FFT functions.

Namespace: Imsl.Math
Assembly: ImslCS (in ImslCS.dll) Version: 6.5.0.0

Syntax

C#
[SerializableAttribute]
public class FFT
Visual Basic (Declaration)
<SerializableAttribute> _
Public Class FFT
Visual C++
[SerializableAttribute]
public ref class FFT

Remarks

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

q_{2m - 1} = \sum\limits_{k = 0}^{n - 1} {p_k } \cos \frac{{2\pi km}}{n} \,\,\,\, m = 1,\; \ldots ,\;n/2

q_{2m - 2} = - \sum\limits_{k = 0}^{n - 1} {p_k } \sin \frac{{2\pi km}}{n} \,\,\,\, m = 1,\; \ldots ,\;n/2 - 1

q_0 = \sum\limits_{k = 0}^{n - 1} {p_k }

If n is odd, q_m 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 \delta seconds starting at time t_0. That is, we have

p_i : = f\left( {t_0 + i\Delta } \right)\, i = 0,\,1, \ldots ,\,n - 1

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 f\left( {t_0 } \right) = f\left( {t_0 + n\Delta } \right). Hence, the period of the function is assumed to be T = n\Delta. We can invert the above transform for p as follows:

p_m = {1 \over n}\left[ {q_0 + 2\sum\limits_{k = 0}^{\left( {n - 3} \right)/2} {\quad q_{2k + 1} } \cos {{2\pi (k+1)m} \over n} - 2\sum\limits_{k = 0}^{\left( {n - 3} \right)/2} {\quad q_{2k + 2} } \sin {{2\pi (k+1)m} \over n}} \right]

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

g\left( t \right) = {1 \over n}\left[{q_0 + 2\sum\limits_{k = 0}^{\left( {n - 3} \right)/2} {\quad q_{2k + 1} } \cos {{2\pi (k+1)\left( {t - t_0 } \right)} \over {n\Delta }} - 2\sum\limits_{k = 0}^{\left( {n - 3} \right)/2} {\quad q_{2k + 2} } \sin {{2\pi (k+1)\left( {t - t_0 } \right)} \over {n\Delta }}} \right]

= {1 \over n}\left[ {q_0 + 2\sum\limits_{k = 0}^{\left( {n - 3} \right)/2} {\quad q_{2k + 1} } \cos {{2\pi (k+1)\left( {t - t_0 } \right)} \over T} - 2\sum\limits_{k = 0}^{\left( {n - 3} \right)/2} {\quad q_{2k + 2} } \sin {{2\pi (k+1)\left( {t - t_0 } \right)} \over T}} \right]

then we have

f\left( {{\rm{t}}_{\rm{0}} + \left( {i - 1} \right)\Delta } \right) = g\left( {{\rm{t}}_{\rm{0}} + \left( {i - 1} \right)} \right)\Delta

Now suppose we want to discover the dominant frequencies, forming the vector P of length (n + 1)/2 as follows:

P_0: = \left| {q_0 } \right|

P_k: = \sqrt {q_{2k - 2}^2 + q_{2k - 1}^2 } \,\,\,\, k = 1,\;2,\; \ldots ,\;\left( {n - 1} \right)/2

These numbers correspond to the energy in the spectrum of the signal. In particular, P_k corresponds to the energy level at frequency

{k \over T} = {k \over {n\Delta }} \,\,\,\,\,\,\,\, k = 0,\;1,\; \ldots ,\;{{n - 1} \over 2}

Furthermore, note that there are only (n + 1)/2 \approx T/(2\Delta) 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

q_m = p_0 + \left( { - 1} \right)^m p_{n - 1} + 2\sum\limits_{k = 0}^{n/2 - 1} {p_{2k + 1} } \cos \frac{{2\pi (k+1)m}}{n} - 2\sum\limits_{k = 0}^{n/2 - 2} {p_{2k + 2} } \sin \frac{{2\pi (k+1)m}}{n}

If n is odd,

q_m = p_0 + 2\sum\limits_{k = 0}^{\left( {n - 3} \right)/2} {\;p_{2k + 1} } \cos {{2\pi (k+1)m} \over n} - 2\sum\limits_{k = 0}^{\left( {n - 3} \right)/2} {\;p_{2k + 2} } \sin {{2\pi (k+1)m} \over n}

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.

Inheritance Hierarchy

System..::.Object
Imsl.Math..::.FFT

See Also