The code calculates a phase shift between a reference sawtooth signal and mixer diode signal. The phase shift is proportional to electron density. A big part of the code are repairing mechanisms used, when the measured signal is damaged, which occurs very often.
(author: L. Lobko)
%matplotlib inline
import numpy as np
import matplotlib.pyplot as plt
from scipy import constants
from scipy import signal as sigproc
from scipy import fftpack
from scipy import ifft
from scipy.fftpack import fft
from scipy.fftpack import next_fast_len
import math
The data is read from ISF files, using the isfreader library. (Can be installed by "!pip3 install isfreader" command.)
import isfreader
shot_no = 44075
ds = np.DataSource('/tmp') # temporary storage for downloaded files
data_URL = 'http://golem.fjfi.cvut.cz/shots/{shot_no}/Diagnostics/Interferometry/DAS_raw_data_dir/ch{ch_id}.isf'
scalars_URL = 'http://golem.fjfi.cvut.cz/shots/{shot_no}/Diagnostics/PlasmaDetection/Results/{name}'
def get_scalar(shot_no, name):
return float(ds.open(scalars_URL.format(shot_no=shot_no, name=name)).read())
t_plasma_start = get_scalar(shot_no, 't_plasma_start')
t_plasma_end = get_scalar(shot_no, 't_plasma_end')
def load_channel(shot_no, chanel_id):
fname = ds.open(data_URL.format(shot_no=shot_no, ch_id=chanel_id)).name
data = isfreader.read_file(fname)
data[:, 0] = data[:, 0] * 1e3
return data
mixer = load_channel(shot_no, 1)
ref_saw = load_channel(shot_no, 3)
# phase_det = load_channel(shot_no, 4)
x, y = mixer.shape
f_s = x / (mixer[-1, 0] - mixer[0, 0]) # ms -> kHz
print('Sampling frequency is {} MHz.'.format(round(f_s / 1000)))
Sampling frequency is 250 MHz.
The mixer signal is a base sine wave (the envelope of the mixing) at a frequency close to 500 kHz. The reference saw-tooth frequency sweeping wave has the same base frequency, but with a trail of harmonics forming the sharp saw-tooth shape.
def calculate_spectrum(signal, target_dfreq=10):
nperseg = int(f_s / target_dfreq)
f, psd = sigproc.welch(signal[:, 1], fs=f_s, nperseg=nperseg, nfft=next_fast_len(nperseg))
return f, psd
ref_saw_f, ref_saw_psd = calculate_spectrum(ref_saw)
mixer_f, mixer_psd = calculate_spectrum(mixer)
f_base = mixer_f[mixer_psd.argmax()]
f_base_ref_saw = ref_saw_f[ref_saw_psd.argmax()]
print('The base frequency of the mixer is {} kHz.'.format(f_base))
print('The base frequency of the ref_saw is {} kHz.'.format(f_base_ref_saw))
The base frequency of the mixer is 540.0001080000216 kHz. The base frequency of the ref_saw is 540.0001080000216 kHz.
fig, ax = plt.subplots(dpi=150)
ax.set(xlabel='frequency [kHz]', ylabel='power spectral density [V$^2$]')
ax.plot(ref_saw_f, ref_saw_psd, label='ref_saw')
ax.plot(mixer_f, mixer_psd, label='mixer')
plt.axvline(f_base, label=f'base frequency f={f_base:.0f} kHz', color='C3')
ax.loglog()
plt.grid()
plt.legend();
<matplotlib.legend.Legend at 0x7fface379e50>
The instantaneous phase and amplitude of the base signal can be inferred only for the baseband, i.e. by removing higher and lower frequencies (i.e. extracting the base sine wave from the reference saw-tooth signal).
