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## D.3 `fftw`

Some functions in the `fftw` fast Fourier transform library are callable by virtual code programs of the form `library('fftw',f)`, where `f` can be one of the following character strings.

`u_fw_dft`

  (uni-dimensional forward Discrete Fourier transform)

`u_bw_dft`

  (uni-dimensional backward Discrete Fourier transform)

`b_fw_dft`

  (bi-dimensional forward Discrete Fourier transform)

`b_bw_dft`

  (bi-dimensional backward Discrete Fourier transform)

`u_dht`

  (uni-dimensional Discrete Hartley transform)

`b_dht`

  (bi-dimensional Discrete Hartley transform)

These stand for the discrete Fourier transform, in one dimension and two dimensions, either backward or forward, and the discrete Hartley transform in one dimension and two dimensions. The `fftw` library documentation (http://www.fftw.org) can give more information about the meaning of these transformations.

The interface is somewhat simplified compared to the API for the `fftw` C library because there are no considerations of memory management or planning, nor any provision for dimensions higher than two.

Furthermore, from the virtual side of the interface, these functions operate on lists rather than arrays. The one dimensional Fourier transforms take a list of complex numbers to a list of complex numbers (see `complex`), and the one dimensional Hartley transforms take a list of reals to a list of reals (see `math`). The two dimensional transforms are analogous but they take a matrix represented as a list of lists. Error messages pertaining to invalid input documented at the beginning of this section (External Libraries) are relevant.

Finally, unlike the native API for `fftw`, these transformations are scaled so that the backward transformation is the inverse of the forward, and the Hartley transformations are their own inverses (subject to roundoff error).

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