比较

	C.Loo	Corazza	Lutz
	非静止轨道卫星	非静止轨道卫星	静止轨道卫星
	乡村	所有	所有
	模拟卫星的直升机	实际卫星	实际卫星

1. C.Loo模型和Corazza模型适用于描述非静止轨道卫星信道的传播特性
   Lutz模型适用于描述静止轨道卫星信道的传播特性；
2. C.Loo模型只适用于乡村信道环境，
   而Corazza模型和Lutz模型均适用所有的卫星移动通信信道环境（公路、乡村、郊区和城市）；
3. C.Loo模型拟合参数时依据的实测数据来自于模拟卫星的直升机所发射的信号；
   而Corazza模型和Lutz模型则采用实际卫星所发射信号的实测数据，对卫星移动通信信道传输特性的反映更为真实。

C. Loo模型：直射阴影，多径不阴影

A statistical model for a land mobile satellite link
Chun Loo, “A statistical model for a land mobile satellite link,” in IEEE Transactions on Vehicular Technology, vol. 34, no. 3, pp. 122-127, Aug. 1985, doi: 10.1109/T-VT.1985.24048.

假设接收信号中只有直射信号分量受到阴影遮蔽的作用而多径信号分量不受阴影遮蔽的作用，因此该模型又称为部分阴影信道模型。
C. Loo模型接收信号包络的概率密度函数
fr(r)=∫0∞fr(r∣z)fz(z)dz=rb02πd0∫0∞1zexp⁡[−(ln⁡z−μ)22d0−(r2+z2)2b0]dz {f_r}\left( r \right) = \int_0^\infty {{f_r}\left( {r\left| z \right.} \right){f_z}\left( z \right)dz = \frac{r}{{{{\text{b}}_{\text{0}}}\sqrt {2\pi {d_0}} }}\int_0^\infty {\frac{1}{z}\exp \left[ { - \frac{{{{\left( {\ln z - \mu } \right)}^2}}}{{2{d_0}}} - \frac{{\left( {{r^2} + {z^2}} \right)}}{{2{b_0}}}} \right]} } {\text{dz }}fr​(r)=∫0∞​fr​(r∣z)fz​(z)dz=b0​2πd0​​r​∫0∞​z1​exp[−2d0​(lnz−μ)2​−2b0​(r2+z2)​]dz 
z表示直射波，
b0是平均散射多径功率，
μ和d0分别是lnz的均值和方差。

Corazza模型：直射和多径都阴影

假设接收信号中的直射信号分量和多径信号分量同时都受到阴影遮蔽的作用，因此该模型又称为全阴影信道模型
接收信号包络的概率密度函数为
fr(r)=∫0∞fr(r∣s)fs(s)= 2(k + 1)rhσs2πexp⁡(−k)∫0∞exp⁡[−(k+1)r2s2−(ln⁡s−μ)22(hσ)2]I0(2rk(k+1)s)ds{f_r}\left( r \right) = \int_0^\infty {{f_r}\left( {r\left| s \right.} \right){f_s}\left( s \right)} {\text{ = }}\frac{{{\text{2}}\left( {{\text{k + 1}}} \right)r}}{{h\sigma s\sqrt {2\pi } }}\exp \left( { - k} \right)\int_0^\infty {\exp \left[ { - \frac{{\left( {k + 1} \right){r^2}}}{{{s^2}}} - \frac{{{{\left( {\ln s - \mu } \right)}^2}}}{{2{{\left( {h\sigma } \right)}^2}}}} \right]} {I_0}\left( {\frac{{2r\sqrt {k\left( {k + 1} \right)} }}{s}} \right)dsfr​(r)=∫0∞​fr​(r∣s)fs​(s) = hσs2π​2(k + 1)r​exp(−k)∫0∞​exp[−s2(k+1)r2​−2(hσ)2(lns−μ)2​]I0​(s2rk(k+1)​​)ds
S(t)表示阴影慢衰落效应，
h=(ln10)/20，
μ和σ2为lns的均值和方差，
k为Rice因子，
r为接收信号包络。
K(α)= k0+ k1α+ k2α2μ(α)= μ0+ μ1α+ μ2α2+ μ3α3σ(α)= σ0+ σ1α{\text{K}}\left( \alpha \right){\text{ = }}{{\text{k}}_{\text{0}}}{\text{ + }}{{\text{k}}_{\text{1}}}\alpha {\text{ + }}{{\text{k}}_{\text{2}}}{\alpha ^{\text{2}}} \\ \mu \left( \alpha \right){\text{ = }}{\mu _{\text{0}}}{\text{ + }}{\mu _{\text{1}}}\alpha {\text{ + }}{\mu _{\text{2}}}{\alpha ^{\text{2}}}{\text{ + }}{\mu _{\text{3}}}{\alpha ^{\text{3}}} \\ \sigma \left( \alpha \right){\text{ = }}{\sigma _{\text{0}}}{\text{ + }}{\sigma _{\text{1}}}\alpha {\text{ }}K(α) = k0​ + k1​α + k2​α2μ(α) = μ0​ + μ1​α + μ2​α2 + μ3​α3σ(α) = σ0​ + σ1​α 

Lutz模型：好坏2个状态

E. Lutz, D. Cygan, M. Dippold, F. Dolainsky and W. Papke, “The land mobile satellite communication channel-recording, statistics, and channel model,” in IEEE Transactions on Vehicular Technology, vol. 40, no. 2, pp. 375-386, May 1991, doi: 10.1109/25.289418.

将卫星移动通信信道分为两个状态——“好状态”和“坏状态”。
当信道为“好状态”时接收信号中的直射信号分量和多径信号分量均不受阴影遮蔽的作用；
当信道为“坏状态”时接收信号中只有受到阴影遮蔽作用的多径信号分量且不存在直射信号分量，
因此该模型又称为两状态信道模型。

好状态下接收信号功率S的归一化概率密度函数为
fs_rician(s)=cexp⁡[−c(s+1)]I0(2cs){f_{s\_rician}}\left( s \right) = c\exp \left[ { - c\left( {s + 1} \right)} \right]{I_0}\left( {2c\sqrt s } \right)fs_rician​(s)=cexp[−c(s+1)]I0​(2cs​)
C为归一化因子，c = 1/(2σ2)

坏状态下接收信号功率S的归一化概率密度函数为
fs_rayl_LN(s)= ∫0∞fs(s∣s0)fs0(s0)ds0= 10σln102π∫0∞1s02exp[- ss0−(10log⁡10s0−μ)22σ2]{f_{s\_rayl\_LN}}\left( {\text{s}} \right){\text{ = }}\int_{\text{0}}^\infty {{{\text{f}}_{\text{s}}}\left( {s\left| {{s_0}} \right.} \right){f_{s0}}\left( {{s_0}} \right)} {\text{d}}{{\text{s}}_{\text{0}}} \\ {\text{ = }}\frac{{{\text{10}}}}{{\sigma {\text{ln10}}\sqrt {{\text{2}}\pi } }}\int_{\text{0}}^\infty {\frac{{\text{1}}}{{{{\text{s}}_{\text{0}}}^{\text{2}}}}} {\text{exp}}\left[ {{\text{ - }}\frac{{\text{s}}}{{{{\text{s}}_{\text{0}}}}} - \frac{{{{\left( {10{{\log }_{10}}{s_0} - \mu } \right)}^2}}}{{2{\sigma ^2}}}} \right]{\text{ }}fs_rayl_LN​(s) = ∫0∞​fs​(s∣s0​)fs0​(s0​)ds0​ = σln102π​10​∫0∞​s0​21​exp[ - s0​s​−2σ2(10log10​s0​−μ)2​] 
S0 为短时平均接收功率

