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[Undescribed Horrific Abuse, One Victim & Survivor of Many]

it might help to think of the overlap

at some point wavidx == N, the recording indices restart
meanwhile, there are 4 frequencies, that are repeatedly and
continuously restarting.

at wavidx == N, we might want all those frequencies to align, so that
they restart and match the same points. but wavidx == N is never
actually sampled, so it could go a different way.

one could also add more modelling to the system, and decide the
waveform is made of specific underlying sinusoids.

--

maybe it could be helpful to consider sample 1

at sample 1, when sampling, a bunch of frequencies are scaled by 1 and
a multiplier.
sample 1 breaks into those N frequencies, multiplied by a sampling multiplier

meanwhile, when reconstructing, sample 1 comes from N frequencies,
which are all multiplied by index 1 ...? things actually happen a
litle more complexly

there are 4 passes of frequencies and indices.
first, the fft in the sampling,
then, the loop in the sampling, for every sample
then the microft
then the microift .

so "sample 1" is repeatedly spread everywhere, and reconsolidated
again, or vice versa.

the waveform is projected in a modular way through the recording, then
back through the microfts.

the sampling and the microft use two different sets of frequencies to
transform the data.
in the sampling, the waveform is considered a sum of even frequencies.
in the microft, the recording is considered a sum of denser frequencies.

is it sufficient for these even frequencies to equate to the same
portions of the waveform that the denser frequencies do? it seems at
least a step. if they are the same frequencies in the waveform, then
they simply need to be combined with the same indices in the waveform.

so it _should eventually work_. because when sampling the waveform
into the recording, these can be the same waveform indices as when
reconstructing it.

so how do we have frequencies in the recording, that equate to the
same regions of the waveform?
it sounds confusing, but it's highpass, so we just need to use the
same actual frequencies, and scale them down to the scaling of the
waveform .  i think!

it's real-valued data, so the negative frequencies in np.fft.fftfreq
can be ignored.
the micro_*fts , however, may want to process these negative
frequencies, a quick approach could be two exp calls