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In addition one may keep the comments of the original code. With FFTs it is necessary to identify (‘reverse engineer’) the trigonometric values that occur in the process in terms of the corresponding argument (rational multiples of π). 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.
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.
R − 1) be a 2-dimensional array of data7 . 77) k=0 h=0 For a m-dimensional array ax (x = (x1 , x2 , x3 , . . , xm ), xi ∈ 0, 1, 2, . . , Si ) the m-dimensional Fourier transform ck (k = (k1 , k2 , k3 , . . , km ), ki ∈ 0, 1, 2, . . k ... x1 =0 x2 =0 where z = e± 2 π i/n , n = S1 S2 . . k where S = (S1 − 1, S2 − 1, . . 79) x=0 The inverse transform is again the one with the minus in the exponent of z. 80) x=0 which shows that the 2-dimensional FT can be accomplished by using 1-dimensional FTs to transform first the rows and then the columns8 .