FFT Class |
Namespace: Imsl.Math
The FFT type exposes the following members.
Name | Description | |
---|---|---|
Backward |
Compute the real 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 real 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.) |
Class FFT computes the discrete Fourier transform of a real 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.
The Forward method computes the forward transform. If n is even, then the forward transform is
If n is odd, is defined as above for m from 1 to (n - 1)/2.
Let f be a real valued function of time. Suppose we sample f at n equally spaced time intervals of length seconds starting at time . That is, we have
We will assume that n is odd for the remainder of this discussion. The class FFT treats this sequence as if it were periodic of period n. In particular, it assumes that . Hence, the period of the function is assumed to be . We can invert the above transform for p as follows:
This formula is very revealing. It can be interpreted in the following manner. The coefficients q produced by FFT determine an interpolating trigonometric polynomial to the data. That is, if we define
then we have
Now suppose we want to discover the dominant frequencies, forming the vector P of length (n + 1)/2 as follows:
These numbers correspond to the energy in the spectrum of the signal. In particular, corresponds to the energy level at frequency
Furthermore, note that there are only resolvable frequencies when n observations are taken. This is related to the Nyquist phenomenon, which is induced by discrete sampling of a continuous signal. Similar relations hold for the case when n is even.
If the Backward method is used, then the backward transform is computed. If n is even, then the backward transform is
If n is odd,
The backward Fourier transform is the unnormalized inverse of the forward Fourier transform.
FFT is based on the real FFT in FFTPACK, which was developed by Paul Swarztrauber at the National Center for Atmospheric Research.