Portability | non-portable |
---|---|

Stability | experimental |

Maintainer | jed@59A2.org |

This module exposes an interface to FFTW, the Fastest Fourier Transform in the West.

These bindings present several levels of interface. All the higher level
functions (`dft`

, `idft`

, `dftN`

, ...) are easily derived from the general
functions (`dftG`

, `dftRCG`

, ...). Only the general functions let you
specify planner flags. The higher levels all set `estimate`

so you should
not have to wait through time consuming planning (see below for more).

The simplest interface is the one-dimensional transforms. If you supply a multi-dimensional array, these will only transform the first dimension. These functions only take one argument, the array to be transformed.

At the next level, we have multi-dimensional transforms where you specify which dimensions to transform in and the array to transform. For instance

b = dftRCN [0,2] a

is the real to complex transform in dimensions 0 and 2 of the array `a`

which
must be at least rank 3. The array `b`

will be complex valued with the same
extent as `a`

in every dimension except `2`

. If `a`

had extent `n`

in
dimension `2`

then the `b`

will have extent `a `

which consists of
all non-negative frequency components in this dimension (the negative
frequencies are conjugate to the positive frequencies because of symmetry
since `div`

2 + 1`a`

is real valued).

The real to real transforms allow different transform kinds in each transformed dimension. For example,

b = dftRRN [(0,DHT), (1,REDFT10), (2,RODFT11)] a

is a Discrete Hartley Transform in dimension 0, a discrete cosine transform
(DCT-2) in dimension 1, and distrete sine transform (DST-4) in dimension 2
where the array `a`

must have rank at least 3.

The general interface is similar to the multi-dimensional interface, takes as
its first argument, a bitwise `.|.`

of planning `Flag`

s. (In the complex
version, the sign of the transform is first.) For example,

b = dftG DFTBackward (patient .|. destroy_input) [1,2] a

is an inverse DFT in dimensions 1 and 2 of the complex array `a`

which has
rank at least 3. It will use the patient planner to generate a (near)
optimal transform. If you compute the same type of transform again, it
should be very fast since the plan is cached.

Inverse transforms are typically normalized. The un-normalized inverse
transforms are `dftGU`

, `dftCRGU`

and `dftCROGU`

. For example

b = dftCROGU measure [0,1] a

is an un-normalized inverse DFT in dimensions 0 and 1 of the complex array
`a`

(representing the non-negative frequencies, where the negative
frequencies are conjugate) which has rank at least 2. Here, dimension 1 is
logically odd so if `a`

has extent `n`

in dimension 1, then `b`

will have
extent `(n - 1) * 2 + 1`

in dimension 1. It is more common that the logical
dimension is even, in which case we would use `dftCRGU`

in which case `b`

would have extent `(n - 1) * 2`

in dimension `1`

.

The FFTW library separates transforms into two steps. First you compute a
plan for a given transform, then you execute it. Often the planning stage is
quite time-consuming, but subsequent transforms of the same size and type
will be extremely fast. The planning phase actually computes transforms, so
it overwrites its input array. For many C codes, it is reasonable to re-use
the same arrays to compute a given transform on different data. This is not
a very useful paradigm from Haskell. Fortunately, FFTW caches its plans so
if try to generate a new plan for a transform size which has already been
planned, the planner will return immediately. Unfortunately, it is not
possible to consult the cache, so if a plan is cached, we may use more memory
than is strictly necessary since we must allocate a work array which we
expect to be overwritten during planning. FFTW can export its cached plans
to a string. This is known as wisdom. For high performance work, it is a
good idea to compute plans of the sizes you are interested in, using
aggressive options (i.e. `patient`

), use `exportWisdomString`

to get a string
representing these plans, and write this to a file. Then for production
runs, you can read this in, then add it to the cache with
`importWisdomString`

. Now you can use the `estimate`

planner so the Haskell
bindings know that FFTW will not overwrite the input array, and you will
still get a high quality transform (because it has wisdom).

