Numeric.FFT.Vector.Unnormalized

Raw, unnormalized versions of the transforms in fftw.

Note that the forwards and backwards transforms of this module are not actually inverses. For example, run idft (run dft v) /= v in general.

For more information on the individual transforms, see http://www.fftw.org/fftw3_doc/What-FFTW-Really-Computes.html.

Creating and executing Plans

run :: (Vector v a, Vector v b, Storable a, Storable b) => Transform a b -> v a -> v b

Create and run a Plan for the given transform.

plan :: (Storable a, Storable b) => Transform a b -> Int -> Plan a b

Create a Plan of a specific size. This function is equivalent to planOfType Estimate.

execute :: (Vector v a, Vector v b, Storable a, Storable b) => Plan a b -> v a -> v b

Run a plan on the given Vector.

If planInputSize p /= length v, then calling execute p v will throw an exception.

Complex-to-complex transforms

dft :: Transform (Complex Double) (Complex Double)

A forward discrete Fourier transform. The output and input sizes are the same (n).

y_k = sum_(j=0)^(n-1) x_j e^(-2pi i j k/n)

idft :: Transform (Complex Double) (Complex Double)

A backward discrete Fourier transform. The output and input sizes are the same (n).

y_k = sum_(j=0)^(n-1) x_j e^(2pi i j k/n)

Real-to-complex transforms

dftR2C :: Transform Double (Complex Double)

A forward discrete Fourier transform with real data. If the input size is n, the output size will be n `div` 2 + 1.

dftC2R :: Transform (Complex Double) Double

A backward discrete Fourier transform which produces real data.

This Transform behaves differently than the others:

• Calling plan dftC2R n creates a Plan whose output size is n, and whose input size is n `div` 2 + 1.
• If length v == n, then length (run dftC2R v) == 2*(n-1).

Real-to-real transforms

The real-even (DCT) and real-odd (DST) transforms. The input and output sizes are the same (n).

dct1 :: Transform Double Double

A type-1 discrete cosine transform.

y_k = x_0 + (-1)^k x_(n-1) + 2 sum_(j=1)^(n-2) x_j cos(pi j k/(n-1))

dct2 :: Transform Double Double

A type-2 discrete cosine transform.

y_k = 2 sum_(j=0)^(n-1) x_j cos(pi(j+1/2)k/n)

dct3 :: Transform Double Double

A type-3 discrete cosine transform.

y_k = x_0 + 2 sum_(j=1)^(n-1) x_j cos(pi j(k+1/2)/n)

dct4 :: Transform Double Double

A type-4 discrete cosine transform.

y_k = 2 sum_(j=0)^(n-1) x_j cos(pi(j+1/2)(k+1/2)/n)

Discrete sine transforms

dst1 :: Transform Double Double

A type-1 discrete sine transform.

y_k = 2 sum_(j=0)^(n-1) x_j sin(pi(j+1)(k+1)/(n+1))

dst2 :: Transform Double Double

A type-2 discrete sine transform.

y_k = 2 sum_(j=0)^(n-1) x_j sin(pi(j+1/2)(k+1)/n)

dst3 :: Transform Double Double

A type-3 discrete sine transform.

y_k = (-1)^k x_(n-1) + 2 sum_(j=0)^(n-2) x_j sin(pi(j+1)(k+1/2)/n)

dst4 :: Transform Double Double

A type-4 discrete sine transform.

y_k = sum_(j=0)^(n-1) x_j sin(pi(j+1/2)(k+1/2)/n)