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

By Arndt J.

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Further let α ≥ 2 be the number of times the data set exceeds the RAM size. 5 one reads from disk (row by row, involving R seeks) the number of colums that just fit into RAM, does the (many, short) column-FFTs3 , writes back (again R seeks) and proceeds to the next block; this happens for α of these blocks, giving a total of 4 α R seeks for steps 1 and 3. In step 2 one has to read (α times) blocks of one or more rows, which lie in contiguous portions of the disk, perform the FFT on the rows and write back to disk, leading to a total of 2 α seeks.

All other weighted convolutions involve complex computations, but it is easy to see how to reduce the work by 50 percent: As the result must be real the data in row number R − r must, because of the symmetries of the real and imaginary part of the (inverse) Fourier transform of real data, be the complex conjugate of the data in row r. Therefore one can use real FFTs (R2CFTs) for all column-transforms for step 1 and half-complex to real FFTs (C2RFTs) for step 3. Let the computational cost of a cyclic (real) convolution be q, then For R even one must perform 1 cyclic (row 0), 1 negacyclic (row R/2) and R/2 − 2 complex (weighted) convolutions (rows 1, 2, .

Careful analysis shows that this idea leads to an algorithm far worse than simply using linear convolution. 1): 5 If you know one, tell me about it! CHAPTER 2. CONVOLUTIONS 45 1. Apply a FFT on each column. 2. R − 1 the index of the row, C the length of each row (or, equivalently the total number columns) 3. Apply a FFT on each column (of the transposed matrix). 21) For the acyclic (or linear) convolution of sequences one can use the cyclic convolution of the zero padded sequences zx := {x0 , x1 , .

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