ComplexFFT (JMSL Numerical Library)

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JMSL^TM Numerical Library 5.0.1

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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
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 , method forward returns

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

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

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

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

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

$s_m = sumlimits_{n = 0}^N {c_n e^{2pi inm/N}}$

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

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

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

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
`ComplexFFT(int n)` Constructs a complex FFT object.

Method Summary
`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

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

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.