vast success of the Fourier transform is amply reflected in the wide applicability it enjoys in a variety of engineering fields such as signal and image processing, control, communications, filtering, geophysics, seismics, optics, acoustics, radar, and sonar signal processing. This explosion in applications was brought about by the introduction of the now famous and ubiquitous fast Fourier transform (FFT) which has transformed the classical discrete Fourier transform (DFT) from being a mere “academic” curiosity, with limited applications, to being a powerful computational tool whose applications continue to grow unabatedly [1]. The original radix-2 structure of the FFT underwent several structural changes all aiming at further increasing the computational speed and/or adapting the original FFT algorithm to various data length characteristics (e.g., prime and composite lengths) [2]. A contemporary view as well as a review of the state of the art of the FFT can be found in [3, 4], respectively. The numerous variations of the original radix-2 FFT algorithm were brought about through the dual use of the exploitation of symmetry properties inherent in the FFT algorithm and the principle of “divide and conquer.” How-
ever, both the original FF
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