System Function DFT
From NARS2000
This function is available in both monadic and dyadic forms
Monadic Function


R is a scalar or vector of Real or Complex numbers.  
Z is a vector of length 2*⌈2⍟≢R of Complex numbers. 
The monadic function is implemented by choosing the Fast Fourier Transform algorithm appropriate to the datatype of R. In particular, if the datatype of R is
 Fixed Precision Integer or Floating Point, the FFT algorithm from Gnu Scientific Library is used
 Multiple Precision Integer/Rational or Floating Point, the FFT algorithm MPFFT is used
 Ball Arithmetic, the FFT algorithm from ARB is used
Because the underlying FFT algorithm is most efficient when the length of R is a power of two, the system function automatically pads its argument with a sufficient number of trailing zeros, and returns a result of the padded length.
For example,
⎕PP←10 ⎕DFT 1 2 3 4 10 ¯2i2 ¯2 ¯2i¯2
Dyadic Function


R is a scalar or vector of Real or Complex numbers.  
L is an integer scalar whose value is either 1 or ¯1.  
Z is a vector of length 2*⌈2⍟≢R of Complex numbers. 
The dyadic function 1 ⎕DFT R returns the Discrete Fourier Transform of R; ¯1 ⎕DFT R returns the Inverse Discrete Fourier Transform. As in the monadic case, the dyadic function chooses the (Inverse) Fast Fourier Transform algorithm appropriate to the datatype of R.
For example,
⎕PP←10 ⎕DFT 1 2 3 4 10 ¯2i2 ¯2 ¯2i¯2 1 ⎕DFT 1 2 3 4 10 ¯2i2 ¯2 ¯2i¯2 ¯1 ⎕DFT ⎕DFT 1 2 3 4 1 2 3 4