{-# OPTIONS -fglasgow-exts -fno-implicit-prelude #-} module Synthesizer.Plain.Analysis where import qualified Synthesizer.Plain.Signal as Sig import qualified Synthesizer.Plain.Control as Ctrl import qualified Synthesizer.Plain.Filter.Recursive.Integration as Integration -- import qualified Algebra.Module as Module -- import qualified Algebra.Transcendental as Trans import qualified Algebra.Algebraic as Algebraic import qualified Algebra.RealField as RealField import qualified Algebra.Field as Field import qualified Algebra.Real as Real import qualified Algebra.Ring as Ring import qualified Algebra.Additive as Additive import qualified Algebra.NormedSpace.Maximum as NormedMax import qualified Algebra.NormedSpace.Euclidean as NormedEuc import qualified Algebra.NormedSpace.Sum as NormedSum import qualified Data.Array as Array import qualified Data.IntMap as IntMap -- import Algebra.Module((*>)) import Data.Array (accumArray) import Data.List (foldl', ) import qualified Prelude as P import PreludeBase import NumericPrelude {- * Notions of volume -} {- | Volume based on Manhattan norm. -} volumeMaximum :: (Real.C y) => Sig.T y -> y volumeMaximum = foldl max zero . rectify -- maximum . rectify {- | Volume based on Energy norm. -} volumeEuclidean :: (Algebraic.C y) => Sig.T y -> y volumeEuclidean = Algebraic.sqrt . volumeEuclideanSqr volumeEuclideanSqr :: (Field.C y) => Sig.T y -> y volumeEuclideanSqr = average . map sqr {- | Volume based on Sum norm. -} volumeSum :: (Field.C y, Real.C y) => Sig.T y -> y volumeSum = average . rectify {- | Volume based on Manhattan norm. -} volumeVectorMaximum :: (NormedMax.C y yv, Ord y) => Sig.T yv -> y volumeVectorMaximum = NormedMax.norm -- maximum . map NormedMax.norm {- | Volume based on Energy norm. -} volumeVectorEuclidean :: (Algebraic.C y, NormedEuc.C y yv) => Sig.T yv -> y volumeVectorEuclidean = Algebraic.sqrt . volumeVectorEuclideanSqr volumeVectorEuclideanSqr :: (Field.C y, NormedEuc.Sqr y yv) => Sig.T yv -> y volumeVectorEuclideanSqr = average . map NormedEuc.normSqr {- | Volume based on Sum norm. -} volumeVectorSum :: (NormedSum.C y yv, Field.C y) => Sig.T yv -> y volumeVectorSum = average . map NormedSum.norm {- | Compute minimum and maximum value of the stream the efficient way. Input list must be non-empty and finite. -} bounds :: Ord y => Sig.T y -> (y,y) bounds [] = error "Analysis.bounds: List must contain at least one element." bounds (x:xs) = foldl' (\(minX,maxX) y -> (min y minX, max y maxX)) (x,x) xs {- * Miscellaneous -} {- histogram: length x = sum (histogramDiscrete x) units: 1) histogram (amplify k x) = timestretch k (amplify (1/k) (histogram x)) 2) histogram (timestretch k x) = amplify k (histogram x) timestretch: k -> (s -> V) -> (k*s -> V) amplify: k -> (s -> V) -> (s -> k*V) histogram: (a -> b) -> (a^ia*b^ib -> a^ja*b^jb) x: (s -> V) 1) => (s^ia*(k*V)^ib -> s^ja*(k*V)^jb) = (s^ia*V^ib*k -> s^ja*V^jb/k) => ib=1, jb=-1 2) => ((k*s)^ia*V^ib -> (k*s)^ja*V^jb) = (s^ia*V^ib -> s^ja*V^jb*k) => ia=0, ja=1 histogram: (s -> V) -> (V -> s/V) histogram': integral (histogram' x) = integral x histogram' (amplify k x) = timestretch k (histogram' x) histogram' (timestretch k x) = amplify k (histogram' x) -> this does only apply if we slice the area horizontally and sum the slice up at each level, we must also restrict to the positive values, this is not quite the usual histogram -} {- | Input list must be finite. List is scanned twice, but counting may be faster. -} histogramDiscreteArray :: Sig.T Int -> (Int, Sig.T Int) histogramDiscreteArray [] = (error "histogramDiscreteArray: no bounds found", []) histogramDiscreteArray x = let hist = accumArray (+) zero (bounds x) (attachOne x) in (fst (Array.bounds hist), Array.elems hist) {- | Input list must be finite. If the input signal is empty, the offset is @undefined@. List is scanned twice, but counting may be faster. The sum of all histogram values is one less than the length of the signal. -} histogramLinearArray :: RealField.C y => Sig.T y -> (Int, Sig.T y) histogramLinearArray [] = (error "histogramLinearArray: no bounds found", []) histogramLinearArray [x] = (floor x, []) histogramLinearArray x = let (xMin,xMax) = bounds x hist = accumArray (+) zero (floor xMin, floor xMax) (meanValues x) in (fst (Array.bounds hist), Array.elems hist) {- | Input list must be finite. If the input signal is empty, the offset is @undefined@. List is scanned once, counting may be slower. -} histogramDiscreteIntMap :: Sig.T Int -> (Int, Sig.T Int) histogramDiscreteIntMap [] = (error "histogramDiscreteIntMap: no bounds found", []) histogramDiscreteIntMap x = let hist = IntMap.fromListWith (+) (attachOne x) in case IntMap.toAscList hist of [] -> error "histogramDiscreteIntMap: the list was non-empty before processing ..." fAll@((fIndex,fHead):fs) -> (fIndex, fHead : concat (zipWith (\(i0,_) (i1,f1) -> replicate (i1-i0-1) zero ++ [f1]) fAll fs)) histogramLinearIntMap :: RealField.C y => Sig.T y -> (Int, Sig.T y) histogramLinearIntMap [] = (error "histogramLinearIntMap: no bounds found", []) histogramLinearIntMap [x] = (floor x, []) histogramLinearIntMap x = let hist = IntMap.fromListWith (+) (meanValues x) -- we can rely on the fact that the keys are contiguous (startKey:_, elems) = unzip (IntMap.toAscList hist) in (startKey, elems) -- This doesn't work, due to a bug in IntMap of GHC-6.4.1 -- in (head (IntMap.keys hist), IntMap.elems hist) {- The bug in IntMap GHC-6.4.1 is: *Synthesizer.Plain.Analysis> IntMap.keys $ IntMap.fromList $ [(0,0),(-1,-1::Int)] [0,-1] *Synthesizer.Plain.Analysis> IntMap.elems $ IntMap.fromList $ [(0,0),(-1,-1::Int)] [0,-1] *Synthesizer.Plain.Analysis> IntMap.assocs $ IntMap.fromList $ [(0,0),(-1,-1::Int)] [(0,0),(-1,-1)] The bug has gone in IntMap as shipped with GHC-6.6. -} histogramIntMap :: (RealField.C y) => y -> Sig.T y -> (Int, Sig.T Int) histogramIntMap binsPerUnit = histogramDiscreteIntMap . quantize binsPerUnit quantize :: (RealField.C y) => y -> Sig.T y -> Sig.T Int quantize binsPerUnit = map (floor . (binsPerUnit*)) attachOne :: Sig.T i -> Sig.T (i,Int) attachOne = map (\i -> (i,one)) meanValues :: RealField.C y => Sig.T y -> [(Int,y)] meanValues x = concatMap spread (zip x (tail x)) spread :: RealField.C y => (y,y) -> [(Int,y)] spread (l0,r0) = let (l,r) = if l0<=r0 then (l0,r0) else (r0,l0) (li,lf) = splitFraction l (ri,rf) = splitFraction r k = recip (r-l) nodes = (li,k*(1-lf)) : zip [li+1 ..] (replicate (ri-li-1) k) ++ (ri, k*rf) : [] in if li==ri then [(li,one)] else nodes {- | Requires finite length. This is identical to the arithmetic mean. -} directCurrentOffset :: Field.C y => Sig.T y -> y directCurrentOffset = average scalarProduct :: Ring.C y => Sig.T y -> Sig.T y -> y scalarProduct xs ys = sum (zipWith (*) xs ys) {- | 'directCurrentOffset' must be non-zero. -} centroid :: Field.C y => Sig.T y -> y centroid xs = scalarProduct (iterate (one+) zero) xs / sum xs centroidAlt :: Field.C y => Sig.T y -> y centroidAlt xs = sum (scanr (+) zero (tail xs)) / sum xs average :: Field.C y => Sig.T y -> y average x = sum x / fromIntegral (length x) rectify :: Real.C y => Sig.T y -> Sig.T y rectify = map abs {- | Detects zeros (sign changes) in a signal. This can be used as a simple measure of the portion of high frequencies or noise in the signal. It ca be used as voiced\/unvoiced detector in a vocoder. @zeros x !! n@ is @True@ if and only if @(x !! n >= 0) \/= (x !! (n+1) >= 0)@. The result will be one value shorter than the input. -} zeros :: (Ord y, Ring.C y) => Sig.T y -> Sig.T Bool zeros xs = let signs = map (>=zero) xs in zipWith (/=) signs (tail signs) data BinaryLevel = Low | High deriving (Eq, Show, Enum) binaryLevelFromBool :: Bool -> BinaryLevel binaryLevelFromBool False = Low binaryLevelFromBool True = High binaryLevelToNumber :: Ring.C a => BinaryLevel -> a binaryLevelToNumber Low = negate one binaryLevelToNumber High = one {- | Detect thresholds with a hysteresis. -} flipFlopHysteresis :: (Ord y) => (y,y) -> BinaryLevel -> Sig.T y -> Sig.T BinaryLevel flipFlopHysteresis (lower,upper) = scanl (\state x -> binaryLevelFromBool $ case state of High -> not(x<lower) Low -> x>upper) {- | Almost naive implementation of the chirp transform, a generalization of the Fourier transform. More sophisticated algorithms like Rader, Cooley-Tukey, Winograd, Prime-Factor may follow. -} chirpTransform :: Ring.C y => y -> Sig.T y -> Sig.T y chirpTransform z xs = let powers = Ctrl.curveMultiscaleNeutral (*) z one powerPowers = map (\zn -> Ctrl.curveMultiscaleNeutral (*) zn one) powers in map (scalarProduct xs) powerPowers binarySign :: Real.C y => Sig.T y -> Sig.T BinaryLevel binarySign = map (binaryLevelFromBool . (zero <=)) {- | The output type could be different from the input type but then we would need a conversion from output to input for feedback. -} deltaSigmaModulation :: Real.C y => Sig.T y -> Sig.T BinaryLevel deltaSigmaModulation x = let y = binarySign (Integration.runInit zero (x - map binaryLevelToNumber y)) in y