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

so what do i have here ...

micro_ift and micro_ft do something similar to a fourier transform in
a highpass way. they can reconstruct a signal that contains those
precise highpass components.

i haven't tried or considered what would make all th eparts correct,
but they're not presently reconstructing a signal that contains
imprecise highpass components, including a strong undersampled high
frequency.

one thing i haven't addressed is whether these strong high frequencies
are being interpolated in the same ways between the functions.
ideally, the data represents the same underlying thing, and is sampled
at the same points, and modeled in the same ways, such that it returns
exactly.

in the micro_*ft functions, a sequence of sinusoids are used to model
the signal, between a minimum and maximum frequency. the minimum
frequency is determined based on a passed maximum period, and this is
linearly scaled to something like the maximum frequency.

i'm guessing that i can discern the correct scaling to reconstruct the
extracted signals, and consider whether or why they will be
reconstructed exactly, by thinking about the frequencies used in
micro_*ft, and the angles of the resulting sinusoids, at the points in
the signal where they are sampled.

each point in the simulated recording has a matching point in the
underlying waveform, that i am trying to reconstruct. for each of
those points in both the recording, and the waveform, there is a set
of sinusoids and angles that are multiplied by the data, to form a new
linear combination of its parts.

if those sinusoids and angles are the same, the data will be reconstructed.

so there's a shuffling challenge here.
4 angles of 2 sinusoids, taken from 4 frequencies at 4 time points and
4 offsets, need to be the same on both sides. it's 32 total sinusoids:
2 for cos and sin, 4 for the indices, and 4 for the frequencies. cos
and sine are just encoding an angle, so it's 16 total angles. 4
indices, and 4 frequencies.

each frequency shifts by the index. if we consider the 1/period form
of the frequency, then the value that must match is like index *
frequency.

so 4 indices, each times 4 different frequencies, must match for both
the input and the output, for the signal to be reconstructed.

the difference is that when the input is sampled, the indices are sequential.
whereas, when the input is reconstructed, the indices (mapped from the
recording back to the waveform) are interleaved.

the different frequency mappings would kind of need to be able to undo
that interleaving using the modulo behavior of sinusoids.