From 5c27a784700ee35fed457cd6f1d3a2180df593d5 Mon Sep 17 00:00:00 2001 From: BlitherBoom812 Date: Mon, 6 May 2024 12:15:28 +0800 Subject: [PATCH] 20240506 --- source/_posts/StaSP.md | 171 ++++++++++++++++++++++++++ source/images/StaSP/1714966200160.png | Bin 0 -> 126637 bytes source/images/StaSP/1714966226811.png | Bin 0 -> 79603 bytes source/images/StaSP/1714966230536.png | Bin 0 -> 79603 bytes 4 files changed, 171 insertions(+) create mode 100644 source/images/StaSP/1714966200160.png create mode 100644 source/images/StaSP/1714966226811.png create mode 100644 source/images/StaSP/1714966230536.png diff --git a/source/_posts/StaSP.md b/source/_posts/StaSP.md index ae4853a1..d4f4bb11 100644 --- a/source/_posts/StaSP.md +++ b/source/_posts/StaSP.md @@ -1109,3 +1109,174 @@ MSE修正: $M\left [ n\mid n\right ] = \left ( 1- K\left [ n\right ] \right ) M\ * 不同时刻的待估计参数并不完全一样,但是存在某些内在联系 * 卡尔曼滤波利用这种联系进行 LMMSE 估计,并减少了运算量 * 如果信号与噪声是高斯的,则卡尔曼滤波在 MMSE 准则下最佳,否则,在 LMMSE 准则下是最佳的。 + +## 信号检测基本准则与方法 + +之前一直在研究连续型的问题(回归/估计),这里研究离散型的问题(分类/检测)。 + +### Neyman-Pearson 准则 + +适用于没有先验信息、代价不好量化的场景。 + +两种假设: + +$$ +\begin{aligned}&H_0:x[n]=w[n],n=0,1,...,N-1\\&H_1:x[n]=s[n]+w[n],n=0,1,...,N-1\end{aligned} +$$ + +$$ +\begin{aligned}&P\left(H_1;H_0\right):\text{ 虚警概率}\left(P_{FA},\text{有时简记为}P_F\right)\\&P\left(H_0;H_1\right):\text{ 漏检概率 }\left(P_M\right)\\&P\left(H_1;H_1\right):\text{ 检测概率 }\left(P_D\right)\end{aligned} +$$ + +要求:在虚警概率一定情况下,使检测概率最大化 + +检测概率和虚警概率之间追求折中,不可能两者都改善。 + +对给定的虚警概率 $P_{FA}=\alpha$ ,使检测概率 $P_D$ 最大的判决为 + +$$L(x)=\frac{p(x;H_1)}{p(x;H_0)}>\gamma $$ + +其中门限由$P_{FA}=\int_{\{\mathbf{x}:L(\mathbf{x})>\gamma\}}p(\boldsymbol{x};H_0)d\boldsymbol{x}=\alpha$决定 + +对于信号检测问题: + +$$ +H_0:\boldsymbol{x}\sim N\left(\boldsymbol{0},\sigma^2\mathbf{I}\right)\\H_1:\boldsymbol{x}\sim N\left(A\mathbf{1},\sigma^2\mathbf{I}\right) +$$ + +NP 检测器: + +$$ +\frac{p\left(\boldsymbol{x};H_1\right)}{p\left(\boldsymbol{x};H_0\right)}=\frac{\frac1{\left(2\pi\sigma^2\right)^{N/2}}\exp\left\{-\frac1{2\sigma^2}\sum_{n=0}^{N-1}\left(x[n]-A\right)^2\right\}}{\frac1{\left(2\pi\sigma^2\right)^{N/2}}\exp\left\{-\frac1{2\sigma^2}\sum_{n=0}^{N-1}x^2[n]\right\}}>\gamma\\\quad\exp\left\{-\frac1{2\sigma^2}\sum_{n=0}^{N-1}\left(x[n]-A\right)^2+\frac1{2\sigma^2}\sum_{n=0}^{N-1}x^2[n]\right\}>\gamma +$$ + +$$ +-\frac1{2\sigma^2}\Bigg(-2A\sum_{n=0}^{N-1}x[n]+NA^2\Bigg)>\ln\gamma\\\frac1N\sum_{n=0}^{N-1}x[n]>\frac{\sigma^2}{NA}\ln\gamma+\frac A2 +$$ + +称为检测统计量。