base_band_hwidth = 50 # kHz
base_band_filter = sigproc.iirfilter(8, [f_base - base_band_hwidth, f_base + base_band_hwidth], fs=f_s, btype='bandpass', output='sos')
def freq_filter(signal, sos_filter):
signal[:, 1] = sigproc.sosfiltfilt(sos_filter, signal[:, 1])
return signal
mixer_filtered = freq_filter(mixer, base_band_filter)
ref_saw_filtered = freq_filter(ref_saw, base_band_filter)
cut 0.1 ms from the beginning and from the end for better signal processing
mixer_filtered = mixer_filtered[(mixer_filtered[:, 0] < (mixer_filtered[-1, 0] - 0.1)) & ((mixer_filtered[0, 0] + 0.1) < mixer_filtered[:, 0])]
ref_saw_filtered = ref_saw_filtered[(ref_saw_filtered[:, 0] < (ref_saw_filtered[-1, 0] - 0.1)) & ((ref_saw_filtered[0, 0] + 0.1) < ref_saw_filtered[:, 0])]
func find_peaks - finds peaks with optimal output array
def find_peaks(data):
peaks_indexes, _ = sigproc.find_peaks(data[:, 1])
return np.vstack((data[peaks_indexes, 0], data[peaks_indexes, 1])).T
func initial_phase_shift - mixer and ref_saw signals are in general a bit phase shifted from each other -> it calculates "initial" phase shift and removes it
def initial_phase_shift(peaks, peaks_ref):
phase_mean = peaks[0, 0] - peaks_ref[0, 0]
peaks_ref[:, 0] += phase_mean
return peaks_ref
func cut_edges - cut first and last data point, which is distorted from the spectral filter
def cut_edges(peaks, peaks_ref):
peaks = peaks[(peaks[0, 0] < peaks[:, 0]) & (peaks[:, 0] < peaks[-1, 0])]
peaks_ref = peaks_ref[(peaks_ref[0, 0] < peaks_ref[:, 0]) & (peaks_ref[:, 0] < peaks_ref[-1, 0])]
return peaks, peaks_ref
func smooth - classic func for signal smoothing
def smooth(y, box_pts):
box = np.ones(box_pts) / box_pts
y_smooth = np.convolve(y, box, mode='same')
return y_smooth
func without_correction - a sum of basic operations
def without_correction(mixer_filtered, ref_saw_filtered):
peaks = find_peaks(mixer_filtered)
peaks_ref = find_peaks(ref_saw_filtered)
peaks_ref = initial_phase_shift(peaks, peaks_ref)
peaks, peaks_ref = cut_edges(peaks, peaks_ref)
return peaks, peaks_ref
func find_nearest - finds the nearest peak of the given one
def find_nearest(array, value):
array = array.copy()
x, y = array.shape
array_data = np.zeros((x, 1))
for i in range(0, x):
array_data[i, 0] = array[i, 0]
idx = (np.abs(array_data[:, 0] - value)).argmin()
return array_data[idx]
func calc_dphase_unchanged - calculates dphase from unrepaired data
def calc_dphase_unchanged(peaks, peaks_ref):
x_peaks, y_peaks = peaks.shape
x_ref_peaks, y_ref_peaks = peaks_ref.shape
dphase = np.ones((min(x_peaks, x_ref_peaks), 2))
for i in range(0, int(len(dphase))):
dphase[i, 0] = peaks[i, 0]
dphase[i, 1] = peaks[i, 0] - peaks_ref[i, 0]
dphase[:, 1] *= 2 * math.pi * f_base
return dphase
func calc_lost_phase - calculates lost phase in the signal - to define how much the signal was damaged
def calc_lost_phase(peaks, peaks_ref):
dphase = calc_dphase_unchanged(peaks, peaks_ref)
time_interval = 0.1 # ms
indexes = np.argwhere(dphase[:, 0] > (dphase[-1, 0] - time_interval))
data = dphase[indexes[:, 0]]
return np.average(data[:, 1]), dphase
func optimizing_cycle - defines the most probable parts of the data, where the signal was damaged and deletes the corresponding waveforms from the reference signal, because these waveforms did not travelled properly through the plasma
def optimizing_cycle(number_of_bad_peaks, deriv_sort, distance, peaks, peaks_ref):
bad_peaks_indexes = np.empty((0, 1))
k = 0 # help variable
l = 0 # help variable
while k < number_of_bad_peaks:
index = np.argwhere(peaks[:, 0] == deriv_sort[l, 0])
if len(bad_peaks_indexes) != 0 and (abs((index[0, 0] - find_nearest(bad_peaks_indexes, index[0, 0]))) < distance): # or (t_plasma_end < peaks[index, 0]) or (peaks[index, 0] < t_plasma_start):
l += 1
else:
bad_peaks_indexes = np.vstack((bad_peaks_indexes, index[0, 0]))
peaks_ref = np.delete(peaks_ref, index, 0)
k += 1
l += 1
return bad_peaks_indexes, peaks_ref
func repair - creates the most plausible repair of the given damaged interferometer data (probably caused by the scatter of the probing wave from the plasma, plasma instability, ...)
def repair(dphase, peaks, peaks_ref):
global dphase_final, bad_peaks_indexes, best_smooth_factor, best_distance
smooth_factors = np.zeros((0, 1))
distances = np.zeros((0, 1))
varieties = np.zeros((0, 1))
for smooth_factor in range(1, 20):
deriv = dphase.copy()
deriv[:, 1] = np.gradient(dphase[:, 1])
deriv[:, 1] = smooth((deriv[:, 1]), smooth_factor)
deriv_sort = deriv.copy()
deriv_sort = deriv_sort[deriv_sort[:, 1].argsort()[::-1]]
for distance in range(1, 20):
bad_peaks_indexes, repaired_ref_peaks = optimizing_cycle(number_of_bad_peaks, deriv_sort, distance, peaks, peaks_ref)
dphase_final = calc_dphase(peaks, repaired_ref_peaks)
dphase_final[:, 1] = smooth(dphase_final[:, 1], 100)
if np.min(dphase_final[:, 1]) > -1:
smooth_factors = np.vstack((smooth_factors, smooth_factor))
distances = np.vstack((distances, distance))
varieties = np.vstack((varieties, calc_curve_length(dphase_final)))
all = np.hstack((smooth_factors, distances, varieties))
if len(varieties) == 0:
print("Cannot repair interferometric data.")
else:
varieties_min = varieties.argmin()
best_smooth_factor = int(all[int(varieties_min), 0])
best_distance = int(all[int(varieties_min), 1])
deriv = dphase.copy()
deriv[:, 1] = np.gradient(dphase[:, 1])
deriv[:, 1] = smooth((deriv[:, 1]), best_smooth_factor)
deriv_sort = deriv.copy()
deriv_sort = deriv_sort[deriv_sort[:, 1].argsort()[::-1]]
bad_peaks_indexes, repaired_ref_peaks = optimizing_cycle(number_of_bad_peaks, deriv_sort, best_distance, peaks, peaks_ref)
dphase_final = calc_dphase(peaks, repaired_ref_peaks)
return dphase_final, bad_peaks_indexes, best_smooth_factor, best_distance
func calc_dphase - calculates dphase
def calc_dphase(mixer_peaks, repaired_ref_peaks):
x_peaks, y_peaks = mixer_peaks.shape
x_ref_peaks, y_ref_peaks = repaired_ref_peaks.shape
dphase_final = np.ones((min(x_peaks, x_ref_peaks), 2))
dphase_final[:, 0] = repaired_ref_peaks[0:len(dphase_final), 0]
dphase_final[:, 1] = (mixer_peaks[0:len(dphase_final), 0] - repaired_ref_peaks[0:len(dphase_final), 0]) * 2 * math.pi * f_base
# for i in range(0, int(len(dphase_final))):
# dphase_final[i, 0] = repaired_ref_peaks[i, 0]
# dphase_final[i, 1] = mixer_peaks[i, 0] - repaired_ref_peaks[i, 0]
# dphase_final[:, 1] *= 2 * math.pi * f_base
return dphase_final
func calc_curve_length - calculates the length of the dphase curve as a key parameter to decide, whether the repair is good enough
def calc_curve_length(dphase_final):
x, y = dphase_final.shape
length = 0
part_length = np.sqrt(((dphase_final[1:-1, 0] - dphase_final[0:-2, 0]) ** 2) + (dphase_final[1:-1, 1] - dphase_final[0:-2, 1]) ** 2)
# for i in range(1, x):
# length += np.sqrt(((dphase_final[i, 0] - dphase_final[i - 1, 0]) ** 2) + (dphase_final[i, 1] - dphase_final[i - 1, 1]) ** 2)
# # distance = np.sqrt((xy_inside[:, 0] - xy_inside[int(((n_i) * (n_i) - 1) / 2), 0]) ** 2 + (xy_inside[:, 1] - xy_inside[int(((n_i) * (n_i) - 1) / 2), 1]) ** 2)
return np.sum(part_length)
# prepare data without any correction
peaks_no_corr, peaks_ref_no_corr = without_correction(mixer_filtered, ref_saw_filtered)
# calculate lost phase and dphase_zero - dphase without any correction
lost_phase, dphase_zero = calc_lost_phase(peaks_no_corr, peaks_ref_no_corr)
repaired_discharge = False
unrepairable_discharge = False
number_of_bad_peaks = round(lost_phase / (2 * math.pi))
# make data repair, if it is necessary
if 1 <= number_of_bad_peaks:
try:
dphase_final, bad_peaks_indexes, best_smooth_factor, best_distance = repair(dphase_zero, peaks_no_corr, peaks_ref_no_corr)
dphase_final[:, 1] = smooth(dphase_final[:, 1], 100)
repaired_discharge = True
except:
unrepairable_discharge = True
print('Cannot repair data.')
The ordinary wave (O-mode) with a carrier frequency $\omega$ traveling through a collisionless plasma with the plasma frequency $\omega_{p} = \sqrt{\frac{n_e e^2}{\epsilon_0 m_e}}$ has a refractive index $$N_O=\sqrt{1-\frac{\omega_p^2}{\omega^2}}$$ Under the assumption that the carrier wave frequency is much larger than the plasma frequency $\omega>>\omega_p$ this formula can be expanded into a Taylor series as $$N_O\approx 1-\frac{\omega_p^2}{2\omega^2}$$ A wave traveling through a medium with a refractive index $N(l)$ accumulates a total phase shift $\varphi = \frac{\omega}{c} \int N(l) \mathrm{d}l$. Therefore, in comparison to a wave traveling in vacuum (or clear air) with $N\approx 1$, the wave traveling through the plasma over a chord with length $L$ accumulates a relative phase shift of $$\Delta \varphi = \frac{e^2}{2\omega c\epsilon_0 m_e}\int\limits_L n_e(l) \mathrm{d}l$$ Therefore, it is possible to estimate the line-averaged density $\bar n_e = \frac{1}{L} \int\limits_L n_e(l) \mathrm{d}l$ from the detected phase shift between the reference and mixer signals.
omega_carrier = 2 * np.pi * 71e9 # 71 GHz microwave generator
a = 0.085 # limiter radius [m]
L = 2 * a # central plasma chord estimate
prop_const = constants.elementary_charge ** 2 / (2 * omega_carrier * constants.speed_of_light * constants.epsilon_0 * constants.m_e)
ne_lav - final line-averaged electron density
if repaired_discharge:
ne_lav = dphase_final.copy()
ne_lav[:, 1] = ne_lav[:, 1] * (1 / (prop_const * L))
# ne_lav = ne_lav[(ne_lav[:, 0] >= 0) & (ne_lav[:, 0] <= (t_plasma_end + 5))]
dphase_zero_raw = dphase_zero.copy()
dphase_zero[:, 1] = smooth(dphase_zero[:, 1], 100)
ne_lav_zero - final line-averaged electron density without any correction
ne_lav_zero = dphase_zero.copy()
ne_lav_zero[:, 1] = ne_lav_zero[:, 1] * (1 / (prop_const * L))
# ne_lav_zero = ne_lav_zero[(ne_lav_zero[:, 0] >= 0) & (ne_lav_zero[:, 0] <= (t_plasma_end + 5))]
ne_lav_zero_raw - final line-averaged electron density without any correction and without any smoothing - used to see more clearly, where the data was damaged and repaired in the last figure
ne_lav_zero_raw = dphase_zero_raw.copy()
ne_lav_zero_raw[:, 1] = ne_lav_zero_raw[:, 1] * (1 / (prop_const * L))
# ne_lav_zero_raw = ne_lav_zero_raw[(ne_lav_zero_raw[:, 0] >= 0) & (ne_lav_zero_raw[:, 0] <= (t_plasma_end + 5))]
if repaired_discharge:
fig, ax = plt.subplots(dpi=200)
ax.plot(ne_lav_zero[:, 0], ne_lav_zero[:, 1] / 1e18, label='damaged signal', linestyle='dotted', color='red')
ax.plot(ne_lav[:, 0], ne_lav[:, 1] / 1e18, label='repaired signal', color='deepskyblue')
if (np.max(ne_lav_zero[:, 1] / 1e18) < 0.1):
ax.set_ylim([-0.1, 5])
else:
