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5.7 Discrete Fourier Transform

(require 'dft) or (require 'Fourier-transform) fft and fft-1 compute the Fast-Fourier-Transforms (O(n*log(n))) of arrays whose dimensions are all powers of 2.

sft and sft-1 compute the Discrete-Fourier-Transforms for all combinations of dimensions (O(n^2)).

— Function: sft array prot
— Function: sft array

array is an array of positive rank. sft returns an array of type prot (defaulting to array) of complex numbers comprising the Discrete Fourier Transform of array.

— Function: sft-1 array prot
— Function: sft-1 array

array is an array of positive rank. sft-1 returns an array of type prot (defaulting to array) of complex numbers comprising the inverse Discrete Fourier Transform of array.

— Function: fft array prot
— Function: fft array

array is an array of positive rank whose dimensions are all powers of 2. fft returns an array of type prot (defaulting to array) of complex numbers comprising the Discrete Fourier Transform of array.

— Function: fft-1 array prot
— Function: fft-1 array

array is an array of positive rank whose dimensions are all powers of 2. fft-1 returns an array of type prot (defaulting to array) of complex numbers comprising the inverse Discrete Fourier Transform of array.

dft and dft-1 compute the discrete Fourier transforms using the best method for decimating each dimension.

— Function: dft array prot
— Function: dft array

dft returns an array of type prot (defaulting to array) of complex numbers comprising the Discrete Fourier Transform of array.

— Function: dft-1 array prot
— Function: dft-1 array

dft-1 returns an array of type prot (defaulting to array) of complex numbers comprising the inverse Discrete Fourier Transform of array.

(fft-1 (fft array)) will return an array of values close to array. 

     (fft '#(1 0+i -1 0-i 1 0+i -1 0-i)) ⇒
     
     #(0.0 0.0 0.0+628.0783185208527e-18i 0.0
       0.0 0.0 8.0-628.0783185208527e-18i 0.0)
     
     (fft-1 '#(0 0 0 0 0 0 8 0)) ⇒
     
     #(1.0 -61.23031769111886e-18+1.0i -1.0 61.23031769111886e-18-1.0i
       1.0 -61.23031769111886e-18+1.0i -1.0 61.23031769111886e-18-1.0i)