前言
本文对雷达检测的内容以思维导图的形式呈现，有关仿真部分进行了讲解实现。
一、雷达检测
思维导图如下图所示，如有需求请到文章末尾端自取。
二、Matlab 仿真
1、高斯和瑞利概率密度函数
瑞利概率密度函数：f ( x ) = x σ 2 e − x 2 2 σ 2 f(x)=\frac{x}{\sigma^2}e^{-\frac{x^2}{2\sigma^2}}f(x)=σ2xe−2σ2x2
高斯概率密度函数：f ( x ) ≈ 1 2 π σ 2 e − ( x − μ ) 2 2 σ 2 f(x) \approx \frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{(x-\mu)^2}{2\sigma^2}}f(x)≈2πσ21e−2σ2(x−μ)2
x xx 是变量，μ \muμ 是均值，σ \sigmaσ 是方差
①、MATLAB 源码
clear all close all xg = linspace(-6,6,1500); % randowm variable between -4 and 4 xr = linspace(0,6,1500); % randowm variable between 0 and 8 mu = 0; % zero mean Gaussain pdf mean sigma = 1.5; % standard deviation (sqrt(variance) ynorm = normpdf(xg,mu,sigma); % use MATLAB funtion normpdf yray = raylpdf(xr,sigma); % use MATLAB function raylpdf plot(xg,ynorm,'k',xr,yray,'k-.'); grid legend('Gaussian pdf','Rayleigh pdf') xlabel('x') ylabel('Probability density') gtext('\mu = 0; \sigma = 1.5') gtext('\sigma =1.5')
②、仿真
高斯和瑞利概率密度
2、归一化门限相对虚警概率的曲线
虚警概率：P f a = e − V T 2 2 ψ 2 P_{fa}=e^{\frac{-V_T^2}{2\psi^2}}Pfa=e2ψ2−VT2
门限电压：V T = 2 ψ 2 l n ( 1 P f a ) V_T=\sqrt{2\psi^2ln(\frac{1}{P_{fa}})}VT=2ψ2ln(Pfa1)
注：V T V_TVT 为门限电压，ψ 2 \psi^2ψ2 为方差
①、MATLAB 源码
close all clear all logpfa = linspace(.01,250,1000); var = 10.^(logpfa ./ 10.0); vtnorm = sqrt( log (var)); semilogx(logpfa, vtnorm,'k') grid
②、仿真
横坐标为 l o g ( 1 / P f a ) log(1/P_{fa})log(1/Pfa)
纵坐标为 V T 2 ψ 2 \frac{V_T}{\sqrt{2\psi^2}}2ψ2VT
归一化检测门限对虚警概率
从图中可以明显看出，P f a P_{fa}Pfa 对门限值的小变化非常敏感
3、检测概率相对于单个脉冲 SNR 的关系曲线
检测概率 P D P_DPD：
Q QQ 称为 M a r c u m Q Marcum QMarcumQ 函数。
①、MATLAB 源码
marcumsq.m
function PD = marcumsq (a,b) % This function uses Parl's method to compute PD max_test_value = 5000.; if (a < b) alphan0 = 1.0; dn = a / b; else alphan0 = 0.; dn = b / a; end alphan_1 = 0.; betan0 = 0.5; betan_1 = 0.; D1 = dn; n = 0; ratio = 2.0 / (a * b); r1 = 0.0; betan = 0.0; alphan = 0.0; while betan < 1000., n = n + 1; alphan = dn + ratio * n * alphan0 + alphan; betan = 1.0 + ratio * n * betan0 + betan; alphan_1 = alphan0; alphan0 = alphan; betan_1 = betan0; betan0 = betan; dn = dn * D1; end PD = (alphan0 / (2.0 * betan0)) * exp( -(a-b)^2 / 2.0); if ( a >= b) PD = 1.0 - PD; end return
prob_snr1.m
% This program is used to produce Fig. 2.4 close all clear all for nfa = 2:2:12 b = sqrt(-2.0 * log(10^(-nfa))); index = 0; hold on for snr = 0:.1:18 index = index +1; a = sqrt(2.0 * 10^(.1*snr)); pro(index) = marcumsq(a,b); end x = 0:.1:18; set(gca,'ytick',[.1 .2 .3 .4 .5 .6 .7 .75 .8 .85 .9 ... .95 .9999]) set(gca,'xtick',[1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18]) loglog(x, pro,'k'); end hold off xlabel ('Single pulse SNR - dB') ylabel ('Probability of detection') grid