for t in (t_plasma_start, t_plasma_end):
plt.axvline(t, color='k', linestyle='--')
ax.set(xlabel='time [ms]', ylabel='$\\bar n_e$ [10$^{18}$ m$^{-3}$]')
plt.title('Line-averaged electron density (repaired waveforms: {})'.format(number_of_bad_peaks))
plt.legend()
plt.grid()
plt.show()
fig, ax = plt.subplots(dpi=200)
if repaired_discharge:
ax.plot(ne_lav[:, 0], ne_lav[:, 1] / 1e18, label='$\\bar n_e$', color='deepskyblue')
else:
ax.plot(ne_lav_zero[:, 0], ne_lav_zero[:, 1] / 1e18, label='$\\bar n_e$', color='deepskyblue')
if (np.max(ne_lav_zero[:, 1] / 1e18) < 0.1):
ax.set_ylim([-0.1, 5])
else:
for t in (t_plasma_start, t_plasma_end):
plt.axvline(t, color='k', linestyle='--')
ax.set(xlabel='time [ms]', ylabel='$\\bar n_e$ [10$^{18}$ m$^{-3}$]')
if unrepairable_discharge:
plt.title('Line-averaged electron density (!!!UNREPAIRED waveforms: {}!!!)'.format(number_of_bad_peaks))
else:
plt.title('Line-averaged electron density (repaired waveforms: {})'.format(number_of_bad_peaks))
plt.legend()
plt.grid()
plt.savefig('icon-fig.png')
plt.show()
if repaired_discharge:
fig, ax = plt.subplots(dpi=200)
ax.plot(ne_lav_zero_raw[:, 0], ne_lav_zero_raw[:, 1] / 1e18, label='damaged signal', color='red')
ax.plot(ne_lav[:, 0], ne_lav[:, 1] / 1e18, label='$\\bar n_e$', color='deepskyblue')
x, y = bad_peaks_indexes.shape
for i in range(0, x):
plt.axvline(peaks_no_corr[int(bad_peaks_indexes[i, 0]), 0], color='orange', linestyle='--')
ax.set(xlabel='time [ms]', ylabel='$\\bar n_e$ [10$^{18}$ m$^{-3}$]')
plt.title('best smooth factor: {}, best distance: {}, repaired waveforms: {}'.format(best_smooth_factor, best_distance, number_of_bad_peaks))
plt.legend()
plt.grid()
plt.show()
file ne_lav.csv - the final line-averaged electron density data file ne_lav_max.txt - max value of the line-averaged electron density file ne_lav_mean.txt - mean value of the line-averaged electron density
if repaired_discharge:
np.savetxt('ne_lav.csv', ne_lav, delimiter=",") # repaired data
ne_lav_plasma = ne_lav[(0.1 < ne_lav[:, 1] / 1e18)]
for m in ('mean', 'max'):
v = getattr(ne_lav_plasma[:, 1], m)()
with open(f'ne_lav_{m}', 'w') as f: f.write(f'{v:.3e}')
else:
np.savetxt('ne_lav.csv', ne_lav_zero, delimiter=",")
if unrepairable_discharge:
for m in ('mean', 'max'):
v = float('NaN') # unrepaired data
with open(f'ne_lav_{m}', 'w') as f: f.write(f'{v:.3e}')
else:
if np.max(ne_lav_zero[:, 1]/1e18) < 0.1:
for m in ('mean', 'max'):
v = 0 # no plasma
with open(f'ne_lav_{m}', 'w') as f:f.write(f'{v:.3e}')
else:
ne_lav_plasma = ne_lav_zero[(0.1 < ne_lav_zero[:, 1] / 1e18)]
for m in ('mean', 'max'):
v = getattr(ne_lav_plasma[:, 1], m)()
with open(f'ne_lav_{m}', 'w') as f: f.write(f'{v:.3e}')