FRFT

The discrete Fourier transform evaluate fft_frft_1.gif.
In some cases we need the fractional Fourier transform (FrFT) fft_frft_2.gif.
This can also be done in O(n log n) computations.
The following algorithm is given by David H. Bailey & Paul N. Swarztrauber.
Again, the built-in function Fourier[] which can do this.
I just write this to serve as a pseudo code so that i can impliment it in C.
Note that the inverse FrFT is unnecessary since α can be negative.

fft_frft_3.gif

77.
-14.272-1.91872 i
-0.210611+8.33817 i
13.1626 +3.66773 i
-11.5849-60.725 i
15.2433 +13.7436 i
7.01947 -8.43697 i
-6.85686-10.1127 i
77.
-14.272-1.91872 i
-0.210611+8.33817 i
13.1626 +3.66773 i
-11.5849-60.725 i
15.2433 +13.7436 i
7.01947 -8.43697 i
-6.85686-10.1127 i
77.
-14.272-1.91872 i
-0.210611+8.33817 i
13.1626 +3.66773 i
-11.5849-60.725 i
15.2433 +13.7436 i
7.01947 -8.43697 i
-6.85686-10.1127 i

fft_frft_4.gif

Spikey Created with Wolfram Mathematica 8.0