Figure: Fourier Waveform and DTFT/DFT

figures/methods/fourier/Makefile

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+.PHONY: all
+
+all: waveforms.pdf
+
+.PHONY: clean
+clean:
+	@rm -v waveforms.*
+	@rm -v spectrum.*
+
+waveforms.% spectrum.% : src/fourier_figure.py
+	$< .

figures/methods/fourier/src/fourier_figure.py

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