若均值大于门限,则判为有信号,否则为无信号。 + +使用方法: + +$$ +\begin{aligned} +&\text{检测统计量:}T(x)=\frac1N\sum_{n=0}^{N-1}x[n]\sim\begin{cases}N\Big(0,\sigma^2\Big/_N\Big),&H_0\\[2ex]N\Big(A,\sigma^2\Big/_N\Big),&H_1\end{cases} \\ +&\text{虚警概率:}P_{FA}=Pr\left(T\left(\boldsymbol{x}\right)>\gamma^{'};H_0\right)=Q\left(\frac{\gamma^{'}}{\sqrt{\sigma^2/N}}\right) \\ +&\text{门限设置:}\gamma^{\prime}=\sqrt{\frac{\sigma^2}N}Q^{-1}\left(P_{FA}\right) \\ +& \begin{aligned}&\text{相应的检测概率:}\\&P_{D}=Pr\Big(T\big(\boldsymbol{x}\big)>\gamma^{'};H_{1}\Big)=Q\Bigg(\frac{\gamma^{'}-A}{\sqrt{\sigma^{2}/N}}\Bigg)=Q\Bigg(\frac{\sqrt{\frac{\sigma^{2}}{N}}Q^{-1}\big(P_{FA}\big)-A}{\sqrt{\sigma^{2}/N}}\Bigg)\end{aligned} \\ +&=Q\Bigg(Q^{-1}\big(P_{F_A}\big)-\sqrt{\frac{NA^2}{\sigma^2}}\Bigg) +\end{aligned} +$$ + +### 检测性能分析 + +接收机工作特性曲线(ROC, receiver operating characteristics) + +![1714966200160](../images/StaSP/1714966200160.png) + +直观理解: + +![1714966230536](../images/StaSP/1714966230536.png) + +多次观测的好处 + +- 从数学角度:不同假设下的pdf分隔更开,更易区分不同假设 +- 从信号处理角度:增加信号预检测积分时间,获得更多的能量用于检测 +- 从信息论角度:多的观测数据带来了新的信息 + +### 最小错误概率准则 + +$$ +\begin{aligned} +P_{e}& =\Pr\left\{\text{判}H_0,H_1\text{为真}\right\}+\Pr\left\{\text{判}H_1,H_0\text{为真}\right\} \\ +&=P\big(H_0,H_1\big)+P\big(H_1,H_0\big) \\ +&=P\big(H_1\big)P\big(H_0|H_1\big)+P\big(H_0\big)P\big(H_1|H_0\big) \\ +&=P\big(H_1\big)\int_{R_0}p\big(\boldsymbol{x}|H_1\big)d\boldsymbol{x}+P\big(H_0\big)\int_{R_1}p\big(\boldsymbol{x}|H_0\big)d\boldsymbol{x} \\ +&=P\big(H_1\big)\Bigg(1-\int_{R_1}p\big(\boldsymbol{x}\mid H_1\big)d\boldsymbol{x}\Bigg)+P\big(H_0\big)\int_{R_1}p\big(\boldsymbol{x}\mid H_0\big)d\boldsymbol{x} \\ +&=P\left(H_1\right)+\int_{R_1}\left\{P\left(H_0\right)p\left(\boldsymbol{x}\mid H_0\right)-P\left(H_1\right)p\left(\boldsymbol{x}\mid H_1\right)\right\}d\boldsymbol{x} +\end{aligned} +$$ + +为了使错误最小,需要在积分式小于零的区域积分,即 + +$$ +\frac{p\left(\boldsymbol{x}\mid H_1\right)}{p\left(\boldsymbol{x}\mid H_0\right)}>\frac{P\left(H_0\right)}{P\left(H_1\right)}\text{ 时,判 }H_1 +$$ + +最小错误概率准则可推导出最大后验概率检测器: + +$$ +\frac{p\left(\boldsymbol{x}\mid