PDFs: modified script from Harm for PhasorSum distributions

simulations/12_noise_phase.py

@@ -0,0 +1,154 @@
+#!/usr/bin/env python3
+
+__doc__ = """
+Phase and Amplitude distributions for a phasor in the presence of noise.
+
+Author: Harm Schoorlemmer
+"""
+
+import numpy as np
+import matplotlib.pyplot as plt
+import matplotlib
+import scipy.stats as stat
+from scipy import special
+
+from lib.util import MethodMappingProxy as MethodProxy
+
+def expectation(x,pdfx):
+    dx = x[1]-x[0]
+    return np.sum(x*pdfx*dx)
+
+def variance(x,pdfx):
+    mu = expectation(x,pdfx)
+    dx = x[1]-x[0]
+    return np.sum((x**2*pdfx*dx))-mu**2
+
+def phase_distribution(theta,sigma,s):
+    theta = np.asarray(theta)
+    ct = np.cos(theta)
+    st = np.sin(theta)
+    k=s/sigma
+    pipi=2*np.pi
+    return (np.exp(-k**2/2)/pipi) + (
+        (pipi**-0.5)*k*np.exp(-(k*st)**2/2)) * (
+        (1.+special.erf(k*ct*2**-0.5))*ct/2)
+
+def phase_distribution_gauss(theta,sigma,s):
+    theta = np.asarray(theta)
+    k=s/sigma
+    return (2*np.pi)**-0.5*k*np.exp(-(k*theta)**2/2)
+
+def amplitude_distribution(a,sigma,s):
+    a = np.asarray(a)
+    k = s/sigma
+    return stat.rice.pdf(a,s/sigma,scale=sigma)
+
+def amplitude_distribution_gauss(a,sigma,s):
+    a = np.asarray(a)
+    k=s/sigma
+    return (2*np.pi)**-0.5*np.exp(-((a-s)/sigma)**2/2)
+
+signal_max= 4
+amp_max = signal_max*2
+thetas = np.linspace(-np.pi,np.pi,500)
+amplitudes = np.linspace(0,amp_max,500)
+signals = np.linspace(0,signal_max,5)
+sigma = 1
+
+
+## figure 1
+if True:
+    fig, ax = plt.subplots(1,2,figsize=(2*8,1*8))
+    _fig1, _ax0 =plt.subplots(1,1, figsize=(1*8, 1*8))
+    _fig2, _ax1 =plt.subplots(1,1, figsize=(1*8, 1*8))
+
+    ax0 = MethodProxy(ax[0], _ax0)
+    ax1 = MethodProxy(ax[1], _ax1)
+    for s in signals:
+        pdfs_label='s/$\sigma$ ='+str(s)
+        phase_vals= phase_distribution(thetas,sigma,s)
+        amp_vals= amplitude_distribution(amplitudes,sigma,s)
+        phase_vals_g = phase_distribution_gauss(thetas,sigma,s)
+
+        ax0.plot(amplitudes,amp_vals, label=pdfs_label)
+        ax1.plot(thetas,phase_vals,   label=pdfs_label)
+    ax0.legend()
+    _ax1.legend()# only in the separate figure
+    ax0.set_xlabel(r'$a$')
+    ax0.set_xlabel(r'$\theta$')
+    ax1.set_ylabel(r'$p(a)$')
+    ax1.set_ylabel(r'$p(\theta)$')
+    # ax[0].grid()
+    # ax[1].grid()
+    MethodProxy(fig, _fig1, _fig2).tight_layout()
+
+    fig.savefig('pdfs.pdf')
+    _fig1.savefig('pdfs-amplitudes.pdf')
+    _fig2.savefig('pdfs-phases.pdf')
+
+    plt.close(_fig1)
+    plt.close(_fig2)
+
+## figure 2
+amplitudes = np.linspace(0,amp_max*5,500)
+signals = np.linspace(0.1,signal_max*5,101)
+if False:
+    fig2, ax2 = plt.subplots(2,2,figsize=(2*8,2*8))
+    ax2 = fig2.get_axes()
+    V_theta = [variance(thetas,phase_distribution(thetas,sigma,s)) for s in signals ]
+    E_theta=[expectation(thetas,phase_distribution(thetas,sigma,s)) for s in signals ]
+    V_theta_g = [variance(thetas,phase_distribution_gauss(thetas,sigma,s)) for s in signals ]
+    E_theta_g=[expectation(thetas,phase_distribution_gauss(thetas,sigma,s)) for s in signals ]
+    V_a = [variance(amplitudes,amplitude_distribution(amplitudes,sigma,s)) for s in signals ]
+    E_a=[expectation(amplitudes,amplitude_distribution(amplitudes,sigma,s)) for s in signals ]
+    V_a_g = [variance(amplitudes,amplitude_distribution_gauss(amplitudes,sigma,s)) for s in signals ]
+    E_a_g=[expectation(amplitudes,amplitude_distribution_gauss(amplitudes,sigma,s)) for s in signals ]
+
+    ax2[0].plot(signals,E_a,label='$p(a)$')
+    ax2[0].plot(signals,E_a_g,ls='dashed',label='Gaussian approx.')
+    ax2[0].set_xscale('log')
+    ax2[0].set_yscale('log')
+    ax2[0].set_ylabel('$\mu_a$')
+    
+    ax2[1].plot(signals,V_a,label='$p(a)$')
+    ax2[1].plot(signals,V_a_g,ls='dashed',label='Gaussian approx.')
+    ax2[1].set_xscale('log')
+    ax2[1].set_ylabel('$\sigma_a^2$')
+    
+    ax2[2].plot(signals,E_theta,label=r'$p(\theta)$')
+    ax2[2].plot(signals,E_theta_g,ls='dashed',label='Gaussian approx.')
+    ax2[2].set_xscale('log')
+    ax2[2].set_ylim(-1.1,1.1)
+    ax2[2].set_ylabel(r'$\mu_\theta$')
+    
+    ax2[3].plot(signals,V_theta,label=r'$p(\theta)$')
+    ax2[3].plot(signals,V_theta_g,ls='dashed',label='Gaussian approx.')
+    ax2[3].set_xscale('log')
+    ax2[3].set_yscale('log')
+    ax2[3].set_ylabel(r'$\sigma_\theta^2$')
+    for a in ax2:
+        a.grid(which='both')
+        a.set_xlabel(r'$s/\sigma$')
+        a.legend()
+    
+    fig2.tight_layout()
+    fig2.savefig('expectation_variance.pdf')
+
+## figure 3, beacon timing accuracy
+if True:
+    fig3, ax3 = plt.subplots(1,1,figsize=(1*8,1*8))
+    ax3 = fig3.get_axes()
+    sigma_t = [variance(thetas,phase_distribution(thetas,sigma,s)) for s in signals ]
+    lfs=np.linspace(np.log10(50.),4,1)
+    for lf in lfs:
+        freq = (10**lf)*1e6
+        sigma_t = np.asarray(sigma_t)**0.5/(2*np.pi*freq)
+        ax3[0].plot(signals,sigma_t/1e-9,'o-')
+        ax3[0].set_ylim(0,2.5)
+        ax3[0].set_xlabel(r'$s/\sigma$')
+        ax3[0].set_ylabel(r'$\Delta t$(ns)')
+
+    fig3.tight_layout()
+    fig3.savefig('timing_accuracy.pdf')
+
+plt.show()