```{- |
This module provides normalized versions of the transforms in @fftw@.

The forwards transforms in this module are identical to those in "Numeric.FFT.Vector.Unnormalized".
The backwards transforms are normalized to be their inverse operations (approximately, due to floating point precision).

For more information on the underlying transforms, see
<http://www.fftw.org/fftw3_doc/What-FFTW-Really-Computes.html>.
-}
module Numeric.FFT.Vector.Invertible(
-- * Creating and executing 'Plan's
run,
plan,
execute,
-- * Complex-to-complex transforms
U.dft,
idft,
-- * Real-to-complex transforms
U.dftR2C,
dftC2R,
-- * Real-to-real transforms
-- \$dct_size
-- ** Discrete cosine transforms
U.dct1,
idct1,
U.dct2,
idct2,
U.dct3,
idct3,
U.dct4,
idct4,
-- ** Discrete sine transforms
U.dst1,
idst1,
U.dst2,
idst2,
U.dst3,
idst3,
U.dst4,
idst4,
) where

import Numeric.FFT.Vector.Base
import qualified Numeric.FFT.Vector.Unnormalized as U
import Data.Complex

-- | A backward discrete Fourier transform which is the inverse of 'U.dft'.  The output and input sizes are the same (@n@).
--
-- @y_k = (1\/n) sum_(j=0)^(n-1) x_j e^(2pi i j k/n)@
idft :: Transform (Complex Double) (Complex Double)
idft = U.idft {normalization = \n -> constMultOutput \$ 1 / toEnum n}

-- | A normalized backward discrete Fourier transform which is the left inverse of
-- 'U.dftR2C'.  (Specifically, @run dftC2R . run dftR2C == id@.)
--
-- This 'Transform' behaves differently than the others:
--
--  - Calling @plan dftC2R n@ creates a 'Plan' whose /output/ size is @n@, and whose
--    /input/ size is @n \`div\` 2 + 1@.
--
--  - If @length v == n@, then @length (run dftC2R v) == 2*(n-1)@.
--
dftC2R :: Transform (Complex Double) Double
dftC2R = U.dftC2R {normalization = \n -> constMultOutput \$ 1 / toEnum n}

-- OK, the inverse of each unnormalized operation.

-- \$dct_size
-- The real-even (DCT) and real-odd (DST) transforms.  The input and output sizes
-- are the same (@n@).

-- | A type-1 discrete cosine transform which is the inverse of 'U.dct1'.
--
-- @y_k = (1\/(2(n-1)) [x_0 + (-1)^k x_(n-1) + 2 sum_(j=1)^(n-2) x_j cos(pi j k\/(n-1))]@
idct1 :: Transform Double Double
idct1 = U.dct1 {normalization = \n -> constMultOutput \$ 1 / toEnum (2 * (n-1))}

-- | A type-3 discrete cosine transform which is the inverse of 'U.dct2'.
--
-- @y_k = (1\/(2n)) [x_0 + 2 sum_(j=1)^(n-1) x_j cos(pi j(k+1\/2)\/n)]@
idct2 :: Transform Double Double
idct2 = U.dct3 {normalization = \n -> constMultOutput \$ 1 / toEnum (2 * n)}

-- | A type-2 discrete cosine transform which is the inverse of 'U.dct3'.
--
-- @y_k = (1\/n) sum_(j=0)^(n-1) x_j cos(pi(j+1\/2)k\/n)@
idct3 :: Transform Double Double
idct3 = U.dct2 {normalization = \n -> constMultOutput \$ 1 / toEnum (2 * n)}

-- | A type-4 discrete cosine transform which is the inverse of 'U.dct4'.
--
-- @y_k = (1\/n) sum_(j=0)^(n-1) x_j cos(pi(j+1\/2)(k+1\/2)\/n)@
idct4 :: Transform Double Double
idct4 = U.dct4 {normalization = \n -> constMultOutput \$ 1 / toEnum (2 * n)}

-- | A type-1 discrete sine transform which is the inverse of 'U.dst1'.
--
-- @y_k = (1\/(n+1)) sum_(j=0)^(n-1) x_j sin(pi(j+1)(k+1)\/(n+1))@
idst1 :: Transform Double Double
idst1 = U.dst1 {normalization = \n -> constMultOutput \$ 1 / toEnum (2 * (n+1))}

-- | A type-3 discrete sine transform which is the inverse of 'U.dst2'.
--
-- @y_k = (1\/(2n)) [(-1)^k x_(n-1) + 2 sum_(j=0)^(n-2) x_j sin(pi(j+1)(k+1\/2)/n)]@
idst2 :: Transform Double Double
idst2 = U.dst3 {normalization = \n -> constMultOutput \$ 1 / toEnum (2 * n)}

-- | A type-2 discrete sine transform which is the inverse of 'U.dst3'.
--
-- @y_k = (1\/n) sum_(j=0)^(n-1) x_j sin(pi(j+1\/2)(k+1)\/n)@
idst3 :: Transform Double Double
idst3 = U.dst2 {normalization = \n -> constMultOutput \$ 1 / toEnum (2 * n)}

-- | A type-4 discrete sine transform which is the inverse of 'U.dst4'.
--
-- @y_k = (1\/(2n)) sum_(j=0)^(n-1) x_j sin(pi(j+1\/2)(k+1\/2)\/n)@
idst4 :: Transform Double Double
idst4 = U.dst4 {normalization = \n -> constMultOutput \$ 1 / toEnum (2 * n)}

```