Data.Array.Accelerate.Fourier.Planned

Like Data.Array.Accelerate.Fourier.Preprocessed this module allows to factor out some preprocessing. Additionally it gives you concrete objects (plans and caches) for sharing preprocessed data between transforms. You cannot only share the preprocessing between transforms of the same size, but across all array sizes. This implementation also has the largest collection of algorithms and thus should be generally fastest among all implementations in this package.

Transforms

type Transform sh a = Acc (Array sh a) -> Acc (Array sh a)

transform :: (Slice sh, Shape sh, RealFloat a, Elt a, IsFloating a) => Sign a -> Int -> Transform (sh :. Int) (Complex a)

Fourier transform of arbitrary size. Sign can be

• forward: from time domain to frequency spectrum
• inverse: from frequency spectrum to time domain

You may share transform sign n between several calls in order to run some preprocessing only once. You must make sure that the length is equal to the extent of the inner dimension of every transformed array.

transformDecompose :: (Slice sh, Shape sh, Elt a, IsFloating a, RealFloat a) => Sign a -> Int -> Transform (sh :. Int) (Complex a)

Transform using only Cooley-Tukey, Good-Thomas, Rader, Split-Radix, but no Bluestein. This is more for testing and benchmarking than for real use.

transformChirp2 :: (Slice sh, Shape sh, Elt a, IsFloating a, RealFloat a) => Sign a -> Int -> Transform (sh :. Int) (Complex a)

Fourier transform for arbitrary lengths based on the Bluestein transform or chirp z-transform on an array with power-of-two size. It may be faster than transform for certain prime factors. Find bad factors e.g. in http://oeis.org/A061092 and http://oeis.org/A059411 and nicer factors in http://oeis.org/A061303.

transformChirp235 :: (Slice sh, Shape sh, Elt a, IsFloating a, RealFloat a) => Sign a -> Int -> Transform (sh :. Int) (Complex a)

Fourier transform for arbitrary lengths based on the Bluestein transform on an array with 5-smooth size. (5-smooth = all prime factors are at most 5)

convolveCyclic :: (Shape sh, Slice sh, a ~ Complex b, Elt b, IsFloating b, RealFloat b) => Int -> Acc (Array (sh :. Int) a) -> Acc (Array (sh :. Int) a) -> Acc (Array (sh :. Int) a)

Signals must have equal size and must not be empty.

Planning

data Plan

Memorize factorizations of the data size and permutation vectors.

Instances

Show Plan

plan :: Integer -> Plan

Plan transform algorithms for a certain array size.

transformWithPlanner :: (Slice sh, Shape sh, Elt a, IsFloating a, RealFloat a) => (Integer -> State PlanMap Plan) -> Sign a -> Int -> Transform (sh :. Int) (Complex a)

type PlanMap = Map Integer Plan

planWithMapUpdate :: Integer -> State PlanMap Plan

Too many nested Rader transformations slow down the transform, up to quadratic time in the worst case. As a heuristic we allow at most nesting depth two, and switch to Bluestein transformation otherwise. We could compute more precise operation counts and base our decision on these, but we found that the actual execution time differs considerably from the operation counts.

planDecomposeWithMapUpdate :: Integer -> State PlanMap Plan

planChirpWithMapUpdate :: (Integer -> Integer) -> Integer -> State PlanMap Plan

smallPlanMap :: PlanMap

Map of primitive transforms. You should use this as the initial map when evaluating a planning sequence using evalState.

Caching

data Cache a

Cache arrays of twiddle factors, i.e. powers of the primitive root of unity.

Instances

Elt a => Show (Cache a)

cache :: (RealFloat a, Elt a, IsFloating a) => Sign a -> Int -> Cache (Complex a)

The expression cache sign len precomputes all data that is needed for Fourier transforms for signals of length len. You can use this cache in transformWithCache.

cacheDuplex :: (a ~ Complex b, RealFloat b, Elt b, IsFloating b) => Int -> (Cache a, Cache a)

It is (cache inverse x, cache forward x) = cacheDuplex x but cacheDuplex shares common data of both caches.

transformWithCache :: (Slice sh, Shape sh, Elt a, IsFloating a) => Cache (Complex a) -> Transform (sh :. Int) (Complex a)

The size and type of the signal must match the parameters, that the cache was generated for.

type CacheMap a = Map (Integer, Direction) (Cache a)

data Direction

Constructors

Forward

Inverse

Instances

Eq Direction

Ord Direction

Show Direction

cacheFromPlanWithMapUpdate :: (a ~ Complex b, RealFloat b, Elt b, IsFloating b) => Plan -> (Direction, Sign b) -> State (CacheMap a) (Cache a)

directionMode :: (Num a, Ord a) => Sign a -> Int -> (Direction, Sign a)

cacheFromPlanWithMapUpdate2 :: (a ~ Complex b, RealFloat b, Elt b, IsFloating b) => (Plan, Plan) -> ((Direction, Sign b), (Direction, Sign b)) -> State (CacheMap a) (Cache a, Cache a)

directionModes :: Num a => Int -> ((Direction, Sign a), (Direction, Sign a))

Miscellaneous

data Sign a

Instances

(Lift Exp a, Elt (Plain a)) => Lift Exp (Sign a)

Elt a => Unlift Exp (Sign (Exp a))

Eq a => Eq (Sign a)

Show a => Show (Sign a)

Num a => Arbitrary (Sign a)

IsTuple (Sign a)

Elt a => Elt (Sign a)

Typeable (* -> *) Sign

type TupleRepr (Sign a) = ((), a)

type EltRepr' (Sign a) = EltRepr' a

type EltRepr (Sign a) = EltRepr a

type Plain (Sign a) = Sign (Plain a)

forward :: Num a => Sign a

inverse :: Num a => Sign a