
Citation: | LIN Min, ZHU Liwen, KONG Huaicong, GUO Kefeng, OUYANG Jian. Outage Performance Analysis of Unmanned Aerial Vehicle Assisted Satellite Communication System with Cooperative Non-Orthogonal Multiple Access[J]. Journal of Electronics & Information Technology, 2022, 44(9): 3033-3042. doi: 10.11999/JEIT211289 |
卫星通信凭借其覆盖范围广、通信距离远、不受地理条件限制等众多优点,已经被广泛应用于偏远地区通信以及导航定位、抗震抢险等领域[1-3],并将成为下一代无线通信的关键技术之一。然而,卫星与地面用户之间存在大时延和大路径损耗以及遮蔽效应导致的视距传输受阻等问题,使得卫星系统的用户体验有时无法得到保证。在这种情况下,基于中继转发的星地协作传输技术被认为是提升卫星通信服务质量的有效手段之一[4]。在大多研究的星地协作传输网络中,通常采用地面中继将接收到的卫星信号转发给地面用户。例如,文献[5]研究了单用户场景下地面中继采用放大转发(Amplify and Forward, AF)协议的星地协作传输网络的误码性能与中断性能。进一步,针对地面多用户的星地协作传输网络,文献[6]在采用最优用户选择方案的情况下,推导得到用户的中断概率闭合表达式。需要指出的是,虽然使用地面中继可以建立卫星与地面用户之间的高质量通信链路,但对人口稀少的偏远地区而言,建造地面中继站存在高成本、低回报等问题,因此需要探索其他更加实用的解决方案[7]。
跟地面中继相比,无人机由于其机动性好、通信组网方式灵活等优势,作为空中中继协助卫星与地面用户通信[8],可以实现增强接收信号功率、提高系统容量等目的,并且得到了学术界和工业界的高度重视。例如,文献[9]研究了基于无人机中继转发的星地协作网络中多用户传输场景下系统的中断性能;文献[10]研究了无人机中继采用AF协议和多用户调度方案下的星地协作网络性能;文献[11]针对无人机辅助卫星通信系统,分析了系统的中断性能。然而,考虑到卫星服务用户数量越来越多,需要进一步提高系统资源利用率以及服务用户通信质量。现有的正交多址 (Orthogonal Multiple Access, OMA)技术已经无法满足上述需求。近年来,非正交多址(Non-Orthogonal Multiple Access, NOMA)技术以其可大大提高系统频谱资源利用率和用户公平性等独特优势,已经成为极具发展前景的新型多址技术[12]。在这种情况下,已经有学者研究如何将NOMA技术应用于卫星通信系统。例如,文献[13,14]研究了两用户场景下基于NOMA的星地协作传输系统性能;文献[15,16]针对基于NOMA的卫星通信系统中多用户传输场景,分析了系统性能;文献[17]针对基于NOMA的无人机辅助卫星通信系统,分析了无人机中继采用AF协议下的系统中断性能。
总的来看,上述文献对星地协作传输网络进行了深入的研究,验证了中继协作技术能够显著提升卫星通信系统存在遮蔽效应下的性能,但是它们主要存在以下问题:一是大多数文献,例如文献[5]仅研究了单用户场景下星地协作传输网络的性能;二是虽然也有相关文献研究多用户场景,但是大多数文献,例如文献[6,10]都是在假设准确信道状态信息(Channel State Information, CSI)已知的情况下,研究了采用用户调度或空分多址(Space Division Multiple Access, SDMA)方案下星地协作传输网络的性能;三是将NOMA技术应用于卫星通信系统的大多数文献中,例如文献[13-16]都是采用地面中继将接收到的卫星信号转发给地面用户;四是虽然也有相关文献提出基于NOMA的无人机辅助卫星通信系统,但是大多数文献,例如文献[17]仅考虑无人机配置单天线且地面用户簇内仅包含远近两个用户的通信场景,没有充分利用空间资源和考虑卫星服务用户数量越来越多的实际情况。在这种情况下,本文针对无人机辅助的卫星通信系统下行链路,研究基于SDMA和协作NOMA相结合的多用户传输系统。首先,配置多根天线的无人机作为中继站辅助卫星通信,采用NOMA技术服务多个地面用户,并得到用户的输出信干噪比(Signal-to-Interference-plus-Noise Ratio, SINR)表达式。然后,建立以平均SINR最大化为准则的优化问题,并在无人机仅已知信道角度信息的情况下,得到无人机对卫星信号的接收波束成形权矢量。进一步,为了降低算法复杂度,提出基于角度信息的迫零波束成形方案,得到无人机对多个地面用户的发射波束成形权矢量。其次,在卫星-无人机链路服从相关阴影莱斯分布,而无人机-地面用户链路服从Nakagami-m分布的条件下,分别推导出系统的中断概率闭合表达式和近似表达式。最后,仿真结果验证了所提方案的优越性和理论分析的正确性。
如图1所示,本文研究无人机辅助的卫星通信系统下行链路,其中静止轨道卫星S通过无人机中继R转发信号,与地面用户D进行通信。假设卫星S采用点波束技术,无人机中继R配置N元的均匀直线阵(Uniform Linear Array, ULA),地面用户D配置单天线。为了实现多用户同时通信,将无人机覆盖范围内的地面用户中信道相关性强和信道增益差异大的用户划分为一簇[18],从而将地面用户划分为L簇,且簇内用户采用NOMA技术提高频谱利用率。跟针对单用户场景以及多用户场景下采用用户调度或SDMA方案的星地协作传输网络的文献相比[6,10],本文的研究更具一般性。为了便于理解,本文将首先介绍信道模型和信号模型,然后对所提的波束成形方案进行描述。
在卫星通信中,为了克服远距离传输导致的大路径损耗,通常采用高增益的点波束技术。在这种情况下,将卫星的波束增益、电波传播过程中的自由空间损耗和小尺度衰落的影响加以考虑之后,卫星到无人机链路,即S-R链路信道矢量可以表示为
hSR=ζSRgSR | (1) |
其中,
GS=GmaxS(J1(us)2us+36J3(us)u3s)2 | (2) |
其中,
此外,式(1)中
gSR=ˉgSR+˜Φ1/122R˜gSR | (3) |
其中,
ˉgSR = ZSRa(θr) | (4) |
其中,
a(θr)=[1,exp(j2πλdesinθr),⋯,exp(j(N−1)2πλdesinθr)]T | (5) |
其中,
跟前面类似,考虑自由空间路径损耗以及小尺度衰落的影响,无人机与地面第
hl,n=L′gl,n | (6) |
其中,
gl,n = ρl,na(θl,n) | (7) |
其中,信道系数
如图1所示,卫星首先将叠加信号
yR=wHSR(hSRxSR+nSR)=wHSRhSRM∑n=1√αl,nPSxl,n⏟l 簇信号+wHSRhSRL∑j=1,j≠lM∑k=1√αj,kPSxj,k⏟其他簇干扰+wHSRnSR | (8) |
其中,
与译码转发协议(Decode and Forward, DF)相比,AF协议是一种更加容易实现的中继协议[20],更加适用于本文基于无人机辅助的卫星通信场景。因此,无人机中继采用AF协议,通过固定增益因子
yl,n=GhHl,nwlwHSRhSRM∑n=1√αl,nPSxl,n⏟l 簇信号+GhHl,nwHSRhSRL∑j=1,j≠lM∑k=1wj√αj,kPSxj,k⏟其他簇干扰+GhHl,nwlwHSRnSR+nl,n | (9) |
其中,
