Safe Haskell	None
Language	Haskell98

Numeric.FFT.Vector.Unitary

Contents

Description

This module provides normalized versions of the transforms in fftw.

All of the transforms are normalized so that

Each transform is unitary, i.e., preserves the inner product and the sum-of-squares norm of its input.
Each backwards transform is the inverse of the corresponding forwards transform.

(Both conditions only hold approximately, due to floating point precision.)

Synopsis

Creating and executing `Plan`s

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

Create and run a Plan for the given transform.

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 Source #

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

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

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

An inverse discrete Fourier transform. The output and input sizes are the same (n).

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

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

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).

Some normalized real-even (DCT). The input and output sizes are the same (n).

A type-2 discrete cosine transform. Its inverse is dct3.

y_k = w(k) sum_(j=0)^(n-1) x_j cos(pi(j+1/2)k/n); where w(0)=1/sqrt n, and w(k)=sqrt(2/n) for k>0.

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

y_k = (-1)^k w(n-1) x_(n-1) + 2 sum_(j=0)^(n-2) w(j) x_j sin(pi(j+1)(k+1/2)/n); where w(0)=1/sqrt(n), and w(k)=1/sqrt(2n) for k>0.

A type-4 discrete cosine transform. It is its own inverse.

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