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In C matters are simple: e± 2 π i/n is a primitive n-th root of unity for arbitrary n. e2 π i/21 is a 21-th root of unity. r = e2 π i/3 is also 21-th root of unity but not a primitive root, because r3 = 1. A primitive n-th root of 1 in Z/mZ is also called an element of order n. The ‘cyclic’ property of the elements r of order n lies in the heart of all FFT algorithms: rn+k = rk . In Z/mZ things are not that simple since primitive roots of unity do not exist for arbitrary n, they exist for some maximal order R only.

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.

X with n zeros appended). ω) B {ω} B is the cc. of C {ω2 } C and therefore every B {} B-term is the cc. of the C {} C-term in the same line. Is there a nice and general scheme for real valued convolutions based on the MFA? Read on for the positive answer. 6 s, d lower half plus/minus higher half of x CHAPTER 2. 6 46 Convolution of real valued data using the MFA For row 0 (which is real after the column FFTs) one needs to compute the (usual) cyclic convolution; for row R/2 (also real after the column FFTs) a negacyclic convolution is needed7 , the code for that task is given on page 62.

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Algorithms for programmers. Ideas and source code by Arndt J.

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