with all this verbosity i might lose the file! attaching it
# AGPL-3 Karl Semich 2022
import numpy as np
#### Notes on wavelets
# I naively tried using other functions than sinusoids for a DFT matrix.
# Given complex multiplication performs a scale + rotation operation on the model samples,
# only cos(t) + sin(t)i works as a coefficient and an underlying wave, without changing
# how the waveforms are evaluated, because it performs a rotation over time.
# Some other ideas include:
# - frequency error/resonance matrix, outputs the error or resonance for each frequency,
# without regard to underlying shape
# - converting between a sinusoid spectrum and one based on wavelets, in frequency space
# - evaluating wavelets forward pass only, and thinking on the result
# - other ideas [note: could look up wavelets]
## parts commented below
# # At the moment, wavelets must be:
# # - Zero at 0.25 and 0.75 f(0.25) == f(0.75) == 0
# # - Even functions f(x) == f(-x)
# # - Odd functions when slid by 90 degrees f(0.25 - x) == -f(0.25 + x)
# # Presently x is passed in the modular range [0,1) but this is arbitrary, just remove it if helpful
# COSINE = lambda x: np.cos(2 * np.pi * x)
# STEP = lambda x: COSINE(x).round()
# def complex_wavelet(wavelet, x):
# real_x = (abs(x) % 1) * np.sign(x)
# x -= 0.25
# imag_x = (abs(x) % 1) * np.sign(x)
# return wavelet(real_x) + wavelet(imag_x) * 1j
def fftfreq(freq_count = None, sample_rate = None,
min_freq = None, max_freq = None,
dc_offset = True, complex = True, sample_time = None,
repetition_time = None, repetition_samples = None,
freq_sample_rate = None, freq_sample_time = None):
'''
Calculates and returns the frequency bins used to convert between the time
and frequency domains with a discrete Fourier transform.
With no optional arguments, this function should be equivalent to numpy.fft.fftfreq .
To specify frequencies in terms of repetitions / sample_count, set
sample_rate to sample_count. If frequencies were in Hz (cycles/sec), this
implies that sample_count should be considered to last 1 second, so the
frequency becomes equal to the cycle count.
Parameters:
- freq_count: the number of frequency bins to generate
- sample_rate: the time-domain sample rate (default: 1 sample/time_unit)
- min_freq: the minimum frequency the signal contains
- max_fraq: the maximum frequency the signal contains
- dc_offset: whether to include a DC offset component (0 Hz)
- complex: whether to generate negative frequencies
- sample_time: sample_rate as the duration of a single sample
- repetition_time: min_freq as a period time, possibly of a subsignal
- repetition_samples: min_freq as a period size in samples
- freq_sample_rate: convert to or from a different frequency-domain sample rate
- freq_sample_time: freq_sample_rate as the duration of a single sample
Returns a vector of sinusoid time scalings that can be used to perform or
analyse a discrete Fourier transform. Multiply this vector by the time-domain
sample rate to get the frequencies.
