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2 n − 1 are the imaginary part. 23) Cyclic, negacyclic and right-angle convolution can be understood as polynomial products modulo the polynomials z n − 1, z n + 1 and z n ± i, respectively (see [Nussb]). cc] The semi-symbolic table (see page 40) for the negacyclic convolution is [fxtbook draft of 2004-May-24] 46 Chapter 2: Algorithms for fast convolution +-| 0: 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14: 15: 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 0- 2 3 4 5 6 7 8 9 10 11 12 13 14 15 01- 3 4 5 6 7 8 9 10 11 12 13 14 15 012- 4 5 6 7 8 9 10 11 12 13 14 15 0123- 5 6 7 8 9 10 11 12 13 14 15 01234- 6 7 8 9 10 11 12 13 14 15 012345- 7 8 9 10 11 12 13 14 15 0123456- 8 9 10 11 12 13 14 15 01234567- 9 10 11 12 13 14 15 012345678- 10 11 12 13 14 15 0123456789- 11 12 13 14 15 012345678910- 12 13 14 15 01234567891011- 13 14 15 0123456789101112- 14 15 012345678910111213- 15 01234567891011121314- Here the products that enter with negative sign are indicated with a postfix minus at the corresponding entry.

An−1 , 0, 0, . . 9) Same for B. 10a) 2 n−1 hτ := Ax Bτ −x τ = 0, 1, 2, . . 10b) 40 Chapter 2: Algorithms for fast convolution As an illustration consider the convolution of the sequence [1, 1, 1, 1] with itself: its linear self convolution is the length-8 sequence [h0 ][h1 ] = [1, 2, 3, 4][3, 2, 1, 0], its cyclic self convolution is [h0 + h1 ] = [4, 4, 4, 4]. A convenient way to illustrate the cyclic convolution of two sequences is the following semi-symbolical table: +-| 0: 1: 2: 3: 0 1 2 3 4 5 6 0 1 2 3 1 2 3 4 2 3 4 5 3 4 5 6 4 5 6 7 5 6 7 8 4: 5: 6: 7: 4 5 6 7 5 6 7 8 9 10 11 12 13 14 15 6 7 7 8 8 9 9 10 8 9 10 11 9 10 11 12 10 11 12 13 11 12 13 14 12 13 14 15 13 14 15 0 14 15 0 1 15 0 1 2 6 7 7 8 8 9 9 10 8 9 10 11 9 10 11 12 10 11 12 13 11 12 13 14 12 13 14 15 13 14 15 0 14 15 0 1 15 0 1 2 8: 9: 10: 11: 8 9 10 11 9 10 11 12 10 11 12 13 11 12 13 14 12 13 14 15 13 14 15 0 14 15 0 1 15 0 1 2 12: 13: 14: 15: 12 13 14 15 13 14 15 0 14 15 0 1 15 0 1 2 0 1 2 3 1 2 3 4 2 3 4 5 7 3 4 5 6 8 0 1 2 3 1 2 3 4 2 3 4 5 3 4 5 6 4 5 6 7 5 6 7 8 6 7 7 8 8 9 9 10 0 1 2 3 1 2 3 4 2 3 4 5 3 4 5 6 4 5 6 7 5 6 7 8 6 7 7 8 8 9 9 10 8 9 10 11 9 10 11 12 10 11 12 13 11 12 13 14 The entries denote where in the convolution the products of the input elements can be found: +-| 0: 1: 2: 3: 0 1 2 0 1 1 3 4 8 ...

51) cn/2 For the imaginary part of the result there are two schemes: Scheme 1 (‘parallel ordering’) is a[n/2 + 1] a[n/2 + 2] a[n/2 + 3] = = = ... 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.

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