总的接收信号功率S的概率密度函数为
fs(s)= (1 - A)fs_rician(s)+ Afs_rayl_LN(s){f_s}\left( {\text{s}} \right){\text{ = }}\left( {{\text{1 - A}}} \right){{\text{f}}_{{\text{s\_rician}}}}\left( {\text{s}} \right){\text{ + A}}{{\text{f}}_{{\text{s\_rayl\_LN}}}}\left( {\text{s}} \right)fs​(s) = (1 - A)fs_rician​(s) + Afs_rayl_LN​(s)

Rayleigh and Rician 信道生成

2.4.2 Rayleigh and Rician fading
(2.54)
Fundamentals of Wireless Communication

Rayleigh channel (NLoS), Rician channel (LoS)
https://blog.csdn.net/weixin_41181312/article/details/123516627

bandwidth
B=180kHzB = 180kHzB=180kHz

Noise power spectral density
N0=10−17010mW/Hz{N_0} = {10^{\frac{{ - 170}}{{10}}}}mW/HzN0​=1010−170​mW/Hz
N0,dBm=−170dBm/Hz{N_{0,dBm}} = - 170dBm/HzN0,dBm​=−170dBm/Hz
Noise power
PN=N0B(mW)=10−17010×180000(mW){P_N} = {N_0}B\left( {mW} \right)= {10^{{{ - 170} \over {10}}}} \times 180000\left( {mW} \right)PN​=N0​B(mW)=1010−170​×180000(mW)
PN,dBm=10log⁡10(PN)=N0,dBm+10log⁡10(B)(dBm)=−170+10log⁡10(180000)(dBm){P_{N,dBm}} = 10{\log _{10}}\left( {{P_N}} \right) = {N_{0,dBm}} + 10{\log _{10}}\left( B \right)\left( {dBm} \right) = - 170 + 10{\log _{10}}\left( {180000} \right)\left( {dBm} \right)PN,dBm​=10log10​(PN​)=N0,dBm​+10log10​(B)(dBm)=−170+10log10​(180000)(dBm)

pathloss
L(d,f)=(4πdfc)2L\left( {d,f} \right) = {\left( {{{4\pi df} \over c}} \right)^2}L(d,f)=(c4πdf​)2
LdB(d,f)=10log⁡10(L(d,f))(dB)L_{dB}\left( {d,f} \right) = 10{\log _{10}}\left( {L\left( {d,f} \right)} \right)\left( {dB} \right)LdB​(d,f)=10log10​(L(d,f))(dB)

Further advancements for E-UTRA physical layer aspects (Release 9). 3GPP TS 36.814, Mar. 2010.
3GPP TS 36.814
the 3GPP propagation environment [52, Table B.1.2.1-1].
LOS
lossdB=28 + 22*log10(dist_) + 20*log10(freq_/10^9);
NLOS
lossdB=22.7 + 36.7*log10(dist_) + 26*log10(freq_/10^9);

attenuation
α(d,f)=L(d,f)−1/PN=10−LdB(d,f)10/10PN,dBm10=10−PN,dBm−LdB(d,f)10\alpha \left( {d,f} \right) = \sqrt {L{{\left( {d,f} \right)}^{ - 1}}/P_N} = \sqrt {{{10}^{{{ - L_{dB}\left( {d,f} \right)} \over {10}}}}/{{10}^{{{P_{N,dBm}} \over {10}}}}} = \sqrt {{{10}^{{{ - P_{N,dBm}- L_{dB}\left( {d,f} \right)} \over {10}}}}}α(d,f)=L(d,f)−1/PN​​=1010−LdB​(d,f)​/1010PN,dBm​​​=1010−PN,dBm​−LdB​(d,f)​​

1 M-antenna transmitter → 1 single-antenna receiver
NLOS
hNLOS=1/2(c+je)∈CM×1{{\bf{h}}_{NLOS}} = \sqrt {1/2} \left( {{\bf{c}} + j{\bf{e}}} \right) \in {{\Bbb C}^{M \times 1}}hNLOS​=1/2​(c+je)∈CM×1
均值为0，方差为var的复高斯分布
sqrt(var/2)*(randn(1,K) +j*randn(1,K))
randn产生均值为0，方差为1的正态分布随机数
Rayleigh
hRayleigh(d,f)=α(d,f)hNLOS∈CM×1{{\bf{h}}_{Rayleigh}}\left( {d,f} \right) = \alpha \left( {d,f} \right){{\bf{h}}_{NLOS}} \in {{\Bbb C}^{M \times 1}}hRayleigh​(d,f)=α(d,f)hNLOS​∈CM×1
Rician
hRician(d,f)=α(d,f)(ε1+εa(θ)+11+εhNLOS)∈CM×1{{\bf{h}}_{Rician}}\left( {d,f} \right) = \alpha \left( {d,f} \right)\left( {\sqrt {\frac{\varepsilon }{{1 + \varepsilon }}} {\bf{a}}\left( \theta \right) + \sqrt {\frac{1}{{1 + \varepsilon }}} {{\bf{h}}_{NLOS}}} \right) \in {{\Bbb C}^{M \times 1}}hRician​(d,f)=α(d,f)(1+εε​​a(θ)+1+ε1​​hNLOS​)∈CM×1

Shadowed-Rician 直射径 散射径

优点
计算量小

Although some mathematical models, such as those of Loo, Barts–Stutzman, and Karasawa et al.,have been developed to describe the satellite channel, the shadowed-Rician model proposed in [19] is a popular model, which provides a significantly less computational burden than other channel models.

A. Abdi, W. C. Lau, M. . -S. Alouini and M. Kaveh, “A new simple model for land mobile satellite channels: first- and second-order statistics,” in IEEE Transactions on Wireless Communications, vol. 2, no. 3, pp. 519-528, May 2003, doi: 10.1109/TWC.2003.811182.

复包络
the lowpass-equivalent complex envelope of the stationary narrowband shadowed Rice single model can be written as
ℜ(t)=A(t)exp⁡[jα(t)]+Z(t)exp⁡(jζ0)\Re (t) = A(t)\exp [j\alpha (t)] + Z(t)\exp (j\zeta_0)ℜ(t)=A(t)exp[jα(t)]+Z(t)exp(jζ0​)
α(t)α(t)α(t) is the stationary random phase process with uniform distribution over [0,2π)[0,2π)[0,2π), 随机，多天线咋办？
ζ0ζ0ζ0 is the deterministic phase of the LOS component. 随机选一个就行，多天线咋办？
The independent stationary random processes A(t)A(t)A(t) and Z(t)Z(t)Z(t), which are also independent of α(t)α(t)α(t), are the amplitudes of the scatter and the LOS components, following Rayleigh and Nakagami distributions, respectively 根据分布及其参数生成
pA(a)=ab0exp⁡(−a22b0),a≥0p_A (a) = {{a}\over {b_0}}\, \exp \left( {{ {-}a^2}\over {2\,b_0}} \right),a \ge 0pA​(a)=b0​a​exp(2b0​−a2​),a≥0
pZ(z)=2mmΓ(m)Ωmz2m−1exp⁡(−mz2Ω),z≥0p_Z (z) = {{2m^m}\over {\Gamma (m)\Omega^m}} z^{2m - 1} \exp \left( {{{-}m\,z^2}\over {\Omega}} \right),\qquad z \ge 0pZ​(z)=Γ(m)Ωm2mm​z2m−1exp(Ω−mz2​),z≥0
Γ(.)Γ(.)Γ(.) is the gamma function,
2b0=E[A2]2b_0 = E[A^2]2b0​=E[A2] is the average power of the scatter component,
m=(E[Z2])2/Var[Z2]≥0m = (E[Z^2])^2 / {\rm Var}[Z^2] \ge 0m=(E[Z2])2/Var[Z2]≥0 is the Nakagami parameter with Var[.]Var[.]Var[.] as the variance,
Ω=E[Z2]\Omega = E[Z^2]Ω=E[Z2] is the average power of the LOS component.

包络
Let us define the envelope as R(t)=∣ℜ(t)∣R(t) = \left\vert \Re (t) \right\vertR(t)=∣ℜ(t)∣
the shadowed Rice PDF for the signal envelope in an LMS channel can be written as
pR(r)=EZ[rb0exp⁡(−r2+Z22b0)I0(Zrb0)],r≥0p_R (r) = E_Z \left[ {{r}\over {b_0}}\, \exp \left( {-}{{r^2 + Z^2}\over {2b_0}} \right)I_0 \left( {{Z\,r}\over {b_0}} \right) \right],\qquad r \ge 0pR​(r)=EZ​[b0​r​exp(−2b0​r2+Z2​)I0​(b0​Zr​)],r≥0
EZ[.]EZ[.]EZ[.] is the expectation with respect to ZZZ
In(.)In(.)In(.) is the nnn th-order modified Bessel function of the first kind.
pR(r)=(2b0m2b0m+Ω)mrb0exp⁡(−r22b0)1F1(m,1,Ωr22b0(2b0m+Ω)),r≥0p_R (r) = \left( {{2b_0 m}\over {2b_0 m + \Omega}} \right)^m {{r}\over {b_0 }}\, \exp \left({-}{{r^2}\over {2b_0}}\right) {\;}_1 F_1 \left( m,\,1,\, {{\Omega r^2}\over {2b_0 (2b_0 m + \Omega)}} \right),r \ge 0pR​(r)=(2b0​m+Ω2b0​m​)mb0​r​exp(−2b0​r2​)1​F1​(m,1,2b0​(2b0​m+Ω)Ωr2​),r≥0
where 1F1(.,.,.)_1F_1 (.,\,.,\,.)1​F1​(.,.,.) is the confluent hypergeometric function [28].
For m=0m=0m=0, (3) simplifies to the Rayleigh PDF in (1), i.e., (r/b0)exp⁡(−r2/2b0)(r/b_0)\exp ({-}r^2 / 2b_0)(r/b0​)exp(−r2/2b0​),
for m=∞m=∞m=∞, it reduces to the Rice PDF (r/b0)exp⁡(−(r2+Ω)/2b0)I0(Ωr/b0)(r/b_0)\exp ({-}(r^2 + \Omega) / 2b_0)I_0 (\sqrt{\Omega} r / b_0)(r/b0​)exp(−(r2+Ω)/2b0​)I0​(Ω​r/b0​).

包络的平方
S(t)=R2(t)S(t) = R^2(t)S(t)=R2(t) is the instantaneous power, and its PDF can be derived from (3)as
pS(s)=(2b0m2b0m+Ω)m12b0exp⁡(−s2b0)1F1(m,1,Ωs2b0(2b0m+Ω)),s≥0p_S (s) = \left( {{2b_0 m}\over {2b_0 m + \Omega}} \right)^m {{1}\over {2b_0 }}\, \exp \left({-}{{s}\over {2b_0}} \right) {\,}_1F_1 \left( m,\, 1,\, {{\Omega s}\over {2b_0 (2b_0 m + \Omega)}} \right),s \ge 0pS​(s)=(2b0​m+Ω2b0​m​)m2b0​1​exp(−2b0​s​)1​F1​(m,1,2b0​(2b0​m+Ω)Ωs​),s≥0

the probability density function (PDF)
f∣h∣2(x)=αexp⁡(−βx)1F1(m;1;δx){f_{|{h}{|^2}}}(x) = {\alpha }\exp {( - {\beta }x)_1}{F_1}({m};1;{\delta }x)f∣h∣2​(x)=αexp(−βx)1​F1​(m;1;δx)
f∣h∣2(x)=αe−βx1F1(m;1;δx)f_{\left |{ {h} }\right |^{2} } \left ({x }\right) = \alpha e^{ - \beta x} {}_{1}F_{1} \left ({{m;1;\delta x} }\right)f∣h∣2​(x)=αe−βx1​F1​(m;1;δx)
α=12b(2bm2bm+Ω)m{\alpha } = \frac{1}{{2{b}}}{\left( {\frac{{2{b}\;{m}}}{{2{b}\;{m} + {\Omega }}}} \right)^{{m}}}α=2b1​(2bm+Ω2bm​)m
β=12b{\beta } = \frac{1}{{2{b}}}β=2b1​
δ=Ω2b(2bm+Ω){\delta } = \frac{{{\Omega }}}{{2{b}(2{b}\;{m} + {\Omega })}}δ=2b(2bm+Ω)Ω​

总结

the confluent hypergeometric function of the first kind [31, Eq. (9.14.1)],
https://blog.csdn.net/gsgbgxp/article/details/126136164
https://www.mathworks.com/help/symbolic/hypergeom.html#bt1nkmw-2
1F1(m;1;δx)=eδx∑k=0m−1(−1)k(1−m)k(δx)k(k!)2_1{F_1}({m};1;{\delta }x) = {e^{{\delta }x}}\sum\nolimits_{k = 0}^{{m} - 1} {\frac{{{{( - 1)}^k}{{(1 - {m})}_k}{{({\delta }x)}^k}}}{{{{(k!)}^2}}}}1​F1​(m;1;δx)=eδx∑k=0m−1​(k!)2(−1)k(1−m)k​(δx)k​
the pochhammer symbol
(x)K = x (x + 1) (x + 2) · · · (x − k + 1)

The SAT is equipped with NtN_tNt​ -antennas
h=b(φ)h~∈CNt×1\mathbf {h}=\sqrt {b(\varphi )}\tilde {\mathbf {h}} \in \mathbb {C}^{N_{t}\times 1}h=b(φ)​h~∈CNt​×1
b(φ)=(J1(u)2u+36J3(u)u3)2b(\varphi )=\left ({\frac {J_{1}(u)}{2u}+36\frac {J_{3}(u)}{u^{3}} }\right )^{2}b(φ)=(2uJ1​(u)​+36u3J3​(u)​)2
u=2.07123sin⁡φsin⁡φ3dBu=2.07123\frac {\sin \varphi }{\sin \varphi _{3\mathrm {dB}}}u=2.07123sinφ3dB​sinφ​
h~=Aexp⁡(jψ)⁣+⁣Zexp⁡(jϕ)\tilde {\mathbf {h}}=A\exp (j\boldsymbol {\psi })\!+\!Z\exp (j\boldsymbol {\phi })h~=Aexp(jψ)+Zexp(jϕ)

