Undescribed Horrific Abuse, One Victim & Survivor of Many gmkarl at gmail.com
Thu Nov 10 22:08:43 PST 2022

notes after a little sleep:

- you can see that this should work by imagining the waveform defined
in analytical frequency space, as a sum of sinusoids at precise
frequencies. [makes a clear an accurate fourier transform via many
different methods]. removes aliasing artefacts. [noise can then be
added to the recording, and lengthening it shown to improve the snr]

- can also see it should work given we have shown denser frequencies
than needed work. we would define the unit circle as the ceiling of
the max period. if the data length is cropped to be an even multiple
of this, then it can be extracted normally.

- using random data to test with without defining its analytical form
complicates the situation, adding unneccessary extra challenges.

- having the recording be an even fraction of the waveform does not
mean that the subsample data is lost: because this does not imply it
is a whole number of samples. this is done by setting the recording
length to a wonky value, and the max_period to an even fraction of
that value. for example, if recording_N is 99, and max_period is 9.9,
then it fits 10 times and offsets its phase by 10% each instance. this
can also work with recording_N 990 and max_period 99.

- one can also see it should be workable by considering those long,
stretched instances, where max_period > waveform_N * 2. considering
this builds reason to test with an analytic shape such as a sine wave
or a square wave. it shows how the interpolation plays out.

- my sense, from past experience, is that if i want this to work
right, i'll want to crop the recording length to be an even multiple
of the max_period. this works if the max_period is an integer.

having some difficulty simplifying the problem space to use an
integral max period and an analytically-defined waveform. i don't seem
to choose to do these things.

still, it's nice to list out two or three ways to show that it should
actually work, and that these ways have clear implementation paths.
struggling with this with result helps engage my algorithmic

most of my success yesterday was in manually making fourier and
inverse fourier transforms, notably with denser frequencies than the
data required.

a next step could be accumulating a fourier transform over extended
data -- a high pass, but such that there is still a complete unit
circle. basically, the max_period would be selected so as to be a
fraction of the whole thing already.

the reason the high pass max_period = (tN / (tN // max_period)) works
is because it is equivalent to summing individual max_period fourier
transforms (where max_period is an integer, or recording_length is
cropped), each one offset in phase by a constant, and each of those
individual transforms would show an individual downsampled copy of the
waveform. this is only the same as analogous successes if max_period
is an integer fraction of the recording length.

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