为了进一步消除簇内用户干扰,采用串行干扰消除(Successive Interference Cancellation, SIC)技术消除干扰。假设第
{\gamma _{l,n}} = \frac{{{{\bar \gamma }_{{\rm{SR}}}}{{\bar \gamma }_{l,n}}{\alpha _{l,n}}{{\left| {{\boldsymbol{w}}_{{\rm{SR}}}^{\rm{H}}{{\boldsymbol{g}}_{{\rm{SR}}}}} \right|}^2}{{\left| {{{\boldsymbol{g}}^{\rm{H}}_{l,n}}{{\boldsymbol{w}}_l}} \right|}^2}}}{{{{\bar \gamma }_{{\rm{SR}}}}{{\bar \gamma }_{l,n}}{{\left| {{\boldsymbol{w}}_{{\rm{SR}}}^{\rm{H}}{{\boldsymbol{g}}_{{\rm{SR}}}}} \right|}^2}\left( {\displaystyle\sum\limits_{i = n + 1}^M {{\alpha _{l,i}}} {{\left| {{{\boldsymbol{g}}^{\rm{H}}_{l,n}}{{\boldsymbol{w}}_l}} \right|}^2} + \displaystyle\sum\limits_{j = 1,j \ne l}^L {\sum\limits_{k = 1}^M {{\alpha _{j,k}}{{\left| {{{\boldsymbol{g}}^{\rm{H}}_{l,n}}{{\boldsymbol{w}}_j}} \right|}^2}} } } \right) + {{\bar \gamma }_{l,n}}{{\left| {{{\boldsymbol{g}}^{\rm{H}}_{l,n}}{{\boldsymbol{w}}_l}} \right|}^2} + {\varLambda _{{{\rm{SR}}} }}}} | (10) |
其中,
在无线通信中,考虑到信道的随机性,一般通过使用户输出平均SINR最大来达到系统传输性能最优的目的。因此根据式(10),本文建立以用户输出平均SINR最大化为准则的优化问题,在数学上可以表示为[19]
\begin{split} & \mathop {\max }\limits_{{{\boldsymbol{w}}_{{\rm{SR}}}},{{\boldsymbol{w}}_l}} {\text{ E}}\left[ {{\gamma _{l,n}}} \right] , \\ & \quad {\rm{s.t.}} {\text{ }}\left\| {{{\boldsymbol{w}}_{{\rm{SR}}}}} \right\| = 1{\text{ and }}\left\| {{{\boldsymbol{w}}_l}} \right\| = 1,l = 1, 2,\cdots ,L \end{split} | (11) |
考虑到式(11)中
\begin{split} & {\text{E}}\left[ {{\gamma _{l,n}}} \right] \approx\\ & \frac{{{{{\rm{E}}}_{{{\boldsymbol{g}}_{\rm{SR}}},{{\boldsymbol{g}}_{l,n}}}}\left[ {{{\bar \gamma }_{\rm{SR}}}{{\bar \gamma }_{l,n}}{\alpha _{l,n}}{\boldsymbol{w}}_{\rm{SR}}^{\rm{H}}{{\boldsymbol{g}}_{\rm{SR}}}{\boldsymbol{g}}_{\rm{SR}}^{\rm{H}}{{\boldsymbol{w}}_{\rm{SR}}}{\boldsymbol{w}}_l^{\rm{H}} {{\boldsymbol{g}}_{l,n}}{\boldsymbol{g}}_{l,n}^{\rm{H}} {{\boldsymbol{w}}_l}} \right]}}{{{{{\rm{E}}}_{{{\boldsymbol{g}}_{\rm{SR}}},{{\boldsymbol{g}}_{l,n}}}}\left[ {{{\bar \gamma }_{{\rm{SR}} }}{{\bar \gamma }_{l,n}}{\boldsymbol{w}}_{\rm{SR}}^{\rm{H}}{{\boldsymbol{g}}_{\rm{SR}}}{\boldsymbol{g}}_{\rm{SR}}^{\rm{H}}{{\boldsymbol{w}}_{\rm{SR}}}\left( {\displaystyle\sum\limits_{i = n + 1}^M {{\alpha _{l,i}}} {\boldsymbol{w}}_l^{\rm{H}} {{\boldsymbol{g}}_{l,n}}{\boldsymbol{g}}_{l,n}^{\rm{H}} {{\boldsymbol{w}}_l} + \sum\limits_{j = 1,j \ne l}^L {\sum\limits_{k = 1}^M {{\alpha _{j,k}}} } {\boldsymbol{w}}_j^{\rm{H}} {{\boldsymbol{g}}_{l,n}}{\boldsymbol{g}}_{l,n}^{\rm{H}} {{\boldsymbol{w}}_j}} \right) + {{\bar \gamma }_{l,n}}{\boldsymbol{w}}_l^{\rm{H}} {{\boldsymbol{g}}_{l,n}}{\boldsymbol{g}}_{l,n}^{\rm{H}} {{\boldsymbol{w}}_l} + {\varLambda _{{\rm{SR}} }}} \right]}} \end{split} | (12) |
于是,优化问题式(11)可以重新表示为
\begin{split} & \mathop {\max }\limits_{{{\boldsymbol{w}}_{\rm{SR}}},{{\boldsymbol{w}}_l}}\\ & \frac{{{{\bar \gamma }_{\rm{SR}}}{{\bar \gamma }_{l,n}}{\alpha _{l,n}}{\boldsymbol{w}}_{\rm{SR}}^{\rm{H}}{{\boldsymbol{\varPhi }}_{{\rm{SR}} }}{{\boldsymbol{w}}_{\rm{SR}}}{\boldsymbol{w}}_l^{\rm{H}} {{\boldsymbol{\varPhi }}_{l,n}}{{\boldsymbol{w}}_l}}}{{{{\bar \gamma }_{\rm{SR}}}{{\bar \gamma }_{l,n}}{\boldsymbol{w}}_{\rm{SR}}^{\rm{H}}{{\boldsymbol{\varPhi }}_{{\rm{SR}} }}{{\boldsymbol{w}}_{\rm{SR}}}\left( {\displaystyle\sum\limits_{i = n + 1}^M {{\alpha _{l,i}}} {\boldsymbol{w}}_l^{\rm{H}} {{\boldsymbol{\varPhi }}_{l,n}}{{\boldsymbol{w}}_l} + \sum\limits_{j = 1,j \ne l}^L {\sum\limits_{k = 1}^M {{\alpha _{j,k}}} } {\boldsymbol{w}}_j^{\rm{H}} {{\boldsymbol{\varPhi }}_{l,n}}{{\boldsymbol{w}}_j}} \right) + {{\bar \gamma }_{l,n}}{\boldsymbol{w}}_l^{\rm{H}} {{\boldsymbol{\varPhi }}_{l,n}}{{\boldsymbol{w}}_l} + {{\bar \gamma }_{\rm{SR}}}{\boldsymbol{w}}_{\rm{SR}}^{\rm{H}}{{\boldsymbol{\varPhi }}_{{\rm{SR}} }}{{\boldsymbol{w}}_{{\rm{SR}} }} + 1}} , \\ & \quad {{\rm{s}}} .