Table of contents

# 1. Algorithm description

The Rayleigh distribution is a stationary narrow-band Gaussian process with a mean of 0 and a variance of σ², and the one-dimensional distribution of its envelope is the Rayleigh distribution. Its expression and probability density are shown in the figure. The Rayleigh distribution is the most common type of distribution used to describe the statistical time-varying characteristics of the received envelope of a flat fading signal or the received envelope of an independent multipath component. The envelope of the sum of two orthogonal Gaussian noise signals follows a Rayleigh distribution.

Rayleigh fading can effectively describe the wireless propagation environment in the presence of obstacles that can largely scatter radio signals. If there is enough scattering in the propagation environment, the impulse signal will appear as a superposition of a large number of statistically independent random variables after reaching the receiver. According to the central limit theorem, the impulse response of this wireless channel will be a Gaussian process. If there is no dominant signal component in this scatter channel, usually this condition means no line-of-sight signal (LoS), then the mean of this process is 0 and the phase follows a uniform distribution from 0 to 2π. That is, the energy or envelope of the channel response follows a Rayleigh distribution. If there is a main component in the channel, such as the direct signal (LoS), the envelope of the channel response obeys the Rice distribution, and the corresponding channel model is the Rice fading channel. Usually, the channel gain is represented by an equivalent baseband signal, that is, a complex number is used to represent the amplitude and phase characteristics of the channel. Rayleigh fading can thus be represented by this complex number whose real and imaginary parts are subject to an independent and identically distributed Gaussian process with zero mean. the

A Rayleigh fading channel is a statistical model of the radio signal propagation environment. This model assumes that after the signal passes through the wireless channel, its signal amplitude is random, that is, "fading", and its envelope obeys the Rayleigh distribution. This channel model is capable of describing shortwave channels reflected by the ionosphere and troposphere, as well as densely built urban environments. Rayleigh fading is only applicable when there is no direct signal (LoS, Line of Sight) from the transmitter to the receiver, otherwise the Rician fading channel should be used as the channel model. the

Rayleigh Fading: In a wireless communication channel, since the signal multipath propagates and reaches the field strength at the receiving point from different propagation paths, the delay time of each path is different, and the superposition of component waves in each direction , and the standing wave field strength is generated, thus forming a fast signal fading called Rayleigh fading. Rayleigh fading is a small-scale fading effect, which is always superimposed on large-scale fading effects such as shadowing and attenuation.

Due to factors such as multipath and mobile station movement, the mobile channel causes dispersion to the transmission signal in time, frequency and angle, such as time dispersion, frequency dispersion, angle dispersion, etc. Therefore, the characteristics of multipath channels have a great impact on communication quality. The crucial influence, and the envelope statistical characteristics of the multipath channel become the focus of our research. According to different wireless environments, the received signal envelope generally obeys several typical distributions, such as Rayleigh distribution.

When there is no strong direct path in the channel, its signal envelope obeys Rayleigh distribution. In mobile wireless channels, Rayleigh distribution is a common type of distribution used to describe the statistical time-varying characteristics of received envelopes of flat fading signals or independent multipath components. It is well known that the envelope of the sum of two orthogonal noise signals obeys the Rayleigh distribution. The probability density function (pdf) for the Rayleigh distribution is:

The block diagram of the channel model is shown in Figure 2:

# 2. Simulation effect preview

The matlab2022a simulation results are as follows:

# 3.MATLAB core program

LengthOfSignal=10240; %signal length(preferably more than twice fc) fm=512; %Maximum Doppler Shift fc=5120; %carrier frequency t=1:LengthOfSignal; % SignalInput=sin(t/100); SignalInput=sin(t/100)+cos(t/65); %signal input delay=[0 31 71 109 173 251]; power=[0 -1 -9 -10 -15 -20]; %dB y_in=[zeros(1,delay(6)) SignalInput]; %zero padding for time shift y_out=zeros(1,LengthOfSignal); %for signal output for i=1:6 Rayl; y_out=y_out+r.*y_in(delay(6)+1-delay(i):delay(6)+LengthOfSignal-delay(i))*10^(power(i)/20); end; figure(1); subplot(2,1,1); plot(SignalInput(delay(6)+1:LengthOfSignal)); %Remove the blank signal caused by time delay title('Signal Input'); subplot(2,1,2); plot(y_out(delay(6)+1:LengthOfSignal)); %Remove the blank signal caused by time delay title('Signal Output'); figure(2); subplot(2,1,1); hist(r,256); title('Amplitude Distribution Of Rayleigh Signal') subplot(2,1,2); hist(angle(r0)); title('Angle Distribution Of Rayleigh Signal'); figure(3); plot(Sf1); title('The Frequency Response of Doppler Filter'); A211

# 4. Complete MATLAB

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