Numeric.FFTW.Rank1

Documentation

fourier :: (Integral n, Real a) => Sign -> Array (Cyclic n) (Complex a) -> Array (Cyclic n) (Complex a)

>>> complexFloatList $ Trafo1.fourier Trafo1.Forward $ Array.fromList (Shape.Cyclic (5::Int)) [1,0,0,0,0]
[1.00:+0.00,1.00:+0.00,1.00:+0.00,1.00:+0.00,1.00:+0.00]
>>> complexFloatList $ Trafo1.fourier Trafo1.Forward $ Array.fromList (Shape.Cyclic (5::Int)) [0,1,0,0,0]
 [1.00:+0.00,0.31:+(-0.95),(-0.81):+(-0.59),(-0.81):+0.59,0.31:+0.95]
>>> complexFloatList $ Trafo1.fourier Trafo1.Backward $ Array.fromList (Shape.Cyclic (5::Int)) [0,1,0,0,0]
[1.00:+0.00,0.31:+0.95,(-0.81):+0.59,(-0.81):+(-0.59),0.31:+(-0.95)]

QC.forAll genCyclicArray1 $ \xs sign-> approxComplex floatTol (adjust xs) (Trafo1.fourier sign (Trafo1.fourier (flipSign sign) xs))

QC.forAll genCyclicArray1 $ \xs sign -> approxComplex doubleTol (adjust xs) (Trafo1.fourier sign (Trafo1.fourier (flipSign sign) xs))

QC.forAll genCyclicArray1 $ \xs sign -> approxComplex floatTol (Trafo1.fourier sign (Array.map Complex.conjugate xs)) (Array.map Complex.conjugate (Trafo1.fourier (flipSign sign) xs))

QC.forAll genCyclicArray1 $ \xs sign -> approxComplex doubleTol (Trafo1.fourier sign (Array.map Complex.conjugate xs)) (Array.map Complex.conjugate (Trafo1.fourier (flipSign sign) xs))

QC.forAll genCyclicArray1 $ \xys sign -> let (xs,ys) = split xys in approxComplex floatTol (Trafo1.fourier sign $ Array.zipWith (+) xs ys) (Array.zipWith (+) (Trafo1.fourier sign xs) (Trafo1.fourier sign ys))

QC.forAll genCyclicArray1 $ \xys sign -> let (xs,ys) = split xys in approxComplex doubleTol (Trafo1.fourier sign $ Array.zipWith (+) xs ys) (Array.zipWith (+) (Trafo1.fourier sign xs) (Trafo1.fourier sign ys))

QC.forAll genCyclicArray1 $ \xys sign -> let (xs,ys) = split xys in Complex.magnitude (scalarProduct (adjust xs) ys - scalarProduct (Trafo1.fourier sign xs) (Trafo1.fourier sign ys)) <= floatTol * normInf Complex.magnitude (adjust xs) * normInf Complex.magnitude ys

QC.forAll genCyclicArray1 $ \xys sign -> let (xs,ys) = split xys in Complex.magnitude (scalarProduct (adjust xs) ys - scalarProduct (Trafo1.fourier sign xs) (Trafo1.fourier sign ys)) <= doubleTol * normInf Complex.magnitude (adjust xs) * normInf Complex.magnitude ys

\sign -> QC.forAll genCyclicArray1 $ immutable (Trafo1.fourier sign) . arrayComplexFloat

\sign -> QC.forAll genCyclicArray1 $ immutable (Trafo1.fourier sign) . arrayComplexDouble

data Sign

Order is chosen such that the numeric sign is (-1) ^ fromEnum sign.

