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(fftw.info)What RFFTW Really Computes

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What RFFTW Really Computes
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   In this section, we define precisely what RFFTW computes.

   The real to complex (`FFTW_REAL_TO_COMPLEX') transform of a real
array X of size n computes an hermitian array Y, where

  Y[i] = sum for j = 0 to (n - 1) of X[j] * exp(-2 pi i j sqrt(-1)/n)

   (That Y is a hermitian array is not intended to be obvious, although
the proof is easy.)  The hermitian array Y is stored in halfcomplex
order (Note: Data Types.).  Currently, RFFTW provides no way to
compute a real to complex transform with a positive sign in the
exponent.

   The complex to real (`FFTW_COMPLEX_TO_REAL') transform of a hermitian
array X of size n computes a real array Y, where

  Y[i] = sum for j = 0 to (n - 1) of X[j] * exp(2 pi i j sqrt(-1)/n)

   (That Y is a real array is not intended to be obvious, although the
proof is easy.)  The hermitian input array X is stored in halfcomplex
order (Note: Data Types.).  Currently, RFFTW provides no way to
compute a complex to real transform with a negative sign in the
exponent.

   Like FFTW, RFFTW computes an unnormalized transform.  In other words,
applying the real to complex (forward) and then the complex to real
(backward) transform will multiply the input by n.