Chris@10: Chris@10: Chris@10: The 1d Real-data DFT - FFTW 3.3.3 Chris@10: Chris@10: Chris@10: Chris@10: Chris@10: Chris@10: Chris@10: Chris@10: Chris@10: Chris@10: Chris@10: Chris@10: Chris@10: Chris@10:
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4.8.2 The 1d Real-data DFT

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The real-input (r2c) DFT in FFTW computes the forward transform Chris@10: Y of the size n real array X, exactly as defined Chris@10: above, i.e. Chris@10:

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This output array Y can easily be shown to possess the Chris@10: “Hermitian” symmetry Chris@10: Yk = Yn-k*,where we take Y to be periodic so that Chris@10: Yn = Y0. Chris@10: Chris@10:

As a result of this symmetry, half of the output Y is redundant Chris@10: (being the complex conjugate of the other half), and so the 1d r2c Chris@10: transforms only output elements 0...n/2 of Y Chris@10: (n/2+1 complex numbers), where the division by 2 is Chris@10: rounded down. Chris@10: Chris@10:

Moreover, the Hermitian symmetry implies that Chris@10: Y0and, if n is even, the Chris@10: Yn/2element, are purely real. So, for the R2HC r2r transform, these Chris@10: elements are not stored in the halfcomplex output format. Chris@10: Chris@10: Chris@10:

The c2r and H2RC r2r transforms compute the backward DFT of the Chris@10: complex array X with Hermitian symmetry, stored in the Chris@10: r2c/R2HC output formats, respectively, where the backward Chris@10: transform is defined exactly as for the complex case: Chris@10:

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The outputs Y of this transform can easily be seen to be purely Chris@10: real, and are stored as an array of real numbers. Chris@10: Chris@10:

Like FFTW's complex DFT, these transforms are unnormalized. In other Chris@10: words, applying the real-to-complex (forward) and then the Chris@10: complex-to-real (backward) transform will multiply the input by Chris@10: n. Chris@10: Chris@10: Chris@10: Chris@10: