197 lines
5.9 KiB
Python
197 lines
5.9 KiB
Python
#!/usr/bin/env python3
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__doc__ = """
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Phase and Amplitude distributions for a phasor in the presence of noise.
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Author: Harm Schoorlemmer
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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
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import scipy.stats as stat
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from scipy import special
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from lib.util import MethodMappingProxy as MethodProxy
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from lib.ft_plot import axis_pi_ticker
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def expectation(x,pdfx):
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dx = x[1]-x[0]
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return np.sum(x*pdfx*dx)
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def variance(x,pdfx):
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mu = expectation(x,pdfx)
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dx = x[1]-x[0]
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return np.sum((x**2*pdfx*dx))-mu**2
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def phase_distribution(theta,sigma,s):
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theta = np.asarray(theta)
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ct = np.cos(theta)
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st = np.sin(theta)
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k=s/sigma
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pipi=2*np.pi
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return (np.exp(-k**2/2)/pipi) + (
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(pipi**-0.5)*k*np.exp(-(k*st)**2/2)) * (
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(1.+special.erf(k*ct*2**-0.5))*ct/2)
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def phase_distribution_gauss(theta,sigma,s):
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theta = np.asarray(theta)
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k=s/sigma
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return (2*np.pi)**-0.5*k*np.exp(-(k*theta)**2/2)
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def amplitude_distribution(a,sigma,s):
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a = np.asarray(a)
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k = s/sigma
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return stat.rice.pdf(a,s/sigma,scale=sigma)
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def amplitude_distribution_gauss(a,sigma,s):
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a = np.asarray(a)
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k=s/sigma
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return (2*np.pi)**-0.5*np.exp(-((a-s)/sigma)**2/2)
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figsize = (8,6)
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if True:
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from matplotlib import rcParams
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#rcParams["text.usetex"] = True
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rcParams["font.family"] = "serif"
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plt.rc('lines',lw=2)
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if True:# small
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figsize = (6, 4)
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rcParams["font.size"] = "15" # 15 at 6,4 looks fine
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elif True: # large
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figsize = (9, 6)
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rcParams["font.size"] = "16" # 15 at 9,6 looks fine
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rcParams["grid.linestyle"] = 'dotted'
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rcParams["figure.figsize"] = figsize
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signal_max= 4
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amp_max = signal_max*2
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thetas = np.linspace(-np.pi,np.pi,500)
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amplitudes = np.linspace(0,amp_max,500)
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signals = np.linspace(0,signal_max,5)
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sigma = 1
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## figure 1
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if True:
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fig, ax = plt.subplots(1,2,figsize=(2*figsize[0],figsize[1]))
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_fig1, _ax0 =plt.subplots(1,1)
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_fig2, _ax1 =plt.subplots(1,1)
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ax0 = MethodProxy(ax[0], _ax0)
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ax1 = MethodProxy(ax[1], _ax1)
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for s in signals:
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pdfs_label='s = '+str(int(s))
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phase_vals= phase_distribution(thetas,sigma,s)
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amp_vals= amplitude_distribution(amplitudes,sigma,s)
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phase_vals_g = phase_distribution_gauss(thetas,sigma,s)
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ax0.plot(amplitudes,amp_vals, label=pdfs_label)
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ax1.plot(thetas,phase_vals, label=pdfs_label)
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ax0.legend()
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ax0.set_xlabel(r'Amplitude $a$')
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ax0.set_ylabel(r'$p(a)$')
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ax1.set_xlabel(r'Phase $\varphi$')
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ax1.set_ylabel(r'$p(\varphi)$')
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_ax1.legend()# only in the separate figure
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[ axis_pi_ticker(ax.xaxis, major_divider=3) for ax in ax1.elements ]
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for a in [ax0, ax1]:
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a.grid()
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MethodProxy(fig, _fig1, _fig2).tight_layout()
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fig.savefig('pdfs.pdf')
