[Target Detection] Radar Target CFAR Detection Algorithm

I. Overview

1. Basic concepts

The detection process of radar can be described by threshold detection. Almost all judgments are based on the comparison between the output of the receiver and a certain threshold level. If the envelope output by the receiver exceeds a certain set threshold, it is considered that there is a target.
Radar will be affected by noise, clutter and interference during detection, so when a fixed threshold is used for target detection, certain false alarms will occur, especially when the clutter background fluctuates and changes, the false alarm rate will increase sharply, seriously affecting the radar performance. Detection performance. Therefore, the detection threshold is dynamically adjusted according to the radar clutter data, and the target detection probability is maximized while the false alarm probability remains unchanged. This method is called Constant False Alarm Rate (CFAR) detection technology.

2. Basic knowledge

During the judgment process of the radar, two types of errors may occur. The first type is to judge that there is a target when there is no target, and this type of error is called a false alarm. The other type is to judge that there is no target when there is a target, and this type of error is called a missed alarm. The above two types of errors occur with a certain probability, which are called false alarm probability and missed alarm probability respectively.

2. CFAR detection algorithm

At present, researchers have proposed many efficient CFAR detection algorithms for various clutter environments. The wave is evenly distributed; the other is the ordered statistical CFAR (OS-CFAR) algorithm, which is designed to deal with the multi-target situation in the neighborhood. Different CFAR detection algorithms have their own advantages and disadvantages, and they are all designed for specific situations. We only need to learn the basic principles of CFAR and be familiar with several typical CFAR detection algorithms.

1. Basic principles

The input of a CFAR detector generally includes a detection unit $Y$ sum$2n$ reference units. The reference unit is located on both sides of the detection unit, and the front and rear are$n$ number. The protection unit is mainly used in the case of a single target to prevent the target energy from leaking to the reference unit and affecting the detection effect. Let the reference threshold level be$V_{TH}$，$V_{TH}=T\times Z$ , of which:$Z$ is an estimate of the total clutter power level,$T$ is the threshold factor, then when$Y>V_{TH}$When , it is considered that there is a target; otherwise, it is considered that there is no target.
The processing flow of the CFAR algorithm is shown in the figure below:

In general, clutter and noise are independent of each other, and after square-law detection, they all satisfy exponential distribution. The probability density function of the reference unit is:

Let $H_0$Expressed as no target, $P[Y>TZ|H_0]$ is expressed as the probability of judging that there is a target under the condition that there is no target, so that the expression of the false alarm probability is:

where$\mu$ is the noise power;$Z$ is a random variable whose distribution depends on the type of CFAR algorithm and the distribution of the reference unit;$M_Z(u{)}_{u=T/2}$is the moment generating function.

2. Several typical CFAR detection algorithms

The difference between different types of CFAR algorithms is mainly reflected in the different processing of reference units, also known as ZZThe choice of$Z\; value\; is\; different.$When the background noise is independent and identically distributed, by determining the constant$T$ to achieve a constant false alarm probability. different detection algorithms which determine the constant$The\; method\; of\; T$ will be different accordingly. The following will deduce the false alarm probability expressions of several typical CFAR algorithms.

(1) CA (Cell Averaging)-CFAR detection algorithm

Background clutter power level $Z$ is calculated as$Sum\; of\; 2n$ reference single elements:
$$Z=\sum _{i=1}^n{X}_i+\sum _{i=n+1}^2{X}_i=\sum _{i=1}^2{X}_i$$
In deriving $P_{fa}$with $Before\; the\; relationship\; of\; T$ , we first give the relevant knowledge of the gamma (Γ) distribution.
$The\; probability\; density\; function\; of\; the\; \Gamma$ distribution is:

where,$\alpha ，\beta$是个parameters，当$a=1，b=2\mu$ , formula (3) degenerates into exponential distribution in formula (1);$\Gamma \left(\alpha \right)$ is the gamma function in mathematics, for positive integer$\alpha$，有$C\left(a\right)=(a-1)！$。
任$G(a,b)$ for$The\; probability\; distribution\; function\; of\; the\; \Gamma$ distribution, if$X$ is subject to$A\; random\; variable\; of\; \Gamma$ distribution, then:

XXThe moment generating function of$X$

is: assuming that each variable in the input signal satisfies the condition of independent and identical distribution, then for$2\; The\; moment\; generating\; function\; of\; the\; sum\; ofn$ random variables is equal to the product of the moment generating functions of each random variable, so there is:

Formula (5) and formula (6) and$u=$Substituting$T$$/2$$\mu$

into formula (2), we can get: the false alarm probability$P_{fa}$with threshold factor $T$ relationship:

(2) GO-CFAR, SO-CFAR detection algorithm

The largest choice GO (Greatest Of)-CFAR is to select the front $The\; sum\; of\; n$ reference units and the followingnnThe larger of the sum of$n$$Z$ ; while the minimum selection SO(Smallest Of)-CFAR is to select the front$The\; sum\; of\; n$ reference units and the followingnnThe smaller of the sum of$n$$Z.$ _

For the GO-CFAR algorithm, the probability density function of Z is$f_Z\left(z\right)$ , and then derive its false alarm probability as$P_{fa,g}$, as the following formula:

Similarly, the probability density function of the SO-CFAR algorithm can be obtained as $f_Z\left(z\right)$ , the false alarm probability is$P_{fa,}$, as follows:

(3) OS-CFAR detection algorithm

The principle of order statistics OS (Order Statistics) CFAR is to sort the reference units from small to large, and then select the $k$ samples as$Z.$ _
Let the false alarm probability of OS-CFAR algorithm be$P_{fa,}$, can be deduced according to the formula in the reference:

$k$ is a parameter in OS-CFAR, and the selection of its value has a great influence on the detection performance of the algorithm.

