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信道相关系数 $R_{HH}$

为什么需要介绍信道相关系数：主要是因为OFDM系统中需要使用导频进行信道估计，工程上常用的信道估计算法为MMSE，而MMSE中很重要的一个参数便是导频间的互相关系数$R_{HH}$,这个预先我们是不知道的，一般都是根据统计信道模型预先模拟好预存下来的

对于给定时间点$t$,假设信道冲击响应为$h(t,\tau )$,输入信号为$x(t)$,则接收到的信号可表示为输入与信道冲击相互卷积：
$$y(t)={\int }_0^{\infty }h(t,\tau )x(t-\tau )d\tau$$
WSS-US(宽平稳非相关散射信道)，相关系数主要就信道冲击响应在多径信道下的相关性进行研究。

Wide sense stationary (WSS,广义宽平稳信道)

该信道模型中认为信道冲击响应被公认为是广义宽平稳的即信道的相关性$R_{hh}(t,\tau )$只与时间距离相关即:
Rhh(Δt,τ1,τ2)=E{h∗(t,τ1)h(t+Δt,τ2)} R_{hh}(\Delta t,\tau_{1},\tau_{2})=E\{h^{*}(t,\tau_{1})h(t+\Delta t,\tau_{2})\} Rhh​(Δt,τ1​,τ2​)=E{h∗(t,τ1​)h(t+Δt,τ2​)}

Uncorrelated scattering(US,不相关散射)

US 要求多径情况下不同时延分量互不相关:
E{h∗(t,τ1)h(t+Δt,τ2)}=0,τ1≠τ2 E\{h^{*}(t,\tau_{1})h(t+\Delta t,\tau_{2})\}=0,\tau_{1}\neq \tau_{2} E{h∗(t,τ1​)h(t+Δt,τ2​)}=0,τ1​=τ2​
则：
Rhh(Δt,τ1,τ2)=E{h(t,τ1)h∗(t+Δt,τ2)}=Rhh(Δt,τ1)δ(τ1−τ2) R_{hh}(\Delta t,\tau_{1},\tau_{2})=E\{h(t,\tau_{1})h^{*}(t+\Delta t,\tau_{2})\}=R_{hh}(\Delta t,\tau_{1})\delta(\tau_{1}-\tau_{2}) Rhh​(Δt,τ1​,τ2​)=E{h(t,τ1​)h∗(t+Δt,τ2​)}=Rhh​(Δt,τ1​)δ(τ1​−τ2​)

宽带平稳散射信道（WSS-US）

宽带平稳散射信道包含了两个含义宽平稳和散射两个特性即：信道是宽平稳的且每一条径在接收机看来都是不相关的所以：

Rhh(Δt,τ)=E{h∗(t,τ)h(t+Δt,τ)}=Rhh(Δt,τ) R_{hh}(\Delta t,\tau)=E\{h^{*}(t,\tau)h(t+\Delta t,\tau)\}=R_{hh}(\Delta t,\tau) Rhh​(Δt,τ)=E{h∗(t,τ)h(t+Δt,τ)}=Rhh​(Δt,τ)

散射函数

散射函数相当于对$R_{hh}(\Delta t,\tau )$ 在变量$\Delta t$的傅里叶变换：
S(τ,v)=F{Rhh(Δt,τ)}=∫−∞+∞Rhh(Δt,τ)e−j2πvΔtdΔt S(\tau,v)=\mathcal{F}\{R_{hh}(\Delta t,\tau)\}=\int_{-\infty}^{+\infty}R_{hh}(\Delta t,\tau)e^{-j2\pi v\Delta t}d\Delta t S(τ,v)=F{Rhh​(Δt,τ)}=∫−∞+∞​Rhh​(Δt,τ)e−j2πvΔtdΔt
散射函数$S(\tau ,v)$表征的是，$h(t,\tau )$在$(v,\tau )$所承载的功率谱密度有:
$$\int \int S(\tau ,v)d\tau dv=E\left[{\int }_{-\infty }^{+\infty }|h(t,\tau ){|}^{2}d\tau \right]=1$$
前面在clark 模型中我们已经证明每一条多径射线服从相同的统计特性：
$$R(\Delta t)=P_r{J}_0(2\pi f_D\tau )$$
则：$R_{hh}(\tau ,\Delta t)=P(\tau )R_{hh}(\Delta t)$，所以：
S(τ,v)=F{Rhh(Δt,τ)}=∫−∞+∞Rhh(Δt,τ)e−j2πvΔtdΔt=F{Rhh(Δt,τ)}=P(τ)F(Rhh(Δt))=:S(τ)S(v) \begin{align*} S(\tau,v)&=\mathcal{F}\{R_{hh}(\Delta t,\tau)\}=\int_{-\infty}^{+\infty}R_{hh}(\Delta t,\tau)e^{-j2\pi v\Delta t}d\Delta t\\ &=\mathcal{F}\{R_{hh}(\Delta t,\tau)\}=P(\tau)\mathcal{F}(R_{hh}(\Delta t))=:S(\tau)S(v) \end{align*} S(τ,v)​=F{Rhh​(Δt,τ)}=∫−∞+∞​Rhh​(Δt,τ)e−j2πvΔtdΔt=F{Rhh​(Δt,τ)}=P(τ)F(Rhh​(Δt))=:S(τ)S(v)​

