Jakes 模型
前面我们介绍了多径信道合成信号可表示为：
r(t)=Re{∑i=0N(t)−1ai(t)u(t−τi(t))ej2πfc(t−τi(t))+ϕDi(t)}
r(t)=Re \left\{\sum_{i=0}^{N(t)-1}a_{i}(t)u(t-\tau_{i}(t))e^{j2\pi f_{c}(t-\tau_{i}(t))+\phi_{D_{i}}(t)} \right\}
r(t)=Re⎩⎨⎧i=0∑N(t)−1ai(t)u(t−τi(t))ej2πfc(t−τi(t))+ϕDi(t)⎭⎬⎫
假设窄带模型这个在OFDM系统中是可以满足的即时延扩展远小于符号长度，这样符号时间内u(t−τi(t))=u(t)u(t-\tau_{i}(t))=u(t)u(t−τi(t))=u(t),为了更好的描述基带信号我们假设发射信号只是相位为ϕ0\phi_{0}ϕ0未调制信号。
r(t)=Re{∑i=0N(t)−1ai(t)u(t)e−j2πfcτi(t)+ϕDi(t)}=∑i=0N(t)−1ai(t)e−j2πfcτi(t)+ϕDi(t)+ϕ0
\begin{align*}r(t)&=Re \left\{\sum_{i=0}^{N(t)-1}a_{i}(t)u(t)e^{-j2\pi f_{c}\tau_{i}(t)+\phi_{D_{i}}(t)}\right\}\\&=\sum_{i=0}^{N(t)-1}a_{i}(t)e^{-j2\pi f_{c}\tau_{i}(t)+\phi_{D_{i}}(t)+\phi_{0}}
\end{align*}
r(t)=Re⎩⎨⎧i=0∑N(t)−1ai(t)u(t)e−j2πfcτi(t)+ϕDi(t)⎭⎬⎫=i=0∑N(t)−1ai(t)e−j2πfcτi(t)+ϕDi(t)+ϕ0
这里我们假设观察时间内变化足够缓慢：ai(t)=ai,τi(t)=τi,fDi(t)=fDi，ϕDi(t)=2πfDita_{i}(t)=a_{i},\tau_{i}(t)=\tau_{i},f_{D_{i}}(t)=f_{D_{i}}，\phi_{D_{i}}(t)=2\pi f_{D_{i}}tai(t)=ai,τi(t)=τi,fDi(t)=fDi，ϕDi(t)=2πfDit，则可使用信道冲击响应替换如下：
h(t)=∑i=0N(t)−1aie−j2πfcτi+2πfDit+ϕ0=∑i=0N(t)−1aie−jϕi(t)
\begin{align*}h(t) &=\sum_{i=0}^{N(t)-1}a_{i}e^{-j2\pi f_{c}\tau_{i}+2\pi f_{D_{i}}t+\phi_{0}}\\&=\sum_{i=0}^{N(t)-1}a_{i}e^{-j\phi_{i}(t)}\\\end{align*}
h(t)=i=0∑N(t)−1aie−j2πfcτi+2πfDit+ϕ0=i=0∑N(t)−1aie−jϕi(t)
其中ϕi(t)=2πfcτi−2πfDit−ϕ0\phi_{i}(t)=2\pi f_{c}\tau_{i}-2\pi f_{D_{i}}t-\phi_{0}ϕi(t)=2πfcτi−2πfDit−ϕ0
Rh(Δt)=E[h(t)h∗(t+Δt)]=E[∑iN(t)−1∑jN(t)−1aie−jϕi(t)aj∗ejϕj(t+Δt)]=E[∑iN(t)−1∑jN(t)−1aiaj∗e−jϕi(t)ejϕj(t+Δt)]=E[∑iN(t)−1∑jN(t)−1aiaj∗e−j(2πfcτi−2πfDit−ϕ0)+j(2πfcτj−2πfDj(t+Δt)−ϕ0)]=E[∑i=0N(t)∣ai∣2e−j2πfDiΔt]=E[∑i=0N(t)∣ai∣2e−j2πfmcosθiΔt]
\begin{align*}
R_{h}(\Delta t)&=E\left[ h(t)h^{*}(t+\Delta t)\right]\\