- data Sign
- data Kind
- destroyInput :: Flag
- preserveInput :: Flag
- estimate :: Flag
- measure :: Flag
- patient :: Flag
- exhaustive :: Flag
- dft :: (FFTWReal r, Ix i, Shapable i) => CArray i (Complex r) -> CArray i (Complex r)
- idft :: (FFTWReal r, Ix i, Shapable i) => CArray i (Complex r) -> CArray i (Complex r)
- dftN :: (FFTWReal r, Ix i, Shapable i) => [Int] -> CArray i (Complex r) -> CArray i (Complex r)
- idftN :: (FFTWReal r, Ix i, Shapable i) => [Int] -> CArray i (Complex r) -> CArray i (Complex r)
- dftG :: (FFTWReal r, Ix i, Shapable i) => Sign -> Flag -> [Int] -> CArray i (Complex r) -> CArray i (Complex r)
- dftGU :: (FFTWReal r, Ix i, Shapable i) => Sign -> Flag -> [Int] -> CArray i (Complex r) -> CArray i (Complex r)
- dftRC :: (FFTWReal r, Ix i, Shapable i) => CArray i r -> CArray i (Complex r)
- dftCR :: (FFTWReal r, Ix i, Shapable i) => CArray i (Complex r) -> CArray i r
- dftCRO :: (FFTWReal r, Ix i, Shapable i) => CArray i (Complex r) -> CArray i r
- dftRCN :: (FFTWReal r, Ix i, Shapable i) => [Int] -> CArray i r -> CArray i (Complex r)
- dftCRN :: (FFTWReal r, Ix i, Shapable i) => [Int] -> CArray i (Complex r) -> CArray i r
- dftCRON :: (FFTWReal r, Ix i, Shapable i) => [Int] -> CArray i (Complex r) -> CArray i r
- dftRCG :: (FFTWReal r, Ix i, Shapable i) => Flag -> [Int] -> CArray i r -> CArray i (Complex r)
- dftCRG :: (FFTWReal r, Ix i, Shapable i) => Flag -> [Int] -> CArray i (Complex r) -> CArray i r
- dftCROG :: (FFTWReal r, Ix i, Shapable i) => Flag -> [Int] -> CArray i (Complex r) -> CArray i r
- dftCRGU :: (FFTWReal r, Ix i, Shapable i) => Flag -> [Int] -> CArray i (Complex r) -> CArray i r
- dftCROGU :: (FFTWReal r, Ix i, Shapable i) => Flag -> [Int] -> CArray i (Complex r) -> CArray i r
- dftRH :: (FFTWReal r, Ix i, Shapable i) => CArray i r -> CArray i r
- dftHR :: (FFTWReal r, Ix i, Shapable i) => CArray i r -> CArray i r
- dht :: (FFTWReal r, Ix i, Shapable i) => CArray i r -> CArray i r
- dct1 :: (FFTWReal r, Ix i, Shapable i) => CArray i r -> CArray i r
- dct2 :: (FFTWReal r, Ix i, Shapable i) => CArray i r -> CArray i r
- dct3 :: (FFTWReal r, Ix i, Shapable i) => CArray i r -> CArray i r
- dct4 :: (FFTWReal r, Ix i, Shapable i) => CArray i r -> CArray i r
- dst1 :: (FFTWReal r, Ix i, Shapable i) => CArray i r -> CArray i r
- dst2 :: (FFTWReal r, Ix i, Shapable i) => CArray i r -> CArray i r
- dst3 :: (FFTWReal r, Ix i, Shapable i) => CArray i r -> CArray i r
- dst4 :: (FFTWReal r, Ix i, Shapable i) => CArray i r -> CArray i r
- dftRHN :: (FFTWReal r, Ix i, Shapable i) => [Int] -> CArray i r -> CArray i r
- dftHRN :: (FFTWReal r, Ix i, Shapable i) => [Int] -> CArray i r -> CArray i r
- dhtN :: (FFTWReal r, Ix i, Shapable i) => [Int] -> CArray i r -> CArray i r
- dct1N :: (FFTWReal r, Ix i, Shapable i) => [Int] -> CArray i r -> CArray i r
- dct2N :: (FFTWReal r, Ix i, Shapable i) => [Int] -> CArray i r -> CArray i r
- dct3N :: (FFTWReal r, Ix i, Shapable i) => [Int] -> CArray i r -> CArray i r
- dct4N :: (FFTWReal r, Ix i, Shapable i) => [Int] -> CArray i r -> CArray i r
- dst1N :: (FFTWReal r, Ix i, Shapable i) => [Int] -> CArray i r -> CArray i r
- dst2N :: (FFTWReal r, Ix i, Shapable i) => [Int] -> CArray i r -> CArray i r
- dst3N :: (FFTWReal r, Ix i, Shapable i) => [Int] -> CArray i r -> CArray i r
- dst4N :: (FFTWReal r, Ix i, Shapable i) => [Int] -> CArray i r -> CArray i r
- dftRRN :: (FFTWReal r, Ix i, Shapable i) => [(Int, Kind)] -> CArray i r -> CArray i r
- dftRRG :: (FFTWReal r, Ix i, Shapable i) => Flag -> [(Int, Kind)] -> CArray i r -> CArray i r
- importWisdomString :: String -> IO Bool
- importWisdomSystem :: IO Bool
- exportWisdomString :: IO String

