【雷达信号优化】第A章 统计信号处理理论基础(数理深化层)

目录

A.1 检测理论数理基础

A.1.1 复合假设检验框架

A.1.1.1 Neyman-Pearson准则与似然比检验

A.1.1.2 广义似然比检验的渐近性能分析

A.1.1.3 Rao检验与Wald检验的雷达目标检测应用

A.1.2 贝叶斯检测与M元假设

A.1.2.1 最小错误概率准则与最大后验概率

A.1.2.2 M元脉冲序列检测

A.1.3 信号检测 in 非高斯噪声

A.1.3.1 球不变随机过程与复合高斯杂波

A.1.3.2 Alpha-stable分布噪声中的鲁棒检测

A.1.3.3 基于Zernike矩的非线性检测器设计

A.2 估计理论下界分析

A.2.1 非随机参数下界

A.2.1.1 Cramér-Rao Lower Bound的矩阵形式推导

A.2.1.2 修正Cramér-Rao Bound for 杂波相关场景

A.2.1.3 Barankin Bound与Hammerley-Chapman-Robbins Bound

A.2.2 贝叶斯估计下界

A.2.2.1 Weiss-Weinstein Bound for 波形选择

A.2.2.2 Bayesian Ziv-Zakai Bound for 时延估计

A.2.2.3 下界在波形自适应中的决策应用

A.2.3 信息几何与统计流形

A.2.3.1 Fisher信息矩阵作为黎曼度量张量

A.2.3.2 统计流形上的测地距离与参数估计效率

A.1 检测理论代码实现

脚本A.1.1.1:Neyman-Pearson准则与似然比检验

脚本A.1.1.2:广义似然比检验渐近性能分析

脚本A.1.1.3:Rao检验与Wald检验雷达目标检测

脚本A.1.2.1:最小错误概率准则与MAP检测

脚本A.1.2.2:M元脉冲序列检测

脚本A.1.3.1:球不变随机过程与复合高斯杂波

脚本A.1.3.2:Alpha-stable分布噪声中的鲁棒检测

脚本A.1.3.3:基于Zernike矩的非线性检测器

A.2 估计理论代码实现

脚本A.2.1.1:Cramér-Rao Lower Bound矩阵形式

脚本A.2.1.2:修正CRLB for 杂波相关场景

脚本A.2.1.3:Barankin Bound与HCR Bound

脚本A.2.2.1:Weiss-Weinstein Bound for 波形选择

脚本A.2.2.2:Bayesian Ziv-Zakai Bound for 时延估计

脚本A.2.2.3:下界在波形自适应中的决策应用

脚本A.2.3.1:Fisher信息矩阵作为黎曼度量张量

脚本A.2.3.2:统计流形上的测地距离与参数估计效率


A 检测理论与估计理论下界分析

A.1 检测理论数理基础

A.1.1 复合假设检验框架

A.1.1.1 Neyman-Pearson准则与似然比检验

二元假设检验构成了雷达信号检测的数学基石。观测样本 $x \in \mathbb{C}^N$ 在噪声干扰下携带目标回波或仅含背景噪声,分别对应假设 $H_1$ 和 $H_0$。概率密度函数 $p(x;H_0)$ 和 $p(x;H_1)$ 刻画了两种状态下的统计特性,检测器设计转化为观测空间的区域划分问题。

Neyman-Pearson 准则在约束虚警概率 $P_{FA} = \int_{R_1} p(x;H_0)dx \le \alpha$ 的条件下最大化检测概率 $P_D = \int_{R_1} p(x;H_1)dx$。拉格朗日泛函构造为:

$$J = P_D + \lambda(P_{FA} - \alpha)$$

变分分析表明最优判决区域由似然比函数决定。检验统计量

$$\Lambda(x) = \frac{p(x;H_1)}{p(x;H_0)}$$

与门限 $\gamma$ 比较形成判决规则。门限值由虚警约束确定,满足:

$$\int_{\{x:\Lambda(x)>\gamma\}} p(x;H_0)dx = \alpha$$

高斯白噪声环境下的线性观测模型 $x = s + w$,其中 $w \sim \mathcal{CN}(0, \sigma^2 I)$,似然比退化为匹配滤波器

船载 OpenHarmony AIS 航海工作台:告警与防碰撞预警系统 船载OpenHarmony AIS航海工作台的告警与防碰撞系统,集成十余类告警(超速、偏航、会遇、SOS等),通过网格空间索引实现高效邻近判定,保障实时性。告警管理器统一调度规则、去重、通知与日志,支持悬浮球、语音、跑马灯、弹窗多渠道触达,确保航行安全。SOS紧急通道独立运行,避免干扰。系统设计强调性能优化与多场景适配,为航海安全提供核心支撑。 阅读详情

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% Dropout fraction opts.numepochs = 1; % Number of full sweeps through dataopts.batchsize = 100; % Take a mean gradient step over this many samplesnn = nntrain(nn, train_x, train_y, opts);[er, bad] = nntest(nn, test_x, test_y);assert(er < 0.1, 'Too big error');%% ex4 neural net with sigmoid activation functionrand('state',0)nn = nnsetup([784 100 10]);nn.activation_function = 'sigm'; % Sigmoid activation functionnn.learningRate = 1; % Sigm require a lower learning rateopts.numepochs = 1; % Number of full sweeps through dataopts.batchsize = 100; % Take a mean gradient step over this many samplesnn = nntrain(nn, train_x, train_y, opts);[er, bad] = nntest(nn, test_x, test_y);assert(er < 0.1, 'Too big error');%% ex5 plotting functionalityrand('state',0)nn = nnsetup([784 20 10]);opts.numepochs = 5; % Number of full sweeps through datann.output = 'softmax'; % use softmax outputopts.batchsize = 1000; % Take a mean gradient step over this many samplesopts.plot = 1; % enable plottingnn = nntrain(nn, train_x, train_y, opts);[er, bad] = nntest(nn, test_x, test_y);assert(er < 0.1, 'Too big error');%% ex6 neural net with sigmoid activation and plotting of validation and training error% split training data into training and validation datavx = train_x(1:10000,:);tx = train_x(10001:end,:);vy = train_y(1:10000,:);ty = train_y(10001:end,:);rand('state',0)nn = nnsetup([784 20 10]); nn.output = 'softmax'; % use softmax outputopts.numepochs = 5; % Number of full sweeps through dataopts.batchsize = 1000; % Take a mean gradient step over this many samplesopts.plot = 1; % enable plottingnn = nntrain(nn, tx, ty, opts, vx, vy); % nntrain takes validation set as last two arguments (optionally)[er, bad] = nntest(nn, test_x, test_y);assert(er < 0.1, 'Too big error');```[![Bitdeli Badge](https://d2weczhvl823v0.cloudfront.net/rasmusbergpalm/deeplearntoolbox/trend.png)](https://bitdeli.com/free "Bitdeli Badge")

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