Synthesizer.Generic.Fourier

Complete implementation for Fast Fourier Transform for any signal length. Although defined for all kinds of signal storage, we need fast access to arbitrary indices.

Documentation

class C y => Element y where

Methods

recipInteger :: Read sig y => sig y -> y

addId :: Read sig y => sig y -> y

multId :: Read sig y => sig y -> y

conjugatePrimitiveRootsOfUnity :: Read sig y => sig y -> (y, y)

It must hold:

 uncurry (*) (conjugatePrimitiveRootsOfUnity n) = 1

 mapPair ((^m), (^m)) (conjugatePrimitiveRootsOfUnity (n*m) y)
    == conjugatePrimitiveRootsOfUnity n y@

since we need for caching that the cache is uniquely determined by singal length and transform direction.

Instances

C a => Element (T a)

(PrimitiveRoot a, C a, Eq a) => Element (T a)

conversion between time and frequency domain (spectrum)

transformForward :: (Element y, Transform sig y) => sig y -> sig y

transformBackward :: (Element y, Transform sig y) => sig y -> sig y

Shall we divide the result values by the length of the signal? Our dimensional wrapper around the Fourier transform does not expect this.

cacheForward :: (Element y, Transform sig y) => sig y -> Cache sig y

The expression cacheForward prototype precomputes all data that is needed for forward Fourier transforms for signals of the type and length prototype. You can use this cache in transformWithCache.

cacheBackward :: (Element y, Transform sig y) => sig y -> Cache sig y

See cacheForward.

cacheDuplex :: (Element y, Transform sig y) => sig y -> (Cache sig y, Cache sig y)

It is (cacheForward x, cacheBackward x) = cacheDuplex x but cacheDuplex shared common data of both caches.

transformWithCache :: (Element y, Transform sig y) => Cache sig y -> sig y -> sig y

The size and type of the signal must match the parameters, that the cache was generated for.

convolution based on Fourier transform

convolveCyclic :: (Element y, Transform sig y) => sig y -> sig y -> sig y

Signal must have equal size and must not be empty.

data Window sig y

Filter window stored as spectrum such that it can be applied efficiently to long signals.

Instances

(Show y, Show (sig y)) => Show (Window sig y)

window :: (Element y, Transform sig y) => sig y -> Window sig y

convolveWithWindow :: (Element y, Transform sig y) => Window sig y -> sig y -> sig y

Efficient convolution of a large filter window with a probably infinite signal.