Gs(θ)g=Gs(θ)∣gˉ/D∣2G_{s}(\theta){g}=G_{s}(\theta){\left |{ {\bar g} }\right /D|^{2}}Gs​(θ)g=Gs​(θ)∣gˉ​/D∣2
g=∣gˉ/D∣2{g} = {\left |{ {\bar g} }\right /D|^{2}}g=∣gˉ​/D∣2
gˉ=Aexp⁡(jϕ)+Zexp⁡(jφ)\bar g = A\exp \left ({{j\phi } }\right) + Z\exp \left ({{j\varphi } }\right)gˉ​=Aexp(jϕ)+Zexp(jφ)
Gs(θ)=Gtx(J1(u)2u+36J3(u)u3)2{G_{s}}\left ({\theta }\right) = G_{tx}{\left ({{\frac {J_{1}\left ({u }\right)}{2u} + 36\frac {J_{3}\left ({u }\right)}{u^{3}}} }\right)^{2}}Gs​(θ)=Gtx​(2uJ1​(u)​+36u3J3​(u)​)2

hSJ=LGt,SJGr,SJKBTBgSJ=C4πfdSJGt,SJGr,SJKBTBgSJ{{\bf{h}}_{SJ}} = \sqrt {L\frac{{{G_{t,SJ}}{G_{r,SJ}}}}{{{K_B}TB}}} {{\bf{g}}_{SJ}} = \frac{C}{{4\pi f{d_{SJ}}}}\sqrt {\frac{{{G_{t,SJ}}{G_{r,SJ}}}}{{{K_B}TB}}} {{\bf{g}}_{SJ}}hSJ​=LKB​TBGt,SJ​Gr,SJ​​​gSJ​=4πfdSJ​C​KB​TBGt,SJ​Gr,SJ​​​gSJ​

y=LGsG(φ)hx+ny = \sqrt {LG_{s}G(\varphi )}h x + ny=LGs​G(φ)​hx+n
L=(c4πfcd)2L = \left ({{\frac {c}{{4\pi f_{c}d}}} }\right)_{}^{2}L=(4πfc​dc​)2​
G(φ)=Gj(J1(u)2u+36J3(u)u3)2G(\varphi ) = G_{j }\left ({{\frac {J_{1}\left ({u }\right)}{2u} + 36\frac {J_{3}\left ({u }\right)}{{u^{3}}}} }\right)_{}^{2}G(φ)=Gj​(2uJ1​(u)​+36u3J3​(u)​)2​
f∣h∣2(x)=αe−βx1F1(m;1;δx)f_{\left |{ {h} }\right |^{2} } \left ({x }\right) = \alpha e^{ - \beta x} {}_{1}F_{1} \left ({{m;1;\delta x} }\right)f∣h∣2​(x)=αe−βx1​F1​(m;1;δx)

Joint CoMP Transmission for UAV-Aided Cognitive Satellite Terrestrial Networks

For the SAT channel links, the channel coefficients from SAT to SU/PU follow shadowed-Rician (SR) model [13], [33], which can be written as
gˉ=Aexp⁡(jϕ)+Zexp⁡(jφ)\bar g = A\exp \left ({{j\phi } }\right) + Z\exp \left ({{j\varphi } }\right)gˉ​=Aexp(jϕ)+Zexp(jφ)

Denote the transmit power of SAT as PsPsPs , which is assumed to be invariant over each time slot.
In addition, the transmit power of BS and UAV within time slot nnn are respectively denoted as pb[n]pb[n]pb[n] and pu[n]pu[n]pu[n] .
Therefore, the achievable rate of SU (bps/Hz) within time slot nnn can be expressed as
Rm[n]=log⁡2(1+Gupu[n]hm[n]+Gbpb[n]fmGs(θ)Psgm+σm2){R_{m}}\left [{ n }\right] = {\log _{2}}\left ({{1 + \frac {{G_{u}{p_{u}}\left [{ n }\right]{h_{m}}\left [{ n }\right] + {G_{b}}{p_{b}}\left [{ n }\right]{f_{m}}}}{{G_{s}(\theta){P_{s}}{g_{m}} + \sigma _{m}^{2}}}} }\right)Rm​[n]=log2​(1+Gs​(θ)Ps​gm​+σm2​Gu​pu​[n]hm​[n]+Gb​pb​[n]fm​​)
where gmg_{m}gm​ represents the channel gain from SAT to SU and can be calculated as g=∣gˉ/D∣2{g} = {\left |{ {\bar g} }\right /D|^{2}}g=∣gˉ​/D∣2 , where
DDD denotes the distance between SAT and ground user.
σm2\sigma _{m}^{2}σm2​ denotes the power of additive Gaussian white noise at SU,
GuG_{u}Gu​ and GbG_{b}Gb​ represent the UAV and BS directional antenna gain,
Gs(θ)G_{s}(\theta)Gs​(θ) denotes the SAT’s transmit antenna gain, which is determined by the angle between SAT and SU, and can be approximately represented as [34]
Gs(θ)=Gtx(J1(u)2u+36J3(u)u3)2{G_{s}}\left ({\theta }\right) = G_{tx}{\left ({{\frac {J_{1}\left ({u }\right)}{2u} + 36\frac {J_{3}\left ({u }\right)}{u^{3}}} }\right)^{2}}Gs​(θ)=Gtx​(2uJ1​(u)​+36u3J3​(u)​)2

Secure Transmission in Cognitive Satellite Terrestrial Networks

K. An, M. Lin, J. Ouyang and W. -P. Zhu, “Secure Transmission in Cognitive Satellite Terrestrial Networks,” in IEEE Journal on Selected Areas in Communications, vol. 34, no. 11, pp. 3025-3037, Nov. 2016, doi: 10.1109/JSAC.2016.2615261.

the SAT, PU, SU and EVE are all equipped with a single antenna
the terrestrial BS has N antennas

the overall channel for satellite link can be modeled as
g=b(φ)gˉg = \sqrt {b\left ({ {\varphi} }\right )} \bar gg=b(φ)​gˉ​

Given a receiver’s position within the satellite spot beam coverage area, the beam gain factor can be approximated as [1], [2]
b(φ)=(J1(u)2u+36J3(u)u3)2b\left ({ \varphi }\right ) = \left ({ {\frac {J_{1}\left ({ u }\right )}{2u} + 36\frac {J_{3}\left ({ u}\right )}{{u^{3}}}} }\right )^{2}b(φ)=(2uJ1​(u)​+36u3J3​(u)​)2
u=2.07123sin⁡φsin⁡φ3dBu = 2.07123\frac {\sin \varphi }{\sin \varphi _{\mathrm{3dB}}}u=2.07123sinφ3dB​sinφ​
φ\varphiφ is the angle between the location of the corresponding receiver and the beam center with respect to the satellite, and φ3dB\varphi _{\mathrm{3dB}}φ3dB​ is the 3-dB angle.

the channel fading coefficient gˉ\bar ggˉ​ is described as
gˉ=Aexp⁡(jψ)+Zexp⁡(jζ)\bar g = A\exp \left ({ {j\psi } }\right ) + Z\exp \left ({ {j\zeta } }\right )gˉ​=Aexp(jψ)+Zexp(jζ)

AAA is the amplitudes of the scattering components, which are independent stationary random processes following Rayleigh distributions.
pA(a)=ab0exp⁡(−a22b0),a≥0p_A (a) = {{a}\over {b_0}}\, \exp \left( {{ {-}a^2}\over {2\,b_0}} \right),a \ge 0pA​(a)=b0​a​exp(2b0​−a2​),a≥0
ψψψ is a stationary random phase vector with elements uniformly distributed over [0,2π)[0,2π)[0,2π) ,

ZZZ is the amplitudes of the LOS components, which are independent stationary random processes following Nakagami-mmm distributions.
pZ(z)=2mmΓ(m)Ωmz2m−1exp⁡(−mz2Ω),z≥0p_Z (z) = {{2m^m}\over {\Gamma (m)\Omega^m}} z^{2m - 1} \exp \left( {{{-}m\,z^2}\over {\Omega}} \right),\qquad z \ge 0pZ​(z)=Γ(m)Ωm2mm​z2m−1exp(Ω−mz2​),z≥0
ζζζ the deterministic phase vector of the LOS component.

The SR fading distribution
gˉ∼SR(Ω,b,m){ \bar g} \sim \mathrm {SR}\left ({ {\Omega ,b,m} }\right )gˉ​∼SR(Ω,b,m)
Ω\OmegaΩ being the average power of the LoS component,
2b2b2b the average power of the multipath component, and
mmm the Nakagami-mmm parameter corresponding to the fading severity.

仿真参数
Omega=0.835
b=0.126
m=10
φ3dB=0.4/180∗π\varphi _{3\mathrm {dB}}=0.4/180*\piφ3dB​=0.4/180∗π

X=γˉb(φ)∣gˉ∣2X={\bar \gamma }{b\left ({ {\varphi } }\right )}\left |{ {\bar {g}} }\right |^{2}X=γˉ​b(φ)∣gˉ​∣2
fX(x)=αb(φ)γˉexp⁡(−βb(φ)γˉx)1F1(m;1;δb(φ)γˉx)f_{X}( x ) = \frac {\alpha }{{{b( {\varphi } )}\bar \gamma }}\exp \left ({ { - \frac {\beta }{{{b( {\varphi } )}\bar \gamma }}x} }\right ){}_{1}{F_{1}}\left ({ {m;1;\frac {\delta }{{{b( {\varphi } )}\bar \gamma }}x} }\right )fX​(x)=b(φ)γˉ​α​exp(−b(φ)γˉ​β​x)1​F1​(m;1;b(φ)γˉ​δ​x)
the probability density function (PDF)
f∣h∣2(x)=αexp⁡(−βx)1F1(m;1;δx){f_{|{h}{|^2}}}(x) = {\alpha }\exp {( - {\beta }x)_1}{F_1}({m};1;{\delta }x)f∣h∣2​(x)=αexp(−βx)1​F1​(m;1;δx)
α=12b(2bm2bm+Ω)m{\alpha } = \frac{1}{{2{b}}}{\left( {\frac{{2{b}\;{m}}}{{2{b}\;{m} + {\Omega }}}} \right)^{{m}}}α=2b1​(2bm+Ω2bm​)m
β=12b{\beta } = \frac{1}{{2{b}}}β=2b1​
δ=Ω2b(2bm+Ω){\delta } = \frac{{{\Omega }}}{{2{b}(2{b}\;{m} + {\Omega })}}δ=2b(2bm+Ω)Ω​

Resource Allocations for Secure Cognitive Satellite-Terrestrial Networks

The SAT is equipped with NtN_tNt​ -antennas
h=b(φ)h~∈CNt×1\mathbf {h}=\sqrt {b(\varphi )}\tilde {\mathbf {h}} \in \mathbb {C}^{N_{t}\times 1}h=b(φ)​h~∈CNt​×1
b(φ)=(J1(u)2u+36J3(u)u3)2b(\varphi )=\left ({\frac {J_{1}(u)}{2u}+36\frac {J_{3}(u)}{u^{3}} }\right )^{2}b(φ)=(2uJ1​(u)​+36u3J3​(u)​)2
u=2.07123sin⁡φsin⁡φ3dBu=2.07123\frac {\sin \varphi }{\sin \varphi _{3\mathrm {dB}}}u=2.07123sinφ3dB​sinφ​
h~=Aexp⁡(jψ)⁣+⁣Zexp⁡(jϕ)\tilde {\mathbf {h}}=A\exp (j\boldsymbol {\psi })\!+\!Z\exp (j\boldsymbol {\phi })h~=Aexp(jψ)+Zexp(jϕ)
where b(φ)b(\varphi )b(φ) is the corresponding beam gain factor, which is determined by their location.
φ\varphiφ is the angle between the corresponding receiver and the beam center, 怎么确定？？
φ3dB=0.4/180∗π\varphi _{3\mathrm {dB}}=0.4/180*\piφ3dB​=0.4/180∗π is the 3-dB angle.
J1(⋅)J_{1}(\cdot )J1​(⋅) and J3(⋅)J_{3}(\cdot )J3​(⋅) represent the first-kind Bessel function of order 1 and 3.
h~∈CNt×1\tilde {\mathbf {h}} \in \mathbb {C}^{N_{t}\times 1}h~∈CNt​×1 denotes the channel fading vector from SAT to the receiver, which include the scattering and the line-of-sight (LOS) components.
ψ∈[0,2π)\boldsymbol {\psi }\in [0,2\pi )ψ∈[0,2π) denotes the stationary random phase 随机，多天线咋办？
ϕ\boldsymbol {\phi }ϕ denotes the deterministic phase of the LOS component. 随机选一个就行，多天线咋办？
AAA and ZZZ are the amplitudes of the scattering and the LOS components. 根据分布及其参数生成
The beam angles from SAT to PU, to Eve and to SU are set as 0.01°, 0.4° and 0.8°, respectively

Outage Analysis of Cognitive Hybrid Satellite-Terrestrial Networks With Hardware Impairments and Multi-Primary Users

MMM terrestrial primary users (single antenna)
secondary satellite (single antenna) → relay (NNN antennas)→ destination (single antenna)

对噪声归一化了？？
hSJ=FSJgSJ=LGt,SJGr,SJKBTBgSJ=C4πfdSJGt,SJGr,SJKBTBgSJ{{\bf{h}}_{SJ}} = {F_{SJ}}{{\bf{g}}_{SJ}} = \sqrt {L\frac{{{G_{t,SJ}}{G_{r,SJ}}}}{{{K_B}TB}}} {{\bf{g}}_{SJ}} = \frac{C}{{4\pi f{d_{SJ}}}}\sqrt {\frac{{{G_{t,SJ}}{G_{r,SJ}}}}{{{K_B}TB}}} {{\bf{g}}_{SJ}}hSJ​=FSJ​gSJ​=LKB​TBGt,SJ​Gr,SJ​​​gSJ​=4πfdSJ​C​KB​TBGt,SJ​Gr,SJ​​​gSJ​

gSJ{{\mathbf{g}}_{SJ}}gSJ​ is the channel coefficient vector, which is often assumed to undergo SR fading [6]–[8],

FSJ{F_{SJ}}FSJ​ a scaling parameter including many practical effects, such as free space loss (FSL) and antenna pattern, which is given by
FSJ=C4πfdSJGt,SJGr,SJKBTB{F_{SJ}} = \frac{C}{{4\pi f{d_{SJ}}}}\sqrt {\frac{{{G_{t,SJ}}{G_{r,SJ}}}}{{{K_B}TB}}}FSJ​=4πfdSJ​C​KB​TBGt,SJ​Gr,SJ​​​ with
CCC being the light speed,
fff the carrier frequency,
dSJd_{SJ}dSJ​ the distance between the satellite and the user,

KB=1.38×10−23J/K{K_{B}} = 1.38 \times {10^{ - 23}}J/KKB​=1.38×10−23J/K the Boltzman constant,
TTT the receiver noise temperature,
BBB the carrier bandwidth.

