ComplexFFT Class

ComplexFFT Class

Complex FFT.

Inheritance Hierarchy

SystemObject
Imsl.MathComplexFFT

Namespace: Imsl.Math
Assembly: ImslCS (in ImslCS.dll) Version: 6.5.2.0

Syntax

C++

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[SerializableAttribute]
public class ComplexFFT

<SerializableAttribute>
Public Class ComplexFFT

[SerializableAttribute]
public ref class ComplexFFT

[<SerializableAttribute>]
type ComplexFFT =  class end

The ComplexFFT type exposes the following members.

Constructors

	Name	Description
	ComplexFFT	Constructs a complex FFT object.

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Methods

	Name	Description
	Backward	Compute the complex periodic sequence from its Fourier coefficients.
	Equals	Determines whether the specified object is equal to the current object. (Inherited from Object.)
	Finalize	Allows an object to try to free resources and perform other cleanup operations before it is reclaimed by garbage collection. (Inherited from Object.)
	Forward	Compute the Fourier coefficients of a complex periodic sequence.
	GetHashCode	Serves as a hash function for a particular type. (Inherited from Object.)
	GetType	Gets the Type of the current instance. (Inherited from Object.)
	MemberwiseClone	Creates a shallow copy of the current Object. (Inherited from Object.)
	ToString	Returns a string that represents the current object. (Inherited from Object.)

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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.

Reference

Imsl.Math Namespace

Other Resources

Example