时频相关系数

信道$h(t,\tau )$的频域响应为$H(t,f)$:
H(t,f)=F{h(t,τ)}=∫−∞+∞h(t,τ)e−j2πfτdτH(t,f)=\mathcal{F}\{h(t,\tau)\}=\int_{-\infty}^{+\infty}h(t,\tau)e^{-j2\pi f \tau}d\tau H(t,f)=F{h(t,τ)}=∫−∞+∞​h(t,τ)e−j2πfτdτ
其在时延$\Delta t$ 和频移$\Delta f$下的自相关性为：
$$\begin{array}{rl}{R}_H(\Delta t,\Delta f)& =E[H^{\ast }(t,f)H(t+\Delta t,f+\Delta f)]\\ & =E[h^{\ast }(t,\tau )h(t+\Delta t,{\tau }^{\prime })]e^{j2\pi (f\tau -(f+\Delta f){\tau }^{\prime })}d\tau d{\tau }^{\prime }\\ \stackrel{WSS-US}{\to }& \\ & =\int \int R_{hh}(\Delta t,\tau )\delta (\tau -{\tau }^{\prime })e^{j2\pi (f\tau -(f+\Delta f){\tau }^{\prime })}d\tau d{\tau }^{\prime }\\ & =\int R_{hh}(\Delta t,\tau )e^{-j2\pi \Delta f\tau }d\tau \end{array}$$
又：
Rhh(Δt,τ)=F−1{S(τ,v)}=∫−∞+∞S(τ,v)ej2πvΔtdvR_{hh}(\Delta t,\tau)=\mathcal{F}^{-1}\{S(\tau,v)\}=\int_{-\infty}^{+\infty}S(\tau,v)e^{j2\pi v\Delta t}dv \\ Rhh​(Δt,τ)=F−1{S(τ,v)}=∫−∞+∞​S(τ,v)ej2πvΔtdv
将其带入：
$$\begin{array}{rl}{R}_H(\Delta t,\Delta f)& =\int \int S(\tau ,v)e^{j2\pi v\Delta t}{e}^{-j2\pi \Delta f\tau }dvd\tau \\ & =\int \int S(\tau ,v)e^{j2\pi v\Delta t}{e}^{-j2\pi \Delta f\tau }dvd\tau \\ & =\int S(v)e^{j2\pi v\Delta t}dv\int S(\tau )e^{-j2\pi \Delta f\tau }d\tau \\ R_t(\Delta t)& =\int S(v)e^{j2\pi v\Delta t}dv,R_f(\Delta f)=\int S(\tau )e^{-j2\pi \Delta f\tau }d\tau \\ R_H(\Delta t,\Delta f)& =R_t(\Delta t)R_f(\Delta f)\end{array}$$

自相关函数与散射函数关系

$$S(\tau ,v)={\mathcal{F}}_{\Delta t}{\mathcal{F}}_{\Delta f}^{-1}[R_H(\Delta t,\Delta f)]={\int }_{-\infty }^{+\infty }{\int }_{-\infty }^{+\infty }{R}_H(\Delta t,\Delta f)e^{-j2\pi v\Delta t}{e}^{2\pi \tau \Delta f}d\Delta td\Delta f$$

前面我们说过$S(v,\tau )$ 表示，$h(t,\tau )$在$(v,\tau )$所承载的功率谱密度所以,功率时延谱(PDP)可以认为散射函数去除多普勒的影响即PDP的傅里叶变换：
$$\begin{array}{rl}S(\tau )& ={\int }_{-\infty }^{+\infty }S(f,\tau )df\\ R(\Delta f)& =\mathcal{F}[S(\tau )]={\int }_{-\infty }^{+\infty }[{\int }_{-\infty }^{+\infty }S(f,\tau )df]e^{-j2\pi \Delta f\tau }d\tau \end{array}$$
同理时域自相关为多普勒谱的逆傅里叶变换为：
$$\begin{array}{rl}S(f)& ={\int }_{-\infty }^{+\infty }S(f,\tau )d\tau \\ R(\Delta t)& ={\mathcal{F}}^{-1}[S(f)]={\int }_{-\infty }^{+\infty }[{\int }_{-\infty }^{+\infty }S(f,\tau )d\tau ]e^{j2\pi f\Delta t}df\end{array}$$

仿真假设参设函数$S(v,\tau )$如下：
$$\begin{array}{rl}S(f,\tau )& =\frac{P_{local-mean}}{4\pi f_m\sqrt{(1-(\frac{f-f_c}{f_m}{)}^2)}}\ast \frac{1}{{\tau }_{rms}}{e}^{-\frac{\tau }{{\tau }_{rms}}}\\ S(f)& ==\frac{P_{local-mean}}{4\pi f_m\sqrt{(1-(\frac{f-f_c}{f_m}{)}^2)}}\\ S(\tau )& =\frac{1}{{\tau }_{rms}}{e}^{-\frac{\tau }{{\tau }_{rms}}}\end{array}$$

综上：计算时域和频域自相关的关键在于得到$S(f,\tau ),S(f),S(\tau )$,也即功率时延谱PDP以及多普勒谱。MMSE 再得到相关系数后会在频域和时域分别对LS结果分别滤波进而获得信道系数。

参考：
[0] Wireless Communication Andrea Goldsmith
[1] A comparative study of pilot-based channel estimators for wireless OFDM
[2] WSS-US信道冲激响应的相关系数
[3] Wireless Communication Systems in Matlab