(4) Supplementary Notes

Threshold factor $T$ is difficult to express in mathematical expressions, and can be solved by iterative operation.
When solving the threshold factor$After\; T$ , it can be substituted into the expression of the detection probability to obtain the change of the detection probability under the condition of different signal-to-noise ratios. The detection probability P d P_{d}
is given in references [4] and [5]$P_d$calculation method.

3. Performance comparison of different CFAR detection algorithms

1. MATLAB simulation results

In order to compare the detection performance of different CFAR algorithms, the relationship curves between the detection probability and the signal-to-noise ratio of the four types of CFAR algorithms, CA, GO, SO, and OS, are established. The signal-to-noise ratio is set to 0~30dB, and the number of reference units is 2 n $2n$ takes 16, 24, 32, 48, 64, 128 respectively, and there are 3 protection units on the left and right respectively, the false alarm probability$P_{fa}=10^{-6}$ .${}^{\_}$
The MATLAB simulation results are shown in the figure below.

2. MATLAB code

% ------ 程序功能：四类CFAR检测算法的检测概率与SNR的关系 %
clc
clear all;
close all;

%% 参数设置
R = 24;                     % 参考单元长度
n = R/2;                    % 半滑窗长度
k = R*3/4;                  % os-cfar的参数
P_fa = 1e-6;                % 虚警概率
SNR_dB = (0:30);            % 信噪比
SNR = 10.^(SNR_dB./10);     % 信号功率与噪声功率的比值
syms T;                     % 门限因子的符号变量
%% CA-CFAR
T_CA = P_fa^(-1/R)-1;           % CA-CFAR的门限因子
Pd_CA = (1+T_CA./(1+SNR)).^(-R);    % CA-CFAR的检测概率

%% SO-CFAR、GO-CFAR
Pfa_SO = 0;
syms T
for i = 0:n-1
    Pfa_SO = Pfa_SO+2*nchoosek(n+i-1,i)*(2+T)^(-(n+i));     % SO-CFAR的虚警概率表达式
end
T1_SO = solve(Pfa_SO == P_fa, T);       % 求解出虚警概率为P_fa时对应的门限因子T
T2_SO = double(T1_SO);
T_SO = T2_SO(T2_SO == abs(T2_SO));      % SO-CFAR的门限因子

Pfa_GO = 2*(1+T)^(-n)-Pfa_SO;           % GO-CFAR的虚警概率表达式
T1_GO = solve(Pfa_GO == P_fa, T);       % 求解出虚警概率为P_fa时对应的门限因子T
T2_GO = double(T1_GO);
T_GO = T2_GO(T2_GO == abs(T2_GO));      % GO-CFAR的门限因子

Pd_SO = 0;
Pd_GO = 0;
for j = 0:n-1
    Pd_SO = Pd_SO+2*nchoosek(n+j-1,j).*(2+T_SO./(1+SNR)).^(-(n+j));     % SO-CFAR的检测概率
    Pd_GO = Pd_GO+2*nchoosek(n+j-1,j).*(2+T_GO./(1+SNR)).^(-(n+j));
end
Pd_GO = 2.*(1+T_GO./(1+SNR)).^(-n)-Pd_GO;         % GO-CFAR的检测概率

%% OS-CFAR
Pfa_OS = k*nchoosek(R,k)*gamma(R-k+1-T)*gamma(k)/gamma(R+T+1);           % OS-CFAR的虚警概率表达式
T1_OS = solve(Pfa_OS == P_fa, T);       % 求解出虚警概率为P_fa时对应的门限因子T
T2_OS = double(T1_OS);
T_OS = T2_OS(T2_OS == abs(T2_OS));      % OS-CFAR的门限因子
Pd_OS = k*nchoosek(R,k)*gamma(R-k+1-T_OS./(1+SNR))*gamma(k)./gamma(R+T_OS./(1+SNR)+1);      % OS-CFAR的检测概率

%% 画图
figure;
plot(SNR_dB,Pd_CA,'r-*');
hold on;
plot(SNR_dB,Pd_SO,'k-^');
plot(SNR_dB,Pd_GO,'b-o');
plot(SNR_dB,Pd_OS,'m-.');
grid on;
xlabel('SNR','FontName','Time New Romans','FontAngle','italic');
ylabel('P_{d}','FontName','Time New Romans','FontAngle','italic');
title(['恒虚警率 P_{fa}= ',num2str(P_fa),'，参考单元 2n= ',num2str(R)]);
legend('CA','SO','GO','OS');

3. Summary of advantages and disadvantages

In practical applications, due to the limited computing resources of radar hardware, the commonly used algorithm is still the CA-CFAR detection algorithm with low complexity.

4. References

[1] Zou Chengxiao, Zhang Haixia, Cheng Yukun. A Review of Radar Constant False Alarm Rate Detection Algorithms [J]. Radar and Countermeasures, 2021, 41(02): 29-35. [2] Liu Chaojun, Zhang Xin, Wang Shouquan. Radar Target Constant False Alarm
Rate Research on Alarm Detection Algorithm[J]. Ship Electronic Engineering, 2008(07): 107-109.
[3] Cui Ning, Luo Yunhua, Yu Zhongjun, et al. A Fast Two-Dimensional CFAR Algorithm Based on Improved Convolution Kernel[J] . Modern Radar, 2020, 42(08): 30-35.
[4] Wang Bei. Research on constant false alarm processing technology based on clutter map [D]. Xidian University, 2018. [
5] Detection loss due to interfering targets in ordered statistics CFAR.