&=E\left[\sum_{i}^{N(t)-1}\sum_{j}^{N(t)-1}a_{i}e^{-j\phi_{i}(t)}a_{j}^{*}e^{j\phi_{j}(t+\Delta t)} \right]\\
&=E\left[\sum_{i}^{N(t)-1}\sum_{j}^{N(t)-1}a_{i}a_{j}^{*}e^{-j\phi_{i}(t)}e^{j\phi_{j}(t+\Delta t)} \right]\\
&=E\left[\sum_{i}^{N(t)-1}\sum_{j}^{N(t)-1}a_{i}a_{j}^{*}e^{-j(2\pi f_{c}\tau_{i}-2\pi f_{D_{i}}t-\phi_{0})+j(2\pi f_{c}\tau_{j}-2\pi f_{D_{j}}(t+\Delta t)-\phi_{0})} \right]\\
&=E\left[ \sum_{i=0}^{N(t)}|a_{i}|^2e^{-j2\pi f_{D_{i}}\Delta t} \right]\\
&=E\left[ \sum_{i=0}^{N(t)}|a_{i}|^2e^{-j2\pi f_{m}\cos \theta_{i}\Delta t} \right]
\end{align*}
Rh(Δt)=E[h(t)h∗(t+Δt)]=Ei∑N(t)−1j∑N(t)−1aie−jϕi(t)aj∗ejϕj(t+Δt)=Ei∑N(t)−1j∑N(t)−1aiaj∗e−jϕi(t)ejϕj(t+Δt)=Ei∑N(t)−1j∑N(t)−1aiaj∗e−j(2πfcτi−2πfDit−ϕ0)+j(2πfcτj−2πfDj(t+Δt)−ϕ0)=Ei=0∑N(t)∣ai∣2e−j2πfDiΔt=Ei=0∑N(t)∣ai∣2e−j2πfmcosθiΔt
当i≠ji\neq ji=j时ai,aj,ϕi,ϕja_{i},a_{j},\phi_{i},\phi_{j}ai,aj,ϕi,ϕj相互独立,fDi=fmcosθif_{D_{i}}=f_{m}\cos\theta_{i}fDi=fmcosθi ，功率归一化E{∣ai∣2}=1E\{|a_{i}|^2\}=1E{∣ai∣2}=1,当存在N条路径N→∞，θ∽(0,2π)N \to \infty，\theta \backsim(0,2\pi)N→∞，θ∽(0,2π)。看文献好像大于8条多径就可以了。
Rh(Δt)=12π∫−ππe−j2πfmcosθΔtdθ=1π∫0πe−j2πfmcosθΔtdθ=J0(2πfmΔt)
\begin{align*}R_{h}(\Delta t)&=\frac{1}{2\pi}\int_{-\pi}^{\pi}e^{-j2\pi f_{m}\cos\theta\Delta t} d\theta\\&=\frac{1}{\pi}\int_{0}^{\pi}e^{-j2\pi f_{m}\cos\theta\Delta t} d\theta\\&=J_{0}(2\pi f_{m}\Delta t)
\end{align*}
Rh(Δt)=2π1∫−ππe−j2πfmcosθΔtdθ=π1∫0πe−j2πfmcosθΔtdθ=J0(2πfmΔt)
0 阶贝塞尔函数性质
J0(x)=1π∫0πe−jxcosθdθJn(−x)=(−1)nJn(x)J0(x)=J0(−x)J_{0}(x)=\frac{1}{\pi}\int_{0}^{\pi}e^{-jx\cos\theta}d\theta\\J_{n}(-x)=(-1)^{n}J_{n}(x)\\J_{0}(x)=J_{0}(-x)
J0(x)=π1∫0πe−jxcosθdθJn(−x)=(−1)nJn(x)J0(x)=J0(−x)
上面我的得到了多径多普勒信道下的时域相关性描述。