{\rm{t}}.{\text{ }}\left\| {{{\boldsymbol{w}}_{\rm{SR}}}} \right\| = 1,\;\left\| {{{\boldsymbol{w}}_l}} \right\| = 1,l = 1,2, \cdots ,L \\[-10pt] \end{split} | (13) |
其中,
\qquad {{\boldsymbol{\varPhi }}_{{\rm{SR}}}} = \left( {{\varOmega _{{\rm{R}}} } + 2b} \right){\boldsymbol{a}}\left( {{\theta _r}} \right){{\boldsymbol{a}}^{{\rm{H}}} }\left( {{\theta _r}} \right) | (14) |
\qquad {{\boldsymbol{\varPhi }}_{l,n}} = {\varOmega _{l,n}}{\boldsymbol{a}}\left( {{\theta _{l,n}}} \right){{\boldsymbol{a}}^{\rm{H}}}\left( {{\theta _{l,n}}} \right) | (15) |
考虑到优化问题式(13)的目标函数随
首先,建立优化问题式(16)获得归一化的无人机接收波束成形权矢量
\begin{split} & \mathop {\max }\limits_{{{\boldsymbol{w}}_{{\rm{SR}}}}} {\text{ }}{\boldsymbol{w}}_{{\rm{SR}}}^{\rm{H}}{{\boldsymbol{\varPhi }}_{{\rm{SR}}}}{{\boldsymbol{w}}_{{\rm{SR}}}} , \\ & \quad {\rm{s.t}}.{\text{ }}\left\| {{{\boldsymbol{w}}_{{\rm{SR}}}}} \right\| = 1 \\ \end{split} | (16) |
利用式(14),对优化问题式(16)求解可以得到无人机接收波束成形权矢量
{\boldsymbol{w}}_{{\rm{SR}}}^* = \frac{{{\boldsymbol{a}}\left( {{\theta _r}} \right)}}{{\sqrt N }} | (17) |
然后,将式(17)代入优化问题式(13),可以将优化问题式(13)简化为
\begin{split} & \mathop {\max }\limits_{{{\boldsymbol{w}}_l}} {\text{ }}\frac{{{{\bar \gamma }_{\rm{SR}}}{{\bar \gamma }_{l,n}}{\alpha _{l,n}}Q{\boldsymbol{w}}_l^{\rm{H}} {{\boldsymbol{\varPhi }}_{l,n}}{{\boldsymbol{w}}_l}}}{{{{\bar \gamma }_{\rm{SR}}}{{\bar \gamma }_{l,n}}Q\left( {\displaystyle\sum\limits_{i = n + 1}^M {{\alpha _{l,i}}} {\boldsymbol{w}}_l^{\rm{H}} {{\boldsymbol{\varPhi }}_{l,n}}{{\boldsymbol{w}}_l} + \sum\limits_{j = 1,j \ne l}^L {\sum\limits_{k = 1}^M {{\alpha _{j,k}}} } {\boldsymbol{w}}_j^{\rm{H}} {{\boldsymbol{\varPhi }}_{l,n}}{{\boldsymbol{w}}_j}} \right) + N{{\bar \gamma }_{l,n}}{\boldsymbol{w}}_l^{\rm{H}} {{\boldsymbol{\varPhi }}_{l,n}}{{\boldsymbol{w}}_l} + {{\bar \gamma }_{\rm{SR}}}Q + N}},\\ & \quad {{\rm{s}}} .{\rm{t}}. {\text{ }}\left\| {{{\boldsymbol{w}}_l}} \right\| = 1,l = 1, 2,\cdots ,L \end{split} | (18) |
其中,
由于优化问题式(18)的目标函数中
\begin{split} & \mathop {\max }\limits_{{{\boldsymbol{w}}_l}} {\text{ }}{\boldsymbol{w}}_l^{{\rm{H}}} {{\boldsymbol{\varPhi }}_{l,n}}{{\boldsymbol{w}}_l} , \\ & \quad{\rm{s.t}}. {\text{ }}{\boldsymbol{w}}_j^{{\rm{H}}} {{\boldsymbol{\varPhi }}_{l,n}}{{\boldsymbol{w}}_j} = 0,\forall j \ne l, \\ & \quad\left\| {{{\boldsymbol{w}}_l}} \right\| = 1,l = 1,2, \cdots ,L \\ \end{split} | (19) |
利用式(15)并根据矩阵零空间理论,对优化问题式(19)进行求解可以得到
{\boldsymbol{w}}_l^*{\text{ = }}\frac{{{{\boldsymbol{\varPi }}_l}{\boldsymbol{a}}\left( {{\theta _{l,n}}} \right)}}{{\left\| {{{\boldsymbol{\varPi }}_l}{\boldsymbol{a}}\left( {{\theta _{l,n}}} \right)} \right\|}} | (20) |
其中,
于是,将式(17)和式(20)代入式(10)后,用户
