Numeric.FFT.Vector.Invertible

This module provides normalized versions of the transforms in fftw.

The forwards transforms in this module are identical to those in Numeric.FFT.Vector.Unnormalized. The backwards transforms are normalized to be their inverse operations (approximately, due to floating point precision).

For more information on the underlying 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 which is the inverse of dft. The output and input sizes are the same (n).

y_k = (1/n) 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 normalized backward discrete Fourier transform which is the left inverse of dftR2C. (Specifically, run dftC2R . run dftR2C == id.)

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))

idct1 :: Transform Double Double

A type-1 discrete cosine transform which is the inverse of dct1.

y_k = (1/(2(n-1)) [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)

idct2 :: Transform Double Double

A type-3 discrete cosine transform which is the inverse of dct2.

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

idct3 :: Transform Double Double

A type-2 discrete cosine transform which is the inverse of dct3.

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

idct4 :: Transform Double Double

A type-4 discrete cosine transform which is the inverse of dct4.

y_k = (1/n) 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))

idst1 :: Transform Double Double

A type-1 discrete sine transform which is the inverse of dst1.

y_k = (1/(n+1)) 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)

idst2 :: Transform Double Double

A type-3 discrete sine transform which is the inverse of dst2.

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

idst3 :: Transform Double Double

A type-2 discrete sine transform which is the inverse of dst3.

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

idst4 :: Transform Double Double

A type-4 discrete sine transform which is the inverse of dst4.

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