com.imsl.math

Class ComplexFFT

java.lang.Object

com.imsl.math.ComplexFFT

All Implemented Interfaces:

Serializable, Cloneable

public class ComplexFFT
extends Object
implements Serializable, Cloneable

Complex FFT.

Class ComplexFFT computes the discrete complex Fourier transform of a complex vector of size N. The method used is a variant of the Cooley-Tukey algorithm, which is most efficient when N is a product of small prime factors. If N satisfies this condition, then the computational effort is proportional to N log N. This considerable savings has historically led people to refer to this algorithm as the "fast Fourier transform" or FFT.

Specifically, given an N-vector \(x\), method forward returns

$$c_m = \sum\limits_{n = 0}^{N - 1} {x_n e^{ - 2\pi inm/N}}$$

Furthermore, a vector of Euclidean norm S is mapped into a vector of norm

$$\sqrt {N}S$$

Finally, note that we can invert the Fourier transform as follows:

$$x_n = \frac{1}{N}\sum_{j=0}^{N-1} c_m e^{2\pi inj/N}$$

This formula reveals the fact that, after properly normalizing the Fourier coefficients, one has the coefficients for a trigonometric interpolating polynomial to the data. An unnormalized inverse is implemented in backward. ComplexFFT is based on the complex FFT in FFTPACK. The package, FFTPACK was developed by Paul Swarztrauber at the National Center for Atmospheric Research.

Specifically, given an N-vector c, backward returns

$$s_m = \sum\limits_{n = 0}^N {c_n e^{2\pi inm/N}}$$

Furthermore, a vector of Euclidean norm S is mapped into a vector of norm

$$\sqrt{N}S$$

Finally, note that we can invert the inverse Fourier transform as follows:

$$c_n = \frac{1}{N}\sum\limits_{m = 0}^{N - 1} {s_m e^{ - 2\pi inm/N}}$$

This formula reveals the fact that, after properly normalizing the Fourier coefficients, one has the coefficients for a trigonometric interpolating polynomial to the data. backward is based on the complex inverse FFT in FFTPACK. The package, FFTPACK was developed by Paul Swarztrauber at the National Center for Atmospheric Research.

See Also:

Example, Serialized Form

Constructor Summary

Constructors

Constructor and Description
ComplexFFT(int n) Constructs a complex FFT object.

Method Summary

Modifier and Type	Method and Description
Complex[]	backward(Complex[] coef) Compute the complex periodic sequence from its Fourier coefficients.
Complex[]	forward(Complex[] seq) Compute the Fourier coefficients of a complex periodic sequence.

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Constructor Detail

ComplexFFT

public ComplexFFT(int n)

Constructs a complex FFT object.

Parameters:

n - is the array size that this object can handle.

Method Detail

forward

public Complex[] forward(Complex[] seq)

Compute the Fourier coefficients of a complex periodic sequence.

Parameters:

seq - is the Complex array containing the sequence to be transformed.

Returns:

a Complex array containing the transformed sequence.

backward

public Complex[] backward(Complex[] coef)

Compute the complex periodic sequence from its Fourier coefficients.

Parameters:

coef - Complex array of Fourier coefficients

Returns:

Complex array containing the periodic sequence