{\gamma _{l,n}} = \frac{{{{\bar \gamma }_{{\rm{SR}}}}{{\bar \gamma }_{l,n}}{\alpha _{l,n}}{{\left| {{\rho _{l,n}}} \right|}^2}{{\left| {{\boldsymbol{a}}^{\rm{H}}{{\left( {{\theta _r}} \right)}}{{\boldsymbol{g}}_{{\rm{SR}}}}} \right|}^2}{{\left| {{\boldsymbol{a}}^{{\rm{H}}}{{\left( {{\theta _{l,n}}} \right)} }{{\boldsymbol{\varPi }}_l}{\boldsymbol{a}}\left( {{\theta _{l,n}}} \right)} \right|}^2}}}{{{{\bar \gamma }_{{\rm{SR}}}}{{\bar \gamma }_{l,n}}{{\left| {{\rho _{l,n}}} \right|}^2} \displaystyle\sum\limits_{i = n + 1}^M {{\alpha _{l,i}}} {{\left| {{\boldsymbol{a}}^{{\rm{H}}}{{\left( {{\theta _r}} \right)} }{{\boldsymbol{g}}_{{\rm{SR}}}}} \right|}^2}{{\left| {{\boldsymbol{a}}^{{\rm{H}}}{{\left( {{\theta _{l,n}}} \right)} }{{\boldsymbol{\varPi }}_l}{\boldsymbol{a}}\left( {{\theta _{l,n}}} \right)} \right|}^2} + N{{\bar \gamma }_{l,n}}{{\left| {{\rho _{l,n}}} \right|}^2}{{\left| {{\boldsymbol{a}}^{\rm{H}}{{\left( {{\theta _{l,n}}} \right)}}{{\boldsymbol{\varPi }}_l}{\boldsymbol{a}}\left( {{\theta _{l,n}}} \right)} \right|}^2} + {{\varLambda '}_{{{\rm{SR}}} }}}} | (21) |
其中,
需要指出的是,跟大多数采用基于准确CSI的波束成形方案,例如文献[10]不同的是,本文利用信道角度信息进行波束成形设计,避免了信道估计和反馈等需要额外消耗无线资源的过程,从而更加适合于无人机通信场景。接下来,将进一步对系统的中断性能进行分析。
中断概率是衡量无线通信服务质量(Quality of Service, QoS)的一项重要指标,定义为信号输出SINR低于某一特定门限值的概率。因此,根据式(21)用户
\begin{split} P_{{{\rm{out}}} }^{l,n} =&\Pr \left( {\frac{{{{\bar \gamma }_{{\rm{SR}} }}{{\bar \gamma }_{l,n}}{\alpha _{l,n}}{{\left| {{\rho _{l,n}}} \right|}^2}{{\left| {{\boldsymbol{a}}^{\rm{H}}{{\left( {{\theta _r}} \right)}}{{\boldsymbol{g}}_{\rm{SR}}}} \right|}^2}{{\left| {{\boldsymbol{a}}^{{\rm{H}}}{{\left( {{\theta _{l,n}}} \right)} }{{\boldsymbol{\varPi }}_l}{\boldsymbol{a}}\left( {{\theta _{l,n}}} \right)} \right|}^2}}}{{{{\bar \gamma }_{\rm{SR}}}{{\bar \gamma }_{l,n}}{{\left| {{\rho _{l,n}}} \right|}^2} \displaystyle\sum\limits_{i = n + 1}^M {{\alpha _{l,i}}} {{\left| {{\boldsymbol{a}}^{\rm{H}}{{\left( {{\theta _r}} \right)}}{{\boldsymbol{g}}_{\rm{SR}}}} \right|}^2}{{\left| {{\boldsymbol{a}}^{{\rm{H}}}{{\left( {{\theta _{l,n}}} \right)} }{{\boldsymbol{\varPi }}_l}{\boldsymbol{a}}\left( {{\theta _{l,n}}} \right)} \right|}^2} + N{{\bar \gamma }_{l,n}}{{\left| {{\rho _{l,n}}} \right|}^2}{{\left| {{\boldsymbol{a}}^{\rm{H}}{{\left( {{\theta _{l,n}}} \right)}}{{\boldsymbol{\varPi }}_l}{\boldsymbol{a}}\left( {{\theta _{l,n}}} \right)} \right|}^2} + {{\varLambda '}_{{\rm{SR}} }}}} \le {\gamma _{{\rm{th}},n}}} \right) \\ = &\underbrace {{F_X}\left( u \right)}_{{I_1}} + \underbrace {\int\limits_u^\infty {{F_{{Y_{l,n}}}} \left[ {{Y_{l,n}} \le \left( {\frac{{{\gamma _{{\rm{th}},n}}{{\left\| {{{\boldsymbol{\varPi }}_l}{\boldsymbol{a}}\left( {{\theta _{l,n}}} \right)} \right\|}^2}\left( {{{\varLambda ''}_{{{\rm{\rm{SR}}}} }} + N} \right)}}{{\left( {{{\bar \gamma }_{\rm{SR}}}{{\bar \gamma }_{l,n}}\left( {{\alpha _{l,n}} - {\gamma _{th,n}}\displaystyle\sum\limits_{i = n + 1}^M {{\alpha _{l,i}}} } \right)x - N{\gamma _{{\rm{th}},n}}{{\bar \gamma }_{l,n}}} \right) {{\left| {{\boldsymbol{a}}^{\rm{H}}{{\left( {{\theta _{l,n}}} \right)}}{{\boldsymbol{\varPi }}_l}{\boldsymbol{a}}\left( {{\theta _{l,n}}} \right)} \right|}^2}}}} \right)} \right]} {f_X} \left( x \right){\rm{d}}x}_{{I_2}}\\ \end{split} | (22) |
其中,
为了进一步得到
{f_X}\left( x \right) = {a_1}\exp \left( { - \frac{x}{{2{b_{{\rm{R}}} }}}} \right){}_1{F_1}\left( {{m_{\rm{R}}},1,{a_2}x} \right) | (23) |
其中,
