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Surveillance performance of satellite-based ADS-B system in co-channel interference environment
LIU Haitao, LI Shaoyang, QIN Dingben, LI Dongxia
Tianjin Key Lab for Advanced Signal Processing, Civil Aviation University of China, Tianjin 300300, China
Abstract: Satellite-based ADS-B system is an important technical scheme for wide-area aircraft surveillance system. To reveal the influence of co-channel interference on the surveillance performance of the satellite-based ADS-B system, a theoretical formula of the surveillance capacity of the system is proposed. Firstly, the model of satellite-based ADS-B system is given. Then, the correct reception probability of ADS-B message, the update interval of aircraft position message, and the update interval of aircraft position report are theoretically derived. Based on these equations, the surveillance capacity of the system is given. Finally, these theoretical formulas are verified by computer simulations using the Space Network Simulation (SNS) software. Our study shows that the surveillance capacity of the satellite-based ADS-B system is jointly determined by the symbol error rate of aircraft-satellite links, the symbol error rate of satellite-earth links, the number of satellites, and the required update interval of the ADS-B application subsystem.
Keywords: satellite-based ADS-B system     co-channel interference     update interval of aircraft position message     update interval of aircraft position report     surveillance capacity

1.2 1 090 MHz共信道干扰模型

2.1 消息到达速率与概率

 ${\lambda _{\rm{B}}} = {N_{\rm{B}}}{v_{\rm{B}}}{\alpha _{\rm{B}}}$ （1）

 ${\lambda _{\rm{S}}} = {N_{\rm{S}}}{v_{\rm{S}}}{\alpha _{\rm{S}}}$ （2）

 ${\lambda _{\rm{A}}} = {N_{\rm{A}}}{v_{\rm{A}}}{\alpha _{\rm{A}}}$ （3）

 ${P_{\rm{B}}}(k,t) = \frac{{{{\left( {{\lambda _{\rm{B}}}t} \right)}^k}}}{{k!}}{{\rm{e}}^{ - {\lambda _{\rm{B}}}t}}$ （4）

 ${P_{\rm{S}}}(k,t) = \frac{{{{\left( {{\lambda _S}t} \right)}^k}}}{{k!}}{{\rm{e}}^{ - {\lambda _{\rm{S}}}t}}$ （5）

 ${P_{\rm{A}}}(k,t) = \frac{{{{\left( {{\lambda _{\rm{A}}}t} \right)}^k}}}{{k!}}{{\rm{e}}^{ - {\lambda _{\rm{A}}}t}}$ （6）

 ${P_{\rm{u}}} = P(A) = \sum\limits_{i = 0}^4 P \left( {A|{B_i}} \right)P\left( {{B_i}} \right)$ （7）

