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Re: Re: Re: FFT
David Withoff wrote: > Regarding the comments about theorems and quantitative work and such, > all of these transforms are fairly standard, and so will satisfy all > relevant theorems, and other than the care that one should always > exercise in quantitative work, no special effort is needed when using > these transforms. > > Dave Withoff > Wolfram Research I have tried to explain these theorems to mathematicians and computer science types, but to no avail. To a mathematician, all mathematically valid definitions of fourier transform are equally valid. To physicists, engineers and mother nature, this is not quite the case. For example, the central ordinate theorem states that the integral of a function is equal to the value of its transform at the origin. For standard fft's, it is equal to 1/(N-1) times that value. This is trivial to correct, but still must be done to every fft that one plans on using. In any event, the standard fft routines do not satisfy several standard properties of 2-D physical transforms as are implemented by nature in the form of lenses and linear propagation systems of E-M radiation. The results of the standard fft are easily modifiable to satisfy all properties of physical fourier transforms by simple linear transformations and I think it is fair to point out to a beginner that the results of an fft are not physically, quantitatively valid without some slight modification or at least with strict attention paid to scale and units of the transformed axes. Note that I am not stating or implying that something is "wrong" with fft's or mathematicas implementation of them.