module Numeric.Transform.Fourier.Rader (fft_rader1, fft_rader2) where
import Data.Array
import Data.Complex
fft_rader1 :: (Ix a, Integral a, RealFloat b) => Array a (Complex b)
-> a
-> Array a (Complex b)
fft_rader1 :: forall a b.
(Ix a, Integral a, RealFloat b) =>
Array a (Complex b) -> a -> Array a (Complex b)
fft_rader1 Array a (Complex b)
f a
n = Array a (Complex b)
f'
where h :: Array a (Complex b)
h = forall i e. Ix i => (i, i) -> [e] -> Array i e
listArray (a
0,a
nforall a. Num a => a -> a -> a
-a
2) [ Array a (Complex b)
fforall i e. Ix i => Array i e -> i -> e
!(a
a forall {t}. (Num t, Eq t) => a -> t -> a
^* (a
nforall a. Num a => a -> a -> a
-(a
1forall a. Num a => a -> a -> a
+a
n'))) | a
n' <- [a
0..(a
nforall a. Num a => a -> a -> a
-a
2)] ]
g :: Array a (Complex b)
g = forall i e. Ix i => (i, i) -> [e] -> Array i e
listArray (a
0,a
nforall a. Num a => a -> a -> a
-a
2) [ Array a (Complex b)
wforall i e. Ix i => Array i e -> i -> e
!(a
a forall {t}. (Num t, Eq t) => a -> t -> a
^* a
n') | a
n' <- [a
0..(a
nforall a. Num a => a -> a -> a
-a
2)] ]
hg :: Array a (Complex b)
hg = forall i e. Ix i => (i, i) -> [e] -> Array i e
listArray (a
0,a
nforall a. Num a => a -> a -> a
-a
2) [ forall (t :: * -> *) a. (Foldable t, Num a) => t a -> a
sum [ Array a (Complex b)
hforall i e. Ix i => Array i e -> i -> e
!a
j forall a. Num a => a -> a -> a
* Array a (Complex b)
gforall i e. Ix i => Array i e -> i -> e
!((a
iforall a. Num a => a -> a -> a
-a
j)forall a. Integral a => a -> a -> a
`mod`(a
nforall a. Num a => a -> a -> a
-a
1)) | a
j <- [a
0..(a
nforall a. Num a => a -> a -> a
-a
2)] ] | a
i <- [a
0..(a
nforall a. Num a => a -> a -> a
-a
2)] ]
f' :: Array a (Complex b)
f' = forall i e. Ix i => (i, i) -> [(i, e)] -> Array i e
array (a
0,a
nforall a. Num a => a -> a -> a
-a
1) ((a
0, forall (t :: * -> *) a. (Foldable t, Num a) => t a -> a
sum [ Array a (Complex b)
fforall i e. Ix i => Array i e -> i -> e
!a
i | a
i <- [a
0..(a
nforall a. Num a => a -> a -> a
-a
1)] ]) forall a. a -> [a] -> [a]
: [ (a
a forall {t}. (Num t, Eq t) => a -> t -> a
^* a
i, Array a (Complex b)
fforall i e. Ix i => Array i e -> i -> e
!a
0 forall a. Num a => a -> a -> a
+ Array a (Complex b)
hgforall i e. Ix i => Array i e -> i -> e
!a
i) | a
i <- [a
0..(a
nforall a. Num a => a -> a -> a
-a
2)] ])
wn :: Complex b
wn = forall a. Floating a => a -> Complex a
cis (-b
2 forall a. Num a => a -> a -> a
* forall a. Floating a => a
pi forall a. Fractional a => a -> a -> a
/ forall a b. (Integral a, Num b) => a -> b
fromIntegral a
n)
w :: Array a (Complex b)
w = forall i e. Ix i => (i, i) -> [e] -> Array i e
listArray (a
0,a
nforall a. Num a => a -> a -> a
-a
1) forall a b. (a -> b) -> a -> b
$ forall a. (a -> a) -> a -> [a]
iterate (forall a. Num a => a -> a -> a
* Complex b
wn) Complex b
1
a
_ ^* :: a -> t -> a
^* t
0 = a
1
a
i ^* t
j = (a
i forall a. Num a => a -> a -> a
* (a
i a -> t -> a
^* (t
jforall a. Num a => a -> a -> a
-t
1))) forall a. Integral a => a -> a -> a
`mod` a
n
a :: a
a = forall a. Integral a => a -> a
generator a
n
{-# specialize fft_rader2 :: Array Int (Complex Float) -> Int -> (Array Int (Complex Float) -> Array Int (Complex Float)) -> Array Int (Complex Float) #-}
{-# specialize fft_rader2 :: Array Int (Complex Double) -> Int -> (Array Int (Complex Double) -> Array Int (Complex Double)) -> Array Int (Complex Double) #-}
fft_rader2 :: (Ix a, Integral a, RealFloat b) => Array a (Complex b)
-> a
-> (Array a (Complex b) -> Array a (Complex b))
-> Array a (Complex b)
fft_rader2 :: forall a b.