H_1\right)}{p\left(\boldsymbol{x}\mid H_0\right)}>\frac{P\left(H_0\right)}{P\left(H_1\right)}\text{ 时,判 }H_1 +$$ + +等价于 + +$$ +\frac{p\left(\boldsymbol{x}\mid H_1\right)P\left(H_1\right)}{p\left(\boldsymbol{x}\right)}>\frac{p\left(\boldsymbol{x}\mid H_0\right)P\left(H_0\right)}{p\left(\boldsymbol{x}\right)}\\p\left(H_1\mid\boldsymbol{x}\right)>p\left(H_0\mid\boldsymbol{x}\right) +$$ + +若先验概率相同,则为最大似然检测器: + +$$ +p\left(x\mid H_1\right)>p\left(x\mid H_0\right) +$$ + +### 二元贝叶斯风险准则 + +引入判错代价 + +$$ +R=C_{01}P\left(H_1\right)P\left(H_0\mid H_1\right)+C_{10}P\left(H_0\right)P\left(H_1\mid H_0\right) +$$ + +进一步泛化: + +$$ +R=C_{00}P\big(H_0\big)P\big(H_0|H_0\big)+C_{10}P\big(H_0\big)P\big(H_1|H_0\big)\\+C_{01}P\big(H_1\big)P\big(H_0|H_1\big)+C_{11}P\big(H_1\big)P\big(H_1|H_1\big) +$$ + +$$ +\begin{aligned}\text{R}&=C_{00}P\left(H_0\right)+C_{01}P\left(H_1\right)\\&+\int_{R_1}\left\{\left(C_{10}-C_{00}\right)P\left(H_0\right)p\left(\boldsymbol{x}\mid H_0\right)-\left(C_{01}-C_{11}\right)P\left(H_1\right)p\left(\boldsymbol{x}\mid H_1\right)\right\}d\boldsymbol{x}\end{aligned} +$$ + +因此,有了最小贝叶斯风险判决准则: + +$$ +\frac{p\left(\boldsymbol{x}\mid H_1\right)}{p\left(\boldsymbol{x}\mid H_0\right)}>\frac{\left(C_{10}-C_{00}\right)P\left(H_0\right)}{\left(C_{01}-C_{11}\right)P\left(H_1\right)}\text{ 时,判 }H_1 +$$ + +风险一致条件下($C_{00}=C_{11}=0,C_{10}=C_{01}=1$),回到最小错误概率准则。 + +### 多元贝叶斯风险准则 + +贝叶斯风险: + +$$ +\begin{aligned} +\text{R}& =\sum_{i=0}^{M-1}\sum_{j=0}^{M-1}C_{ij}P\Big(H_i,H_j\Big) \\ +&=\sum_{i=0}^{M-1}\sum_{j=0}^{M-1}C_{ij}\int_{R_i}P\Big(x,H_j\Big)dx \\ +&=\sum_{i=0}^{M-1}\int_{R_i}\sum_{j=0}^{M-1}C_{ij}P\Big(x,H_j\Big)dx \\ +&=\sum_{i=0}^{M-1}\int_{R_i}\sum_{j=0}^{M-1}C_{ij}P\Big(H_j\mid x\Big)p(x)dx +\end{aligned} +$$ + + +应选择使平均风险$C_i\left(\boldsymbol{x}\right)=\sum_{j=0}^{M-1}C_{ij}P\left(H_j\mid\boldsymbol{x}\right)$最小的假设 + +在风险一致条件下 + +$$ +C_{ij}=\begin{cases}0,&i=j\\1,&i\neq j\end{cases} +$$ + +$$ +\begin{aligned}C_{i}\left(\boldsymbol{x}\right)&=\sum_{j=0\atop j\neq i}^{M-1}P\Big(H_j\mid\boldsymbol{x}\Big)\\&=\sum_{j=0}^{M-1}P\Big(H_j\mid\boldsymbol{x}\Big)-\underline{P\Big(H_i\mid\boldsymbol{x}\Big)}_{最大化这个}\end{aligned} +$$ + +因此等价于最大后验准则: + +$$ 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