.Chris@19: Next: 1d Real-even DFTs (DCTs), Chris@19: Previous: The 1d Discrete Fourier Transform (DFT), Chris@19: Up: What FFTW Really Computes Chris@19:
The real-input (r2c) DFT in FFTW computes the forward transform
Chris@19: Y of the size n real array X, exactly as defined
Chris@19: above, i.e.
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.As a result of this symmetry, half of the output Y is redundant Chris@19: (being the complex conjugate of the other half), and so the 1d r2c Chris@19: transforms only output elements 0...n/2 of Y Chris@19: (n/2+1 complex numbers), where the division by 2 is Chris@19: rounded down. Chris@19: Chris@19:
Moreover, the Hermitian symmetry implies that
Chris@19: Y0and, if n is even, the
Chris@19: Yn/2element, are purely real. So, for the R2HC r2r transform, these
Chris@19: elements are not stored in the halfcomplex output format.
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The c2r and H2RC r2r transforms compute the backward DFT of the
Chris@19: complex array X with Hermitian symmetry, stored in the
Chris@19: r2c/R2HC output formats, respectively, where the backward
Chris@19: transform is defined exactly as for the complex case:
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.Y of this transform can easily be seen to be purely
Chris@19: real, and are stored as an array of real numbers.
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Chris@19: Like FFTW's complex DFT, these transforms are unnormalized. In other Chris@19: words, applying the real-to-complex (forward) and then the Chris@19: complex-to-real (backward) transform will multiply the input by Chris@19: n. Chris@19: Chris@19: Chris@19: Chris@19: