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

By Arndt J.

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Spr] It is assumed that the procedure fht() does no normalization. cc] Cf. n-1] input, result { // transform data: fht(x[], n) // convolution in transformed domain: j := n-1 for i:=1 to n/2-1 { ci := x[i] cj := x[j] t1 := ci*cj // = cj*ci t2 := 1/2*(ci*ci-cj*cj) // = -1/2*(cj*cj-ci*ci) x[i] := t1 + t2 x[j] := t1 - t2 j := j-1 } x[0] := x[0]*x[0] if n>1 then x[n/2] := x[n/2]*x[n/2] // transform back: fht(x[], n) CHAPTER 3. spr] For odd n replace the line for i:=1 to n/2-1 by for i:=1 to (n-1)/2 and omit the line if n>1 then x[n/2] := x[n/2]*x[n/2] in both procedures above.

Apply a (length C) FFT on each row (of the transposed matrix). (memory access is without skips) 6. Multiply each matrix element (index r, c) by exp(∓2 π i r c/n). 7. Apply a (length R) FFT on each column (of the transposed matrix). (memory access with C-skips) Note that steps 3, 4 and 5 constitute a length-C convolution. e. convolutions of data sets that do not fit into the RAM workspace. An important consideration is the Minimization of the number of disk seeks The number of disk seeks has to be kept minimal because these are slow operations which, if occur too often, degrade performance unacceptably.

Cc] Vice versa: Let T be the operator corresponding to the fht_fft_conversion, T is its own inverse: T = T −1 , or, equivalently T · T = 1. 16) CHAPTER 3. n-1] input,result { fft(c[], n) fht_fft_conversion(c[], n, is) } or the same thing with swapped lines. Of course the same ideas also work for separate real- and imaginaryparts. n-1] input,result { fht(a[], n) for i:=1 to n/2-1 { t := n - i u := a[i] v := a[t] a[i] := 1/2 * (u+v) a[t] := 1/2 * (u-v) } } At the end of this procedure the ordering of the output data c ∈ C is a[0] a[1] a[2] = = = c0 c1 c2 a[n/2] a[n/2 + 1] a[n/2 + 2] a[n/2 + 3] ...

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

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