# Data types

Determine which direction of DFT to execute.

Real to Real transform kinds.

# Planner flags

## Algorithm restriction flags

Allows FFTW to overwrite the input array with arbitrary data; this can sometimes allow more efficient algorithms to be employed.

Setting this flag implies that two memory allocations will be done, one for
work space, and one for the result. When `estimate`

is not set, we will be
doing two memory allocations anyway, so we set this flag as well (since we
don't retain the work array anyway).

`preserveInput`

specifies that an out-of-place transform must not change
its input array. This is ordinarily the default, except for complex to real
transforms for which `destroyInput`

is the default. In the latter cases,
passing `preserveInput`

will attempt to use algorithms that do not destroy
the input, at the expense of worse performance; for multi-dimensional complex
to real transforms, however, no input-preserving algorithms are implemented
so the Haskell bindings will set `destroyInput`

and do a transform with two
memory allocations.

## Planning rigor flags

`estimate`

specifies that, instead of actual measurements of different
algorithms, a simple heuristic is used to pick a (probably sub-optimal) plan
quickly. With this flag, the input/output arrays are not overwritten during
planning.

This is the only planner flag for which a single memory allocation is possible.

`patient`

is like `measure`

, but considers a wider range of algorithms and
often produces a more optimal plan (especially for large transforms), but
at the expense of several times longer planning time (especially for large
transforms).

`exhaustive`

is like `patient`

but considers an even wider range of
algorithms, including many that we think are unlikely to be fast, to
produce the most optimal plan but with a substantially increased planning
time.

# DFT of complex data

## DFT in first dimension only

dft :: (FFTWReal r, Ix i, Shapable i) => CArray i (Complex r) -> CArray i (Complex r)Source

1-dimensional complex DFT.

idft :: (FFTWReal r, Ix i, Shapable i) => CArray i (Complex r) -> CArray i (Complex r)Source

1-dimensional complex inverse DFT. Inverse of `dft`

.

## Multi-dimensional transforms

dftN :: (FFTWReal r, Ix i, Shapable i) => [Int] -> CArray i (Complex r) -> CArray i (Complex r)Source

Multi-dimensional forward DFT.

idftN :: (FFTWReal r, Ix i, Shapable i) => [Int] -> CArray i (Complex r) -> CArray i (Complex r)Source

Multi-dimensional inverse DFT.

