The new files are leaner, and I removed some detailed comments/rambling. Pasting here for the record.
From FrequencyBand:
Extends the interval class.
has a lower_bound, upper_bound, central_frequency and method for Fourier
coefficient indices
Some thoughts 20210617:
TLDR:
For simplicity, I'm going with Half open, df/2 intervals when they are
prscribed by FC indexes, and half_open gates_and_fenceposts when they
are not. The gates and fenceposts can be converted to the precribed form by
mapping to emtf_band_setup_form and then mapping to FCIndex form.
A 3dB point correction etc maybe done in a later version.
ON DEFAULT FREQUENCY BAND CONFIGURATIONS
Because these are Interval()s there is a little complication:
If we use closed intervals we can have an issue with the same Fourier
coefficient being in more than one band [a,b],[b,c] if b corresponds to a harmonic.
Honestly this is not a really big deal, but it feels sloppy. The default
behaviour should partition the frequency axis, not break it into sets with
non-zero overlap, even though the overlapping sets are of measure zero
analytically, in digital land this matters.
On the other hand, it is common enough (at low frequency) to have bands
which are only 1 Harmonic wide, and if we dont use closed intervals we
can wind up with intervals like [a,a), which is the empty set.
However, the harmonic frequencies we are going to interact with in digital
processing will be a collections of discrete values, separated by df.
Given the context of df (= 1/(N*dt)) where N is number of
samples in the original time series and dt is the sample interval,
then we can use half-open intervals with width df centered at the
frequencies under consideration.
I.e since the actual picking of Fourier coefficients and indexes will
always occur in the context of a sampling rate and a frequency axis and
we will know df and therefore we can pad the frequency bounds by +/-
df/2.
In that case we can use open, half open, or closed intervals, it really
doesn't matter, so we will choose half open
[f_i-df/2, f_i+df/2) to get the satisfying property of covering the
frequency axis completely but ensure no accidental double coverage.
This is just a default convention. There is no rule against
using closed intervals, nor having overlapping bands.
The df/2 trick also protects us from numeric roundoff errors resulting in
edge frequencies landing in a bin other than that which is intended.
There is one little oddity which accompanies this scheme. Consider the
case where you have a 1-harmonic wide band, say at 10Hz. And df for
arguments sake is 0.05 Hz. The center frequency harmonically will not
evaluate to 10Hz exactly, rather it will evaluate to sqrt((
9.95*10.05))=9.9999874, and not 10. This is a little bit unsatisfying
but I take solace in two things:
1. The user is welcome to use their own convention, [10.0, 10.0], or even
[10.0-epsilon , 10.0+epsilon] if worried about numeric ghosts which
asymptotes to 10.0
2. I'm not actually 100% sure that the geometric center frequency of the
harmonic at 10Hz is truly 10.0. Afterall a band has finite width even if
the harmonic is a Dirac spike.
At the end of the day we need to choose something, so its half-open,
lower-closed intervals by default.
Here's a good webpage with formulas if we want to get really fancy with
3dB band edges.
http://www.sengpielaudio.com/calculator-cutoffFrequencies.htm
The new files are leaner, and I removed some detailed comments/rambling. Pasting here for the record.
From FrequencyBand: