{-# LANGUAGE NoImplicitPrelude #-} {- | Copyright : (c) Henning Thielemann 2006-2009 License : GPL Maintainer : synthesizer@henning-thielemann.de Stability : provisional Portability : requires multi-parameter type classes -} module Synthesizer.Plain.Filter.NonRecursive where import qualified Synthesizer.Plain.Control as Ctrl import qualified Synthesizer.Plain.Signal as Sig 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 (mapPair, swap, ) import Data.List.HT (sliceVertical, ) import Data.List (tails, ) import qualified Data.List.Match as Match -- import Control.Monad.Trans.State (StateT) -- import Control.Monad.Trans.Writer (WriterT) import NumericPrelude.Base import NumericPrelude.Numeric {- * Envelope application -} amplify :: (Ring.C a) => a -> Sig.T a -> Sig.T a amplify v = map (v*) amplifyVector :: (Module.C a v) => a -> Sig.T v -> Sig.T v amplifyVector = (*>) envelope :: (Ring.C a) => Sig.T a {-^ the envelope -} -> Sig.T a {-^ the signal to be enveloped -} -> Sig.T a envelope = zipWith (*) envelopeVector :: (Module.C a v) => Sig.T a {-^ the envelope -} -> Sig.T v {-^ the signal to be enveloped -} -> Sig.T v envelopeVector = zipWith (*>) fadeInOut :: (Field.C a) => Int -> Int -> Int -> Sig.T a -> Sig.T a fadeInOut tIn tHold tOut xs = let leadIn = take tIn $ Ctrl.linearMultiscale ( recip (fromIntegral tIn)) 0 leadOut = take tOut $ Ctrl.linearMultiscale (- recip (fromIntegral tOut)) 1 (partIn, partHoldOut) = splitAt tIn xs (partHold, partOut) = splitAt tHold partHoldOut in envelope leadIn partIn ++ partHold ++ envelope leadOut partOut fadeInOutAlt :: (Field.C a) => Int -> Int -> Int -> Sig.T a -> Sig.T a fadeInOutAlt tIn tHold tOut = zipWith id ((map (\x y -> y * fromIntegral x / fromIntegral tIn) [0..tIn-1]) ++ (replicate tHold id) ++ (map (\x y -> y * fromIntegral x / fromIntegral tOut) [tOut-1,tOut-2..0])) {- * Shift -} delay :: Additive.C y => Int -> Sig.T y -> Sig.T y delay = delayPad zero delayPad :: y -> Int -> Sig.T y -> Sig.T y delayPad z n = if n<0 then drop (negate n) else (replicate n z ++) {- * Smoothing -} {-| Unmodulated non-recursive filter -} generic :: Module.C a v => Sig.T a -> Sig.T v -> Sig.T v generic m x = let mr = reverse m xp = delay (pred (length m)) x in map (linearComb mr) (init (tails xp)) {-| Unmodulated non-recursive filter Output has same length as the input. It is elegant but leaks memory. -} genericAlt :: Module.C a v => Sig.T a -> Sig.T v -> Sig.T v genericAlt m x = map (linearComb m) (tail (scanl (flip (:)) [] x)) propGeneric :: (Module.C a v, Eq v) => Sig.T a -> Sig.T v -> Bool propGeneric m x = -- generic m x == genericAlt m x and $ zipWith (==) (generic m x) (genericAlt m x) {- | @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. -} 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 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@. -} ratioFreqToVariance :: (Trans.C a) => a -> a -> a ratioFreqToVariance ratio freq = sqrt (-2 * log ratio) / (2*pi*freq) -- inverse of the fourier transformed gaussian binomial1 :: (Additive.C v) => Sig.T v -> Sig.T v binomial1 xt@(x:xs) = x : (xs + xt) binomial1 [] = [] {- | Moving (uniformly weighted) average in the most trivial form. This is very slow and needs about @n * length x@ operations. -} sums :: (Additive.C v) => Int -> Sig.T v -> Sig.T v sums n = map (sum . take n) . init . tails 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. Example 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 = let aux 1 ys = ys aux 2 ys = ys + tail ys -- binomial1 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 {- | Compute the sum of the values from index l to (r-1). (I.e. somehow a right open interval.) This can be used for implementation of a moving average filter. However, its counterpart 'sumRangeFromPyramid' is much faster for large windows. -} sumRange :: (Additive.C v) => Sig.T v -> (Int,Int) -> v sumRange = sumRangePrepare $ \ (l,r) -> sum . take (r-l) . drop l pyramid :: (Additive.C v) => Sig.T v -> [Sig.T v] pyramid = iterate sumsDownsample2 unitSizesFromPyramid :: [signal] -> [Int] unitSizesFromPyramid pyr = reverse $ Match.take pyr $ iterate (2*) 1 sumRangePrepare :: (Additive.C v) => ((Int,Int) -> source -> v) -> (source -> (Int,Int) -> v) sumRangePrepare f pyr (l,r) = case compare l r of LT -> f (l,r) pyr GT -> negate $ f (r,l) pyr