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_fig1.savefig('pdfs-amplitudes.pdf')
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_fig2.savefig('pdfs-phases.pdf')
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plt.close(_fig1)
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plt.close(_fig2)
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## figure 2
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amplitudes = np.linspace(0,amp_max*5,500)
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signals = np.linspace(0.1,signal_max*5,101)
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if True:
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V_theta = [variance(thetas,phase_distribution(thetas,sigma,s)) for s in signals ]
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E_theta=[expectation(thetas,phase_distribution(thetas,sigma,s)) for s in signals ]
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V_theta_g = [variance(thetas,phase_distribution_gauss(thetas,sigma,s)) for s in signals ]
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E_theta_g=[expectation(thetas,phase_distribution_gauss(thetas,sigma,s)) for s in signals ]
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V_a = [variance(amplitudes,amplitude_distribution(amplitudes,sigma,s)) for s in signals ]
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E_a=[expectation(amplitudes,amplitude_distribution(amplitudes,sigma,s)) for s in signals ]
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V_a_g = [variance(amplitudes,amplitude_distribution_gauss(amplitudes,sigma,s)) for s in signals ]
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E_a_g=[expectation(amplitudes,amplitude_distribution_gauss(amplitudes,sigma,s)) for s in signals ]
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fig2, _ax2 = plt.subplots(2,2,figsize=(2*figsize[0],2*figsize[1]))
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ax2 = fig2.get_axes()
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if True:
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_figs = []
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_axs = []
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for i, ax in enumerate(ax2):
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_f, _a = plt.subplots(1,1)
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_figs.append(_f)
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_axs.append(_a)
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ax2[i] = MethodProxy(ax, _a)
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ax2[0].plot(signals,E_a,label='$p(a)$')
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ax2[0].plot(signals,E_a_g,ls='dashed',label='Gaussian approx.')
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ax2[0].set_xscale('log')
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ax2[0].set_yscale('log')
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ax2[0].set_ylabel('$\mu_a$')
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ax2[1].plot(signals,V_a,label='$p(a)$')
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ax2[1].plot(signals,V_a_g,ls='dashed',label='Gaussian approx.')
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ax2[1].set_xscale('log')
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ax2[1].set_ylabel('$\sigma_a^2$')
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ax2[2].plot(signals,E_theta,label=r'$p(\varphi)$')
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ax2[2].plot(signals,E_theta_g,ls='dashed',label='Gaussian approx.')
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ax2[2].set_xscale('log')
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ax2[2].set_ylim(-1.1,1.1)
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ax2[2].set_ylabel(r'$\mu_\varphi$')
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ax2[3].plot(signals,V_theta,label=r'$p(\varphi)$')
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ax2[3].plot(signals,V_theta_g,ls='dashed',label='Gaussian approx.')
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ax2[3].set_xscale('log')
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ax2[3].set_yscale('log')
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ax2[3].set_ylabel(r'$\sigma_\varphi^2$')
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for a in ax2:
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a.grid(which='both')
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a.set_xlabel(r'$s/\sigma$')
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a.legend()
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fig2.tight_layout()
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fig2.savefig('expectation_variance.pdf')
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for i, _f in enumerate(_figs):
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fnames = [
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'amplitude_mean',
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'amplitude_sigma',
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'phase_mean',
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'phase_sigma',
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][i]
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_f.tight_layout()
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_f.savefig(fnames+'.pdf')
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plt.close(_f)
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## figure 3, beacon timing accuracy
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if True:
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fig3, ax3 = plt.subplots(1,1)
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ax3 = fig3.get_axes()
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sigma_t = [variance(thetas,phase_distribution(thetas,sigma,s)) for s in signals ]
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lfs=np.linspace(np.log10(50.),4,1)
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for lf in lfs:
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freq = (10**lf)*1e6
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sigma_t = np.asarray(sigma_t)**0.5/(2*np.pi*freq)
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ax3[0].plot(signals,sigma_t/1e-9,'o-')
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ax3[0].set_ylim(0,2.5)
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ax3[0].set_xlabel(r'$s/\sigma$')
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ax3[0].set_ylabel(r'$\Delta t$(ns)')
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fig3.tight_layout()
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fig3.savefig('timing_accuracy.pdf')
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plt.show()
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