Advertisement

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

  • L ChededEmail author
  • S Akhtar
Open Access
Research Article
  • 748 Downloads

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.

Keywords

Fourier Transform Basis Function Quantum Information Computational Efficiency Digital Signal Processing 

References

  1. 1.
    Cooley JW, Tukey JW: An algorithm for the machine computation of complex Fourier series. Mathematics of Computation 1965, 19(90):297–301. 10.1090/S0025-5718-1965-0178586-1MathSciNetCrossRefGoogle Scholar
  2. 2.
    Burrus CS, Parks TW: DFT/FFT and Convolution Algorithms. John Wiley & Sons, New York, NY, USA; 1985. see also C. S. Burrus: "Notes on the FFT", https://doi.org/www.dsp.rice.eduzbMATHGoogle Scholar
  3. 3.
    Ganapathiraju A, Hamaker J, Picone J, Skjellum A: Contemporary view of FFT algorithms. Proceedings of the IASTED International Conference on Signal and Image Processing (SIP '98), October 1998, Las Vegas, Nev, USA 130–133.Google Scholar
  4. 4.
    Duhamel P, Vetterli M: Fast Fourier transforms: a tutorial review and a state of the art. Signal Processing 1990, 19(4):259–299. 10.1016/0165-1684(90)90158-UMathSciNetCrossRefGoogle Scholar
  5. 5.
    Kuo SM, Gan W-SS: Digital Signal Processors: Architectures, Implementations, and Applications. Prentice-Hall, Upper Saddle River, NJ, USA; 2005.Google Scholar
  6. 6.
    Cheded L: Exact recovery of higher order moments. IEEE Transactions on Information Theory 1998, 44(2):851–858. 10.1109/18.661534MathSciNetCrossRefGoogle Scholar
  7. 7.
    Gray RM, Stockham TG Jr.: Dithered quantizers. IEEE Transactions on Information Theory 1993, 39(3):805–812. 10.1109/18.256489CrossRefGoogle Scholar
  8. 8.
    Wannamaker RA, Lipshitz SP, Vanderkooy J, Wright JN: A theory of nonsubtractive dither. IEEE Transactions on Signal Processing 2000, 48(2):499–516. 10.1109/78.823976CrossRefGoogle Scholar
  9. 9.
    Cheded L, Akhtar S: On the FFT of 1-bit dither-quantized signals. Proceedings of 10th IEEE Technical Exchange Meeting (TEM '03), March 2003, Dhahran, Saudi ArabiaGoogle Scholar
  10. 10.
    Cheded L: On the exact recovery of the FFT of noisy signals using a non-subtractively dither-quantized input channel. Proceedings of 7th International Symposium on Signal Processing and Its Applications (ISSPA '03), July 2003, Paris, France 2: 539–542.Google Scholar
  11. 11.
    Cheded L, Akhtar S: A new, fast and low-cost FFT estimation scheme of signals using 1-bit non-subtractive dithered quantization. Proceedings of the 6th Nordic Signal Processing Symposium (NORSIG '04), June 2004, Espoo, Finland 236–239.Google Scholar
  12. 12.
    Cheded L, Akhtar S: A novel and fast 1-bit FFT scheme with two dither-quantized channels. Proceedings of 12th European Signal Processing Conference (EUSIPCO '04), September 2004, Vienna, AustriaGoogle Scholar
  13. 13.
    Cheded L: On the exact recovery of cumulants. Proceedings of 4th International Conference on Signal Processing (ICSP '98), October 1998, Beijing, China 1: 423–426.CrossRefGoogle Scholar
  14. 14.
    Cheded L, Akhtar S: A new and fast frequency response estimation technique for noisy systems. Proceedings of 35th Asilomar Conference on Signals, Systems and Computers (Asilomar '01), November 2001, Pacific Grove, Calif, USA 2: 1374–1378.Google Scholar
  15. 15.
    Cheded L: Theory for fast and cost-effective frequency response estimation of systems. IEE Proceedings - Vision, Image, & Signal Processing 2004, 151(6):467–479. 10.1049/ip-vis:20040606CrossRefGoogle Scholar
  16. 16.
    Stark H, Woods JW: Probability, Random Processes, and Estimation Theory for Engineers. 2nd edition. Prentice-Hall, Englewood Cliffs, NJ, USA; 1994.Google Scholar

Copyright information

© Cheded and Akhtar 2006

Authors and Affiliations

  1. 1.Systems Engineering DepartmentKing Fahd University of Petroleum and MineralsDhahranSaudi Arabia

Personalised recommendations