Improve this question. Papiro Papiro 1, 5 5 gold badges 29 29 silver badges 43 43 bronze badges. Else, does Huang quote this paper: cis. It's called the Fast Fourier transform because its a fast method of calculating a Fourier transform. What else do you want from this question? Is it really a question? Show 1 more comment.
Active Oldest Votes. Improve this answer. Francois Ziegler Francois Ziegler 28k 4 4 gold badges silver badges bronze badges. Also, there is a paper from L. Alsop and A. Unfortunatelly, I have no access to this paper. On page , we have The data were analyzed not only with the 'fast Fourier transform' but also with a Fourier analysis program prepared by one of the authors A.
N , and The amplitudes obtained by the two programs with the known amplitudes are comparedin Table 1. The fast Fourier transform is slightly more accurate This paper was published on November 15, but received on July 11, These are combined into 4 point arrays, then 8 point arrays, then 16 point arrays and so forth until you get a single array.
The number of times you have to pass through the data is log base 2 of N, not the square of N. The total number of complex calculations is n log base 2 of n. If there are points, the FFT is about times faster. If there are 1,, points, the FFT is about , times faster.
If you need to do Fourier analysis you also need the FFT to get the job done in a reasonable amount of time. This is what the FFT look like in "C. That means, for example, that the contents of locations and are swapped. The result of an FFT is in normal sequence. There are other ways to do it. A separate array is not needed to store the result thus halving memory requirements.
The FFT, in its most natural form, starts with complex data and it produces a complex result. However, some applications produce only real data, the imaginary data being set to zero. The result of an FFT in that case is an array with complex conjugate symmetry.
That means that half of the array can be easily reconstructed from the other half. This fact can be exploited to reduce memory requirements and computation times by half. The need for speed The need to perform Fourier analysis really fast was very apparent in Cooley-Tukey and removing the redundant parts of the computation, saving roughly a factor of two in time and memory.
Bruun's algorithm above is another method that was initially proposed to take advantage of real inputs, but it has not proved popular. A few "FFT" algorithms have been proposed, however, that compute the DFT approximately , with an error that can be made arbitrarily small at the expense of increased computations.
Such algorithms trade the approximation error for increased speed or other properties. For example, an approximate FFT algorithm by Edelman et al. Another algorithm for approximate computation of a subset of the DFT outputs is due to Shentov et al. Only the Edelman algorithm works equally well for sparse and non-sparse data, however, since it is based on the compressibility rank deficiency of the Fourier matrix itself rather than the compressibility sparsity of the data. Even the "exact" FFT algorithms have errors when finite-precision floating-point arithmetic is used, but these errors are typically quite small; most FFT algorithms, e.
Cooley-Tukey, have excellent numerical properties. These results, however, are very sensitive to the accuracy of the twiddle factors used in the FFT i. Moreover, even achieving this accuracy requires careful attention to scaling in order to minimize the loss of precision, and fixed-point FFT algorithms involve rescaling at each intermediate stage of decompositions like Cooley-Tukey.
Equivalently, it is simply the composition of a sequence of d one-dimensional DFTs, performed along one dimension at a time in any order. This compositional viewpoint immediately provides the simplest and most common multidimensional DFT algorithm, known as the row-column algorithm after the two-dimensional case, below.
That is, one simply performs a sequence of d one-dimensional FFTs by any of the above algorithms : first you transform along the k 1 dimension, then along the k 2 dimension, and so on or actually, any ordering will work. This method is easily shown to have the usual O N log N complexity, where is the total number of data points transformed.
In two dimensions, the can be viewed as an matrix , and this algorithm corresponds to first performing the FFT of all the rows and then of all the columns or vice versa , hence the name.
0コメント