Lyons R. G. Understanding Digital Signal Processing / REFERENCES

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Understanding Digital Signal Processing
Authors: Lyons R. G.
Published year: 2004
Pages: 45-46/183

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REFERENCES

[1] Bracewell, R. "The Fourier Transform," Scientific American , June 1989.

[2] Struik, D. A Concise History of Mathematics , Dover Publications Inc., New York, 1967, p. 142.

[3] Williams, C. S. Designing Digital Filters . Section 8.6, Prentice-Hall, Englewood Cliffs, New Jersey, 1986, p. 122.

[4] Press, W., et al. Numerical Recipes—The Art of Scientific Computing . Cambridge University Press, 1989, p. 426.

[5] Geckinli, N. C., and Yavuz, D. "Some Novel Windows and a Concise Tutorial Comparison of Window Families," IEEE Trans. on Acoust. Speech, and Signal Proc. , Vol. ASSP-26, No. 6, December 1978. (By the way, on page 505 of this paper, the phrase "such that W(f) f" indicates that W(f) is never negative. The symbol means "for all.")

[6] O'Donnell, J. "Looking Through the Right Window Improves Spectral Analysis," EDN , November 1984.

[7] Kaiser, J. F. "Digital Filters," in System Analysis by Digital Computer . Ed. by F. F. Kuo and J. F. Kaiser, John Wiley and Sons, New York, 1966, pp. 218–277.

[8] Rabiner, L. R., and Gold, B. The Theory and Application of Digital Signal Processing . Prentice-Hall, Englewood Cliffs, New Jersey, 1975, p. 88.

[9] Schoenwald, J. "The Surface Acoustic Wave Filter: Window Functions," RF Design , March 1986.

[10] Harris, F. "On the Use of Windows for Harmonic Analysis with the Discrete Fourier Transform," Proceedings of the IEEE , Vol. 66, No. 1, January 1978.

[11] Nuttall, A. H. "Some Windows with Very Good Sidelobe Behavior," IEEE Trans. on Acoust. Speech, and Signal Proc. , Vol. ASSP-29, No. 1, February 1981.

[12] Yanagimoto, Y. "Receiver Design for a Combined RF Network and Spectrum Analyzer," Hewlett-Packard Journal , October, 1993.

[13] Gullemin, E. A. The Mathematics of Circuit Analysis . John Wiley and Sons, New York, 1949, p. 511.

[14] Lanczos, C. Discourse on Fourier Series , Chapter 1, Hafner Publishing Co., New York, 1966, p. 7–47.

[15] Freeny, S. "TDM/FDM Translation As an Application of Digital Signal Processing," IEEE Communications Magazine , January 1980.

[16] Oppenheim, A., et al., Discrete-Time Signal Processing , 2nd ed. Prentice-Hall, Upper Saddle River, New Jersey, 1999, pp. 48–51.


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Chapter Four. The Fast Fourier Transform

Although the DFT is the most straightforward mathematical procedure for determining the frequency content of a time-domain sequence, it's terribly inefficient. As the number of points in the DFT is increased to hundreds, or thousands, the amount of necessary number crunching becomes excessive. In 1965 a paper was published by Cooley and Tukey describing a very efficient algorithm to implement the DFT[1]. That algorithm is now known as the fast Fourier transform (FFT). ^[

] Before the advent of the FFT, thousand-point DFTs took so long to perform that their use was restricted to the larger research and university computer centers. Thanks to Cooley, Tukey, and the semiconductor industry, 1024-point DFTs can now be performed in a few seconds on home computers.

^[

] Actually, the FFT has an interesting history. While analyzing X-ray scattering data, a couple of physicists in the 1940s were taking advantage of the symmetries of sines and cosines using a mathematical method based on a technique published in the early 1900s. Remarkably, over 20 years passed before the FFT was (re) discovered . Reference [2] tells the full story.

Volumes have been written about the FFT, and, like no other innovation, the development of this algorithm transformed the discipline of digital signal processing by making the power of Fourier analysis affordable. In this chapter, we'll show why the most popular FFT algorithm (called the radix-2 FFT) is superior to the classical DFT algorithm, present a series of recommendations to enhance our use of the FFT in practice, and provide a list of sources for FFT routines in various software languages. We conclude this chapter, for those readers wanting to know the internal details, with a derivation of the radix-2 FFT and introduce several different ways in which this FFT is implemented.