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Sun Nov 13 06:15:57 PST 2022

oh :D I need to transpose the matrix when passing it to
np.linalg.solve , because np.linalg.solve does Ax = b right-to-left,
not xA = b left-to-right like I have been doing.
0857 .

Now it finishes and produces the exact random data, and it's just like
the fourier.py test .

The original test I had written had a smaller waveform size than
recording size (16 vs 256).
If I want to process the recording all at once, this could mean a
rectangular matrix.

Maybe I'll just try doing that, and see how it goes.

0900
I bumped the recording size back to 256, and changed np.linalg.solve
to np.linalg.lstsq and made no other changes, and got this:
(Pdb) cont
Traceback (most recent call last):
  File "/usr/local/lib/python3.10/pdb.py", line 1726, in main
    pdb._runscript(mainpyfile)
  File "/usr/local/lib/python3.10/pdb.py", line 1586, in _runscript
    self.run(statement)
  File "/usr/local/lib/python3.10/bdb.py", line 597, in run
    exec(cmd, globals, locals)
  File "<string>", line 1, in <module>
  File "/shared/src/scratch/test2_upsamplish.py", line 22, in <module>
    assert np.allclose(np.fft.fft(waveform) @
waveform_freq_2_recording_time, recording)
  File "<__array_function__ internals>", line 200, in allclose
  File "/home/user/.local/lib/python3.10/site-packages/numpy/core/numeric.py",
line 2270, in allclose
    res = all(isclose(a, b, rtol=rtol, atol=atol, equal_nan=equal_nan))
  File "<__array_function__ internals>", line 200, in isclose
  File "/home/user/.local/lib/python3.10/site-packages/numpy/core/numeric.py",
line 2380, in isclose
    return within_tol(x, y, atol, rtol)
  File "/home/user/.local/lib/python3.10/site-packages/numpy/core/numeric.py",
line 2361, in within_tol
    return less_equal(abs(x-y), atol + rtol * abs(y))
ValueError: operands could not be broadcast together with shapes (16,) (256,)

It's interesting, it looks like the failure is inside numpy. It's
actually inside the allclose call.

I guess I can figure this out.

(Pdb) list
 17     recording = sample_sinusoids_funny(waveform, sample_idcs, max_period)
 18
 19     recording_rate = 1
 20     waveform_rate = (waveform_N / max_period) * recording_rate
 21     waveform_freq_2_recording_time =
fourier.create_freq2time(waveform_rate, recording_rate, waveform_N,
recording_N)
 22  -> assert np.allclose(np.fft.fft(waveform) @
waveform_freq_2_recording_time, recording)
 23     waveform_freq_reconstructed =
np.linalg.lstsq(waveform_freq_2_recording_time.T, recording)
 24     waveform_freq_np = np.fft.fft(waveform)
 25     assert np.allclose(waveform_freq_reconstructed, waveform_freq_np)
 26     waveform_reconstructed = np.fft.ifft(waveform_freq_reconstructed)
 27     assert np.allclose(waveform, waveform_reconstructed)
(Pdb) p recording.shape
(256,)
(Pdb) p (np.fft.fft(waveform) @ waveform_freq_2_recording_time).shape
(16,)

Looks like I have my dimensions sideways when they are not equal, in
fourier.py .

(Pdb) p waveform_freq_2_recording_time.shape
(16, 16)
No ... looks like I am using a frequency dimension where I should be
using a time dimension.

On to fourier.py !
0903

# AGPL-3 Karl Semich 2022
import numpy as np

def create_freq2time(freq_rate, time_rate, freq_count, time_count):
    freqs = np.fft.fftfreq(freq_count)
    offsets = np.arange(freq_count) * freq_rate / time_rate
    mat = np.exp(2j * np.pi * np.outer(freqs, offsets))
    return mat / freq_count # scaled to match numpy convention

It's making a square matrix, and I want a rectangular one.

Here it is:
    offsets = np.arange(freq_count) * freq_rate / time_rate
I'll need as many sample offsets as there are samples in the time
domain, to generate them.
0904

That change passed that assertion, and now it is failing with np.lstsq
. This might mean I get to engage more theory to do it better!

