```{-# LANGUAGE NoImplicitPrelude #-}
{-# LANGUAGE MultiParamTypeClasses #-}
{-# LANGUAGE FlexibleInstances #-}
{- |
Copyright   :  (c) Henning Thielemann 2008

Maintainer  :  synthesizer@henning-thielemann.de
Stability   :  provisional
Portability :  requires multi-parameter type classes

All recursive filters with real coefficients
can be decomposed into first order and second order filters with real coefficients.
This follows from the Fundamental theorem of algebra.
-}
module Synthesizer.Plain.Filter.Recursive.SecondOrder where

import Synthesizer.Plain.Filter.Recursive (Passband(Lowpass,Highpass))
import qualified Synthesizer.Plain.Signal   as Sig
import qualified Synthesizer.Plain.Modifier as Modifier
-- import qualified Synthesizer.Plain.Control as Ctrl

import qualified Synthesizer.Interpolation.Class as Interpol
import Synthesizer.ApplicativeUtility (liftA4, liftA5, )

import qualified Synthesizer.Causal.Process as Causal

-- import qualified Algebra.VectorSpace           as VectorSpace
import qualified Algebra.Module                as Module
-- import qualified Algebra.Transcendental        as Trans
import qualified Algebra.Field                 as Field
import qualified Algebra.Ring                  as Ring

import Algebra.Module((*>))

import Data.List (zipWith6)

import Foreign.Storable (Storable(..))
import qualified Foreign.Storable.Record as Store

import qualified Prelude as P
import PreludeBase
import NumericPrelude

{- | Parameters for a general recursive filter of 2nd order. -}
data Parameter a =
Parameter {c0, c1, c2, d1, d2 :: !a}
deriving Show

data Status a =
Status {u1, u2, y1, y2 :: !a}
deriving Show

zeroStatus :: Additive.C a => Status a
zeroStatus =
Status
{u1 = zero, u2 = zero,
y1 = zero, y2 = zero}

instance Interpol.C a v => Interpol.C a (Parameter v) where
{-# INLINE scaleAndAccumulate #-}
scaleAndAccumulate =
Interpol.runMac \$
liftA5 Parameter
(Interpol.element c0)
(Interpol.element c1)
(Interpol.element c2)
(Interpol.element d1)
(Interpol.element d2)

instance Storable a => Storable (Parameter a) where
sizeOf    = Store.sizeOf storeParameter
alignment = Store.alignment storeParameter
peek      = Store.peek storeParameter
poke      = Store.poke storeParameter

storeParameter ::
Storable a => Store.Dictionary (Parameter a)
storeParameter =
Store.run \$
liftA5 Parameter
(Store.element c0)
(Store.element c1)
(Store.element c2)
(Store.element d1)
(Store.element d2)

instance Storable a => Storable (Status a) where
sizeOf    = Store.sizeOf storeStatus
alignment = Store.alignment storeStatus
peek      = Store.peek storeStatus
poke      = Store.poke storeStatus

storeStatus ::
Storable a => Store.Dictionary (Status a)
storeStatus =
Store.run \$
liftA4 Status
(Store.element u1)
(Store.element u2)
(Store.element y1)
(Store.element y2)

{- |
Given a function which computes the filter parameters of a lowpass filter
for a given frequency,
turn that into a function which generates highpass parameters,
if requested filter type is Highpass.
-}
Passband -> (a -> Parameter a) -> (a -> Parameter a)
case kind of
Lowpass  -> comp f
Highpass ->
let p = comp (0.5-f)
in  Parameter (c0 p) (- c1 p) (c2 p) (- d1 p) (d2 p)

{-# INLINE step #-}
step :: (Ring.C a, Module.C a v) =>
Parameter a -> v -> State (Status v) v
step c u0 = state \$ \s ->
let y0 =
c0 c *> u0   +
c1 c *> u1 s + d1 c *> y1 s +
c2 c *> u2 s + d2 c *> y2 s
in  (y0, Status
{u1 = u0, u2 = u1 s,
y1 = y0, y2 = y1 s})

{-# INLINE modifierInit #-}
modifierInit :: (Ring.C a, Module.C a v) =>
Modifier.Initialized (Status v) (Status v) (Parameter a) v v
modifierInit =
Modifier.Initialized id step

{-# INLINE modifier #-}
modifier :: (Ring.C a, Module.C a v) =>
Modifier.Simple (Status v) (Parameter a) v v
modifier =
Sig.modifierInitialize modifierInit zeroStatus

{-# INLINE causal #-}
causal :: (Ring.C a, Module.C a v) =>
Causal.T (Parameter a, v) v
causal =
Causal.fromSimpleModifier modifier

{-# INLINE runInit #-}
runInit :: (Ring.C a, Module.C a v) =>
Status v -> Sig.T (Parameter a) -> Sig.T v -> Sig.T v
runInit sInit control input =
let u0s = input
u1s = u1 sInit : u0s
u2s = u2 sInit : u1s
y1s = y1 sInit : y0s
y2s = y2 sInit : y1s
y0s = zipWith6
(\c u0_ u1_ u2_ y1_ y2_ ->
c0 c *> u0_ +
c1 c *> u1_ + d1 c *> y1_ +
c2 c *> u2_ + d2 c *> y2_)
control u0s u1s u2s y1s y2s
in  y0s

{-# INLINE run #-}
run :: (Ring.C a, Module.C a v) =>
Sig.T (Parameter a) -> Sig.T v -> Sig.T v
run =
runInit zeroStatus
```