By J. Arndt
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Additional resources for Algorithms for Programmers - Ideas, Source Code
Cc]. cc]. 6 Split radix Fourier transforms (SRFT) The idea underlying the split radix FFT is to use both radix-2 and radix-4 decompositions at the same time. 24b) we use the first, the one for the even indices. 36d, slightly reordered). 39a) Now we have expressed the length-N = 2n FFT as one length-N/2 and two length-N/4 FFTs. A nice feature is that the operation count of the split radix FFT is actually lower than that of the radix-4 FFT. cc] uses a DIF core as above (which was given in ). cc] uses an DIF or DIT core, depending on the sign of the transform.
In convolution computations it is straightforward to save the transpositions by using the MFA followed by the TMFA. (The data is assumed to be in memory as row0 , row1 , . . ) For the sake of simplicity auto convolution is considered here. The matrix FFT convolution algorithm: 1. Apply a (length R) FFT on each column. (memory access with C-skips) 2. Multiply each matrix element (index r, c) by exp(+σ 2 π i r c/n). 3: Mass storage convolution using the MFA 43 3. Apply a (length C) FFT on each row.
Minimization of the number of disk seeks The number of disk seeks has to be kept minimal because these are slow operations which, if occur too often, degrade performance unacceptably. √ The crucial modification of the use of the MFA is not to choose R and C as close as possible to n as is usually done. Instead one chooses R to be minimal so that the row length C corresponds to the biggest data set that fits into the available RAM. We now analyze how the number of seeks depends on the choice of R and C: In what follows it is assumed that the data lies in memory as row0 , row1 , .
Algorithms for Programmers - Ideas, Source Code by J. Arndt