Calculation of Misjudgment Probability in Receiver Composition Diversity System
topic
The same incident electromagnetic wave is received by M receivers, and the signal received by each receiver is:
![](https://img-blog.csdnimg.cn/img_convert/fd8a9996cfdcee2c286b59f0af428ea5.png)
{ ni ( t )} is a Gaussian white noise with a mean value of 0 and a power spectral density of N0/2 .
Ai , Bi are Rayleigh distributed random variables
![](https://img-blog.csdnimg.cn/img_convert/c7268fb49e1b1473e32cc59e78871af3.png)
It is known that P ( Hi )=1/2, i =0,1. 1/ f 1≪ T , 1/ f 2< T , | ω 1- ω 0| is very large. Define SNR
SNR=
![](https://img-blog.csdnimg.cn/img_convert/b89dcebc9d0cecef29a5d405da7e3686.png)
The above M receivers constitute a diversity system, and the minimum error probability criterion is used to calculate M = 1, 2, 4, 6, 8, 16. The range of SNR is 0-60 decibels, and the probability of misjudgment when the RF pulse of each channel is 1, 5, 10.
Simulation principle
The average probability of error for a binary hypothesis test is
![](https://img-blog.csdnimg.cn/img_convert/7f68d48c093f640251f37b5412fb2f14.png)
The minimum error probability criterion is to find the most suitable decision threshold th ', so that the average error probability of wrong judgment P e can be minimized. Let the derivative of P e with respect to th ' be 0, then th ' that makes P e reach the minimum value can be obtained, that is
![](https://img-blog.csdnimg.cn/img_convert/23475e02a85c50bc9447cf9bb3f7da79.png)
![](https://img-blog.csdnimg.cn/img_convert/302ba8e4d02d728aa83b23bcc34321bc.png)
Obviously, ᵆ5<ᵆ1ℎ′ judges ᵃb0 to be true, and ᵆ5≥ᵆ1ℎ' judges ᵃb1 to be true. So the decision rule under the minimum error probability criterion is
![](https://img-blog.csdnimg.cn/img_convert/e9f4df40b298407787198922afc38a78.png)
For a multi-receiver system, the internal noise of each receiver is independent and identically distributed. Judgment is made according to the signal received by the M receivers to determine which known signal it is. x (t)=[x1(t),⋯ ,xM(t)]T , the likelihood ratio is
![](https://img-blog.csdnimg.cn/img_convert/5fb2133a833b49d496f73fc9810426f9.png)
The derived decision criterion is as follows
![](https://img-blog.csdnimg.cn/img_convert/e25057ac53331a5f8339dcaa60666862.png)
Simulation of Diversity System Composed of Receivers
![](https://img-blog.csdnimg.cn/img_convert/70f5cf19ca1536bade1f17bea4acf3e2.png)
It can be seen from the above figure that as the number of M increases, the bit error rate tends to decrease, and as the signal-to-noise ratio increases, the bit error rate gradually decreases to 0. This is consistent with the reality, indicating that our simulation is more reasonable.
M = [1,2,4,6,8,16];
SNR = 0:60;
Pf = zeros(length(M),length(SNR));
for i = 1:length(M)
for j = 1:length(SNR)
% 计算概率密度函数
pdf_A = (A0./(A0.^2)).*exp(-A0.^2./(2.*(SNR(i).*N0).^2));
pdf_B = (A0./(A0.^2)).*exp(-A0.^2./(2.*(SNR(i).*N0).^2));
Ax = sqrt((-2*A0)*log(1-rand(1)));
phix = 2*pi*rand(1);
x = Ax*sin(w0*i+phix)+randn(1,length(i));
f0(i) = ((x*sin(w0*i)')*dt)^2+((x*cos(w0*i)')*dt)^2;
f1(i) = ((x*sin(w1*i)')*dt)^2+((x*cos(w1*i)')*dt)^2;
% 计算错误概率
Pf(i,j) = 1 - (1-qfunc(sqrt(2*10^(SNR(j)/10)))).^M(i);
end
end
plot(SNR,Pf);
xlabel('SNR(dB)');
ylabel('Pf');
legend('M=1','M=2','M=4','M=6','M=8','M=16');
(code is not exactly correct)