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Figure: Fourier Waveform and DTFT/DFT
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figures/methods/fourier/Makefile
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figures/methods/fourier/Makefile
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.PHONY: all
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all: waveforms.pdf
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.PHONY: clean
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clean:
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@rm -v waveforms.*
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@rm -v spectrum.*
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waveforms.% spectrum.% : src/fourier_figure.py
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$< .
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figures/methods/fourier/spectrum.pdf
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figures/methods/fourier/spectrum.pdf
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figures/methods/fourier/src/fourier_figure.py
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figures/methods/fourier/src/fourier_figure.py
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#!/usr/bin/env python3
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# vim: fdm=marker fmr=<<<,>>>
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__doc__ = \
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"""
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Create one/two figures exemplifiying the fourier transform of noisy sine wave.
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"""
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import numpy as np
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import matplotlib.pyplot as plt
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import matplotlib.gridspec as gridspec
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import scipy.fft as ft
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rng = np.random.default_rng()
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def fft_spectrum( signal, sample_rate, fft=None, freq=None):
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N_samples = len(signal)
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real_signal = np.isrealobj(signal)
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if fft is None:
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if real_signal:
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fft = ft.rfft
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freq = ft.rfftfreq
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else:
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fft = ft.fft
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freq = ft.fftfreq
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if freq is None:
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freq = ft.fftfreq
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spectrum = fft(signal) / len(signal)
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if real_signal:
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spectrum *= 2
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freqs = freq(N_samples, 1/sample_rate)
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return spectrum, freqs
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def dtft_spectrum(signal, time, freqs):
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freqtime = np.outer(freqs, time)
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# determine all coefficients in one go
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c_k = 2*np.cos(2*np.pi*freqtime) / len(signal)
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s_k = -2*np.sin(2*np.pi*freqtime) / len(signal)
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# dot product gives the dtft
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result = np.dot(c_k, signal) + 1j*np.dot(s_k, signal)
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return result, freqs
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## From https:
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def multiple_formatter(denominator=2, number=np.pi, latex='\pi'):
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def gcd(a, b):
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while b:
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a, b = b, a%b
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return a
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def _multiple_formatter(x, pos):
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den = denominator
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num = int(np.rint(den*x/number))
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com = gcd(num,den)
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(num,den) = (int(num/com),int(den/com))
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if den==1:
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if num==0:
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return r'$0$'
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if num==1:
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return r'$%s$'%latex
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elif num==-1:
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return r'$-%s$'%latex
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else:
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return r'$%s%s$'%(num,latex)
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else:
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if num==1:
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return r'$\frac{%s}{%s}$'%(latex,den)
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elif num==-1:
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return r'$\frac{-%s}{%s}$'%(latex,den)
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else:
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return r'$\frac{%s%s}{%s}$'%(num,latex,den)
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return _multiple_formatter
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#
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def phase_plot(ax, phase_on_xaxis=False, limits=(-1*np.pi, +1*np.pi), major_ticks_div=1, minor_ticks_div=4):
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set_label = ax.set_ylabel
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set_lims = ax.set_ylim
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axis = ax.yaxis
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if phase_on_xaxis:
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set_label = ax.set_xlabel
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set_lims = ax.set_ylim
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axis = ax.xaxis
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set_label("Phase")
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if limits is not None:
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set_lims(*limits)
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ax.margins(y=0.2)
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if major_ticks_div:
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axis.set_major_locator(plt.MultipleLocator(np.pi / major_ticks_div))
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if minor_ticks_div:
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axis.set_minor_locator(plt.MultipleLocator(np.pi / minor_ticks_div))
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axis.set_major_formatter(plt.FuncFormatter(multiple_formatter()))
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return ax
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def main(
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f_sine=0.05153, # GHz