Constructors

Backward

Forward

Instances

Enum Sign

Defined in Numeric.FFTW.Private

Methods

succ :: Sign -> Sign

pred :: Sign -> Sign

toEnum :: Int -> Sign

fromEnum :: Sign -> Int

enumFrom :: Sign -> [Sign]

enumFromThen :: Sign -> Sign -> [Sign]

enumFromTo :: Sign -> Sign -> [Sign]

enumFromThenTo :: Sign -> Sign -> Sign -> [Sign]

Eq Sign

Defined in Numeric.FFTW.Private

Methods

(==) :: Sign -> Sign -> Bool

(/=) :: Sign -> Sign -> Bool

Ord Sign

Defined in Numeric.FFTW.Private

Methods

compare :: Sign -> Sign -> Ordering

(<) :: Sign -> Sign -> Bool

(<=) :: Sign -> Sign -> Bool

(>) :: Sign -> Sign -> Bool

(>=) :: Sign -> Sign -> Bool

max :: Sign -> Sign -> Sign

min :: Sign -> Sign -> Sign

Show Sign

Defined in Numeric.FFTW.Private

Methods

showsPrec :: Int -> Sign -> ShowS

show :: Sign -> String

showList :: [Sign] -> ShowS

Arbitrary Sign

Defined in Numeric.FFTW.Private

Methods

arbitrary :: Gen Sign

shrink :: Sign -> [Sign]

flipSign :: Sign -> Sign

fourierRC :: (Integral n, Real a) => Array (Cyclic n) a -> Array (Half n) (Complex a)

QC.forAll genCyclicArray1 $ \xs -> approxReal floatTol (adjust xs) (Trafo1.fourierCR (Trafo1.fourierRC xs))

QC.forAll genCyclicArray1 $ \xs -> approxReal doubleTol (adjust xs) (Trafo1.fourierCR (Trafo1.fourierRC xs))

QC.forAll genCyclicArray1 $ immutable Trafo1.fourierRC . arrayFloat

QC.forAll genCyclicArray1 $ immutable Trafo1.fourierRC . arrayDouble

fourierCR :: (Integral n, Real a) => Array (Half n) (Complex a) -> Array (Cyclic n) a

>>> floatList $ Trafo1.fourierCR $ Array.fromList (Spectrum.Half (5::Int)) [0,1,0]
[2.00,0.62,-1.62,-1.62,0.62]

QC.forAll (fmap Trafo1.fourierRC genCyclicArray1) $ immutable Trafo1.fourierCR . arrayComplexFloat

QC.forAll (fmap Trafo1.fourierRC genCyclicArray1) $ immutable Trafo1.fourierCR . arrayComplexDouble

cosine :: (Shift shiftTime, Shift shiftSpectrum, Integral n, Real a) => Array (Symmetric Even shiftTime shiftSpectrum n) a -> Array (Symmetric Even shiftSpectrum shiftTime n) a

Symmetric Even Halfway Exact yields _the_ DCT, Symmetric Even Exact Halfway yields _the_ inverse DCT.

QC.forAll (genSymmetric Even Exact Exact `QC.suchThat` ((>=2) . Shape.size . Array.shape)) $ \xs -> approxReal floatTol (adjustSymmetric xs) (Trafo1.cosine (Trafo1.cosine xs))

QC.forAll (genSymmetric Even Exact Exact `QC.suchThat` ((>=2) . Shape.size . Array.shape)) $ \xs -> approxReal doubleTol (adjustSymmetric xs) (Trafo1.cosine (Trafo1.cosine xs))

QC.forAll (genSymmetric Even Halfway Exact) $ \xs -> approxReal floatTol (adjustSymmetric xs) (Trafo1.cosine (Trafo1.cosine xs))

QC.forAll (genSymmetric Even Halfway Exact) $ \xs -> approxReal doubleTol (adjustSymmetric xs) (Trafo1.cosine (Trafo1.cosine xs))

QC.forAll (genSymmetric Even Exact Halfway) $ \xs -> approxReal floatTol (adjustSymmetric xs) (Trafo1.cosine (Trafo1.cosine xs))

QC.forAll (genSymmetric Even Exact Halfway) $ \xs -> approxReal doubleTol (adjustSymmetric xs) (Trafo1.cosine (Trafo1.cosine xs))

QC.forAll (genSymmetric Even Halfway Halfway) $ \xs -> approxReal floatTol (adjustSymmetric xs) (Trafo1.cosine (Trafo1.cosine xs))

QC.forAll (genSymmetric Even Halfway Halfway) $ \xs -> approxReal doubleTol (adjustSymmetric xs) (Trafo1.cosine (Trafo1.cosine xs))

