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

>>> wonky_points = signal(np.array([0,1,2,3])*1.1+1.1) * np.exp(np.array([0,1,2,3]) * 1.1 * 2j * np.pi * np.fft.fftfreq(4)[1])

what's going on here is the product of two complex sinusoids.

>>> abs(zero_based_wonky_points), np.angle(zero_based_wonky_points)*180//np.pi
(array([1., 1., 1., 1.]), array([   0.,  -63., -126.,  170.]))

Just looking it, it seems like a solution would be to scale the angles.

The internet confirms that the product of two complex numbers has the
sum of their angles. I'm immediately having difficulty confirming
that:

>>> np.angle(np.exp(np.array([30, 40]) / 360 * 2j * np.pi)) * 180 // np.pi
array([29., 40.])
>>> np.product(np.angle(np.exp(np.array([30, 40]) / 360 * 2j * np.pi))) * 180 // np.pi
20.0

>>> (0.8660254 +0.5j) * (0.76604444+0.64278761j)
(0.342020137568776+0.939692617065294j)
>>> np.angle((0.8660254 +0.5j) * (0.76604444+0.64278761j)) * 180 // np.pi
70.0

It looks like that is true, and np.product simply doesn't do what I
wanted it to.

Ok, so if the angles of the products of these sinusoids are increasing
by a constant, this means the sums of the angles of the parts are
increasing by a constant.

Multiplying the signal sinusoid, by the fourier sinusoid, adds their
angles together. I'll look at this again:

>>> zero_based_wonky_points = signal(np.array([0,1,2,3])*1.1) * np.exp(np.array([0,1,2,3]) * 1.1 * 2j * np.pi * np.fft.fftfreq(4)[1])
>>> abs(zero_based_wonky_points), np.angle(zero_based_wonky_points)*180//np.pi
(array([1., 1., 1., 1.]), array([   0.,  -63., -126.,  170.]))
>>> np.angle(signal(np.array([0,1,2,3])*1.1)) * 180 // np.pi
array([   0., -162.,   36., -127.])
>>> np.angle(np.exp(np.array([0,1,2,3]) * 1.1 * 2j * np.pi * np.fft.fftfreq(4)[1])) * 180 // np.pi
array([   0.,   99., -162.,  -64.])

The fourier signal is advancing at half the rate of the measured signal.
0652

When the measured signal is at -162, the fourier signal is at 90.
When the measured signal is at 36, the fourier signal is at -162.

>>> -162 - 0
-162
>>> 36 - (-162+360)
-162
>>> -127 - 36
-163

Because the sampling is offset, the measured signal is advancing by
-162 degrees every sample.
Meanwhile, the fourier signal is advancing by 99 degrees:
>>> 99 - 0
99
>>> -162 - (99 - 360)
99
>>> -64 - -162
98

I'm missing something: why does the angular rate of the fourier signal
not appear to be half the angular rate of the measured signal? Some
aliasing effect?

Oh, actually it is:
>>> 99 * 2
198
>>> -162 + 360
198

So the faster signal is advancing by 198 degrees/sample, and the
slower signal is advancing by 99 degrees/sample.

When we multiply them, we get a sequence of unit vectors that advance
by -63 degrees/sample. This is the same as +297 degrees, which is 198
degrees + 99 degrees:

>>> -63+360
297
>>> 99+198
297

When the signals align, the faster frequency advances by 180 degrees,
and the sum advances by 0 degrees. The slower frequency advances by 90
degrees, and the sum advances by 270 degrees, which results in it
hitting -90, 180, and 90 with modulo 360.

If the samples are aligned, the sum advances by 270 degrees. Here, in
the disalignment, the sum instead advances by 297 degrees. Performing
a transform on this row that converts 270 degree angle advancements to
297 degree advancements could help.

Further considering of combinations of these could help figure out a
formula for that transform. But is it even reasonable to scale a
sequence of complex numbers such that they express a rotation at a
different rate?

I'm thinking that could be done a further complex multiplication.
0704

If these all advance at 297 degrees, then maybe I could divide them by
appropriate exponentiated complex numbers to make them align to 0
degrees, then multiply them again to bring them into 270 degrees. Not
sure.

I'm thinking of how to design an exponent that increases by 297
degrees each time. I know I just wrote this to display it, but I'm
still confused around it.

If I want the complex number in the base of the exponent, then the
number I would use would be the one at 297 degrees.

>>> np.angle(np.exp(297 * np.pi / -180j) ** np.array([0,1,2,3])) * 180 // np.pi
array([   0.,  -64., -127.,  170.])

I'm using power operator to sort my thoughts out, but of course it
would be more efficient to simply multiply the angle before taking the
complex sin with the exp function.

Another question is, where does this angle 297 come from? This must
relate to the frequency difference of the original data.

That puts all the parts together. There was something I wanted to
consider to ensure the approach had a reasonable likelihood of being
valid.

>>> zero_based_wonky_points = signal(np.array([0,1,2,3])*1.1) * np.exp(np.array([0,1,2,3]) * 1.1 * 2j * np.pi * np.fft.fftfreq(4)[1])
>>> abs(zero_based_wonky_points), np.angle(zero_based_wonky_points)*180//np.pi
(array([1., 1., 1., 1.]), array([   0.,  -63., -126.,  170.]))
>>> np.angle(signal(np.array([0,1,2,3])*1.1)) * 180 // np.pi
array([   0., -162.,   36., -127.])
>>> np.angle(np.exp(np.array([0,1,2,3]) * 1.1 * 2j * np.pi * np.fft.fftfreq(4)[1])) * 180 // np.pi
array([   0.,   99., -162.,  -64.])

It looks like it's quite possible to adjust the phase offsets of the
results of multiplying a unit signal by a unit fourier component so as
to recover destructive interference.
In reality there are multiple signals at once, multiple components at
once, and notable the signal does not have unit magnitude.

A first test could be to give the signal a different magnitude, and
see what the result looks like.

>>> zero_based_wonky_points = signal(np.array([0,1,2,3])*1.1)*1.1 * np.exp(np.array([0,1,2,3]) * 1.1 * 2j * np.pi * np.fft.fftfreq(4)[1])
>>> abs(zero_based_wonky_points), np.angle(zero_based_wonky_points) * 180 // np.pi
(array([1.1, 1.1, 1.1, 1.1]), array([   0.,  -63., -126.,  170.]))

It looks like magnitude is retained intact, so that much is great.

Then, real data would have more than one signal going on at once, and
more than one fourier component going on at once.
0712

What does that mean? Well, for one thing I could try a further
combination. I can imagine in my mind that changing the fourier
frequency here, or the signal frequency, will just change the angle
advancement, and can be similarly countered.

What if there are multiple signals?

Here, there is just a DC component and a single signal.

It seems it could be reasonable to look briefly at 6-valued data, so
as to consider the interference of 2 disaligned signals at once. It
seems like that might require a more complex transform. The fftfreqs
of 6-valued data have 3 different absolute frequencies in them, in
addition to the DC offset.

I imagine the negative frequencies have an arithmetic equivalence, but
it's a little more confusing for me to consider what that is.

0714