'''
assert not sample_time or not sample_rate
assert not freq_sample_time or not freq_sample_rate
assert (min_freq, repetition_time, repetition_samples).count(None) >= 2
assert freq_count or min_freq or repetition_time or repetition_samples
if sample_time is not None:
sample_rate = 1 / sample_time
if freq_sample_time is not None:
freq_sample_rate = 1 / freq_sample_time
sample_rate = sample_rate or freq_sample_rate or 1
freq_sample_rate = freq_sample_rate or sample_rate or 1
if repetition_time:
min_freq = 1 / repetition_time
elif repetition_samples:
min_freq = freq_sample_rate / repetition_samples
# here freq_count adopts the complex + DC offset value, for calculations
# it might be clearer to use e.g. sample_count, at least consolidate the cases with lower
if freq_count is None:
freq_count = int(np.ceil(freq_sample_rate / min_freq))
#if (freq_count % 2) == 1 and not complex:
# freq_count += 1
else:
if not dc_offset:
freq_count += 1
if not complex:
freq_count = int((freq_count - 1) * 2)
if not min_freq:
min_freq = freq_sample_rate / freq_count
if not max_freq:
#max_freq = freq_sample_rate / 2
max_freq = freq_count * min_freq / 2
if freq_count % 2 != 0:
#max_freq -= freq_sample_rate / (2 * freq_count)
max_freq -= min_freq / 2
min_freq /= sample_rate
max_freq /= sample_rate
if freq_count % 2 == 0:
if complex:
neg_freqs = np.linspace(-max_freq, -min_freq, num=freq_count // 2, endpoint=True)
pos_freqs = -neg_freqs[:0:-1]
else:
pos_freqs = np.linspace(min_freq, max_freq, num=freq_count // 2, endpoint=True)
neg_freqs = pos_freqs[:0]
else:
pos_freqs = np.linspace(min_freq, max_freq, num=freq_count // 2, endpoint=True)
neg_freqs = -pos_freqs[::-1] if complex else pos_freqs[:0]
if not complex:
import warnings
warnings.warn('An odd-valued complex frequency count was generated for real-valued frequencies. Be sure to pass odd_repetition_samples=True when using the frequencies.', stacklevel=2)
return np.concatenate([
np.array([0] if dc_offset else []),
pos_freqs,
neg_freqs
])
def create_freq2time(time_count = None, freqs = None, odd_repetition_samples = None):
'''
Creates a matrix that will perform an inverse discrete Fourier transform
when it post-multiplies a vector of complex frequency magnitudes.
Example
time_data = spectrum @ create_freq2time(len(spectrum))
Parameters:
- time_count: size of the output vector, defaults to the frequency bincount
- freqs: frequency bins to convert, defaults to traditional FFT frequencies
Returns:
- an inverse discrete Fourier matrix of shape (len(freqs), time_count)
'''
assert (time_count is not None) or (freqs is not None)
if freqs is None:
assert odd_repetition_samples is None
odd_repetition_samples = (time_count % 2 == 1)
freqs = fftfreq(time_count)
is_complex = (freqs < 0).any()
has_dc_offset = bool(freqs[0] == 0)
freq_count = len(freqs)
if not has_dc_offset:
freq_count += 1
if not is_complex:
#has_nyquist = bool(freqs[-1] == 0.5)
#has_nyquist = True
has_nyquist = (bool(odd_repetition_samples) != has_dc_offset)
freq_count = (freq_count - 1) * 2 + (not has_nyquist)
if time_count is None:
time_count = freq_count
offsets = np.arange(time_count)
if is_complex:
angles = np.outer(2 * np.pi * freqs, offsets)
mat = np.sin(angles) * 1j
mat.real = np.cos(angles)
else:
head_idx = 1 if has_dc_offset else 0
tail_idx = len(freqs) - has_nyquist
# TODO: note that a negative frequency produces a negative imaginary part
# so some things could be simplified here
complex_freqs = np.concatenate([
freqs[:tail_idx],
-freqs[tail_idx:],
-freqs[head_idx:tail_idx][::-1]
])
complex_angles = np.outer(2 * np.pi * complex_freqs, offsets)
complex_mat = np.sin(complex_angles) * 1j
complex_mat.real = np.cos(complex_angles)
# assuming the input's real and complex components are broken out into separate elements
interim_mat = complex_mat.copy()
interim_mat[head_idx:tail_idx] += interim_mat[len(freqs):][::-1].conj()
mat = np.stack([
interim_mat[:len(freqs)].real.T,
-interim_mat[:len(freqs)].imag.T
], axis=-1).reshape(time_count, len(freqs)*2).T
return mat / freq_count # scaled to match numpy convention
def create_time2freq(time_count = None, freqs = None, odd_repetition_samples = None):
'''
Creates a matrix that will perform a forward discrete Fourier transform
when it post-multiplies a vector of time series data.
If time_count is too small or large, the minimal least squares solution
over all the data passed will be produced.
This function is equivalent to calling .pinv() on the return value of
create_freq2time. If the return value is single-use, it is more efficient and
accurate to use numpy.linalg.lstsq .