EQ -> zero {- | This function should be much faster than 'sumRange' but slower than the recursively implemented @movingAverage@. However in contrast to @movingAverage@ it should not suffer from cancelation. -} sumRangeFromPyramid :: (Additive.C v) => [Sig.T v] -> (Int,Int) -> v sumRangeFromPyramid = sumRangePrepare $ \lr0 pyr0 -> sum $ getRangeFromPyramid pyr0 lr0 symmetricRangePrepare :: ((Int,Int) -> source -> v) -> (source -> (Int,Int) -> v) symmetricRangePrepare f pyr lr = f (if uncurry (<) lr then lr else swap lr) pyr minRange :: (Ord v) => Sig.T v -> (Int,Int) -> v minRange = symmetricRangePrepare $ \ (l,r) -> minimum . take (r-l) . drop l getRangeFromPyramid :: [Sig.T v] -> (Int,Int) -> [v] getRangeFromPyramid pyr0 lr0 = case pyr0 of [] -> error "empty pyramid" (ps0:pss) -> foldr (\psNext k ((l,r), ps) -> let (lh,ll) = - divMod (-l) 2 (rh,rl) = divMod r 2 ls = if ll==0 then [] else [ps!!l] rs = if rl==0 then [] else [ps!!(r-1)] in case r-l of 0 -> [] 1 -> [ps!!l] _ -> ls ++ rs ++ k ((lh,rh), psNext)) (\((l,r), ps) -> take (r-l) $ drop l ps) pss (lr0, ps0) {- mapAccumL cannot work since the pyramid might be infinitely high sum $ takeWhileJust $ snd $ mapAccumL () lr0 $ zip pyr0 $ tail (Match.replicate pyr0 False ++ [True]) -} sumRangeFromPyramidRec :: (Additive.C v) => [Sig.T v] -> (Int,Int) -> v sumRangeFromPyramidRec = let recourse s (l,r) pyr = case pyr of (ps:[]) -> s + (sum $ take (r-l) $ drop l ps) (ps:pss) -> let (lh,ll) = - divMod (-l) 2 (rh,rl) = divMod r 2 ls = if ll==0 then zero else ps!!l rs = if rl==0 then zero else ps!!(r-1) in case r-l of 0 -> s 1 -> s+ps!!l _ -> recourse (s + ls + rs) (lh,rh) pss [] -> error "empty pyramid" in sumRangePrepare (recourse zero) sumRangeFromPyramidFoldr :: (Additive.C v) => [Sig.T v] -> (Int,Int) -> v sumRangeFromPyramidFoldr = sumRangePrepare $ \lr0 pyr0 -> case pyr0 of [] -> error "empty pyramid" (ps0:pss) -> foldr (\psNext k (l,r) ps s -> case r-l of 0 -> s 1 -> s + ps!!l _ -> let (lh,ll) = - divMod (-l) 2 (rh,rl) = divMod r 2 {-# INLINE inc #-} inc b x = if b==0 then id else (x+) in k (lh,rh) psNext $ inc ll (ps!!l) $ inc rl (ps!!(r-1)) $ s) (\(l,r) ps s -> s + (sum $ take (r-l) $ drop l ps)) pss lr0 ps0 zero sumsPosModulated :: (Additive.C v) => Sig.T (Int,Int) -> Sig.T v -> Sig.T v sumsPosModulated ctrl xs = zipWith sumRange (tails xs) ctrl {- | Moving average, where window bounds must be always non-negative. The laziness granularity is @2^height@. -} sumsPosModulatedPyramid :: (Additive.C v) => Int -> Sig.T (Int,Int) -> Sig.T v -> Sig.T v sumsPosModulatedPyramid height ctrl xs = let blockSize = 2 ^ fromIntegral height pyr0 = take (1+height) $ pyramid xs sizes = unitSizesFromPyramid pyr0 pyrStarts = iterate (zipWith drop sizes) pyr0 ctrlBlocks = map (zipWith (\d -> mapPair ((d+), (d+))) $ iterate (1+) 0) $ sliceVertical blockSize ctrl in concat $ zipWith (\pyr -> map (sumRangeFromPyramid pyr)) pyrStarts ctrlBlocks {- | The first argument is the amplification. The main reason to introduce it, was to have only a Module constraint instead of Field. This way we can also filter stereo signals. A control value @n@ corresponds to filter window size @2*n+1@. -} movingAverageModulatedPyramid :: (Field.C a, Module.C a v) => a -> Int -> Int -> Sig.T Int -> Sig.T v -> Sig.T v movingAverageModulatedPyramid amp height maxC ctrl xs = zipWith (\c x -> (amp / fromIntegral (2*c+1)) *> x) ctrl $ sumsPosModulatedPyramid height (map (\c -> (maxC - c, maxC + c + 1)) ctrl) (delay maxC xs) {- * Filter operators from calculus -} {- | Forward difference quotient. Shortens the signal by one. Inverts 'Synthesizer.Plain.Filter.Recursive.Integration.run' in the sense that @differentiate (zero : integrate x) == x@. The signal is shifted by a half time unit. -} differentiate :: Additive.C v => Sig.T v -> Sig.T v differentiate x = zipWith subtract x (tail 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 -} differentiateCenter :: Field.C v => Sig.T v -> Sig.T v differentiateCenter x = map ((1/2)*) $ zipWith subtract x (tail (tail x)) {- | Second derivative. It is @differentiate2 == differentiate . differentiate@ but 'differentiate2' should be faster. -} differentiate2 :: Additive.C v => Sig.T v -> Sig.T v differentiate2 xs0 = let xs1 = tail xs0 xs2 = tail xs1 in zipWith3 (\x0 x1 x2 -> x0+x2-(x1+x1)) xs0 xs1 xs2