Gr,SJ{{G_{r,{SJ}}}}Gr,SJ​ denotes the receive gain,
Gt,SJ{{G_{t,{SJ}}}}Gt,SJ​ the satellite beam gain, which can be approximately written as
Gt,SJ=Gmax⁡(J1(u)2u+36J3(u)u3)2{G_{t,{SJ}}} = {G_{\max }}{\left({{\frac {J_{1}(u)}{2u} + 36\frac {J_{3}(u)}{u^{3}}} }\right)^{2}}Gt,SJ​=Gmax​(2uJ1​(u)​+36u3J3​(u)​)2 [20],
Gmax⁡{{\mathrm{G}}_{\max }}Gmax​ being the maximal beam gain
u=2.07123sin⁡φsin⁡φ3dBu = 2.07123\frac {\sin \varphi }{{\sin {\varphi _{3{\mathrm{dB}}}}}}u=2.07123sinφ3dB​sinφ​ ,
φ\varphiφ is the angle between the location of the corresponding receiver and the beam center with respect to the satellite, and φ3dB{{\varphi _{3{\mathrm{dB}}}}}φ3dB​ is the 3 – dB angle.

The system parameters are given as the GEO, f=2f=2f=2 GHz, φ3dB=0.8∘{\varphi _{3{\mathrm{dB}}}} = {0.8^\circ }φ3dB​=0.8∘ , Gmax⁡=48dB{G_{\max }}=48dBGmax​=48dB , Gr,SJ=4dB{{G_{r,{SJ}}}}=4dBGr,SJ​=4dB , B=15B=15B=15 MHz, and T=300∘KT={300^\circ }\text{K}T=300∘K .
Without loss of generality, we set δ2R=δ2D=1δ2R=δ2D=1δ2R=δ2D=1 and in all plots we denote γ¯¯¯SR=γ¯¯¯SP=γ¯¯¯RD=γ¯¯¯RP=γ¯¯¯γ¯¯¯SR=γ¯¯¯SP=γ¯¯¯RD=γ¯¯¯RP=γ¯¯¯γ¯¯¯SR=γ¯¯¯SP=γ¯¯¯RD=γ¯¯¯RP=γ¯¯¯ , kSR=KRD=kSP=kRP=k=kSP=kRP=kkSR=KRD=kSP=kRP=k=kSP=kRP=kkSR=KRD=kSP=kRP=k=kSP=kRP=k and mU=1mU=1mU=1 , bU=0.063bU=0.063bU=0.063 , ΩU=0.0007ΩU=0.0007ΩU=0.0007 .

The Application of Power-Domain Non-Orthogonal Multiple Access in Satellite Communication Networks

an entire link budget of satellite→→→ User jjj (j∈1,2,…,M,j∈{1,2,…,M},j∈1,2,…,M, ),
including propagation loss, beam gain, channel statistical property, and receive antenna gain,
can be given by
Qj=LjGjGs(φj)h^j2Q_{j}=L_{j}G_{j}G_{s}(\varphi _{j})\hat h_{j}^{2}Qj​=Lj​Gj​Gs​(φj​)h^j2​

Lj : Propagation loss which is mainly caused by attenuation or free space loss.
It is a function of distance ddd from the satellite to the User jjj , the light speed ccc , and the frequency fcfcfc of the transmission.
Gj : The received antenna gain at the User jjj .
Gs(φj)Gs(φj)Gs(φj) : Received beam gain at the User jjj which is related to the angle between User jjj and the beam center with respect to the satellite φjφjφj and the maximum beam gain at the on-board boresight GSmaxGSmaxGSmax [10].
h^j\hat h _{j}h^j​ : The estimated channel coefficient of link satellite→→→ User jjj can be given by h^j⁣ ⁣ ⁣=⁣ ⁣ ⁣hj⁣ ⁣+⁣ ⁣ej\hat h _{j}\! \!\! = \! \!\! h_{j} \!\!+ \!\!e_{j}h^j​=hj​+ej​ with hjhjhj and ejejej denoting the perfect channel coefficient and the estimated channel error caused by user mobility and/or path loss, respectively [11].
Without loss of generality, we assume that the channel power gain ∣hj∣2|h_{j}|^{2}∣hj​∣2 follows a Shadowed-Rice (SR) model [12], which is flexible to fit data and evaluate performance for satellite propagation environments by characterizing a wide range of elevation angle θjθjθj , and has been widely applied in various frequency bands such as the UHF-band, L-band, S-band, and Ka-band.

Performance Analysis of NOMA-Based Land Mobile Satellite Networks

All nodes in the proposed model are also assumed to equip with a single antenna

The satellite channel model including beam gain, fading model, and free space loss (FSL) is described in the following.
Gj(φj)=Gj(J1(uj)2uj+36J3(uj)uj3)2G_{j}(\varphi _{j}) = G_{j }\left ({{\frac {J_{1}\left ({u_{j} }\right)}{2u_{j}} + 36\frac {J_{3}\left ({u_{j} }\right)}{{u_{j}^{3}}}} }\right)_{}^{2}Gj​(φj​)=Gj​(2uj​J1​(uj​)​+36uj3​J3​(uj​)​)2​
f∣hj∣2(x)=αje−βjx1F1(mj;1;δjx)f_{\left |{ {h_{j}} }\right |^{2} } \left ({x }\right) = \alpha _{j} e^{ - \beta _{j} x} {}_{1}F_{1} \left ({{m_{j};1;\delta _{j} x} }\right)f∣hj​∣2​(x)=αj​e−βj​x1​F1​(mj​;1;δj​x)
Lj=(c4πfcdj)2L_{j} = \left ({{\frac {c}{{4\pi f_{c}d_{j}}}} }\right)_{}^{2}Lj​=(4πfc​dj​c​)2​
yj=LjGsGj(φj)hjx+njy_{j} = \sqrt {L_{j}G_{s}G_{j}(\varphi _{j})}h_{j} x + n_{j}yj​=Lj​Gs​Gj​(φj​)​hj​x+nj​

Specifically, the channel parameters depending on the referred shadowing scenario for satellite links are given in Table I [27].
Moreover, we set the carrier frequency to be 1.6 GHz,
φp=0.1∘φp=0.1∘φp=0.1∘ , φq=0.6∘φq=0.6∘φq=0.6∘ , φp3dB=φq3dB=0.4∘{\varphi _{p{{\text {3dB}}}}}= {\varphi _{q{{\text {3dB}}}}} = {0.4^ \circ }φp3dB​=φq3dB​=0.4∘ , and
Gp=Gq=3.5G_{p} =G_{q}= 3.5Gp​=Gq​=3.5 dBi, Gs=24.3Gs=24.3Gs=24.3 dBi,
ξ=2ξ=2ξ=2 , and Pint=50Pint=50Pint=50 W [7], [40].
The label (LS/HS) denotes the link shadowing severity of User-ppp / User-qqq .