Example
spectrum = time_data @ create_time2freq(len(time_data))
Parameters:
- time_count: size of the input vector, defaults to the frequency bincount
- freqs: frequency bins to produce, defaults to traditional FFT frequencies
Returns:
- a discrete Fourier matrix of shape (time_count, len(freqs))
'''
assert (time_count is not None) or (freqs is not None)
if freqs is None:
assert odd_repetition_samples is None
odd_repetition_samples = (time_count % 2 == 1)
freqs = fftfreq(time_count)
is_complex = (freqs < 0).any()
# TODO: provide for DC offsets and nyquists not at index 0 by excluding items in (0,0.5)
has_dc_offset = bool(freqs[0] == 0)
if not is_complex:
#has_nyquist = bool(freqs[-1] == 0.5)
#has_nyquist = True
has_nyquist = (bool(odd_repetition_samples) != has_dc_offset)
head_idx = 1 if has_dc_offset else 0
tail_idx = len(freqs)-1 if has_nyquist else len(freqs)
neg_start_idx = len(freqs)
freqs = np.concatenate([
freqs[:head_idx],
freqs[head_idx:tail_idx],
-freqs[tail_idx:neg_start_idx],
-freqs[head_idx:tail_idx][::-1]
])
inverse_mat = create_freq2time(time_count, freqs, odd_repetition_samples = odd_repetition_samples)
forward_mat = np.linalg.pinv(inverse_mat)
if not is_complex:
forward_mat = np.concatenate([
forward_mat[:,:tail_idx],
forward_mat[:,tail_idx:neg_start_idx].conj()
], axis=1)
return forward_mat
def peak_pair_idcs(freq_data, dc_offset=True, sum=True):
dc_offset = int(dc_offset)
freq_heights = abs(freq_data) # squares and sums the components
if sum:
while len(freq_heights).shape > 1:
freq_heights = freq_height.sum(axis=0)
paired_heights = freq_heights[...,dc_offset:-1] + freq_heights[...,dc_offset+1:]
peak_idx = paired_heights.argmax(axis=-1, keepdims=True) + dc_offset
return np.concatenate(peak_idx, peak_idx + 1, axis=-1)
def improve_peak(time_data, min_freq, max_freq):
freqs = fftfreq(time_data.shape[-1], min_freq = min_freq, max_freq = max_freq)
freq_data = time_data @ np.linalg.inv(create_freq2time(freqs))
left, right = peak_pair_idcs(freq_data)
return freq_data[left], freq_data[right]
def test():
np.random.seed(0)
randvec = np.random.random(16)
freqs16 = fftfreq(16)
ift16 = create_freq2time(16, freqs=freqs16)
ft16 = create_time2freq(16)
randvec2time = randvec @ ift16
randvec2freq = randvec @ ft16
randvec2ifft = np.fft.ifft(randvec)
randvec2fft = np.fft.fft(randvec)
assert np.allclose(freqs16, np.fft.fftfreq(16))
assert np.allclose(randvec2ifft, randvec2time)
assert np.allclose(randvec2fft, randvec2freq)
assert np.allclose(randvec2ifft, np.linalg.solve(ft16.T, randvec))
assert np.allclose(randvec2fft, np.linalg.solve(ift16.T, randvec))
rfreqs15t = fftfreq(repetition_samples=15, complex=False)
rfreqs15f = fftfreq(15, complex=False)
irft_from_15 = create_freq2time(freqs=rfreqs15f)
rft_from_15 = create_time2freq(15, freqs=rfreqs15t, odd_repetition_samples=True)
irft_to_15 = create_freq2time(15, freqs=rfreqs15t, odd_repetition_samples=True)
randvec2rtime15 = randvec[:15].astype(np.complex128).view(float) @ irft_from_15
randvec2rfreq15 = randvec[:15] @ rft_from_15
randvec2irfft = np.fft.irfft(randvec[:15])
randvec2rfft = np.fft.rfft(randvec[:15])
randvec2rfreq152randvec = randvec2rfreq15.view(float) @ irft_to_15
randvec2rfreq152irfft = np.fft.irfft(randvec2rfreq15, 15)
assert np.allclose(rfreqs15t, np.fft.rfftfreq(15))
assert np.allclose(rfreqs15f, np.fft.rfftfreq(28))