## General transform

dftG :: (FFTWReal r, Ix i, Shapable i) => Sign -> Flag -> [Int] -> CArray i (Complex r) -> CArray i (Complex r)Source

Normalized general complex DFT

## Un-normalized general transform

dftGU :: (FFTWReal r, Ix i, Shapable i) => Sign -> Flag -> [Int] -> CArray i (Complex r) -> CArray i (Complex r)Source

Complex to Complex DFT, un-normalized.

# DFT of real data

## DFT in first dimension only

dftRC :: (FFTWReal r, Ix i, Shapable i) => CArray i r -> CArray i (Complex r)Source

1-dimensional real to complex DFT.

dftCR :: (FFTWReal r, Ix i, Shapable i) => CArray i (Complex r) -> CArray i rSource

1-dimensional complex to real DFT with logically even dimension. Inverse of `dftRC`

.

dftCRO :: (FFTWReal r, Ix i, Shapable i) => CArray i (Complex r) -> CArray i rSource

1-dimensional complex to real DFT with logically odd dimension. Inverse of `dftRC`

.

## Multi-dimensional transforms

dftRCN :: (FFTWReal r, Ix i, Shapable i) => [Int] -> CArray i r -> CArray i (Complex r)Source

Multi-dimensional forward DFT of real data.

dftCRN :: (FFTWReal r, Ix i, Shapable i) => [Int] -> CArray i (Complex r) -> CArray i rSource

Multi-dimensional inverse DFT of Hermitian-symmetric data (where only the non-negative frequencies are given).

dftCRON :: (FFTWReal r, Ix i, Shapable i) => [Int] -> CArray i (Complex r) -> CArray i rSource

Multi-dimensional inverse DFT of Hermitian-symmetric data (where only the non-negative frequencies are given) and the last transformed dimension is logically odd.

## General transform

dftRCG :: (FFTWReal r, Ix i, Shapable i) => Flag -> [Int] -> CArray i r -> CArray i (Complex r)Source

Real to Complex DFT.

dftCRG :: (FFTWReal r, Ix i, Shapable i) => Flag -> [Int] -> CArray i (Complex r) -> CArray i rSource

Normalized general complex to real DFT where the last transformed dimension is logically even.

dftCROG :: (FFTWReal r, Ix i, Shapable i) => Flag -> [Int] -> CArray i (Complex r) -> CArray i rSource

Normalized general complex to real DFT where the last transformed dimension is logicall odd.

## Un-normalized general transform

dftCRGU :: (FFTWReal r, Ix i, Shapable i) => Flag -> [Int] -> CArray i (Complex r) -> CArray i rSource

Complex to Real DFT where last transformed dimension is logically even.

dftCROGU :: (FFTWReal r, Ix i, Shapable i) => Flag -> [Int] -> CArray i (Complex r) -> CArray i rSource

Complex to Real DFT where last transformed dimension is logically odd.

# Real to real transforms (all un-normalized)

## Transforms in first dimension only

dftRH :: (FFTWReal r, Ix i, Shapable i) => CArray i r -> CArray i rSource

1-dimensional real to half-complex DFT.

dftHR :: (FFTWReal r, Ix i, Shapable i) => CArray i r -> CArray i rSource

1-dimensional half-complex to real DFT. Inverse of `dftRH`

after normalization.

dht :: (FFTWReal r, Ix i, Shapable i) => CArray i r -> CArray i rSource

1-dimensional Discrete Hartley Transform. Self-inverse after normalization.

dct1 :: (FFTWReal r, Ix i, Shapable i) => CArray i r -> CArray i rSource

1-dimensional Type 1 discrete cosine transform.

dct2 :: (FFTWReal r, Ix i, Shapable i) => CArray i r -> CArray i rSource

1-dimensional Type 2 discrete cosine transform. This is commonly known as *the* DCT.

dct3 :: (FFTWReal r, Ix i, Shapable i) => CArray i r -> CArray i rSource

1-dimensional Type 3 discrete cosine transform. This is commonly known as *the* inverse DCT.

