By J. Arndt

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13) That is, use a convolution algorithm with one of the input sequences reversed. 14) For the computation of self-correlation the latter relation is the only reasonable way to go: First transform the input sequence, then multiply each element by its complex conjugate and finally transform back. The semi-symbolical table for a (cyclic) correlation is +-| 0: 1: 2: 3: 0 4: 5: 6: 7: 4 5 6 7 1 2 3 4 5 0 15 14 13 1 0 15 14 2 1 0 15 3 2 1 0 12 13 14 15 11 12 13 14 6 7 10 9 11 10 12 11 13 12 3 4 5 6 2 3 4 5 1 2 3 4 0 15 14 13 1 0 15 14 2 1 0 15 3 2 1 0 8: 9: 10: 11: 8 7 9 8 10 9 11 10 6 7 8 9 5 6 7 8 4 5 6 7 12: 13: 14: 15: 12 13 14 15 11 12 13 14 10 9 11 10 12 11 13 12 8 9 10 11 8 7 9 8 10 9 11 10 12 13 14 15 11 12 13 14 6 7 8 9 5 6 7 8 10 9 11 10 12 11 13 12 3 4 5 6 2 3 4 5 1 2 3 4 0 15 14 13 1 0 15 14 2 1 0 15 3 2 1 0 8 7 9 8 10 9 11 10 6 7 8 9 5 6 7 8 4 5 6 7 3 4 5 6 2 3 4 5 1 2 3 4 12 13 14 15 4 5 6 7 3 4 5 6 2 3 4 5 1 2 3 4 8 7 9 8 10 9 11 10 6 7 8 9 5 6 7 8 12 13 14 15 11 12 13 14 10 9 11 10 12 11 13 12 0 15 14 13 1 0 15 14 2 1 0 15 3 2 1 0 [fxtbook draft of 2004-May-24] 42 Chapter 2: Algorithms for fast convolution while the acyclic counterpart is: +-| 0: 1: 2: 3: 0 4: 5: 6: 7: 4 5 6 7 1 2 3 4 5 6 7 8 0 31 30 29 1 0 31 30 2 1 0 31 3 2 1 0 28 29 30 31 27 28 29 30 26 27 28 29 25 26 27 28 24 25 26 27 23 24 25 26 22 23 24 25 21 22 23 24 20 21 22 23 19 20 21 22 18 19 20 21 17 18 19 20 28 29 30 31 27 28 29 30 26 27 28 29 25 26 27 28 24 25 26 27 23 24 25 26 22 23 24 25 21 22 23 24 28 29 30 31 27 28 29 30 26 27 28 29 25 26 27 28 3 4 5 6 2 3 4 5 1 2 3 4 0 31 30 29 1 0 31 30 2 1 0 31 3 2 1 0 8: 9: 10: 11: 8 7 9 8 10 9 11 10 6 7 8 9 5 6 7 8 4 5 6 7 12: 13: 14: 15: 12 13 14 15 11 12 13 14 10 9 11 10 12 11 13 12 9 10 11 3 4 5 6 2 3 4 5 1 2 3 4 0 31 30 29 1 0 31 30 2 1 0 31 3 2 1 0 8 7 9 8 10 9 11 10 6 7 8 9 5 6 7 8 4 5 6 7 3 4 5 6 2 3 4 5 1 2 3 4 12 13 14 15 0 31 30 29 1 0 31 30 2 1 0 31 3 2 1 0 Note that bucket 16 does not appear, it is always zero.

Note the vx that entered: the weighted transform with vx = √1n ∀x is just the usual Fourier transform. 18c) Obviously all vx have to be invertible. That Wv Wv−1 [a] is also identity is apparent from the definitions. Given an implemented FFT it is trivial to set up a weighted Fourier transform. 4: Weighted Fourier transforms and convolutions 45 procedure weighted_ft(a[], v[], n, is) { for x:=0 to n-1 { a[x] := a[x] * v[x] } fft(a[], n, is) } The inverse is essentially identical. cc]. 8a. It is not hard to see why this is: Up to the final division by the weight sequence, the weighted convolution is just the cyclic convolution of the two weighted sequences, which is for the element with index τ equal to (ax V x ) (by V y ) x+y≡τ ax bτ −x V τ + = x≤τ mod n ax bn+τ −x V n+τ Final division of this element (by V τ ) gives h(0) + V n h(1) as stated.

52) cn/2−1 Scheme 2 (‘antiparallel ordering’) is a[n/2 + 1] a[n/2 + 2] a[n/2 + 3] = = = ... 53) c1 cn/2 which are always zero. Real valued FT via wrapper routines A simple way to use a complex length-n/2 FFT for a real length-n FFT (n even) is to use some postand preprocessing routines. For a real sequence a one feeds the (half length) complex sequence f = a(even) + i a(odd) into a complex FFT. Some post-processing is necessary. This is not the most elegant real FFT available, but it is directly usable to turn complex FFTs of any (even) length into a real-valued FFT.