{-# LANGUAGE NoImplicitPrelude #-} {- | Copyright : (c) Henning Thielemann 2008 License : GPL Maintainer : synthesizer@henning-thielemann.de Stability : provisional Portability : requires multi-parameter type classes -} module Synthesizer.State.Filter.NonRecursive where import qualified Synthesizer.State.Signal as Sig import qualified Synthesizer.State.Filter.Delay as Delay import qualified Synthesizer.State.Control as Ctrl import qualified Algebra.Transcendental as Trans 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( {- linearComb, -} (*>)) import Data.Function.HT (nest, ) import Data.Tuple.HT (mapFst, ) import NumericPrelude.Base import NumericPrelude.Numeric {- * Envelope application -} {-# INLINE amplify #-} amplify :: (Ring.C a) => a -> Sig.T a -> Sig.T a amplify v = Sig.map (v*) {-# INLINE amplifyVector #-} amplifyVector :: (Module.C a v) => a -> Sig.T v -> Sig.T v amplifyVector v = Sig.map (v*>) {-# INLINE envelope #-} envelope :: (Ring.C a) => Sig.T a {-^ the envelope -} -> Sig.T a {-^ the signal to be enveloped -} -> Sig.T a envelope = Sig.zipWith (*) {-# INLINE envelopeVector #-} envelopeVector :: (Module.C a v) => Sig.T a {-^ the envelope -} -> Sig.T v {-^ the signal to be enveloped -} -> Sig.T v envelopeVector = Sig.zipWith (*>) {-# INLINE fadeInOut #-} fadeInOut :: (Field.C a) => Int -> Int -> Int -> Sig.T a -> Sig.T a fadeInOut tIn tHold tOut = let leadIn = Sig.take tIn $ Ctrl.linear ( recip (fromIntegral tIn)) zero leadOut = Sig.take tOut $ Ctrl.linear (- recip (fromIntegral tOut)) one hold = Sig.replicate tHold one in envelope (leadIn `Sig.append` hold `Sig.append` leadOut) {- * Smoothing -} {-| Unmodulated non-recursive filter -} {-# INLINE generic #-} generic :: (Module.C a v) => Sig.T a -> Sig.T v -> Sig.T v generic m x = let mr = Sig.reverse m xp = Delay.staticPos (pred (Sig.length m)) x in Sig.mapTails (Sig.linearComb mr) xp {- genericSlow :: Module.C a v => Sig.T a -> Sig.T v -> Sig.T v genericSlow m x = let mr = Sig.reverse m xp = delay (pred (Sig.length m)) x in Sig.fromList (map (Sig.linearComb mr) (init (Sig.tails xp))) -} {- {- | @eps@ is the threshold relatively to the maximum. That is, if the gaussian falls below @eps * gaussian 0@, then the function truncated. -} gaussian :: (Trans.C a, RealField.C a, Module.C a v) => a -> a -> a -> Sig.T v -> Sig.T v gaussian eps ratio freq = let var = ratioFreqToVariance ratio freq area = var * sqrt (2*pi) gau t = exp (-(t/var)^2/2) / area width = ceiling (var * sqrt (-2 * log eps)) -- inverse gau gauSmp = map (gau . fromIntegral) [-width .. width] in drop width . generic gauSmp -} {- GNUPlot.plotList [] (take 1000 $ gaussian 0.001 0.5 0.04 (Filter.Test.chirp 5000) :: [Double]) The filtered chirp must have amplitude 0.5 at 400 (0.04*10000). -} {- We want to approximate a Gaussian by a binomial filter. The latter one can be implemented by a convolutional power. However we still require a number of operations per sample which is proportional to the variance. -} {-# INLINE binomial #-} binomial :: (Trans.C a, RealField.C a, Module.C a v) => a -> a -> Sig.T v -> Sig.T v binomial ratio freq = let width = ceiling (2 * ratioFreqToVariance ratio freq ^ 2) in Sig.drop width . nest (2*width) ((asTypeOf 0.5 freq *>) . binomial1) {- exp (-(t/var)^2/2) / area *> cis (2*pi*f*t) == exp (-(t/var)^2/2 +: 2*pi*f*t) / area == exp ((-t^2 +: 2*var^2*2*pi*f*t) / (2*var^2)) / area == exp ((t^2 - i*2*var^2*2*pi*f*t) / (-2*var^2)) / area == exp (((t^2 - i*var^2*2*pi*f)^2 + (var^2*2*pi*f)^2) / (-2*var^2)) / area == exp (((t^2 - i*var^2*2*pi*f)^2 / (-2*var^2) - (var*2*pi*f)^2/2)) / area sumMap (\t -> exp (-(t/var)^2/2) / area *> cis (2*pi*f*t)) [-infinity..infinity] ~ sumMap (\t -> exp (-(t/var)^2/2)) [-infinity..infinity] * exp (-(var*2*pi*f)^2/2) / area = exp (-(var*2*pi*f)^2/2) -} {- | Compute the variance of the Gaussian such that its Fourier transform has value @ratio@ at frequency @freq@. -} {-# INLINE ratioFreqToVariance #-} ratioFreqToVariance :: (Trans.C a) => a -> a -> a ratioFreqToVariance ratio freq = sqrt (-2 * log ratio) / (2*pi*freq) -- inverse of the fourier transformed gaussian {-# INLINE binomial1 #-} binomial1 :: (Additive.C v) => Sig.T v -> Sig.T v binomial1 = Sig.mapAdjacent (+) {- | Moving (uniformly weighted) average in the most trivial form. This is very slow and needs about @n * length x@ operations. -} {-# INLINE sums #-} sums :: (Additive.C v) => Int -> Sig.T v -> Sig.T v sums n = Sig.mapTails (Sig.sum . Sig.take n) {- sumsDownsample2 :: (Additive.C v) => Sig.T v -> Sig.T v sumsDownsample2 (x0:x1:xs) = (x0+x1) : sumsDownsample2 xs sumsDownsample2 xs = xs downsample2 :: Sig.T a -> Sig.T a downsample2 (x0:_:xs) = x0 : downsample2 xs downsample2 xs = xs {- | Given a list of numbers and a list of sums of (2*k) of successive summands, compute a list of the sums of (2*k+1) or (2*k+2) summands. Eample for 2*k+1 @ [0+1+2+3, 2+3+4+5, 4+5+6+7, ...] -> [0+1+2+3+4, 1+2+3+4+5, 2+3+4+5+6, 3+4+5+6+7, 4+5+6+7+8, ...] @ Example for 2*k+2 @ [0+1+2+3, 2+3+4+5, 4+5+6+7, ...] -> [0+1+2+3+4+5, 1+2+3+4+5+6, 2+3+4+5+6+7, 3+4+5+6+7+8, 4+5+6+7+8+9, ...] @ -} sumsUpsampleOdd :: (Additive.C v) => Int -> Sig.T v -> Sig.T v -> Sig.T v sumsUpsampleOdd n {- 2*k -} xs ss = let xs2k = drop n xs in (head ss + head xs2k) : concat (zipWith3 (\s x0 x2k -> [x0+s, s+x2k]) (tail ss) (downsample2 (tail xs)) (tail (downsample2 xs2k))) sumsUpsampleEven :: (Additive.C v) => Int -> Sig.T v -> Sig.T v -> Sig.T v sumsUpsampleEven n {- 2*k -} xs ss = sumsUpsampleOdd (n+1) xs (zipWith (+) ss (downsample2 (drop n xs))) sumsPyramid :: (Additive.C v) => Int -> Sig.T v -> Sig.T v sumsPyramid n xs = let aux 1 ys = ys aux 2 ys = ys + tail ys aux m ys = let ysd = sumsDownsample2 ys in if even m then sumsUpsampleEven (m-2) ys (aux (div (m-2) 2) ysd) else sumsUpsampleOdd (m-1) ys (aux (div (m-1) 2) ysd) in aux n xs propSums :: Bool propSums = let n = 1000 xs = [0::Double ..] naive = sums n xs rec = drop (n-1) $ sumsRec n xs pyramid = sumsPyramid n xs in and $ take 1000 $ zipWith3 (\x y z -> x==y && y==z) naive rec pyramid -} {- | This is inverse to frequency modulation. If all control values in @ctrl@ are above one, then it holds: @ frequencyModulation ctrl (inverseFrequencyModulationFloor ctrl xs) == xs @. Otherwise 'inverseFrequencyModulationFloor' is lossy. For the precise property we refer to "Test.Sound.Synthesizer.Plain.Interpolation". The modulation uses constant interpolation. Other interpolation types are tricky to implement, since they would need interpolation on non-equidistant grids. Ok, at least linear interpolation could be supported with acceptable effort, but perfect reconstruction would be hard to get. The process is not causal in any of its inputs, however control and input are aligned. If you use interpolation for resampling or frequency modulation, you may want to smooth the signal before resampling according to the local resampling factor. However you cannot simply use the resampling control to also control the smoothing, because of the subsequent distortion by resampling. Instead you have to stretch the control inversely to the resampling factor. This is the task of this function. It may be applied like: > frequencyModulation ctrl (smooth (inverseFrequencyModulationFloor ctrl ctrl) xs) -} {-# INLINE inverseFrequencyModulationFloor #-} inverseFrequencyModulationFloor :: (Ord t, Ring.C t) => Sig.T t -> Sig.T v -> Sig.T v inverseFrequencyModulationFloor = inverseFrequencyModulationGen (<1) {- Also a sensible implementation, but incompatible with relative interpolation / frequency modulation. -} {-# INLINE inverseFrequencyModulationCeiling #-} inverseFrequencyModulationCeiling :: (Ord t, Ring.C t) => Sig.T t -> Sig.T v -> Sig.T v inverseFrequencyModulationCeiling = inverseFrequencyModulationGen (<=0) {-# INLINE inverseFrequencyModulationGen #-} inverseFrequencyModulationGen :: (Ord t, Ring.C t) => (t -> Bool) -> Sig.T t -> Sig.T v -> Sig.T v inverseFrequencyModulationGen p ctrl xs = Sig.runSwitchL (Sig.zip ctrl xs) (\switch -> switch Sig.empty (curry $ Sig.unfoldR (let go (c,x) cxs = if p c then switch Nothing (go . mapFst (c+)) cxs else Just (x, ((c-1,x),cxs)) in uncurry go))) {- * Filter operators from calculus -} {- | Forward difference quotient. Shortens the signal by one. Inverts 'Synthesizer.State.Filter.Recursive.Integration.run' in the sense that @differentiate (zero : integrate x) == x@. The signal is shifted by a half time unit. -} {-# INLINE differentiate #-} differentiate :: Additive.C v => Sig.T v -> Sig.T v differentiate x = Sig.mapAdjacent subtract x {- | Central difference quotient. Shortens the signal by two elements, and shifts the signal by one element. (Which can be fixed by prepending an appropriate value.) For linear functions this will yield essentially the same result as 'differentiate'. You obtain the result of 'differentiateCenter' if you smooth the one of 'differentiate' by averaging pairs of adjacent values. ToDo: Vector variant -} {- We use mapAdjacent in order to avoid recomputation of the input signal -} {-# INLINE differentiateCenter #-} differentiateCenter :: Field.C v => Sig.T v -> Sig.T v differentiateCenter = Sig.mapAdjacent (\(x0,_) (_,x1) -> (x1 - x0) * (1/2)) . Sig.mapAdjacent (,) {- differentiateCenter :: Field.C v => Sig.T v -> Sig.T v differentiateCenter x = Sig.map ((1/2)*) $ Sig.zipWith subtract x (Sig.tail (Sig.tail x)) -} {- | Second derivative. It is @differentiate2 == differentiate . differentiate@ but 'differentiate2' should be faster. -} {-# INLINE differentiate2 #-} differentiate2 :: Additive.C v => Sig.T v -> Sig.T v differentiate2 = differentiate . differentiate {- differentiate2 :: Additive.C v => Sig.T v -> Sig.T v differentiate2 xs0 = let xs1 = Sig.tail xs0 xs2 = Sig.tail xs1 in Sig.zipWith3 (\x0 x1 x2 -> x0+x2-(x1+x1)) xs0 xs1 xs2 -}