By Arndt J
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Extra info for Algorithms for Programmers
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 .
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.
It may be a surprise that no such algorithm seems to be known currently5 . Here is a clumsy attempt to find h(0) alone: Use the weighted transform with the weight sequence vx = V x where V n is very small. Then h(1) will in the result be multiplied with a small number and we hope to make it almost disappear. 000}. At least for integer sequences one could choose V n (more than two times) bigger than biggest possible value in h(1) and use rounding to nearest integer to isolate h(0) . Alas, even for modest sized arrays numerical overflow and underflow gives spurious results.