②、仿真
检测概率相对于单个脉冲 S N R SNRSNR 的关系曲线对于 P f a P_{fa}Pfa的个数值:
6 条曲线的 P f a P_{fa}Pfa 从左到右依次是 1 0 − 2 ， 1 0 − 4 ， 1 0 − 6 ， 1 0 − 8 ， 1 0 − 10 ， 1 0 − 12 10^{-2}，10^{-4}，10^{-6}，10^{-8}，10^{-10}，10^{-12}10−2，10−4，10−6，10−8，10−10，10−12，可以看到随着 SNR 信噪比的增加，检测概率逐渐增大，此外，虚警概率越小，随着信噪比的增加，检测概率增加的越快。
4、改善因子和积累损失相对于非相干积累脉冲数的关系曲线
I ( n p ) I(n_p)I(np) 称为积累改善因子
①、改善因子相对于非相干积累脉冲数的关系曲线
1）MATLAB 源码
improv_fac.m
function impr_of_np = improv_fac (np, pfa, pd) % This function computes the non-coherent integration improvment % factor using the empirical formula defind in Eq. (2.48) fact1 = 1.0 + log10( 1.0 / pfa) / 46.6; fact2 = 6.79 * (1.0 + 0.253 * pd); fact3 = 1.0 - 0.14 * log10(np) + 0.0183 * (log10(np))^2; impr_of_np = fact1 * fact2 * fact3 * log10(np); return
fig2_6a.m
% This program is used to produce Fig. 2.6a % It uses the function "improv_fac" clear all close all pfa1 = 1.0e-2; pfa2 = 1.0e-6; pfa3 = 1.0e-10; pfa4 = 1.0e-13; pd1 = .5; pd2 = .8; pd3 = .95; pd4 = .999; index = 0; for np = 1:1:1000 index = index + 1; I1(index) = improv_fac (np, pfa1, pd1); I2(index) = improv_fac (np, pfa2, pd2); I3(index) = improv_fac (np, pfa3, pd3); I4(index) = improv_fac (np, pfa4, pd4); end np = 1:1:1000; semilogx (np, I1, 'k', np, I2, 'k--', np, I3, 'k-.', np, I4, 'k:') %set (gca,'xtick',[1 2 3 4 5 6 7 8 10 20 30 100]); xlabel ('Number of pulses'); ylabel ('Improvement factor in dB') legend ('pd=.5, nfa=e+2','pd=.8, nfa=e+6','pd=.95, nfa=e+10','pd=.999, nfa=e+13'); grid
2）仿真
改善因子相对于非相干积累脉冲数的关系曲线
可以看到随着非相干积累脉冲数的增多，改善因子逐渐增大；在同一脉冲数的情况下，随着检测概率和虚警概率的增大，则改善因子也会逐渐增大
②、积累损失相对于非相干积累脉冲数的关系曲线
1）MATLAB 源码
% This program is used to produce Fig. 2.6b % It uses the function "improv_fac". clear all close all pfa1 = 1.0e-12; pfa2 = 1.0e-12; pfa3 = 1.0e-12; pfa4 = 1.0e-12; pd1 = .5; pd2 = .8; pd3 = .95; pd4 = .99; index = 0; for np = 1:1:1000 index = index+1; I1 = improv_fac (np, pfa1, pd1); i1 = 10.^(0.1*I1); L1(index) = -1*10*log10(i1 ./ np); I2 = improv_fac (np, pfa2, pd2); i2 = 10.^(0.1*I2); L2(index) = -1*10*log10(i2 ./ np); I3 = improv_fac (np, pfa3, pd3); i3 = 10.^(0.1*I3); L3(index) = -1*10*log10(i3 ./ np); I4 = improv_fac (np, pfa4, pd4); i4 = 10.^(0.1*I4); L4 (index) = -1*10*log10(i4 ./ np); end np = 1:1:1000; semilogx (np, L1, 'k', np, L2, 'k--', np, L3, 'k-.', np, L4, 'k:') axis tight xlabel ('Number of pulses'); ylabel ('Integration loss - dB') legend ('pd=.5, nfa=e+12','pd=.8, nfa=e+12','pd=.95, nfa=e+12','pd=.99, nfa=e+12'); grid
2）仿真
积累损失相对于非相干积累脉冲数的关系曲线
可以看到随着非相干积累脉冲数的增多，积累损失逐渐增大；在同一脉冲数的情况下，随着检测概率的增大，则积累损失会逐渐减小
雷达检测及MATLAB仿真（二）https://developer.aliyun.com/article/1472359
