{-# LANGUAGE NoImplicitPrelude #-} {- | Special interpolations defined in terms of Module operations. -} module Synthesizer.Interpolation.Module ( T, constant, linear, cubic, cubicAlt, piecewise, piecewiseConstant, piecewiseLinear, piecewiseCubic, function, ) where import qualified Synthesizer.State.Signal as Sig import qualified Synthesizer.Plain.Control as Ctrl import Synthesizer.Interpolation ( T, cons, getNode, fromPrefixReader, constant, ) import qualified Algebra.Module as Module -- import qualified Algebra.RealField as RealField import qualified Algebra.Field as Field import qualified Algebra.Ring as Ring import qualified Algebra.Additive as Additive import Algebra.Module((*>)) import Control.Applicative (liftA2, ) import Synthesizer.ApplicativeUtility (liftA4, ) import Synthesizer.Utility (affineComb, ) import NumericPrelude.Base import NumericPrelude.Numeric {-| Consider the signal to be piecewise linear. -} {-# INLINE linear #-} linear :: (Module.C t y) => T t y linear = fromPrefixReader "linear" 0 (liftA2 (\x0 x1 phase -> affineComb phase (x0,x1)) getNode getNode) {- | Consider the signal to be piecewise cubic, with smooth connections at the nodes. It uses a cubic curve which has node values x0 at 0 and x1 at 1 and derivatives (x1-xm1)/2 and (x2-x0)/2, respectively. You can see how it works if you evaluate the expression for t=0 and t=1 as well as the derivative at these points. -} {-# INLINE cubic #-} cubic :: (Field.C t, Module.C t y) => T t y cubic = fromPrefixReader "cubic" 1 (liftA4 (\xm1 x0 x1 x2 t -> let lipm12 = affineComb t (xm1,x2) lip01 = affineComb t (x0, x1) three = 3 `asTypeOf` t in lip01 + (t*(t-1)/2) *> (lipm12 + (x0+x1) - three *> lip01)) getNode getNode getNode getNode) {- | The interpolators for module operations do not simply compute a straight linear combination of some vectors. Instead they add then scale, then add again, and so on. This is efficient whenever scaling and addition is cheap. In this case they might save multiplications. I can't say much about numeric cancellations, however. -} {-# INLINE cubicAlt #-} cubicAlt :: (Field.C t, Module.C t y) => T t y cubicAlt = fromPrefixReader "cubicAlt" 1 (liftA4 (\xm1 x0 x1 x2 t -> let half = 1/2 `asTypeOf` t in cubicHalf t x0 (half *> (x1-xm1)) + cubicHalf (1-t) x1 (half *> (x0-x2))) getNode getNode getNode getNode) {- | @\t -> cubicHalf t x x'@ has a double zero at 1 and at 0 it has value x and slope x'. -} {-# INLINE cubicHalf #-} cubicHalf :: (Module.C t y) => t -> y -> y -> y cubicHalf t x x' = (t-1)^2 *> ((1+2*t)*>x + t*>x') {-** Interpolation based on piecewise defined functions -} {-# INLINE piecewise #-} piecewise :: (Module.C t y) => Int -> [t -> t] -> T t y piecewise center ps = cons (length ps) (center-1) (\t -> Sig.linearComb (Sig.fromList (map ($t) (reverse ps)))) {-# INLINE piecewiseConstant #-} piecewiseConstant :: (Module.C t y) => T t y piecewiseConstant = piecewise 1 [const 1] {-# INLINE piecewiseLinear #-} piecewiseLinear :: (Module.C t y) => T t y piecewiseLinear = piecewise 1 [id, (1-)] {-# INLINE piecewiseCubic #-} piecewiseCubic :: (Field.C t, Module.C t y) => T t y piecewiseCubic = piecewise 2 $ Ctrl.cubicFunc (0,(0,0)) (1,(0,1/2)) : Ctrl.cubicFunc (0,(0,1/2)) (1,(1,0)) : Ctrl.cubicFunc (0,(1,0)) (1,(0,-1/2)) : Ctrl.cubicFunc (0,(0,-1/2)) (1,(0,0)) : [] {- GNUPlot.plotList [] $ take 100 $ interpolate (Zero 0) piecewiseCubic (-2.3 :: Double) (repeat 0.1) [2,1,2::Double] -} {-** Interpolation based on arbitrary functions -} {- | with this wrapper you can use the collection of interpolating functions from Donadio's DSP library -} {-# INLINE function #-} function :: (Module.C t y) => (Int,Int) {- ^ @(left extent, right extent)@, e.g. @(1,1)@ for linear hat -} -> (t -> t) -> T t y function (left,right) f = let len = left+right ps = Sig.take len $ Sig.iterate pred (pred right) -- ps = Sig.reverse $ Sig.take len $ Sig.iterate succ (-left) in cons len left (\t -> Sig.linearComb $ Sig.map (\x -> f (t + fromIntegral x)) ps) {- GNUPlot.plotList [] $ take 300 $ interpolate (Zero 0) (function (1,1) (\x -> exp (-6*x*x))) (-2.3 :: Double) (repeat 0.03) [2,1,2::Double] -}