Syntax

C#
[SerializableAttribute] public class ComplexFFT

Visual Basic (Declaration)
<SerializableAttribute> _ Public Class ComplexFFT

Visual C++
[SerializableAttribute] public ref class ComplexFFT

Remarks

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

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

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.

Inheritance Hierarchy

System..::.Object
Imsl.Math..::.ComplexFFT

Syntax

Remarks

Inheritance Hierarchy

See Also