By Arndt J.

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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) where sx dx sy := x(0/2) + x(1/2) := x(0/2) − x(1/2) := y (0/2) + y (1/2) dy := y (0/2) − y (1/2) For the acyclic (or linear) convolution of sequences one can use the cyclic convolution of the zero padded sequences zx := {x0 , x1 , . . , nn−1 , 0, 0, . . e. x with n zeros appended). 23) CHAPTER 2. ω) B {ω} B is the cc.

CHAPTER 1. THE FOURIER TRANSFORM 29 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] a[n − 1] = = = ... = c1 c2 c3 = = = ... 93) c1 cn/2 which are 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.

The actual values should be inlined to some greater precision than actually needed, thereby one avoids the generation of multiple copies of the (logically) same value with differences only due to numeric inaccuracies. 382683432365089771728459984029; // // // // == == == == cos(Pi*1/16) sin(Pi*1/16) cos(Pi*2/16) sin(Pi*2/16) == == == == cos(Pi*1/16) sin(Pi*1/16) cos(Pi*1/8) sin(Pi*1/8) Automatic verification of the generated codes against the original is a mandatory part of the process. A level of abstraction for the array indices is of great use: When the print statements in the generator emit some function of the index instead of its plain value it is easy to generate modified versions of the code for permuted input.

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