(Pdb) cont
/shared/src/scratch/test2_upsamplish.py:23: FutureWarning: `rcond`
parameter will change to the default of machine precision times
``max(M, N)`` where M and N are the input matrix dimensions.
To use the future default and silence this warning we advise to pass
`rcond=None`, to keep using the old, explicitly pass `rcond=-1`.
  waveform_freq_reconstructed =
np.linalg.lstsq(waveform_freq_2_recording_time.T, recording)
Traceback (most recent call last):
  File "/usr/local/lib/python3.10/pdb.py", line 1726, in main
    pdb._runscript(mainpyfile)
  File "/usr/local/lib/python3.10/pdb.py", line 1586, in _runscript
    self.run(statement)
  File "/usr/local/lib/python3.10/bdb.py", line 597, in run
    exec(cmd, globals, locals)
  File "<string>", line 1, in <module>
  File "/shared/src/scratch/test2_upsamplish.py", line 25, in <module>
    assert np.allclose(waveform_freq_reconstructed, waveform_freq_np)
  File "<__array_function__ internals>", line 200, in allclose
  File "/home/user/.local/lib/python3.10/site-packages/numpy/core/numeric.py",
line 2270, in allclose
    res = all(isclose(a, b, rtol=rtol, atol=atol, equal_nan=equal_nan))
  File "<__array_function__ internals>", line 200, in isclose
  File "/home/user/.local/lib/python3.10/site-packages/numpy/core/numeric.py",
line 2363, in isclose
    x = asanyarray(a)
ValueError: setting an array element with a sequence. The requested
array has an inhomogeneous shape after 1 dimensions. The detected
shape was (4,) + inhomogeneous part.

I don't really know what this failure means.
0906
0907
oh, it's actually failing within another allclose call. the lstsq call
completed successfully.

 23     waveform_freq_reconstructed =
np.linalg.lstsq(waveform_freq_2_recording_time.T, recording)
 24     waveform_freq_np = np.fft.fft(waveform)
 25  -> assert np.allclose(waveform_freq_reconstructed, waveform_freq_np)
 26     waveform_reconstructed = np.fft.ifft(waveform_freq_reconstructed)

(Pdb) p waveform_freq_reconstructed.shape
*** AttributeError: 'tuple' object has no attribute 'shape'
(Pdb) p waveform_freq_reconstructed
(array([ 9.13008935-2.05511125e-15j, -1.13139897-1.72422981e-01j,
        0.11932716-6.18590050e-01j, -0.47619445-1.42077247e+00j,
        0.60106394-6.17441233e-01j,  0.4321097 -8.80843476e-01j,
        0.92222665+1.15314024e+00j, -0.4839133 -2.10053038e-01j,
       -0.31551473-6.30467900e-14j, -0.4839133 +2.10053038e-01j,
        0.92222665-1.15314024e+00j,  0.4321097 +8.80843476e-01j,
        0.60106394+6.17441233e-01j, -0.47619445+1.42077247e+00j,
        0.11932716+6.18590050e-01j, -1.13139897+1.72422981e-01j]),
array([2.16596661e-24]), 16, array([1.03318085, 1.03318084,
1.03317917, 1.03309466, 1.03124288,
       1.01732177, 0.99161519, 0.98123564, 0.9801968 , 0.98016181,
       0.98016142, 0.98016142, 0.98016142, 0.98016142, 0.98016142,
       0.98016142]))

The frequencies look correct, but it's returning a tuple rather than a
matrix. It looks like it's showing that the need to process the return
value of lstsq correctly.

It's exciting there's more information than just the least squares
solution. As I was adding it, I was wondering how I would discern
error efficiently.

>>> help(np.linalg.lstsq)

    Returns
    -------
    x : {(N,), (N, K)} ndarray
        Least-squares solution. If `b` is two-dimensional,
        the solutions are in the `K` columns of `x`.
    residuals : {(1,), (K,), (0,)} ndarray
        Sums of squared residuals: Squared Euclidean 2-norm for each column in
        ``b - a @ x``.
        If the rank of `a` is < N or M <= N, this is an empty array.
        If `b` is 1-dimensional, this is a (1,) shape array.
        Otherwise the shape is (K,).
    rank : int
        Rank of matrix `a`.
    s : (min(M, N),) ndarray
        Singular values of `a`.

It returns x, residuals, rank, s.
So it found the right solution, the error is 2e-24 which is very low,
there are 16 data elements, and it also gives 16 singular values. I'm
not sure what a singular value is, although I think I learned once,
but it sounds a little important.
What's immediately interesting is the solution and the error.

In a larger work, the error could indicate background signals that
have not been modeled, maybe.
draft saved at 0911 am.

It works now, using np.lstsq to exactly find the 16 random samples
repeated within a 256 sample long recording.

A further avenue might be to add background noise and see how the
noise magnitude relates to the recording duration needed to completely
remove the noise, and the error reported by the solution. Ideally
you'd figure this in theory, and then verify by testing.
A different further avenue might be to do it while blind to the
repeating frequency of the waveform in the data. (note: it is the only
signal here).

Attached is fourier.py with the dimension bug fixed.
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