Algorithms for Programmers by Arndt J PDF

By Arndt J

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Division is not needed, except for division by n for the final normalization after transform and backtransform. Division by n is multiplication by the inverse of n. 2) Cf. [1], [3], [14] or [2] and books on number theory. 1 Prime modulus: Z/pZ = Fp If the modulus is a prime p then Z/pZ is the field Fp : All elements except 0 have inverses and ‘division is possible’ in Z/pZ. Thereby the second condition is trivially fulfilled for all FFT lengthes n < p: a prime p is coprime to all integers n < p.

Apply a (length C) FFT on each row. 4. Transpose the matrix. Note the elegance! 8 (transposed matrix Fourier algorithm) The (TMFA) for the FFT: transposed matrix Fourier algorithm 1. Transpose the matrix. 2. Apply a (length C) FFT on each column (transposed row). 3. Multiply each matrix element (index r, c) by exp(±2 π i r c/n). 4. Apply a (length R) FFT on each row (transposed column). e. g. in unit strides). In radix 2 (or 2n ) algorithms one even has skips of powers of 2, which is particularly bad on computer systems that use direct mapped cache memory: One piece of cache memory is responsible for caching addresses that lie apart by some power of 2.

For the reason given above the computation of the column FFTs should not be done in place. One can insert additional transpositions in the algorithm to have the columns lie in contiguous memory when they are worked upon. The easy way is to use an additional scratch space for the column FFTs, then only the copying from and to the scratch space will be slow. If one interleaves the copying back with the exp()multiplications (to let the CPU do some work during the wait for the memory access) the performance should be ok.

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Algorithms for Programmers by Arndt J


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