QC.forAll (genSymmetric Even Exact Exact `QC.suchThat` ((>=2) . Shape.size . Array.shape)) $ immutable Trafo1.cosine . arrayFloat

QC.forAll (genSymmetric Even Exact Exact `QC.suchThat` ((>=2) . Shape.size . Array.shape)) $ immutable Trafo1.cosine . arrayDouble

QC.forAll (genSymmetric Even Halfway Exact) $ immutable Trafo1.cosine . arrayFloat

QC.forAll (genSymmetric Even Halfway Exact) $ immutable Trafo1.cosine . arrayDouble

QC.forAll (genSymmetric Even Exact Halfway) $ immutable Trafo1.cosine . arrayFloat

QC.forAll (genSymmetric Even Exact Halfway) $ immutable Trafo1.cosine . arrayDouble

QC.forAll (genSymmetric Even Halfway Halfway) $ immutable Trafo1.cosine . arrayFloat

QC.forAll (genSymmetric Even Halfway Halfway) $ immutable Trafo1.cosine . arrayDouble

sine :: (Shift shiftTime, Shift shiftSpectrum, Integral n, Real a) => Array (Symmetric Odd shiftTime shiftSpectrum n) a -> Array (Symmetric Odd shiftSpectrum shiftTime n) a

QC.forAll (genSymmetric Odd Exact Exact) $ \xs -> approxReal floatTol (adjustSymmetric xs) (Trafo1.sine (Trafo1.sine xs))

QC.forAll (genSymmetric Odd Exact Exact) $ \xs -> approxReal doubleTol (adjustSymmetric xs) (Trafo1.sine (Trafo1.sine xs))

QC.forAll (genSymmetric Odd Halfway Exact) $ \xs -> approxReal floatTol (adjustSymmetric xs) (Trafo1.sine (Trafo1.sine xs))

QC.forAll (genSymmetric Odd Halfway Exact) $ \xs -> approxReal doubleTol (adjustSymmetric xs) (Trafo1.sine (Trafo1.sine xs))

QC.forAll (genSymmetric Odd Exact Halfway) $ \xs -> approxReal floatTol (adjustSymmetric xs) (Trafo1.sine (Trafo1.sine xs))

QC.forAll (genSymmetric Odd Exact Halfway) $ \xs -> approxReal doubleTol (adjustSymmetric xs) (Trafo1.sine (Trafo1.sine xs))

QC.forAll (genSymmetric Odd Halfway Halfway) $ \xs -> approxReal floatTol (adjustSymmetric xs) (Trafo1.sine (Trafo1.sine xs))

QC.forAll (genSymmetric Odd Halfway Halfway) $ \xs -> approxReal doubleTol (adjustSymmetric xs) (Trafo1.sine (Trafo1.sine xs))

QC.forAll (genSymmetric Odd Exact Exact) $ immutable Trafo1.sine . arrayFloat

QC.forAll (genSymmetric Odd Exact Exact) $ immutable Trafo1.sine . arrayDouble

QC.forAll (genSymmetric Odd Halfway Exact) $ immutable Trafo1.sine . arrayFloat

QC.forAll (genSymmetric Odd Halfway Exact) $ immutable Trafo1.sine . arrayDouble

QC.forAll (genSymmetric Odd Exact Halfway) $ immutable Trafo1.sine . arrayFloat

QC.forAll (genSymmetric Odd Exact Halfway) $ immutable Trafo1.sine . arrayDouble

QC.forAll (genSymmetric Odd Halfway Halfway) $ immutable Trafo1.sine . arrayFloat

QC.forAll (genSymmetric Odd Halfway Halfway) $ immutable Trafo1.sine . arrayDouble

hartley :: (Integral n, Real a) => Array (Cyclic n) a -> Array (Cyclic n) a

QC.forAll genCyclicArray1 $ \xs -> approxReal floatTol (adjust xs) (Trafo1.hartley (Trafo1.hartley xs))

QC.forAll genCyclicArray1 $ \xs -> approxReal doubleTol (adjust xs) (Trafo1.hartley (Trafo1.hartley xs))

QC.forAll genCyclicArray1 $ immutable Trafo1.hartley . arrayFloat

QC.forAll genCyclicArray1 $ immutable Trafo1.hartley . arrayDouble