assert np.allclose(randvec2rtime15, randvec2irfft)
assert np.allclose(randvec2rfreq15, randvec2rfft)
assert np.allclose(randvec2rfreq152randvec, randvec2rfreq152irfft)
assert np.allclose(randvec2rfreq152randvec, randvec[:15])
rfreqs16t = fftfreq(repetition_samples=16, complex=False)
rfreqs16f = fftfreq(16, complex=False)
irft16 = create_freq2time(freqs=rfreqs16f)
rft16 = create_time2freq(16, freqs=rfreqs16t)
randvec2rtime16 = randvec[:16].astype(np.complex128).view(float) @ irft16
randvec2rfreq16 = randvec[:16] @ rft16
randvec2irfft = np.fft.irfft(randvec[:16])
randvec2rfft = np.fft.rfft(randvec[:16])
assert np.allclose(rfreqs16t, np.fft.rfftfreq(16))
assert np.allclose(rfreqs16f, np.fft.rfftfreq(30))
assert np.allclose(randvec2rtime16, randvec2irfft)
assert np.allclose(randvec2rfreq16, randvec2rfft)
# [ 0, 1, 2, 3, 4, 3, 2, 1]
#irft9_16 = create_freq2time(16, fftfreq(9, complex = False))
#rft16_30 = create_time2freq(16, fftfreq(30, complex = False))
#randvec2
#assert np.allclose((randvec @ rft16) @ irft16, randvec)
# sample data at a differing rate
time_rate = np.random.random() * 2
freq_rate = 1.0
freqs = np.fft.fftfreq(len(randvec))
rescaling_freqs = fftfreq(len(randvec), freq_sample_rate = freq_rate, sample_rate = time_rate)
rescaling_ift = create_freq2time(freqs = rescaling_freqs)
rescaling_ft = create_time2freq(freqs = rescaling_freqs)
rescaled_time_data = np.array([
np.mean([
randvec[freqidx] * np.exp(2j * np.pi * freqs[freqidx] * sampleidx / time_rate)
for freqidx in range(len(randvec))
])
for sampleidx in range(len(randvec))
])
assert np.allclose(rescaled_time_data, randvec @ rescaling_ift)
assert np.allclose(rescaled_time_data, np.linalg.solve(rescaling_ft.T, randvec))
unscaled_freq_data = rescaled_time_data @ rescaling_ft
unscaled_time_data = unscaled_freq_data @ ift16
assert np.allclose(unscaled_freq_data, randvec)
assert np.allclose(unscaled_time_data, randvec2time)
assert np.allclose(np.linalg.solve(rescaling_ift.T, rescaled_time_data), randvec)
# extract a repeating wave
longvec = np.empty(len(randvec)*100)
short_count = np.random.random() * 4 + 1
short_duration = len(longvec) / short_count
for idx in range(len(longvec)):
longvec[idx] = randvec[int((idx / len(longvec) * short_duration) % len(randvec))]
shortspace_freqs = fftfreq(None, complex = False, dc_offset = True, repetition_samples = len(randvec))
longspace_freqs = fftfreq(len(shortspace_freqs), complex = False, dc_offset = True, repetition_samples = short_duration)
assert np.allclose(longspace_freqs, shortspace_freqs / (short_duration / len(randvec)))
inserting_ift = create_freq2time(len(longvec), longspace_freqs)#, wavelet = STEP)
extracting_ft = create_time2freq(len(longvec), longspace_freqs)#, wavelet = STEP)
extracting_ift = create_freq2time(freqs = shortspace_freqs)#, wavelet = STEP)
unextracting_ft = create_time2freq(freqs = shortspace_freqs)#, wavelet = STEP)
inserting_spectrum = randvec @ unextracting_ft
inserting_spectrum = inserting_spectrum.view(float) # todo: do this out in time2freq?
assert np.allclose(inserting_spectrum @ extracting_ift, randvec)
assert np.allclose(longvec, inserting_spectrum @ inserting_ift)
extracted_freqs = longvec @ extracting_ft
extracted_randvec = extracted_freqs @ extracting_ift
assert np.allclose(extracted_randvec, randvec)
if __name__ == '__main__':
test()