On the Performance of Cognitive Satellite-Terrestrial Networks

a multi-beam satellite
BSs (equipped with multiple antennas)

the beam gain can be approximated as [3]
Gii=Lmax⁡Gs,iGr,i(J1(x)2x+36J3(x)x3)2G_{ii} = \mathcal {L}_{\max } G_{s,i} G_{r,i} \left ({\dfrac {J_{1}(x)}{2 x}+36 \dfrac {J_{3}(x)}{x^{3}}}\right )^{2}Gii​=Lmax​Gs,i​Gr,i​(2xJ1​(x)​+36x3J3​(x)​)2
where LmaxLmaxLmax is the free space loss [24]

The overall channel gain between the j th beam and i th user of the satellite can be given as
hppij=hppjGij(ϕij)1/2,i,j=1,…,K.h_{pp}^{ij} = h_{pp}^{j} \, {G_{ij}\left ({\phi _{ij}}\right )}^{1/2},\,\,\,\, i, j =1,\dotsc ,K.hppij​=hppj​Gij​(ϕij​)1/2,i,j=1,…,K.

The received signal at i th beam user can be formulated as
yi=PsiGiihppixpi+∑j∈ΦU,j≠iPsjGijhppixpj+IBS+ωiy_{i} = \sqrt {P_{si}}\,G_{ii}\, h_{pp}^{i}\, x_{p}^{i} + \sum \limits _{j \in \Phi _{U},j \neq i} \sqrt {P_{sj}}\,G_{ij}\, h_{pp}^{i} \, x_{p}^{j} + \mathcal {I}_{ \textrm {BS}} + \omega _{i}yi​=Psi​​Gii​hppi​xpi​+j∈ΦU​,j=i∑​Psj​​Gij​hppi​xpj​+IBS​+ωi​

测试代码：已知信道包络平方的概率密度函数，生成信道包络平方

close all
clearparameter_Shadowed_Rician=[0.158,19.4,1.29; % Infrequent light shadowing0.063,0.739,8.97*10^(-4); % Frequent heavy shadowing0.251,5.21,0.278; % Overall results0.126,10.1,0.835];% Average shadowing
shadowing_scenario = 4;
bb=parameter_Shadowed_Rician(shadowing_scenario,1);
mm=parameter_Shadowed_Rician(shadowing_scenario,2);
omega_=parameter_Shadowed_Rician(shadowing_scenario,3);
prob_density_f=@(xx) prob_density_Shadowed_Rician_power(bb,mm,omega_,xx);range_begin=0;
range_end=5;
hh=linspace(range_begin,range_end,50);
ff=prob_density_f(hh);
intergra_prob_density=trapz(hh,ff);  %计算整个区间概率密度的积分
ff=ff/intergra_prob_density;         %归一化概率密度figure
plot(hh,ff);%根据公式画概率密度曲线
legend('prob density');
xlabel('The square of the channel envelope');
% ylabel('prob density');
hold on;
grid onnum_rand=1000; %需要随机数的个数
rand_from_prob_density = gene_rand_from_prob_density(prob_density_f,...range_begin,range_end,intergra_prob_density,num_rand);% 统计检验随机数列是否符合分布
num_interval=100; %直方图的区间个数
[counts,centers]=hist(rand_from_prob_density,num_interval);    %统计不同区间出现的个数
range_interval=(range_end-range_begin)/num_interval;        %区间大小
counts=counts/num_rand/range_interval;         %根据统计结果计算概率密度
bar(centers,counts,1);   %根据统计结果画概率密度直方图
% title(['生成了',num2str(num_rand),'个随机数，并分了',num2str(num_interval),'个区间画直方图'])
title(['Generate ',num2str(num_rand),' random numbers, and divide ',num2str(num_interval),' intervals to draw histogram'])function rand_from_prob_density = gene_rand_from_prob_density(prob_density_f,...range_begin,range_end,intergra_prob_density,num_rand)
% 舍选法 (Acceptance-Rejection Method)
rand_from_prob_density = zeros(num_rand,1);
nn=1;
while nn<=num_rand%生成[range_begin,range_end]均匀分布随机数xx=rand(1)*(range_end-range_begin)+range_begin;%生成[0,1]均匀分布随机数rand_prob=rand(1);  %如果随机数r小于密度函数值f(t)，接纳该t并加入序列a中if rand_prob<=prob_density_f(xx)/intergra_prob_density     rand_from_prob_density(nn) = xx;nn = nn+1;end
end
endfunction prob_den_Shadowed_Rician_power = prob_density_Shadowed_Rician_power(bb,mm,omega_,hsquare)
alpha = (2*bb*mm/(2*bb*mm+omega_))^mm / (2*bb);
beta = 1 / (2*bb);
delta = omega_/(2*bb*(2*bb*mm+omega_));
% https://www.mathworks.com/help/symbolic/hypergeom.html#bt1nkmw-2
prob_den_Shadowed_Rician_power = alpha .* exp(-beta*hsquare) .* hypergeom(mm,1,delta.*hsquare);
end

the generalized-K model

K. P. Peppas, " Accurate closed-form approximations to generalised- $K$ sum distributions and applications in the performance analysis of equal-gain combining receivers ", IET Commun., vol. 5, no. 7, pp. 982-989, May 2011.
I. S. Ansari, S. Al-Ahmadi, F. Yilmaz, M.-S. Alouini, and H. Yanikomeroglu, “A new formula for the BER of binary modulations with dual-branch selection over generalized-K composite fading channels,” IEEE Trans. Commun., vol. 59, no. 10, pp. 2654–2658, Oct. 2011.

Energy Efficient Adaptive Transmissions in Integrated Satellite-Terrestrial Networks With SER Constraints

Y. Ruan, Y. Li, C. -X. Wang, R. Zhang and H. Zhang, “Power Allocation in Cognitive Satellite-Vehicular Networks From Energy-Spectral Efficiency Tradeoff Perspective,” in IEEE Transactions on Cognitive Communications and Networking, vol. 5, no. 2, pp. 318-329, June 2019, doi: 10.1109/TCCN.2019.2905199.

Ka频段 自由空间损耗 雨衰 波束增益

Physical Layer Security in Multibeam Satellite Systems
Ka频段

Z. Lin, M. Lin, J. -B. Wang, T. de Cola and J. Wang, “Joint Beamforming and Power Allocation for Satellite-Terrestrial Integrated Networks With Non-Orthogonal Multiple Access,” in IEEE Journal of Selected Topics in Signal Processing, vol. 13, no. 3, pp. 657-670, June 2019, doi: 10.1109/JSTSP.2019.2899731. 不太一样自由空间损耗 雨衰 波束增益 阵列控制向量
mmWave

M. Lin, C. Yin, Z. Lin, J. -B. Wang, T. de Cola and J. Ouyang, “Combined Beamforming with NOMA for Cognitive Satellite Terrestrial Networks,” ICC 2019 - 2019 IEEE International Conference on Communications (ICC), 2019, pp. 1-6, doi: 10.1109/ICC.2019.8761139.
Ka频段

