SECTION 5 FAST FOURIER TRANSFORMS （第五节快速傅里叶变换）.pdfVIP

FAST FOURIER TRANSFORMS SECTION 5 FAST FOURIER TRANSFORMS The Discrete Fourier Transform The Fast Fourier Transform FFT Hardware Implementation and Benchmarks DSP Requirements for Real Time FFT Applications Spectral Leakage and Windowing 5.a FAST FOURIER TRANSFORMS 5.b FAST FOURIER TRANSFORMS SECTION 5 FAST FOURIER TRANSFORMS Walt Kester THE DISCRETE FOURIER TRANSFORM In 1807 the French mathematician and physicist Jean Baptiste Joseph Fourier presented a paper to the Institut de France on the use of sinusoids to represent temperature distributions. The paper made the controversial claim that any continuous periodic signal could be represented by the sum of properly chosen sinusoidal waves. Among the publication review committee were two famous mathematicians: Joseph Louis Lagrange, and Pierre Simon de Laplace. Lagrange objected strongly to publication on the basis that Fourier’s approach would not work with signals having discontinuous slopes, such as square waves. Fourier’s work was rejected, primarily because of Lagrange’s objection, and was not published until the death of Lagrange, some 15 years later. In the meantime, Fourier’s time was occupied with political activities, expeditions to Egypt with Napoleon, and trying to avoid the guillotine after the French Revolution! (This bit of history extracted from Reference 1, p.141). It turns out that both Fourier and Lagrange were at least partially correct. Lagrange was correct that a summation of sinusoids cannot exactly form a signal with a corner. However, you can get very close if enough sinusoids are used. (This is described by the Gibbs effect, and is well understood by scientists, engineers, and mathematicians today). Fo