\begin{split} {a_1} =& {{{\left( {{{2{b_{{\rm{R}}} }{m_{\rm{R}}}}/{\left( {2{b_{\rm{R}}}{m_{{\rm{R}}} } + {{\left| {{\boldsymbol{a}}^{{\rm{H}}}{{\left( {{\theta _r}} \right)} }{\boldsymbol{a}}\left( {{\theta _r}} \right)} \right|}^2}{\varOmega _{{\rm{R}}} }} \right)}}} \right)}^{{m_{\rm{R}}}}}} \\ & \bigr/{\left( {2{b_{\rm{R}}}} \right)}\\[-10pt] \end{split} | (24) |
\begin{split} {a_2} =& {{{\left| {{\boldsymbol{a}}^{\rm{H}}{{\left( {{\theta _r}} \right)}}{\boldsymbol{a}}\left( {{\theta _r}} \right)} \right|}^2}{\varOmega _{{\rm{R}}} }}\\ & \Bigr/{\left( {2{b_{\rm{R}}}\left( {2{b_{\rm{R}}}{m_{{\rm{R}}} } + {{\left| {{\boldsymbol{a}}^{{\rm{H}}}{{\left( {{\theta _r}} \right)} }{\boldsymbol{a}}\left( {{\theta _r}} \right)} \right|}^2}{\varOmega _{{\rm{R}}} }} \right)} \right)} \end{split} | (25) |
进一步,可以得到随机变量
\begin{split} {F_X}\left( x \right) =& {a_1}\sum\limits_{k = 0}^{{m_{\rm{R}}} - 1} {\frac{{{{\left( {1 - {m_{{\rm{R}}} }} \right)}_k}{{\left( { - {a_2}} \right)}^k}}}{{{{\left( \delta \right)}^{k + 1}}{{\left( 1 \right)}_k}}}}\\ & \cdot\left( {1 - \exp \left( { - \delta x} \right)\sum\limits_{n = 0}^k {\frac{{{{\left( \delta \right)}^n}}}{{n!}}{x^n}} } \right) \end{split} | (26) |
其中,
另一方面,由文献[21]中式(3.351.3)和式(23),可将式(22)中
{\varLambda ''_{{{\rm{SR}}} }} = N + {\bar \gamma _{{\rm{SR}}}}{a_1}\sum\limits_{{t_2} = 0}^{{m_{\rm{R}}} - 1} {\frac{{{{\left( {1 - {m_{{\rm{R}}} }} \right)}_{{t_2}}}{{\left( { - {a_2}} \right)}^{{t_2}}}\left( {{t_2}{\text{ + }}1} \right)}}{{{{\left( 1 \right)}_{{t_2}}}{{\left( {{1/{\left( {2{b_{{\rm{R}}} }} \right) - {a_2}}}} \right)}^{{t_2} + 2}}}}} | (27) |
接下来,为了计算
{F_{{Y_{l,n}}}}\left( y \right) = {b_n}\sum\limits_{h = 0}^{M - n} {\left( \begin{gathered} M - n \\ h \\ \end{gathered} \right)} \frac{{{{\left( { - 1} \right)}^h}}}{{n + h}}{\left[ {{F_{{{\left| {{\rho _{l,n}}} \right|}^2}}}\left( y \right)} \right]^{n + h}} | (28) |
其中,
{F_{{{\left| {{\rho _{l,n}}} \right|}^2}}}\left( y \right) = 1 - \exp \left( { - \frac{{{m_{l,n}}y}}{{{\varOmega _{l,n}}}}} \right)\sum\limits_{l = 0}^{{m_{l,n}} - 1} {\frac{1}{{l!}}} {\left( {\frac{{{m_{l,n}}y}}{{{\varOmega _{l,n}}}}} \right)^l} | (29) |
将式(29)代入式(28)中,
\begin{split} {F_{{Y_{l,n}}}}\left( y \right)=& {b_n}\sum\limits_{h = 0}^{M - n} {\left( \begin{gathered} M - n \\ h \\ \end{gathered} \right)} \frac{{{{\left( { - 1} \right)}^h}}}{{n + h}}\sum\limits_{j = 0}^{n + h} \left( \begin{gathered} n + h \\ j \\ \end{gathered} \right)\\ & \cdot{{\left( { - 1} \right)}^j}\exp \left( { - \frac{{{m_{l,n}}yj}}{{{\varOmega _{l,n}}}}} \right)\\ & \cdot\left( {\sum\limits_{l = 0}^{{m_{l,n}} - 1} {\frac{1}{{l!}}} {{\left( {\frac{{{m_{l,n}}y}}{{{\varOmega _{l,n}}}}} \right)}^l}} \right) ^j\\[-25pt] \end{split} | (30) |
将式(30)和式(23)(代入式(22)中,
\begin{split} {I_2} =& {b_n}\sum\limits_{h = 0}^{M - n} {\left( \begin{gathered} M - n \\ h \\ \end{gathered} \right)} \frac{{{{\left( { - 1} \right)}^h}}}{{n + h}}{\sum\limits_{j = 0}^{n + h} {\left( \begin{gathered} n + h \\ j \\ \end{gathered} \right)\left( { - 1} \right)} ^j}\\ & \cdot \int\limits_u^\infty {\exp \left( { - \frac{{{m_{l,n}}{y_1}j}}{{{\varOmega _{l,n}}}}} \right)} \times {\left[ {\sum\limits_{l = 0}^{{m_{l,n}} - 1} {\frac{1}{{l!}}} {{\left( {\frac{{{m_{l,n}}{y_1}}}{{{\varOmega _{l,n}}}}} \right)}^l}} \right]^j}{a_1}\\ & \cdot\exp \left( { - \frac{x}{{2{b_{\rm{R}}}}}} \right){}_1{F_1}\left( {{m_{{\rm{R}}} },1,{a_2}x} \right){\rm{d}}x \\[-18pt] \end{split} | (31) |
