for your info: ch. 11 mbe: fast fourier transforms
Subject: for your info: ch. 11 mbe: fast fourier transforms
From: dlakelan
Date: Tue, 05 May 2009 11:21:19 -0700
Robert Dodier wrote:
> Well, I dunno. The computed results match the stated definition --
> I've verified that directly. We could pick a different definition, I guess,
> if it could make the results more useful or comprehensible.
Yes perhaps this is the thing to do. I think the point is to help the
users of the fft routine when doing something relatively standard, such
as fft transforming a real function, and trying to manipulate the
individual frequencies. The manual's description of sin/cos terms shows
how to eliminate the redundancy that you get when you start with a real
array and do a complete complex transform. But if you want to do
something to the frequency components and then inverse transform it is
not helpful to work with the sin or cos coefficients as described in the
maxima manual, since there is no inverse sin/cos transform function.
perhaps an alternative would be to provide a "rfft" and "irfft" that
handles purely real data in a "nice" way for the user?
If you start with a real array and fourier transform it with the
existing fft function, the frequencies greater than %pi radians per
sample can be interpreted as frequencies starting at -%pi radians per
sample and moving toward 0 radians per sample (the whole thing is a
circle from either 0 to 2pi or 0 to pi followed by -pi to 0)
The interpretation of the frequencies as [0..pi],[-pi..0] (with the
appropriate handling of the corner cases, as for example there is only 1
coefficient corresponding to 0 frequency and one corresponding to pi)
allows for the user to interpret what is going on in such a way as to
get the right calculation of the derivative and the integral of a set of
real samples. It also makes it more clear how to say filter out all the
high frequencies (the high frequencies are in the "center" of the array,
not all out at the "end", the "end" of the array is low, negative
frequencies)
I confess to being a bit baffled about how to describe what I mean here.
Is this helpful at all?
Dan