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E. 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.

The equivalent table for a (cyclic) correlation is +-| 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 CHAPTER 2. CONVOLUTIONS 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 15 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 14 15 0 1 2 3 4 5 6 7 8 9 10 11 12 13 13 14 15 0 1 2 3 4 5 6 7 8 9 10 11 12 12 13 14 15 0 1 2 3 4 5 6 7 8 9 10 11 11 12 13 14 15 0 1 2 3 4 5 6 7 8 9 10 40 10 11 12 13 14 15 0 1 2 3 4 5 6 7 8 9 9 10 11 12 13 14 15 0 1 2 3 4 5 6 7 8 8 9 10 11 12 13 14 15 0 1 2 3 4 5 6 7 7 8 9 10 11 12 13 14 15 0 1 2 3 4 5 6 6 7 8 9 10 11 12 13 14 15 0 1 2 3 4 5 5 6 7 8 9 10 11 12 13 14 15 0 1 2 3 4 4 5 6 7 8 9 10 11 12 13 14 15 0 1 2 3 3 4 5 6 7 8 9 10 11 12 13 14 15 0 1 2 2 3 4 5 6 7 8 9 10 11 12 13 14 15 0 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 0 while the acyclic counterpart is: +-| 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 31 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 30 31 0 1 2 3 4 5 6 7 8 9 10 11 12 13 29 30 31 0 1 2 3 4 5 6 7 8 9 10 11 12 28 29 30 31 0 1 2 3 4 5 6 7 8 9 10 11 27 28 29 30 31 0 1 2 3 4 5 6 7 8 9 10 26 27 28 29 30 31 0 1 2 3 4 5 6 7 8 9 25 26 27 28 29 30 31 0 1 2 3 4 5 6 7 8 24 25 26 27 28 29 30 31 0 1 2 3 4 5 6 7 23 24 25 26 27 28 29 30 31 0 1 2 3 4 5 6 22 23 24 25 26 27 28 29 30 31 0 1 2 3 4 5 21 22 23 24 25 26 27 28 29 30 31 0 1 2 3 4 20 21 22 23 24 25 26 27 28 29 30 31 0 1 2 3 19 20 21 22 23 24 25 26 27 28 29 30 31 0 1 2 18 19 20 21 22 23 24 25 26 27 28 29 30 31 0 1 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 0 Note that bucket 16 does not appear, it is always zero.

The transposing back at the end of the routine can be avoided if a backtransform will follow9 , the backtransform must then be called with R and C swapped. The generalization to higher dimensions is straight forward. 10 The matrix Fourier algorithm (MFA) The matrix Fourier algorithm10 (MFA) works for (composite) data lengths n = R C. Consider the input array as a R × C-matrix (R rows, C columns). 7 (matrix Fourier algorithm) The matrix Fourier algorithm (MFA) for the FFT: 1. Apply a (length R) FFT on each column.

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