WikiSysop: Created page with "This function is available in both monadic and dyadic forms ==Monadic Function== {{BoxStart|Z←⎕DFT R |returns the Discrete Fourier Transform of R..."

2019-02-05T21:28:17Z

Created page with "This function is available in both monadic and dyadic forms ==Monadic Function== {{BoxStart|<apll>Z←⎕DFT R</apll> |returns the Discrete Fourier Transform of <apll>R</apll>..."

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This function is available in both monadic and dyadic forms
==Monadic Function==
{{BoxStart|<apll>Z←⎕DFT R</apll>
|returns the Discrete Fourier Transform of <apll>R</apll>.}}
{{BoxLine|<apll>R</apll> is a scalar or vector of Real or Complex numbers.}}
{{BoxEnd|<apll>Z</apll> is a vector of length <apll>2*⌈2⍟≢R</apll> of Complex numbers.}}
 
The monadic function is implemented by choosing the Fast Fourier Transform algorithm appropriate to the datatype of <apll>R</apll>. In particular, if the datatype of <apll>R</apll> is

<ul>
<li>Fixed Precision Integer or Floating Point, the FFT algorithm from Gnu Scientific Library is used</li>
<li>Multiple Precision Integer/Rational or Floating Point, the FFT algorithm MPFFT is used</li>
<li>Ball Arithmetic, the FFT algorithm from ARB is used</li>
</ul>

Because the underlying FFT algorithm is most efficient when the length of <apll>R</apll> 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,

<apll><pre>
⎕PP←10
⎕DFT 1 2 3 4
10 ¯2<_i/>2 ¯2 ¯2<_i/>¯2</pre></apll>

==Dyadic Function==
{{BoxStart|<apll>Z←L ⎕DFT R</apll>
|returns the Discrete Fourier Transform of <apll>R</apll> or its Inverse depending upon the value of <apll>L</apll>.}}
{{BoxLine|<apll>R</apll> is a scalar or vector of Real or Complex numbers.}}
{{BoxLine|<apll>L</apll> is an integer scalar whose value is either <apll>1</apll> or <apll>¯1</apll>.}}
{{BoxEnd|<apll>Z</apll> is a vector of length <apll>2*⌈2⍟≢R</apll> of Complex numbers.}}
 
The dyadic function <apll>1 ⎕DFT R</apll> returns the Discrete Fourier Transform of <apll>R</apll>; <apll>¯1 ⎕DFT R</apll> 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,

<apll><pre>
⎕PP←10
⎕DFT 1 2 3 4
10 ¯2<_i/>2 ¯2 ¯2<_i/>¯2
1 ⎕DFT 1 2 3 4
10 ¯2<_i/>2 ¯2 ¯2<_i/>¯2
¯1 ⎕DFT ⎕DFT 1 2 3 4
1 2 3 4</pre></apll>

System Function DFT - Revision history

WikiSysop: Created page with "This function is available in both monadic and dyadic forms ==Monadic Function== {{BoxStart|Z←⎕DFT R |returns the Discrete Fourier Transform of R..."