Real-to-Real Transform Kinds

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4.3.6 Real-to-Real Transform Kinds

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cannam@95: FFTW currently supports 11 different r2r transform kinds, specified by cannam@95: one of the constants below. For the precise definitions of these cannam@95: transforms, see What FFTW Really Computes. For a more colloquial cannam@95: introduction to these transform kinds, see More DFTs of Real Data. cannam@95: cannam@95:

For dimension of size n, there is a corresponding “logical” cannam@95: dimension N that determines the normalization (and the optimal cannam@95: factorization); the formula for N is given for each kind below. cannam@95: Also, with each transform kind is listed its corrsponding inverse cannam@95: transform. FFTW computes unnormalized transforms: a transform followed cannam@95: by its inverse will result in the original data multiplied by N cannam@95: (or the product of the N's for each dimension, in cannam@95: multi-dimensions). cannam@95: cannam@95:

FFTW_R2HC computes a real-input DFT with output in cannam@95: “halfcomplex” format, i.e. real and imaginary parts for a transform of cannam@95: size n stored as: cannam@95:
cannam@95: r₀, r₁, r₂, ..., r_n/2, i_(n+1)/2-1, ..., i₂, i₁ cannam@95:
(Logical N=n, inverse is FFTW_HC2R.) cannam@95: cannam@95:
FFTW_HC2R computes the reverse of FFTW_R2HC, above. cannam@95: (Logical N=n, inverse is FFTW_R2HC.) cannam@95: cannam@95:
FFTW_DHT computes a discrete Hartley transform. cannam@95: (Logical N=n, inverse is FFTW_DHT.) cannam@95: cannam@95:
FFTW_REDFT00 computes an REDFT00 transform, i.e. a DCT-I. cannam@95: (Logical N=2*(n-1), inverse is FFTW_REDFT00.) cannam@95: cannam@95:
FFTW_REDFT10 computes an REDFT10 transform, i.e. a DCT-II (sometimes called “the” DCT). cannam@95: (Logical N=2*n, inverse is FFTW_REDFT01.) cannam@95: cannam@95:
FFTW_REDFT01 computes an REDFT01 transform, i.e. a DCT-III (sometimes called “the” IDCT, being the inverse of DCT-II). cannam@95: (Logical N=2*n, inverse is FFTW_REDFT=10.) cannam@95: cannam@95:
FFTW_REDFT11 computes an REDFT11 transform, i.e. a DCT-IV. cannam@95: (Logical N=2*n, inverse is FFTW_REDFT11.) cannam@95: cannam@95:
FFTW_RODFT00 computes an RODFT00 transform, i.e. a DST-I. cannam@95: (Logical N=2*(n+1), inverse is FFTW_RODFT00.) cannam@95: cannam@95:
FFTW_RODFT10 computes an RODFT10 transform, i.e. a DST-II. cannam@95: (Logical N=2*n, inverse is FFTW_RODFT01.) cannam@95: cannam@95:
FFTW_RODFT01 computes an RODFT01 transform, i.e. a DST-III. cannam@95: (Logical N=2*n, inverse is FFTW_RODFT=10.) cannam@95: cannam@95:
FFTW_RODFT11 computes an RODFT11 transform, i.e. a DST-IV. cannam@95: (Logical N=2*n, inverse is FFTW_RODFT11.) cannam@95: cannam@95:

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