其中,
{\left[ {\sum\limits_{l = 0}^{{m_{l,n}} - 1} {\frac{1}{{l!}}} {{\left( {\frac{{{m_{l,n}}{y_1}}}{{{\varOmega _{l,n}}}}} \right)}^l}} \right]^j}{\text{ = }}\sum\limits_{l = 0}^{j\left( {{m_{l,n}} - 1} \right)} {\left( {c_l^j{z^l}} \right)} | (32) |
其中,
c_0^j = 1,c_1^j = j,c_{j\left( {{m_{l,n}} - 1} \right)}^j = {\left( {\frac{1}{{\left( {{m_{l,n}} - 1} \right)!}}} \right)^j} | (33) |
c_l^j = \frac{1}{l}\sum\limits_{{t_3} = 1}^{{J_0}} {\frac{{{t_3}\left( {j + 1} \right) - l}}{{{t_3}!}}} c_{l - {t_3}}^j | (34) |
{J_0} = \min \left( {l,{m_{l,n}} - 1} \right),2 \le l \le j\left( {{m_{l,n}} - 1} \right) - 1 | (35) |
将式(32)代入式(31)中,
\begin{split} {I_2} =& {b_n}\sum\limits_{h = 0}^{M - n} {\left( \begin{gathered} M - n \\ h \\ \end{gathered} \right)} \frac{{{{\left( { - 1} \right)}^h}}}{{n + h}}{\sum\limits_{j = 0}^{n + h} {\left( \begin{gathered} n + h \\ j \\ \end{gathered} \right)\left( { - 1} \right)} ^j}\\ & \cdot \int\limits_u^\infty {\exp \left( { - \frac{{{m_{l,n}}{y_1}j}}{{{\varOmega _{l,n}}}}} \right)} \sum\limits_{l = 0}^{j\left( {{m_{l,n}} - 1} \right)} {\left( {c_l^j{z^l}} \right)} \\ & \cdot {a_1}\exp \left( {\delta x} \right)\sum\limits_{{t_1} = 0}^{{m_{R} } - 1} {\frac{{{{\left( {1 - {m_R}} \right)}_{{t_1}}}{{\left( { - {a_2}x} \right)}^{{t_1}}}}}{{{t_1}!{{\left( 1 \right)}_{{t_1}}}}}} {\rm{d}}x \end{split} | (36) |
令式(36)中
\begin{split} {I_2} =& 2{a_1}{b_n}\sum\limits_{h = 0}^{M - n} {\sum\limits_{j = 0}^{n + h} {\sum\limits_{l = 0}^{j\left( {{m_{l,n}} - 1} \right)} {\left( \begin{gathered} M - n \\ h \\ \end{gathered} \right)} } } \left( \begin{gathered} n + h \\ j \\ \end{gathered} \right)\frac{{{{\left( { - 1} \right)}^{j + h}}c_l^j}}{{n + h}}{\left( {\frac{{{m_{l,n}}{\gamma _{{\rm{th}},n}}{{\varLambda ''}_{{{\rm{SR}}} }}{Z_a}}}{{{\varOmega _{l,n}}}}} \right)^l} \\ & \cdot\sum\limits_{{t_1} = 0}^{{m_R} - 1} {\sum\limits_{p = 0}^{{t_1}} {\left( \begin{gathered} {t_1} \\ p \\ \end{gathered} \right)} } \exp \left( { - \frac{{N{\gamma _{{\rm{th}},n}}{{\bar \gamma }_{l,n}}\delta }}{A}} \right)\frac{{{{\left( {1 - {m_{{\rm{R}}} }} \right)}_{{t_1}}}{{\left( { - {a_2}} \right)}^{{t_1}}}{{\left( {N{\gamma _{{\rm{th}},n}}{{\bar \gamma }_{l,n}}} \right)}^p}}}{{{t_1}!{{\left( 1 \right)}_{{t_1}}}{A^{{t_1} + 1}}}} \\ &\cdot {\left( {\frac{{jA{m_{l,n}}{\gamma _{{\rm{th}},n}}{\varLambda ''_{{{\rm{SR}}} }}{Z_{a} }}}{{{\varOmega _{l,n}}\delta }}} \right)^{\frac{{{t_1} - l - p}}{2}}}{K_{{t_1} - l - p}}\left( {2\sqrt {\frac{{j{m_{l,n}}{\gamma _{{\rm{th}},n}}{\varLambda ''_{{{\rm{SR}}} }}\delta {Z_a}}}{{{\varOmega _{l,n}}A}}} } \right) \end{split} | (37) |
将式(23)、式(26)和式(37)代入式(22),用户
\begin{split} P_{{{\rm{out}}} }^{l,n} =& {a_1}\sum\limits_{k = 0}^{{m_{\rm{R}}} - 1} {\frac{{{{\left( {1 - {m_{\rm{R}}}} \right)}_{{\rm{R}}} }{{\left( { - {a_2}} \right)}^k}}}{{{\delta ^{k + 1}}{{\left( 1 \right)}_k}}}} \times \left( {1 - \exp \left( { - \frac{{N{\gamma _{{\rm{th}},n}}\delta }}{{{{\bar \gamma }_{{{\rm{SR}}} }}\left( {{\alpha _{l,n}} - {\gamma _{{\rm{th}},n}}\displaystyle\sum\limits_{i = n + 1}^M {{\alpha _{l,i}}} } \right)}}} \right)} \right. \\ & \left. { \times \sum\limits_{n = 0}^k {\frac{1}{{n!