Channel论文阅读笔记7-3之multipath interference by Jakes

Level Crossing Rates

水平交叉率 N(R)N(R)N(R) 指信号包络以正方向穿过水平线 RRR 的速率，定义为：
N(R)=∫0∞r˙p(R,r˙)dr˙,(1.3-32)N(R)=\int_0^\infty \dot r p(R,\dot r)d\dot r,\tag {1.3-32}N(R)=∫0∞r˙p(R,r˙)dr˙,(1.3-32)
由概率密度函数 p(r,r˙,θ,θ˙)=r24π2b0b2exp⁡[−12(r2b0+r˙2b2+r2θ˙2b2)](1.3-33)p(r,\dot r,\theta, \dot\theta)={r^2\over 4\pi^2b_0b_2}\exp \left[-{1\over 2}\left({r^2\over b_0}+{\dot r^2\over b_2}+{r^2\dot\theta^2\over b_2}\right)\right]\tag {1.3-33}p(r,r˙,θ,θ˙)=4π2b0b2r2exp[−21(b0r2+b2r˙2+b2r2θ˙2)](1.3-33) 得到
p(r,r˙)=∫02πdθ∫−∞∞p(r,r˙,θ,θ˙)dθ˙=rb0e−r2/2b012πb2e−r˙2/2b2(1.3-34)p(r,\dot r)=\int_0^{2\pi}d\theta\int_{-\infty}^\infty p(r,\dot r,\theta, \dot\theta)d\dot\theta ={r\over b_0}e^{-r^2/ 2b_0}{1\over\sqrt{2\pi b_2}}e^{-\dot r^2/ 2b_2}\tag {1.3-34}p(r,r˙)=∫02πdθ∫−∞∞p(r,r˙,θ,θ˙)dθ˙=b0re−r2/2b02πb21e−r˙2/2b2(1.3-34)
证明（1.3-33）：
设 x=(x,x˙,y,y˙)T\mathrm x=(x, \dot x, y,\dot y)^Tx=(x,x˙,y,y˙)T, xxT=(σx20000σx˙20000σx20000σx˙2)=(b00000b20000b00000b2)\mathrm x\mathrm x^T=\begin{pmatrix}\sigma_x^2&0&0&0\\0&\sigma_{\dot x}^2&0&0\\0&0&\sigma_x^2&0\\0&0&0&\sigma_{\dot x}^2\end{pmatrix}=\begin{pmatrix}b_0&0&0&0\\0&b_2&0&0\\0&0&b_0&0\\0&0&0&b_2\end{pmatrix}xxT=⎝⎜⎜⎛σx20000σx˙20000σx20000σx˙2⎠⎟⎟⎞=⎝⎜⎜⎛b00000b20000b00000b2⎠⎟⎟⎞
由
p(x,x˙,y,y˙)=14π2b0b2exp⁡(x2+y2b0+x˙2+y˙2b2),p(x, \dot x, y,\dot y)={1\over 4\pi^2 b_0b_2}\exp({x^2+y^2\over b_0}+{\dot x^2+\dot y^2\over b_2}),p(x,x˙,y,y˙)=4π2b0b21exp(b0x2+y2+b2x˙2+y˙2), dxdx˙dydy˙=r2drdr˙dθdθ˙dxd\dot xdyd\dot y=r^2drd\dot rd\theta d\dot\thetadxdx˙dydy˙=r2drdr˙dθdθ˙ 得到（1.3-33）。证毕。

水平交叉率 N(R)=p(R)2πb2∫0∞r˙e−r˙2/2b2dr˙=b2πb0ρe−ρ2(1.3-35)N(R)={p(R)\over\sqrt{2\pi b_2}}\int_0^\infty \dot r e^{-\dot r^2/ 2b_2}d\dot r=\sqrt{b_2\over \pi b_0}\rho e^{-\rho^2}\tag {1.3-35}N(R)=2πb2p(R)∫0∞r˙e−r˙2/2b2dr˙=πb0b2ρe−ρ2(1.3-35) 这里 ρ=R/2b0\rho=R/\sqrt{2b_0}ρ=R/2b0 。对于电场分量 EzE_zEz， b2/b0=ωm2/2b_2/b_0=\omega_m^2/2b2/b0=ωm2/2。

Duration of Fades

平均衰落区间指信号包络在 RRR 以下的平均时间。设第 iii 的衰落时间为 τi\tau_iτi，那么在时间 TTT 衰落的概率为 p(r<R)=1T∑τi(1.3-40)p(r<R)={1\over T}\sum \tau_i \tag {1.3-40}p(r<R)=T1∑τi(1.3-40) 平均衰落时间 τˉ=1TN(R)∑τi=1N(R)p(r<R)(1.3-41)\bar\tau={1\over T N(R)}\sum \tau_i ={1\over N(R)}p(r<R) \tag {1.3-41}τˉ=TN(R)1∑τi=N(R)1p(r<R)(1.3-41)

Ref.

Microwave Mobile Communicatiaons, Ch1.
W. B. Davenport, Jr., and W. L. Root, An Introduction to the Theory of Randum Signals and Noise, McGraw-Hill, New York, 1958.