fft-0.1.7.2: Bindings to the FFTW library.

Portability non-portable experimental jed@59A2.org Safe-Infered

Math.FFT

Description

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 div 2 + 1` which consists of all non-negative frequency components in this dimension (the negative frequencies are conjugate to the positive frequencies because of symmetry since `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).

Synopsis

# Data types

data Sign Source

Determine which direction of DFT to execute.

Constructors

 DFTForward DFTBackward

Instances

 Eq Sign Show Sign

data Kind Source

Real to Real transform kinds.

Constructors

 R2HC HC2R DHT REDFT00 REDFT10 REDFT01 REDFT11 RODFT00 RODFT01 RODFT10 RODFT11

Instances

 Eq Kind Show Kind

# 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.

`measure` tells FFTW to find an optimized plan by actually computing several FFTs and measuring their execution time. Depending on your machine, this can take some time (often a few seconds). `measure` is the default planning option.

`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

Add wisdom to the FFTW cache. Returns `True` if it is successful.

Tries to import wisdom from a global source, typically `etcfftw/wisdom`. Returns `True` if it was successful.

Queries the FFTW cache. The `String` can be written to a file so the wisdom can be reused on a subsequent run.