```{-# LANGUAGE BangPatterns, FlexibleContexts #-}
-- |
-- Module    : Statistics.Transform
-- Copyright : (c) 2011 Bryan O'Sullivan
--
-- Maintainer  : bos@serpentine.com
-- Stability   : experimental
-- Portability : portable
--
-- Fourier-related transformations of mathematical functions.
--
-- These functions are written for simplicity and correctness, not
-- speed.  If you need a fast FFT implementation for your application,
-- you should strongly consider using a library of FFTW bindings

module Statistics.Transform
(
-- * Type synonyms
CD
-- * Discrete cosine transform
, dct
, dct_
, idct
, idct_
-- * Fast Fourier transform
, fft
, ifft
) where

import Data.Bits             (shiftL, shiftR)
import Data.Complex          (Complex(..), conjugate, realPart)
import Numeric.SpecFunctions (log2)
import qualified Data.Vector.Generic         as G
import qualified Data.Vector.Generic.Mutable as M
import qualified Data.Vector.Unboxed         as U

type CD = Complex Double

-- | Discrete cosine transform (DCT-II).
dct :: U.Vector Double -> U.Vector Double
{-# INLINE dct #-}
dct = dctWorker . G.map (:+0)

-- | Discrete cosine transform (DCT-II). Only real part of vector is
--   transformed, imaginary part is ignored.
dct_ :: U.Vector CD -> U.Vector Double
{-# INLINE dct_ #-}
dct_ = dctWorker . G.map (\(i :+ _) -> i :+ 0)

dctWorker :: U.Vector CD -> U.Vector Double
dctWorker xs
-- length 1 is special cased because shuffle algorithms fail for it.
| G.length xs == 1 = G.map ((2*) . realPart) xs
| vectorOK xs      = G.map realPart \$ G.zipWith (*) weights (fft interleaved)
| otherwise        = error "Statistics.Transform.dct: bad vector length"
where
interleaved = G.backpermute xs \$ G.enumFromThenTo 0 2 (len-2) G.++
G.enumFromThenTo (len-1) (len-3) 1
weights = G.cons 2 . G.generate (len-1) \$ \x ->
2 * exp ((0:+(-1))*fi (x+1)*pi/(2*n))
where n = fi len
len = G.length xs

-- | Inverse discrete cosine transform (DCT-III). It's inverse of
-- 'dct' only up to scale parameter:
--
-- > (idct . dct) x = (* length x)
idct :: U.Vector Double -> U.Vector Double
{-# INLINE idct #-}
idct = idctWorker . G.map (:+0)

-- | Inverse discrete cosine transform (DCT-III). Only real part of vector is
--   transformed, imaginary part is ignored.
idct_ :: U.Vector CD -> U.Vector Double
{-# INLINE idct_ #-}
idct_ = idctWorker . G.map (\(i :+ _) -> i :+ 0)

idctWorker :: U.Vector CD -> U.Vector Double
idctWorker xs
| vectorOK xs = G.generate len interleave
| otherwise   = error "Statistics.Transform.dct: bad vector length"
where
interleave z | even z    = vals `G.unsafeIndex` halve z
| otherwise = vals `G.unsafeIndex` (len - halve z - 1)
vals = G.map realPart . ifft \$ G.zipWith (*) weights xs
weights
= G.cons n
\$ G.generate (len - 1) \$ \x -> 2 * n * exp ((0:+1) * fi (x+1) * pi/(2*n))
where n = fi len
len = G.length xs

-- | Inverse fast Fourier transform.
ifft :: U.Vector CD -> U.Vector CD
ifft xs
| vectorOK xs = G.map ((/fi (G.length xs)) . conjugate) . fft . G.map conjugate \$ xs
| otherwise   = error "Statistics.Transform.ifft: bad vector length"

-- | Radix-2 decimation-in-time fast Fourier transform.
fft :: U.Vector CD -> U.Vector CD
fft v | vectorOK v  = G.create \$ do mv <- G.thaw v
mfft mv
return mv
| otherwise   = error "Statistics.Transform.fft: bad vector length"

-- Vector length must be power of two. It's not checked
mfft :: (M.MVector v CD) => v s CD -> ST s ()
mfft vec = bitReverse 0 0
where
bitReverse i j | i == len-1 = stage 0 1
| otherwise  = do
when (i < j) \$ M.swap vec i j
let inner k l | k <= l    = inner (k `shiftR` 1) (l-k)
| otherwise = bitReverse (i+1) (l+k)
inner (len `shiftR` 1) j
stage l !l1 | l == m    = return ()
| otherwise = do
let !l2 = l1 `shiftL` 1
!e  = -6.283185307179586/fromIntegral l2
flight j !a | j == l1   = stage (l+1) l2
| otherwise = do
let butterfly i | i >= len  = flight (j+1) (a+e)
| otherwise = do
let i1 = i + l1
xi1 :+ yi1 <- M.read vec i1
let !c = cos a
!s = sin a
d  = (c*xi1 - s*yi1) :+ (s*xi1 + c*yi1)
M.write vec i1 (ci - d)
M.write vec i (ci + d)
butterfly (i+l2)
butterfly j
flight 0 0
len = M.length vec
m   = log2 len

fi :: Int -> CD
fi = fromIntegral

halve :: Int -> Int
halve = (`shiftR` 1)

vectorOK :: U.Unbox a => U.Vector a -> Bool
{-# INLINE vectorOK #-}
vectorOK v = (1 `shiftL` log2 n) == n where n = G.length v
```