" Dangerous and HIStory were more Michael's life story." A product of Jackson switching up his sound to keep up with the R&B of the Nineties, the title track to Dangerous is stark and driving, with vocals that tilt between anger and terror, and lyrics about lust passing over into a "web of sin." The track evolved out of a Bad-era outtake called "Streetwalker" that he revisited and retitled during the Dangerous sessions with co-writer Bill Bottrell. Algorithms that recursively factorize the DFT into smaller operations other than DFTs include the Bruun and QFT algorithms.Off the Wall and Thriller and Bad were more entertainment," recalls longtime Jackson engineer Bruce Swedien. The Rader–Brenner algorithm (1976) is a Cooley–Tukey-like factorization but with purely imaginary twiddle factors, reducing multiplications at the cost of increased additions and reduced numerical stability it was later superseded by the split-radix variant of Cooley–Tukey (which achieves the same multiplication count but with fewer additions and without sacrificing accuracy). There are FFT algorithms other than Cooley–Tukey.įor N = N 1 N 2 with coprime N 1 and N 2, one can use the prime-factor (Good–Thomas) algorithm (PFA), based on the Chinese remainder theorem, to factorize the DFT similarly to Cooley–Tukey but without the twiddle factors. Main articles: Prime-factor FFT algorithm, Bruun's FFT algorithm, Rader's FFT algorithm, Chirp Z-transform, and hexagonal fast Fourier transform Also, because the Cooley–Tukey algorithm breaks the DFT into smaller DFTs, it can be combined arbitrarily with any other algorithm for the DFT, such as those described below. Although the basic idea is recursive, most traditional implementations rearrange the algorithm to avoid explicit recursion. These are called the radix-2 and mixed-radix cases, respectively (and other variants such as the split-radix FFT have their own names as well). The best known use of the Cooley–Tukey algorithm is to divide the transform into two pieces of size N/2 at each step, and is therefore limited to power-of-two sizes, but any factorization can be used in general (as was known to both Gauss and Cooley/Tukey ). This method (and the general idea of an FFT) was popularized by a publication of Cooley and Tukey in 1965, but it was later discovered that those two authors had independently re-invented an algorithm known to Carl Friedrich Gauss around 1805 (and subsequently rediscovered several times in limited forms). As a result, it manages to reduce the complexity of computing the DFT from O ( N 2 ) multiplications by complex roots of unity traditionally called twiddle factors (after Gentleman and Sande, 1966 ). An FFT rapidly computes such transformations by factorizing the DFT matrix into a product of sparse (mostly zero) factors. This operation is useful in many fields, but computing it directly from the definition is often too slow to be practical. The DFT is obtained by decomposing a sequence of values into components of different frequencies. Fourier analysis converts a signal from its original domain (often time or space) to a representation in the frequency domain and vice versa. A discrete Fourier analysis of a sum of cosine waves at 10, 20, 30, 40, and 50 HzĪ fast Fourier transform ( FFT) is an algorithm that computes the discrete Fourier transform (DFT) of a sequence, or its inverse (IDFT).
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