The file import capabilities of Mathematica are made just to handle such cases. Probably you need a bit of experimentation and for this it would probably be helpful to have a smaller testversion of your data ( 1MB?) available. If you should find obstacles, let me know. I'm confident to be able to help.

Dear Ulrich Mutze,

Thank you very much for your answer. But I think the size of the importing file is not at all a problem for me. I just want to know is how make FFT in MATHEMATICA using datas from external source file.

Sorry,

I assumed that you have access to Mathematica and its help menu. If this would be the case your question would answer itself.

The function call for getting the data in is Import["filename"],

the call for making the FFT of a list of complex numbers is Fourier[theList].

There are also discrete cosine and sine transformations under names FourierDCT and FourierDST

The procedure: open a notebook file (extension .nb) type Fourier, place the curser on this word and press the F1 key. You will get a window that will show you, or will let you access, everything you have to know.

The experimentation I mentioned will probably be needed to let the result of function Import fit the input syntax of the function Fourier. Also, you will probably need to create a graphical representation of your results to ensure that all syntax was right. The information obtainable with the F1 key always contains examples for simple and natural types of usage.

Dear Ulrich,

Thank you very much for your answer. First I'll take an experiment. I find get struck I'll back to you.

As noted by Ulrich, Mathematica has a suite of built-in functions for doing Fourier Analysis. Here is a PDF of a Mathematica notebook that I used in an undergraduate Chemical Engineering class (of highly motivated seniors) on numerical methods. One of the topics we discussed was Fourier Transforms. This should help you get started.

You mentioned the 250MB file with data. If we assume that one data stored as ASCII with delimiters will take 20 bytes, resulting vector will contain 12.5 million of data. It is assumed that all data are equally distant (uniform sampling).

Usual approach is that you do the Fourier transform to a block of data, say N samples (its best that N=2^n, ex. N=4096). To reduce edge effect when you cut the part from long string of data, "windowing" is used by multiplying selected data with selected windowing function (Hann, Hamming, ...). Then Fourier transform is done. Next N samples block of data starts (usually) from some intermediate position of previous block (overlapping between blocks), and same way till end. In this way it can be seen how frequency content of the signal changes with time, sometimes plotted as 3-D plot.

Hi Sudharsan, I often do Fourier transforms to get power spectra. For this I take the auto- or cross-correlation function of the time series and make a normal Fourier transformation. I never have met the need to apply FFT, which was possibly a necessity in past times when computers were slow etc., but I am almost sure that the common Fourier trafo will do it, as noted implicitely by Ulrich and Brian.

Helmut, Mathematicas Fourier is the discrete Fourier transform (it takes lists of complex numbers as argument). The implementation is of course one of the many variants of FFT. Mathematica has also a symbolic (non-discrete) verion named FourierTransform. Your idea that modern fast computers do Fourier transforms that don't use the FFT principle is a misconception.

Some notes: 1) the absolute value of the DFT (or FFT) is the periodogram. It is but one of an infinity of estimates of the power spectrum. 2) The DFT and FFT produce identical complex results but with different work scaling (n^2 vs n ln n). 3) FourierTransform is associated with integrable functions that do not have finite period. For functions with finite period you use Fourier Series (functions) or DFT (discrete data). Continuous functions have well defined power spectra. But for discretely sampled data the periodogram (what most people mistakenly call the power spectrum) is at one end of the variance bias tradeoff curve. If you want an optimal tradeoff you need to use multi-taper spectral analysis as in the SAPA book by Percival and Walden.

Also, be aware that Mathematica returns an FFT of the same length as the original time series even though half the results are redundant. Nyquist is at N/2 for an even number of data points in the time series. So you might as well just ListPlot from 1 (DC) to Nyquist. It would make more sense to have DC at the 0 sample with negative frequencies on the left and positive on the right, but Mathematica reserves the 0th element.