dct4 :: (FFTWReal r, Ix i, Shapable i) => CArray i r -> CArray i rSource

1-dimensional Type 4 discrete cosine transform.

dst1 :: (FFTWReal r, Ix i, Shapable i) => CArray i r -> CArray i rSource

1-dimensional Type 1 discrete sine transform.

dst2 :: (FFTWReal r, Ix i, Shapable i) => CArray i r -> CArray i rSource

1-dimensional Type 2 discrete sine transform.

dst3 :: (FFTWReal r, Ix i, Shapable i) => CArray i r -> CArray i rSource

1-dimensional Type 3 discrete sine transform.

dst4 :: (FFTWReal r, Ix i, Shapable i) => CArray i r -> CArray i rSource

1-dimensional Type 4 discrete sine transform.

## Multi-dimensional transforms with the same transform type in each dimension

dftRHN :: (FFTWReal r, Ix i, Shapable i) => [Int] -> CArray i r -> CArray i rSource

Multi-dimensional real to half-complex transform. The result is not normalized.

dftHRN :: (FFTWReal r, Ix i, Shapable i) => [Int] -> CArray i r -> CArray i rSource

Multi-dimensional half-complex to real transform. The result is not normalized.

dhtN :: (FFTWReal r, Ix i, Shapable i) => [Int] -> CArray i r -> CArray i rSource

Multi-dimensional Discrete Hartley Transform. The result is not normalized.

dct1N :: (FFTWReal r, Ix i, Shapable i) => [Int] -> CArray i r -> CArray i rSource

Multi-dimensional Type 1 discrete cosine transform.

dct2N :: (FFTWReal r, Ix i, Shapable i) => [Int] -> CArray i r -> CArray i rSource

Multi-dimensional Type 2 discrete cosine transform. This is commonly known
as *the* DCT.

dct3N :: (FFTWReal r, Ix i, Shapable i) => [Int] -> CArray i r -> CArray i rSource

Multi-dimensional Type 3 discrete cosine transform. This is commonly known
as *the* inverse DCT. The result is not normalized.

dct4N :: (FFTWReal r, Ix i, Shapable i) => [Int] -> CArray i r -> CArray i rSource

Multi-dimensional Type 4 discrete cosine transform.

dst1N :: (FFTWReal r, Ix i, Shapable i) => [Int] -> CArray i r -> CArray i rSource

Multi-dimensional Type 1 discrete sine transform.

dst2N :: (FFTWReal r, Ix i, Shapable i) => [Int] -> CArray i r -> CArray i rSource

Multi-dimensional Type 2 discrete sine transform.

dst3N :: (FFTWReal r, Ix i, Shapable i) => [Int] -> CArray i r -> CArray i rSource

Multi-dimensional Type 3 discrete sine transform.

dst4N :: (FFTWReal r, Ix i, Shapable i) => [Int] -> CArray i r -> CArray i rSource

Multi-dimensional Type 4 discrete sine transform.

## Multi-dimensional transforms with possibly different transforms in each dimension

dftRRN :: (FFTWReal r, Ix i, Shapable i) => [(Int, Kind)] -> CArray i r -> CArray i rSource

Multi-dimensional real to real transform. The result is not normalized.

## General transforms

dftRRG :: (FFTWReal r, Ix i, Shapable i) => Flag -> [(Int, Kind)] -> CArray i r -> CArray i rSource

Real to Real transforms.

# Wisdom

importWisdomString :: String -> IO BoolSource

Add wisdom to the FFTW cache. Returns `True`

if it is successful.

importWisdomSystem :: IO BoolSource

Tries to import wisdom from a global source, typically

.
Returns *etc*fftw/wisdom`True`

if it was successful.

exportWisdomString :: IO StringSource

Queries the FFTW cache. The `String`

can be written to a file so the
wisdom can be reused on a subsequent run.