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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@42: Y of the size n real array X, exactly as defined Chris@42: above, i.e. Chris@42:

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This output array Y can easily be shown to possess the Chris@42: “Hermitian” symmetry Chris@42: Chris@42: Yk = Yn-k*,where we take Y to be periodic so that Chris@42: Yn = Y0.

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As a result of this symmetry, half of the output Y is redundant Chris@42: (being the complex conjugate of the other half), and so the 1d r2c Chris@42: transforms only output elements 0…n/2 of Y Chris@42: (n/2+1 complex numbers), where the division by 2 is Chris@42: rounded down. Chris@42:

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Moreover, the Hermitian symmetry implies that Chris@42: Y0and, if n is even, the Chris@42: Yn/2element, are purely real. So, for the R2HC r2r transform, the Chris@42: halfcomplex format does not store the imaginary parts of these elements. Chris@42: Chris@42: Chris@42: Chris@42:

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The c2r and H2RC r2r transforms compute the backward DFT of the Chris@42: complex array X with Hermitian symmetry, stored in the Chris@42: r2c/R2HC output formats, respectively, where the backward Chris@42: transform is defined exactly as for the complex case: Chris@42:

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The outputs Y of this transform can easily be seen to be purely Chris@42: real, and are stored as an array of real numbers. Chris@42:

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Like FFTW’s complex DFT, these transforms are unnormalized. In other Chris@42: words, applying the real-to-complex (forward) and then the Chris@42: complex-to-real (backward) transform will multiply the input by Chris@42: n. Chris@42:

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