cannam@127: cannam@127: cannam@127: cannam@127: cannam@127: cannam@127: FFTW 3.3.5: The 1d Real-data DFT cannam@127: cannam@127: cannam@127: cannam@127: cannam@127: cannam@127: cannam@127: cannam@127: cannam@127: cannam@127: cannam@127: cannam@127: cannam@127: cannam@127: cannam@127: cannam@127: cannam@127: cannam@127: cannam@127: cannam@127: cannam@127:
cannam@127:

cannam@127: Next: , Previous: , Up: What FFTW Really Computes   [Contents][Index]

cannam@127:
cannam@127:
cannam@127: cannam@127:

4.8.2 The 1d Real-data DFT

cannam@127: cannam@127:

The real-input (r2c) DFT in FFTW computes the forward transform cannam@127: Y of the size n real array X, exactly as defined cannam@127: above, i.e. cannam@127:

.
This output array Y can easily be shown to possess the cannam@127: “Hermitian” symmetry cannam@127: cannam@127: Yk = Yn-k*,where we take Y to be periodic so that cannam@127: Yn = Y0.

cannam@127:

As a result of this symmetry, half of the output Y is redundant cannam@127: (being the complex conjugate of the other half), and so the 1d r2c cannam@127: transforms only output elements 0…n/2 of Y cannam@127: (n/2+1 complex numbers), where the division by 2 is cannam@127: rounded down. cannam@127:

cannam@127:

Moreover, the Hermitian symmetry implies that cannam@127: Y0and, if n is even, the cannam@127: Yn/2element, are purely real. So, for the R2HC r2r transform, the cannam@127: halfcomplex format does not store the imaginary parts of these elements. cannam@127: cannam@127: cannam@127: cannam@127:

cannam@127: cannam@127:

The c2r and H2RC r2r transforms compute the backward DFT of the cannam@127: complex array X with Hermitian symmetry, stored in the cannam@127: r2c/R2HC output formats, respectively, where the backward cannam@127: transform is defined exactly as for the complex case: cannam@127:

.
The outputs Y of this transform can easily be seen to be purely cannam@127: real, and are stored as an array of real numbers. cannam@127:

cannam@127: cannam@127:

Like FFTW’s complex DFT, these transforms are unnormalized. In other cannam@127: words, applying the real-to-complex (forward) and then the cannam@127: complex-to-real (backward) transform will multiply the input by cannam@127: n. cannam@127:

cannam@127: cannam@127: cannam@127: cannam@127: cannam@127: