{-# LANGUAGE NoImplicitPrelude #-} {- | Copyright : (c) Henning Thielemann 2008 License : GPL Maintainer : synthesizer@henning-thielemann.de Stability : provisional Portability : requires multi-parameter type classes -} module Synthesizer.Generic.Filter.NonRecursive where import qualified Synthesizer.Generic.Signal as SigG import qualified Synthesizer.Generic.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 PreludeBase import NumericPrelude {- * Envelope application -} {-# INLINE negate #-} negate :: (Additive.C a, SigG.Transform sig a) => sig a -> sig a negate = SigG.map Additive.negate {-# INLINE amplify #-} amplify :: (Ring.C a, SigG.Transform sig a) => a -> sig a -> sig a amplify v = SigG.map (v*) {-# INLINE amplifyVector #-} amplifyVector :: (Module.C a v, SigG.Transform sig v) => a -> sig v -> sig v amplifyVector v = SigG.map (v*>) {-# INLINE envelope #-} envelope :: (Ring.C a, SigG.Transform sig a) => sig a {-^ the envelope -} -> sig a {-^ the signal to be enveloped -} -> sig a envelope = SigG.zipWith (*) {-# INLINE envelopeVector #-} envelopeVector :: (Module.C a v, SigG.Read sig a, SigG.Transform sig v) => sig a {-^ the envelope -} -> sig v {-^ the signal to be enveloped -} -> sig v envelopeVector = SigG.zipWith (*>) {-# INLINE fadeInOut #-} fadeInOut :: (Field.C a, SigG.Write sig a) => Int -> Int -> Int -> sig a -> sig a fadeInOut tIn tHold tOut xs = let slopeIn = recip (fromIntegral tIn) slopeOut = Additive.negate (recip (fromIntegral tOut)) {- Since we use the size only for the internal envelope no laziness effect can be observed outside the function. We could also create the envelope as State.Signal. But I assume that concatenating chunks of an envelope is more efficient than concatenating generator loops. However, our intermediate envelope is still observable, because we have to use SigG.Write class. -} leadIn = SigG.take tIn $ Ctrl.linear SigG.defaultLazySize slopeIn 0 leadOut = SigG.take tOut $ Ctrl.linear SigG.defaultLazySize slopeOut 1 (partIn, partHoldOut) = SigG.splitAt tIn xs (partHold, partOut) = SigG.splitAt tHold partHoldOut in envelope leadIn partIn `SigG.append` partHold `SigG.append` envelope leadOut partOut {- * Smoothing -} {-# INLINE delay #-} delay :: (Additive.C y, SigG.Write sig y) => Int -> sig y -> sig y delay = delayPad zero {-# INLINE delayPad #-} delayPad :: (SigG.Write sig y) => y -> Int -> sig y -> sig y delayPad z n = if n<0 then SigG.drop (Additive.negate n) else SigG.append (SigG.replicate SigG.defaultLazySize n z) {-# INLINE delayPos #-} delayPos :: (Additive.C y, SigG.Write sig y) => Int -> sig y -> sig y delayPos n = SigG.append (SigG.replicate SigG.defaultLazySize n zero) {-# INLINE delayNeg #-} delayNeg :: (SigG.Transform sig y) => Int -> sig y -> sig y delayNeg = SigG.drop {-# INLINE delayLazySize #-} delayLazySize :: (Additive.C y, SigG.Write sig y) => SigG.LazySize -> Int -> sig y -> sig y delayLazySize size = delayPadLazySize size zero {- | The pad value @y@ must be defined, otherwise the chunk size of the padding can be observed. -} {-# INLINE delayPadLazySize #-} delayPadLazySize :: (SigG.Write sig y) => SigG.LazySize -> y -> Int -> sig y -> sig y delayPadLazySize size z n = if n<0 then SigG.drop (Additive.negate n) else SigG.append (SigG.replicate size n z) {-# INLINE delayPosLazySize #-} delayPosLazySize :: (Additive.C y, SigG.Write sig y) => SigG.LazySize -> Int -> sig y -> sig y delayPosLazySize size n = SigG.append (SigG.replicate size n zero) {-| Unmodulated non-recursive filter -} {-# INLINE generic #-} generic :: (Module.C a v, SigG.Transform sig a, SigG.Write sig v) => sig a -> sig v -> sig v generic m x = let mr = SigG.reverse m xp = delayPos (pred (SigG.length m)) x in SigG.mapTails (SigG.linearComb mr) 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 v -> sig 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, SigG.Transform sig v) => a -> a -> sig v -> sig v binomial ratio freq = let width = ceiling (2 * ratioFreqToVariance ratio freq ^ 2) in SigG.drop width . nest (2*width) (amplifyVector (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 (Additive.negate (2 * log ratio)) / (2*pi*freq) -- inverse of the fourier transformed gaussian {-# INLINE binomial1 #-} binomial1 :: (Additive.C v, SigG.Transform sig v) => sig v -> sig v binomial1 = SigG.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, SigG.Transform sig v) => Int -> sig v -> sig v sums n = SigG.mapTails (SigG.sum . SigG.take n) {- sumsDownsample2 :: (Additive.C v) => sig v -> sig v sumsDownsample2 (x0:x1:xs) = (x0+x1) : sumsDownsample2 xs sumsDownsample2 xs = xs downsample2 :: sig a -> sig 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 v -> sig v -> sig 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 v -> sig v -> sig v sumsUpsampleEven n {- 2*k -} xs ss = sumsUpsampleOdd (n+1) xs (zipWith (+) ss (downsample2 (drop n xs))) sumsPyramid :: (Additive.C v) => Int -> sig v -> sig 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 -} {- * Filter operators from calculus -} {- | Forward difference quotient. Shortens the signal by one. Inverts 'Synthesizer.Generic.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, SigG.Transform sig v) => sig v -> sig v differentiate x = SigG.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 -} {- This implementation is a bit cumbersome, but it fits both StorableVector and State.Signal (since it avoids recomputation). -} {-# INLINE differentiateCenter #-} differentiateCenter :: (Field.C v, SigG.Transform sig v) => sig v -> sig v differentiateCenter = SigG.drop 2 . SigG.crochetL (\x0 (x1,x2) -> Just ((x2-x0)/2, (x0,x1))) (zero,zero) {- | Second derivative. It is @differentiate2 == differentiate . differentiate@ but 'differentiate2' should be faster. -} {-# INLINE differentiate2 #-} differentiate2 :: (Additive.C v, SigG.Transform sig v) => sig v -> sig v differentiate2 = differentiate . differentiate