}}} {{\left( {\frac{{N{\gamma _{{\rm{th}},n}}\delta }}{{{{\bar \gamma }_{{{\rm{SR}}} }}\left( {{\alpha _{l,n}} - {\gamma _{{\rm{th}},n}}\displaystyle\sum\limits_{i = n + 1}^M {{\alpha _{l,i}}} } \right)}}} \right)}^n}} \right) + 2{a_1}{b_n}\displaystyle\sum\limits_{h = 0}^{M - n} {\sum\limits_{j = 0}^{n + h} {\displaystyle\sum\limits_{l = 0}^{j\left( {{m_{l,n}} - 1} \right)} {\left( \begin{gathered} M - n \\ h \\ \end{gathered} \right)} } } \\ & \times \left( \begin{gathered} n + h \\ j \\ \end{gathered} \right)\frac{{{{\left( { - 1} \right)}^{j + h}}c_l^j}}{{n + h}}{\left( {\frac{{{m_{l,n}}{\gamma _{{\rm{th}},n}}{\varLambda ''_{{{\rm{SR}}} }}{Z_a}}}{{{\varOmega _{l,n}}}}} \right)^l}\exp \left( { - \frac{{{\gamma _{{\rm{th}},n}}{\varLambda ''_{{{\rm{SR}}} }}\delta }}{A}} \right) \\ & \times \sum\limits_{{t_1} = 0}^{{m_{\rm{R}}} - 1} {\sum\limits_{p = 0}^{{t_1}} {\frac{{{{\left( {1 - {m_{\rm{R}}}} \right)}_{{t_1}}}{{\left( { - {a_2}} \right)}^{{t_1}}}{{\left( {{\gamma _{{\rm{th}},n}}{{\bar \gamma }_{l,n}}} \right)}^p}}}{{{t_1}!{{\left( 1 \right)}_{{t_1}}}{A^{{t_1} + 1}}}}} } {\left( {\frac{{jA{m_{l,n}}{\gamma _{{\rm{th}},n}}{\varLambda ''_{{{\rm{SR}}} }}{Z_{a} }}}{{{\varOmega _{l,n}}\delta }}} \right)^{\frac{{{t_1} - l - p}}{2}}} \\ & \times {K_{{t_1} - l - p}}\left( {2\sqrt {\frac{{j{m_{l,n}}{\gamma _{{\rm{th}},n}}{\varLambda ''_{{{\rm{SR}}} }}\delta {Z_a}}}{{{\varOmega _{l,n}}A}}} } \right) \end{split} | (38) |
为了进一步揭示关键参数与系统性能之间的影响,在得到中断概率闭合表达式的基础上,进一步讨论高信噪比下中断概率的渐进表达式。用户
P_{{{\rm{out}}} }^\infty = \underbrace {F_X^\infty \left( u \right)}_{I_1^\infty } + \underbrace {\int\limits_u^\infty {F_{{Y_{l,n}}}^\infty \left[ {{Y_{l,n}} \le \left( {\frac{{{\gamma _{{\rm{th}},n}}{Z_a}\left( {{\varLambda ''_{{{\rm{SR}}} }} + N} \right)}}{{\left( {{{\bar \gamma }_{{\rm{SR}}}}{{\bar \gamma }_{l,n}}\left( {{\alpha _{l,n}} - {\gamma _{{\rm{th}},n}}\displaystyle\sum\limits_{i = n + 1}^M {{\alpha _{l,i}}} } \right)x - N{\gamma _{{\rm{th}},n}}{{\bar \gamma }_{l,n}}} \right)}}} \right)} \right]} f_X^\infty \left( x \right){\rm{d}}x}_{I_2^\infty } | (39) |
对式(23)进行级数展开,可以得到
f_X^\infty \left( x \right) \approx {a_1} | (40) |
进一步得到
F_X^\infty \left( x \right) = \int\limits_0^x {f_X^\infty \left( t \right)} {\rm{d}}t \approx {a_1}x | (41) |
由式(29),
F_{{{\left| {{\rho _{l,n}}} \right|}^2}}^\infty \left( y \right) \approx {\left( {\frac{{{m_{l,n}}y}}{{{\varOmega _{l,n}}}}} \right)^{{m_{l,n}}}}\frac{1}{{\varGamma \left( {{m_{l,n}} + 1} \right)}} | (42) |
将式(42)代入式(28)中,可以进一步得到
F_{{Y_{l,n}}}^\infty \left( y \right) \approx \frac{{{b_n}}}{n}\frac{1}{{{\varGamma ^n}\left( {{m_{l,n}} + 1} \right)}}{\left( {\frac{{{m_{l,n}}y}}{{{\varOmega _{l,n}}}}} \right)^{{m_{l,n}} + n}} | (43) |
根据式(40)和式(43),
\begin{split} I_2^\infty \approx & \frac{{{b_n}}}{n}\frac{{{a_1}}}{{{\varGamma ^n}\left( {{m_{l,n}} + 1} \right)}}{\left( {\frac{{{m_{l,n}}{\gamma _{{\rm{th}},n}}{{\varLambda ''}_{{{\rm{SR}}} }}{Z_a}}}{{{\varOmega _{l,n}}}}} \right)^{{m_{l,n}} + n}}\\ & \int\limits_u^\infty {{{\left( {Ax - N{\gamma _{{\rm{th}},n}}{{\bar \gamma }_{l,n}}} \right)}^{ - {m_{l,n}} - n}}{\rm{d}}x}\\[-21pt] \end{split} | (44) |
其中,
