考虑正弦信号检测问题,幅度A,相位fai未知且A大于0。H0假设下z[k]=n[k],n[k]为0均值高斯白噪声,方差为sigma=1,数据长度为20。H1假设下z[n]=Acos(2pif+fai)+n[k],f=0.1. 用matlab语言,蒙特卡洛仿真,写出正交形式检测。提示z[n]分别与cos(2pifk),sin(2pifk)相乘,求和,平方,平均,与门限gama比较,绘制出仿真的ROC曲线与理论计算的ROC曲线。提示cos(2pifk),sin(2pifk),给出完整代码
时间: 2023-06-18 11:08:34 浏览: 238
以下是一个可能的MATLAB代码:
```matlab
% 参数设置
A = 2;
f = 0.1;
sigma = 1;
N = 20;
num_trials = 10000;
gamma = 0:0.1:5;
% 生成H0和H1下的样本
H0_samples = randn(num_trials, N) * sigma;
H1_samples = A * cos(2*pi*f*(0:N-1)' + rand(num_trials, 1)*2*pi) + randn(num_trials, N) * sigma;
% 计算检测统计量
cos_term = cos(2*pi*f*(0:N-1)');
sin_term = sin(2*pi*f*(0:N-1)');
z_cos_H0 = H0_samples * cos_term;
z_sin_H0 = H0_samples * sin_term;
z_cos_H1 = H1_samples * cos_term;
z_sin_H1 = H1_samples * sin_term;
statistic_H0 = z_cos_H0.^2 + z_sin_H0.^2;
statistic_H1 = z_cos_H1.^2 + z_sin_H1.^2;
% 计算Pfa和Pd
Pfa = zeros(size(gamma));
Pd = zeros(size(gamma));
for i = 1:length(gamma)
Pfa(i) = sum(statistic_H0 > gamma(i)) / num_trials;
Pd(i) = sum(statistic_H1 > gamma(i)) / num_trials;
end
% 绘制ROC曲线
figure;
plot(Pfa, Pd);
hold on;
plot(0:0.01:1, 0:0.01:1, 'r--');
xlabel('False Alarm Rate');
ylabel('Detection Rate');
title('ROC Curve');
legend('Simulation', 'Theory');
% 理论计算
syms x
Pfa_theory = double(int(sqrt(2)*exp(-x^2/2) / (2*pi)^(N/2), x, gamma, Inf));
Pd_theory = double(int(sqrt(2)*exp(-(x-A)^2/2) / (2*pi)^(N/2), x, gamma, Inf));
hold on;
plot(Pfa_theory, Pd_theory, 'g');
```
这里的主要思路是生成一定数量的H0和H1下的样本,分别计算样本与cos(2pifk)和sin(2pifk)的内积,并根据内积的平方和得到检测统计量。然后在一定的门限范围内,计算出对应的Pfa和Pd,绘制ROC曲线。最后,利用高斯积分函数求出理论计算的Pfa和Pd,并将它们也绘制到图中,以便与仿真结果进行比较。
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