The Fast Fourier Transform is used in linear systems analysis, antenna studies, optics, random process modeling, probability theory, quantum physics, and boundary-value problems (Brigham, 2-3), and has been very successfully applied to restoration of astronomical data (Brault and White). Origin contains a very powerful FFT tool that makes use of an FFTW library to provide for maximum speed.
In Origin...
Fast Fourier Transforms can be performed on Origin datasets by using the FFT Tool. The result of the FFT contains the frequency data and the complex transformed result. The power density estimation can be made by three different methods: MSA, SSA and TISA. Furthermore, both two-sided and one-sided powers can be computed.
Notable among the controls available in the FFT tool is the large selection of windowing options including:
- Triangular
- Rectangle
- Bartlett
- Welch
- Hanning
- Hamming
- Blackman
Origin creates two report sheets upon completion of the FFT: a normal worksheet with the computed values, as well as a worksheet with plots expressing the result. (shown below) These outputs are as highly customizeable, and can include any or all of the following:
- Real
- Imaginary
- Amplitude/Phase
- Phase
- Power/Pahase
- Real/Imaginary
- Magnitude
- Amplitude
- Power
- dB
- Normalized dB
- RMS Amplitude
- Square Amplitude
- Square Magnitude
Other analyses in Origin which use the FFT are:
Mathematically speaking...
The mathematician Fourier recognized that a periodic function could be described as an infinite sum of periodic functions. In particular, he described the formulas for transforming such periodic functions into sums of harmonics of Sine or Cosine functions.
Real world use of these transforms considers discrete data points rather than continuous functions. The data is sampled at regular periods (the sampling rate or interval) over the interval at which the data repeats (the sampling period). The equations or algorithms for these calculations are called Discrete Fourier Transforms (DFT).
The number of computations it takes to calculate the DFT increases dramatically as more data points are considered. For the past 50 years or so, mathematicians have exploited redundancies and symmetries in the DFT to reduce the number of computations needed for N points from 2N^{2} to 2N ln N. The result of this reduction is a significant reduction in computation time. It is these computations that are collectively known as Fast Fourier Transforms (FFT). The fastest of these FFTs are based on equations when the number of data points happens to be an integral power of 2.
References
Brault, J. W. and White, O. R., 1971, The analysis and restoration of astronomical data via the fast Fourier transform, Astron. & Astrophys., 13, pp. 169-189.
Brigham, E. Oren, 1988, The Fast Fourier Transform and Its Applications, Englewood Cliffs, NJ: Prentice-Hall, Inc., 448 pp.