Joint Beamforming for Secure Communication in Cognitive Satellite Terrestrial Networks
mmWave

the free space loss (FSL)
CL=(λ4πd)2{C_{L}} = {\left ({{\frac {\lambda }{{4\pi d }}} }\right)^{2}}CL​=(4πdλ​)2

the rain attenuation fading vector r∈CNs×1{\mathbf{r}} \in {C^{N_{s} \times 1}}r∈CNs​×1 between the satellite antenna array and the ground user
r=ξ−12e−jp{\mathbf{r}} = {\xi ^{ - \frac {1}{2}}}{e^{ - j{\mathbf{p}}}}r=ξ−21​e−jp
p{\mathbf{p}}p is the Ns×1{N_{s}} \times 1Ns​×1 sized phase vector with its components uniformly distributed over [0,2π)[0,2π)[0,2π) 随机
According to ITU-R Recommendation P.1853 [39], the power gain due to rain attenuation in dB, ξdB=20log⁡10(ξ){\xi _{{\mathrm{dB}}}} = 20{\log _{10}}\left ({\xi }\right)ξdB​=20log10​(ξ) , commonly follows a lognormal distribution, namely, ln(ξdB)∼CN(μ,σ2){\mathrm{ln}}\left ({{{\xi _{{\mathrm{dB}}}}} }\right) \sim {\mathcal{ C}}{\mathcal{ N}}\left ({{\mu,{\sigma ^{2}}} }\right)ln(ξdB​)∼CN(μ,σ2) , 随机
where μ\muμ and σ\sigmaσ , both expressed in dB, are the lognormal location and scale parameter, respectively.
μ=−3.125\mu=-3.125μ=−3.125
σ=1.591\sigma=1.591σ=1.591

the antenna gain from the mmm -th on-board beam to the user
bm=bmax⁡(J1(um)2um+36J3(um)um3)2{b_{m}} = {b_{\max }}{\left ({{\frac {{J_{1}\left ({{u_{m}} }\right)}}{{2{u_{m}}}} + 36\frac {{J_{3}\left ({{u_{m}} }\right)}}{u_{m}^{3}}} }\right)^{2}}bm​=bmax​(2um​J1​(um​)​+36um3​J3​(um​)​)2
bmax⁡{b_{\max }}bmax​ is the maximal satellite antenna gain of the mmm -th beam,
um=2.07123sin⁡ϕm/2.07123sin⁡ϕmsin⁡(ϕ3dB)msin⁡(ϕ3dB)m{u_{m}} = {{2.07123\sin {\phi _{m}}} \mathord {\left /{ {\vphantom {{2.07123\sin {\phi _{m}}} {\sin {{\left ({{{\phi _{{\mathrm{3dB}}}}} }\right)}_{m}}}}} }\right. } {\sin {{\left ({{{\phi _{{\mathrm{3dB}}}}} }\right)}_{m}}}}um​=2.07123sinϕm​/2.07123sinϕm​sin(ϕ3dB​)m​sin(ϕ3dB​)m​ , 怎么确定？？
J1(⋅){J_{1}}\left ({\cdot }\right)J1​(⋅) and J3(⋅){J_{3}}\left ({\cdot }\right)J3​(⋅) the first-kind Bessel functions of order 1 and 3, respectively.

the overall satellite channel
f=CLr⊙b12{\mathbf{f}} = \sqrt {C_{L}} {\mathbf{r}} \odot {{\mathbf{b}}^{\frac {1}{2}}}f=CL​​r⊙b21​
b=[b1,⋯,bNs]T∈CNs×1{\mathbf{b}} = {\left [{ {b_{1}, \cdots,{b_{N_{s}}}} }\right]^{T}} \in {C^{N_{s} \times 1}}b=[b1​,⋯,bNs​​]T∈CNs​×1 is the beam gain vector

f=20GHz
bmax⁡=52dB{b_{\max }}=52dBbmax​=52dB
φ3dB=0.4/180∗π\varphi _{3\mathrm {dB}}=0.4/180*\piφ3dB​=0.4/180∗π
噪声带宽=50MHz

L个散射体的阵列响应

Adaptive scheduling for millimeter wave multi-beam satellite communication systems

Secure Satellite-Terrestrial Transmission Over Incumbent Terrestrial Networks via Cooperative Beamforming
J. Du, C. Jiang, H. Zhang, X. Wang, Y. Ren and M. Debbah, “Secure Satellite-Terrestrial Transmission Over Incumbent Terrestrial Networks via Cooperative Beamforming,” in IEEE Journal on Selected Areas in Communications, vol. 36, no. 7, pp. 1367-1382, July 2018, doi: 10.1109/JSAC.2018.2824623.
mmWave

the channel vector hn∈CNs×1{{\mathbf {h}}_{n}}\in {{\mathbb {C}}^{{N_{s}}\times 1}}hn​∈CNs​×1 of FSS terminal nnn (n∈N≜{1,2,⋯,N}n \in \mathsf {\mathcal {N}} \triangleq \left \{{ 1,2, \cdots,N }\right \}n∈N≜{1,2,⋯,N} )
hn=NsL∑l=1Lδn,lα(θn,l),∀n∈N{{\mathbf {h}}_{n}}=\sqrt {\frac {{N_{s}}}{L}}\sum \nolimits _{l=1}^{L}{{\delta _{n,l}}\boldsymbol {\alpha }\left ({{\theta _{n,l}} }\right)},\quad \forall n\in \mathsf {\mathcal {N}}hn​=LNs​​​∑l=1L​δn,l​α(θn,l​),∀n∈N
where LLL is the number of scatters,
δn,l{\delta _{n,l}}δn,l​ and θn,l{\theta _{n,l}}θn,l​ are the complex gain and normalized direction of the LOS path for FSS nnn , respectively.
In addition, δn,l2∼CN(0,σ02){\delta _{n,l}^{2}}\sim \mathsf {\mathcal {C}\mathcal {N}}\left ({0,\sigma _{0}^{2} }\right)δn,l2​∼CN(0,σ02​) is independent identically distributed (i.i.d.) complex Gaussian distribution with zero-mean, and covariance σ02=1\sigma _{0}^{2}=1σ02​=1 , which indicates Rician factor, and 随机
θn,l∼U[−1,1]{\theta _{n,l}}\sim U\left [{ -1,1 }\right]θn,l​∼U[−1,1] is i.i.d. uniformly distributed. 随机
L=2 [36]

Moreover, when a uniform linear array (ULA) is adopted, the normalized array response α(θ)α(θ)α(θ) is given by
α(θ)=1Ns[1,e−j2πλdsin⁡(φ),⋯,e−j2πλ(Ns−1)dsin⁡(φ)]T\boldsymbol {\alpha }\left ({\theta }\right) = \frac {1}{\sqrt {{N_{s}}}}{{\left [{ 1,{e^{-j\frac {2\pi }{\lambda }d\sin \left ({\varphi }\right)}}, \cdots,{e^{-j\frac {2\pi }{\lambda }\left ({{N_{s}} - 1 }\right)d\sin \left ({\varphi }\right)}} }\right]}^{T}}α(θ)=Ns​​1​[1,e−jλ2π​dsin(φ),⋯,e−jλ2π​(Ns​−1)dsin(φ)]T
normalized direction θnθnθn is related to the physical azimuth angle of departure (AoD) of φ∈[−π/2,π/2]φ∈[−π/2,π/2]φ∈[−π/2,π/2] as θ=(2d/λ)sin(φ)θ=(2d/λ)sin(φ)θ=(2d/λ)sin(φ) , 怎么确定？？
where ddd is the antenna spacing (i.e., the distance between the two adjacent antennas), and
λλλ is the carrier wavelength.
In this work, we assume the critically sampled environment, i.e, d/λ=0.5d/λ=0.5d/λ=0.5 , considering that the normalized AoD is the sine function of the actual AoD.

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