[Ifeffit] Re: Independent data poins in Artemis

[Scott Calvin]

Thanks, Jeremy. Good point.

As a practical matter, what you actually get when a Fourier transform 
algorithm is applied to a data set consisting of a single point will 
depend on the algorithm. If the algorithm pads with 0's, it gets a 
transform that starts to look more and more "white," that is, like a 
constant function in transform space. If it doesn't pad with 0's, but 
instead takes the usual Fourier integral and let the limits of 
integration tend toward 0, so that it only includes the non-zero 
point, the transform approaches a delta function located at 0 in 
transform space. Those results are as different as they could be, and 
they depend entirely on how the truncation is handled. (A typical 
ifeffit calculation, as I understand it, lies between these two 
extremes. It integrates over a finite interval, but pads with 0's to 
get to a convenient number of points.) I think that nicely 
demonstrates that you can't extract information from the "Fourier 
transform" of a single point.

--Scott Calvin
Sarah Lawrence College

At 03:20 PM 1/4/2007, you wrote:
>Rigorously speaking, a single point is not a delta function.  To have a
>delta function, only one point has a non-zero value, all other points
>are zero.  If you have only one data point and you assume all other
>points are zero, you are adding information not contained in your data.
>
>Jeremy Kropf