\zeta \text=\left\{\begin{aligned} & \frac{1}{\left({\alpha }_{l,n}-{\gamma }_{{\rm{th}},n}{\displaystyle \sum _{i=n+1}^{N}{\alpha }_{l,i}}\right){\overline{\gamma }}_{{\rm{SR}}}},\qquad\qquad\qquad\qquad\; n+{m}_{l,n}-1=1\\ &\frac{{\overline{\gamma }}_{l,n}{}^{n+{m}_{l,n}-2}}{\left({\alpha }_{l,n}-{\gamma }_{{\rm{th}},n}{\displaystyle \sum _{i=n+1}^{N}{\alpha }_{l,i}}\right){\overline{\gamma }}_{\mathrm{SR}}}\cdot\dfrac{1}{\left(n+{m}_{l,n}-1\right)},\text{ }n+{m}_{l,n}-1 > 1\end{aligned} \right. | (45) |
将式(41)、式(44)和式(45)代入式(39)中,得到中断概率渐进表达式为
\begin{split} P_{{{\rm{out}}} }^\infty =& \frac{{{a_1}N{\gamma _{{\rm{th}},n}}}}{{\left( {{\alpha _{l,n}} - {\gamma _{{\rm{th}},n}}\displaystyle\sum\limits_{i = n + 1}^N {{\alpha _{l,i}}} } \right){{\bar \gamma }_{{\rm{SR}}}}}}\\ & {\text{ + }}\frac{{{b_n}}}{n}\frac{{{a_1}\zeta }}{{{\varGamma ^n}\left( {{m_{l,n}} + 1} \right)}}{\left( {\frac{{{m_{l,n}}{\gamma _{th,n}}{{\varLambda ''}_{{{\rm{SR}}} }}{Z_{{\rm{a}}} }}}{{{\varOmega _{l,n}}}}} \right)^{{m_{l,n}} + n}} \end{split} | (46) |
假设
P_{{{\rm{out}}} }^\infty \left( {{\gamma _{{\rm{th}},n}}} \right){\text{ = }}{\left( {{G_{\rm{a}}}\bar \gamma } \right)^{ - {G_{\rm{d}}}}} + o\left( {{{\bar \gamma }^{ - {G_{\rm{d}}}}}} \right) | (47) |
其中,
P_{{{\rm{out}}} }^\infty {\text{ = }}\varTheta \left( {\frac{1}{{\bar \gamma }}} \right){\text{ + }}o\left( {\frac{1}{{\bar \gamma }}} \right) | (48) |
其中
\varTheta \text=\left\{\begin{aligned} & \frac{{a}_{1}}{\left({\alpha }_{l,n}-{\gamma }_{{\rm{th}},n}{\displaystyle \sum _{i=n+1}^{N}{\alpha }_{l,i}}\right)}\left(1\text+\frac{{\gamma }_{{\rm{th}},n}{b}_{n}}{{\varGamma }^{n}\left({m}_{l,n}+1\right)n}{\left(\frac{{m}_{l,n}{\gamma }_{{\rm{th}},n}{Z}_{a}}{{\varOmega }_{l,n}}\right)}^{{m}_{l,n}+n}\right),\; n+{m}_{l,n}-1=1\\ & \frac{{a}_{1}N{\gamma }_{{\rm{th}},n}}{\left({\alpha }_{l,n}-{\gamma }_{{\rm{th}},n}{\displaystyle \sum _{i=n+1}^{N}{\alpha }_{l,i}}\right)},\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad n+{m}_{l,n}-1 > 1\end{aligned} \right. | (49) |
于是,系统的分集度和阵列增益分别为
{G_{\rm{d}}} = 1 | (50) |
{G_{\rm{a}}} = \varTheta | (51) |
从以上可以看出,卫星到无人机中继以及无人机中继到地面用户的链路信道参数只影响系统的阵列增益,而对系统的分集度不产生影响。
本节通过计算机仿真验证理论分析的正确性,同时定量分析了系统参数对卫星通信系统性能的影响。此外,为了验证本文所提传输方案的优越性,还跟传统的OMA方案和文献[17]中的方案进行对比。其中OMA方案是一个用户独占一个时间/频率资源块,文献[17]采用无人机配置单天线且簇内仅包含远近两个用户的下行NOMA传输方案。仿真中考虑系统包含6个用户,将所有地面用户分为两个簇,每个簇包含3个用户。假设S-R链路经历中度阴影衰落,信道参数为
参数 | 数值 |
卫星波束最大增益G_{\rm{S}}^{{\rm{max}}}{\text{ } }\left( { {\text{dB} } } \right) | 53 |
3 dB角度{\theta _{ { {\rm{S} } } ,3\;{\text{dB} } } }(o) | 0.4 |
带宽B{\text{ }}\left( {{\text{MHz}}} \right) | 5 |
无人机与地面用户间距离{d_{l,n}}{\text{ }}\left( {\text{m}} \right) | 500 |
噪声温度T{\text{ }}\left( {\text{K}} \right) | 300 |
图2给出了第2个簇中3个用户的中断概率随发射功率
图3给出了不同发射功率
图4为改变无人机配置天线数的条件下,用户
图5为用户
针对无人机辅助的卫星通信系统下行链路,分析了基于SDMA和协作NOMA相结合的多用户传输系统的中断性能。首先,配置多根天线的无人机作为中继站辅助卫星通信,采用NOMA技术服务多个地面用户,得到地面用户的输出SINR表达式。然后,基于用户平均SINR最大化准则,提出了利用信道角度信息的波束成形方案。接着,进一步推导得到系统的中断概率闭合表达式和高信噪比下系统的中断概率近似表达式。最后,计算机仿真验证了本文所提方案的优越性以及理论分析的正确性,并且定量分析了相关参数对用户性能的影响,为进一步探索NOMA技术在无人机辅助卫星通信系统中的应用提供了有益的参考。
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|
参数 | 数值 |
卫星波束最大增益G_{\rm{S}}^{{\rm{max}}}{\text{ } }\left( { {\text{dB} } } \right) | 53 |
3 dB角度{\theta _{ { {\rm{S} } } ,3\;{\text{dB} } } }(o) | 0.4 |
带宽B{\text{ }}\left( {{\text{MHz}}} \right) | 5 |
无人机与地面用户间距离{d_{l,n}}{\text{ }}\left( {\text{m}} \right) | 500 |
噪声温度T{\text{ }}\left( {\text{K}} \right) | 300 |