(Ix a, Integral a, RealFloat b) =>
Array a (Complex b)
-> a
-> (Array a (Complex b) -> Array a (Complex b))
-> Array a (Complex b)
fft_rader2 Array a (Complex b)
f a
n Array a (Complex b) -> Array a (Complex b)
fft = Array a (Complex b)
f'
where h :: Array a (Complex b)
h = forall i e. Ix i => (i, i) -> [e] -> Array i e
listArray (a
0,a
nforall a. Num a => a -> a -> a
-a
2) [ Array a (Complex b)
fforall i e. Ix i => Array i e -> i -> e
!(a
a forall {t}. (Num t, Eq t) => a -> t -> a
^* (a
nforall a. Num a => a -> a -> a
-(a
1forall a. Num a => a -> a -> a
+a
n'))) | a
n' <- [a
0..(a
nforall a. Num a => a -> a -> a
-a
2)] ]
g :: Array a (Complex b)
g = forall i e. Ix i => (i, i) -> [e] -> Array i e
listArray (a
0,a
nforall a. Num a => a -> a -> a
-a
2) [ Array a (Complex b)
wforall i e. Ix i => Array i e -> i -> e
!(a
a forall {t}. (Num t, Eq t) => a -> t -> a
^* a
n') | a
n' <- [a
0..(a
nforall a. Num a => a -> a -> a
-a
2)] ]
h' :: Array a (Complex b)
h' = Array a (Complex b) -> Array a (Complex b)
fft Array a (Complex b)
h
g' :: Array a (Complex b)
g' = Array a (Complex b) -> Array a (Complex b)
fft Array a (Complex b)
g
hg' :: Array a (Complex b)
hg' = forall i e. Ix i => (i, i) -> [e] -> Array i e
listArray (a
0,a
nforall a. Num a => a -> a -> a
-a
2) [ Array a (Complex b)
h'forall i e. Ix i => Array i e -> i -> e
!a
i forall a. Num a => a -> a -> a
* Array a (Complex b)
g'forall i e. Ix i => Array i e -> i -> e
!a
i | a
i <- [a
0..(a
nforall a. Num a => a -> a -> a
-a
2)] ]
hg :: Array a (Complex b)
hg = Array a (Complex b) -> Array a (Complex b)
ifft Array a (Complex b)
hg'
f' :: Array a (Complex b)
f' = forall i e. Ix i => (i, i) -> [(i, e)] -> Array i e
array (a
0,a
nforall a. Num a => a -> a -> a
-a
1) ((a
0, forall (t :: * -> *) a. (Foldable t, Num a) => t a -> a
sum [ Array a (Complex b)
fforall i e. Ix i => Array i e -> i -> e
!a
i | a
i <- [a
0..(a
nforall a. Num a => a -> a -> a
-a
1)] ]) forall a. a -> [a] -> [a]
: [ (a
a forall {t}. (Num t, Eq t) => a -> t -> a
^* a
i, Array a (Complex b)
fforall i e. Ix i => Array i e -> i -> e
!a
0 forall a. Num a => a -> a -> a
+ Array a (Complex b)
hgforall i e. Ix i => Array i e -> i -> e
!a
i) | a
i <- [a
0..(a
nforall a. Num a => a -> a -> a
-a
2)] ])
wn :: Complex b
wn = forall a. Floating a => a -> Complex a
cis (-b
2 forall a. Num a => a -> a -> a
* forall a. Floating a => a
pi forall a. Fractional a => a -> a -> a
/ forall a b. (Integral a, Num b) => a -> b
fromIntegral a
n)
w :: Array a (Complex b)
w = forall i e. Ix i => (i, i) -> [e] -> Array i e
listArray (a
0,a
nforall a. Num a => a -> a -> a
-a
1) forall a b. (a -> b) -> a -> b
$ forall a. (a -> a) -> a -> [a]
iterate (forall a. Num a => a -> a -> a
* Complex b
wn) Complex b
1
a
_ ^* :: a -> t -> a
^* t
0 = a
1
a
i ^* t
j = (a
i forall a. Num a => a -> a -> a
* (a
i a -> t -> a
^* (t
jforall a. Num a => a -> a -> a
-t
1))) forall a. Integral a => a -> a -> a
`mod` a
n
a :: a
a = forall a. Integral a => a -> a
generator a
n
ifft :: Array a (Complex b) -> Array a (Complex b)
ifft Array a (Complex b)
b = forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmap (forall a. Fractional a => a -> a -> a
/ forall a b. (Integral a, Num b) => a -> b
fromIntegral (a
nforall a. Num a => a -> a -> a
-a
1)) forall a b. (a -> b) -> a -> b
$ forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmap forall {a}. Complex a -> Complex a
swap forall a b. (a -> b) -> a -> b
$ Array a (Complex b) -> Array a (Complex b)
fft forall a b. (a -> b) -> a -> b
$ forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmap forall {a}. Complex a -> Complex a
swap Array a (Complex b)
b
swap :: Complex a -> Complex a
swap (a
x:+a
y) = (a
yforall a. a -> a -> Complex a
:+a
x)
{-# specialize generator :: Int -> Int #-}
generator :: (Integral a) => a -> a
generator :: forall a. Integral a => a -> a
generator a
p = a -> a
findgen a
1
where findgen :: a -> a
findgen a
0 = forall a. HasCallStack => [Char] -> a
error [Char]
"rader: generator: no primitive root?"
findgen a
x | (forall {a}. Num a => a -> a -> a
period a
x a
x) forall a. Eq a => a -> a -> Bool
== (a
p forall a. Num a => a -> a -> a
- a
1) = a
x
| Bool
otherwise = a -> a
findgen ((a
x forall a. Num a => a -> a -> a
+ a
1) forall a. Integral a => a -> a -> a
`mod` a
p)
period :: a -> a -> a
period a
_ a
1 = a
1
period a
x a
prod = a
1 forall a. Num a => a -> a -> a
+ (a -> a -> a
period a
x (a
prod forall a. Num a => a -> a -> a
* a
x forall a. Integral a => a -> a -> a
`mod` a
p))