Undescribed Horrific Abuse, One Victim & Survivor of Many gmkarl at gmail.com
Sat Nov 12 06:41:50 PST 2022

Further avenues on that approach could include an analysis of the
noise to account for it immediately, or an anlysis of the recursive
solution, which would involve discerning solving for the measurement
correction given the error after combining with an imperfect
destructive wave.

Another approach is to consider the system as linear algebra. We write
an equation for the signal given how we want to represent it, and
simply solve for the coefficients of the matrix involved.


So, a fourier transform models a signal as being made of as many
sinusoids as it has samples. Here, we have 6 sinusoids. Mostly because
the data is complex; only half as many are needed for real-valued

Each of these 6 sinusoids has a real and a complex component. It's
written like exp(2j * pi * freq * sample_point). There are 6
frequencies, and 6 sample points, so this makes a 6x6 matrix.

This should somehow make a system of equations that has 6 unknowns.

I'm thinking the 6x6 matrix has the evaluation of the sin and cosine
at each row (or column), spread out for each different sample.

When you apply the fourier transform, you multiply a matrix like this
by the signal. But rather here, the matrix is the data. More like an
inverse fourier transform, the inverse wave is the variables here, and
we set it equal to the data.

data = F @ x

We can then solve that linear system of equations for x, and x becomes
the frequency representation.

This linear equation should work for any selection of frequencies, and
any selection of wavelets. So much simpler!

Man I was asking my teachers to teach me this in high school, but it
never happened. I never figured out what classes to take in college to
learn this either. I dunno. But I bet that this is how wavelets are
normally cast in engineering.

I _think_ this should work. And it explains the validity of taking the
matrix inverse later.

Next step is to try it.

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