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Understanding digital signal processing - Chapter 4

Tài liệu tham khảo giáo trình tìm hiểu xử lý tín hiệu số bằng tiếng anh ( Understanding digital signal processing ) Chương 4. Chuyển đổi Fourier nhanh | 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. 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. 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 EFT was re discovered. Reference 2 tells the full story. 129 130 The Fast Fourier Transform 4.1 Relationship of the FFT to the DFT Although many different FFT algorithms have been developed in this section we ll see why .