When a signal is propagating through multiple paths each signal component in each path will be in different phase and of different strength when received. Should there be no line-of-sight component present, the additive signal will fade according to the Rayleigh fading.
For example, a simple sine wave
can after some multi-path propagation sound something like
assuming that the receiver is moving at a constant speed, so that variation in multi-path sources Doppler shift will cause the aggregate signal to vary randomly in time.
If we add some Gaussian white noise
we notice that the original signal is somehow recognizable from the noise (headphone users, watch out!)
but the faded signal
will sometimes bury completely under the noise – these events are referred to as deep fades.
Here is a GNU/Octave or Matlab code for the used Rayleigh simulator:
%Rayleigh simulator. Jakes model. In Octave remember to load the statistics package. close all clear all tic N = 20; %Number multipaths. T= linspace(0,10000,100000); %Time. v = 0.001; %Speed of the receiver. randan = pi*rand(1,N); %Random angles w.r.t. receiver. rI = 1000*rand(1,N); %Random distances of the sources. %Geometrical stuff. Check for the Jakes model in the reference. An = @(t) (atan(sin(randan).*rI./(cos(randan).*rI-v*t))).*(cos(randan).*rI>v*t)+... (pi-atan(sin(randan).*rI./(v*t-cos(randan).*rI))).*(cos(randan).*rI<=v*t); phis = 2*pi.*(rand(1,N)); wc =pi/2; %Frequency of the signal. Beta = 2*pi/(1/(wc/(2*pi))); theta = @(t) cos(An(t)).*Beta*v.*t+phis; powers = rand(1,N); %Random powers of the signals in the multipaths. powers = powers./sum(powers); %Normalize the powers. Ez = @(t) sum(powers.*cos(wc*t+theta(t))); EZ = ; %Faded signal. REF = ; %Original signal. NOISE = ; %Additional Gaussian noise. for t =T EZ = [EZ Ez(t)]; REF = [REF cos(wc*t)]; NOISE = [NOISE 0.9*stdnormal_rnd(1)]; end %Write the audio files at sampling rate 8000. audiowrite('EZ.wav',EZ,8000) audiowrite('EZNOISE.wav',1/2*(EZ+NOISE),8000) audiowrite('REFNOISE.wav', 1/2*(REF+NOISE), 8000) toc plot(T,EZ)
And the same in Python:
import numpy as np import math import matplotlib.pyplot as plt import sounddevice as sd import time #Jakes Rayleigh simulator. Please check for the reference in this site for further details. N = 20 #Number of multipaths. T = np.linspace(0, 10000, 100000) #Time vector. v = 0.001 #Speed of the receiver. randan = np.random.rand(1, N) * math.pi #Random angles w.r.t. receiver. rI = np.random.rand(1, N) * 1000 #Random distances of the sources. phis = np.random.rand(1, N) * 2 * math.pi def An(t): return (np.arctan(np.sin(randan) * rI / (np.cos(randan) * rI - v * t))) * ( np.cos(randan) * rI > v * t ) + (np.pi - np.arctan(np.sin(randan) * rI / (v * t - np.cos(randan) * rI))) * ( np.cos(randan) * rI <= v * t ) wc = np.pi/2 #Frequency of the signal. Beta = 2 * math.pi / (1 / (wc / (2 * math.pi))) def theta(t): return np.cos(An(t)) * Beta * v * t +phis powers = np.random.rand(1,N)*2 #Random powers of the signals in the multipaths. powers = powers/np.sum(powers) #Normalize powers.. def Ez(t): return np.sum(powers * np.cos(wc * t + theta(t))) EZ = np.vectorize(Ez)(T) REF = np.vectorize(lambda time : 2*np.cos(wc * time))(T) #Play and plot. fs = 8000 sd.play(EZ,fs,blocking = True) #sd.play(REF,fs,blocking = True) plt.plot(T, EZ) plt.show()