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noise_sigma = 0.5,
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sine_amplitude = 1,
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phase=0.4,
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):
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t_fine = np.linspace(0, 200, 500) # ns
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t_sampled = np.linspace(0, 200, 50) # ns
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sine_fine = sine_amplitude * np.cos( 2*np.pi * f_sine * t_fine + phase)
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sine_sampled = sine_amplitude * np.cos( 2*np.pi * f_sine * t_sampled + phase)
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combined = noise_sigma*rng.normal(size=len(sine_sampled)) + sine_sampled
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figs = []
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# time domain
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if True:
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fig, ax = plt.subplots()
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ax.set_xlabel("Time [ns]")
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ax.set_ylabel("Amplitude")
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ax.plot(t_fine, sine_fine, label="Clean Signal")
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ax.plot(t_sampled, combined, label="+ Noise", marker='o')
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ax.legend()
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ax.grid()
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figs.append(ax.get_figure())
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# frequency domain
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if True:
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fft_spec, fft_freqs = fft_spectrum(combined, 1/(t_sampled[1] - t_sampled[0]))
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dtft_freqs = np.linspace(fft_freqs[0], fft_freqs[-1], 500)
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dtft_spec, dtft_freqs = dtft_spectrum(combined, t_sampled, dtft_freqs)
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fig2 = plt.figure()
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gs = gridspec.GridSpec(2, 1, figure=fig2, height_ratios=[3,1], hspace=0)
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ax1 = fig2.add_subplot(gs[:-1, -1])
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ax2 = fig2.add_subplot(gs[-1, -1], sharex=ax1)
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axes = np.array([ax1, ax2])
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ax2.set_xlabel("Frequency [GHz]")
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if True:
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dtft_freqs *= 1e3
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fft_freqs *= 1e3
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f_sine *= 1e3
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ax2.set_xlabel("Frequency [MHz]")
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ax1.xaxis.tick_top()
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[label.set_visible(False) for label in ax1.get_xticklabels()]
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# indicate f_sine
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for ax in axes:
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ax.axvline(f_sine, label="$f_\mathrm{sin}$", ls='dashed', color='red')
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# plot amplitudes
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ax1.set_ylabel("Amplitude")
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ax1.set_ylim(0, None)
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ax1.yaxis.get_major_ticks()[0].set_visible(False)# suppress 0.0
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ax1.grid()
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ax1.plot(fft_freqs, np.abs(fft_spec), marker='o', label='DFT')
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ax1.plot(dtft_freqs, np.sqrt(np.abs(dtft_spec)**2), label='DTFT')
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ax1.legend()
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# plot phases
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phase_plot(ax2, limits=[-1.1*np.pi, +1.1*np.pi])
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ax2.grid()
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ax2.plot(fft_freqs, np.angle(fft_spec), marker='o', label='DFT')
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ax2.plot(dtft_freqs, np.angle(dtft_spec), label='DTFT')
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figs.append(fig2)
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return figs
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if __name__ == "__main__":
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from argparse import ArgumentParser
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import os.path as path
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import os
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import sys
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# Append parent directory to import path so pyfiglib can be found
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sys.path.append(os.path.dirname(os.path.dirname(os.path.dirname(os.path.dirname(os.path.abspath(__file__))))))
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import pyfiglib as pfl
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parser = ArgumentParser(description=__doc__)
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parser.add_argument("fname", metavar="path/to/figure[/]", nargs="*", help="Location for generated figure, will append __file__ if a directory. If not supplied, figure is shown.", default=os.getcwd())
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args = parser.parse_args()
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default_extensions = ['pdf', 'png']
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default_names = ['waveforms', 'spectrum']
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if args.fname == 'none':
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args.fname = None
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pfl.rcParams['font.size'] = 20
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pfl.rcParams['figure.figsize'] = (8,6)
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pfl.rcParams['figure.constrained_layout.use'] = True
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###
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figs = main()
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### Save or show figures
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if not args.fname:
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# empty list, False, None
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plt.show()
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else:
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for i, f in enumerate(figs):
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if len(args.fname) == 1 and len(figs) != 1:
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for ext in default_extensions:
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f.savefig(path.join(args.fname[0], default_names[i]) + "." + ext, transparent=True)
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else:
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f.savefig(args.fname[i], transparent=True)
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figures/methods/fourier/waveforms.pdf
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figures/methods/fourier/waveforms.pdf
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