 $P\left( {A|{B_i}} \right)P\left( {{B_i}} \right) = \left\{ {\begin{array}{*{20}{l}} {{{\left( {1 - {P_{{\rm{e}},{\rm{u}}}}} \right)}^n}{{\rm{e}}^{ - \left[ {{\lambda _{\rm{B}}}2{\tau _{\rm{B}}} + {\lambda _{\rm{S}}}\left( {{\tau _{\rm{S}}} + {\tau _{\rm{B}}}} \right) + {\lambda _{\rm{A}}}\left( {{\tau _{\rm{A}}} + {\tau _{\rm{B}}}} \right)} \right]}}}&{i = 0}\\ {0.89{{\left( {1 - {P_{{\rm{e}},{\rm{u}}}}} \right)}^n}{\lambda _{\rm{A}}}\left( {{\tau _{\rm{A}}} + {\tau _{\rm{B}}}} \right){{\rm{e}}^{ - \left[ {{\lambda _{\rm{B}}}2{\tau _{\rm{B}}} + {\lambda _{\rm{S}}}\left( {{\tau _{\rm{S}}} + {\tau _{\rm{B}}}} \right) + {\lambda _{\rm{A}}}\left( {{\tau _{\rm{A}}} + {\tau _{\rm{B}}}} \right)} \right]}}}&{i = 1}\\ {0.64{{\left( {1 - {P_{{\rm{e}},{\rm{u}}}}} \right)}^n}\frac{{{{\left[ {{\lambda _{\rm{A}}} \cdot \left( {{\tau _{\rm{A}}} + {\tau _{\rm{B}}}} \right)} \right]}^2}}}{2}{{\rm{e}}^{ - \left[ {{\lambda _{\rm{B}}}2{\tau _{\rm{B}}} + {\lambda _{\rm{S}}}\left( {{\tau _{\rm{S}}} + {\tau _{\rm{B}}}} \right) + {\lambda _{\rm{A}}}\left( {{\tau _{\rm{A}}} + {\tau _{\rm{B}}}} \right)} \right]}}}&{i = 2}\\ {0.52{{\left( {1 - {P_{{\rm{e}},{\rm{u}}}}} \right)}^n}\frac{{{{\left[ {{\lambda _{\rm{A}}} \cdot \left( {{\tau _{\rm{A}}} + {\tau _{\rm{B}}}} \right)} \right]}^3}}}{6}{{\rm{e}}^{ - \left[ {{\lambda _{\rm{B}}}2{\tau _{\rm{B}}} + {\lambda _{\rm{S}}}\left( {{\tau _{\rm{S}}} + {\tau _{\rm{B}}}} \right) + {\lambda _{\rm{A}}}\left( {{\tau _{\rm{A}}} + {\tau _{\rm{B}}}} \right)} \right]}}}&{i = 3}\\ 0&{i{\rm{ }} = {\rm{ }}4} \end{array}} \right.$ （8）

 $\begin{array}{l} {P_{\rm{u}}} = {\left( {1 - {P_{{\rm{e}},{\rm{u}}}}} \right)^n}\left\{ {1 + 0.89{\lambda _{\rm{A}}}\left( {{\tau _{\rm{A}}} + {\tau _{\rm{B}}}} \right) + } \right.\\ \;\;\;\;\left. {\frac{{0.64}}{2}{{\left[ {{\lambda _{\rm{A}}}\left( {{\tau _{\rm{A}}} + {\tau _{\rm{B}}}} \right)} \right]}^2} + \frac{{0.52}}{6}{{\left[ {{\lambda _{\rm{A}}}\left( {{\tau _{\rm{A}}} + {\tau _{\rm{B}}}} \right)} \right]}^3}} \right\} \cdot \\ \;\;\;\;{{\rm{e}}^{ - \left[ {{\lambda _{\rm{B}}}2{\tau _{\rm{B}}} + {\lambda _{\rm{S}}}\left( {{\tau _{\rm{S}}} + {\tau _{\rm{B}}}} \right) + {\lambda _{\rm{A}}}\left( {{\tau _{\rm{A}}} + {\tau _{\rm{B}}}} \right)} \right]}} \end{array}$ （9）

 $\begin{array}{l} {P_{\rm{u}}} = {\left( {1 - {P_{{\rm{e}},{\rm{u}}}}} \right)^n}{{\rm{e}}^{ - \left[ {{\lambda _{\rm{B}}}2{\tau _{\rm{B}}} + {\lambda _{\rm{S}}}\left( {{\tau _{\rm{S}}} + {\tau _{\rm{B}}}} \right) + {\lambda _{\rm{A}}}\left( {{\tau _{\rm{A}}} + {\tau _{\rm{B}}}} \right)} \right]}} \approx \\ \;\;\;\;\;\;\;\left( {1 - n{P_{{\rm{e}},{\rm{u}}}}} \right){{\rm{e}}^{ - \left[ {{\lambda _{\rm{B}}}2{\tau _{\rm{B}}} + {\lambda _{\rm{S}}}\left( {{\tau _{\rm{S}}} + {\tau _{\rm{B}}}} \right) + {\lambda _{\rm{A}}}\left( {{\tau _{\rm{A}}} + {\tau _{\rm{B}}}} \right)} \right]}} = \\ \;\;\;\;\;\;\;\left( {1 - n{P_{{\rm{e}},{\rm{u}}}}} \right) \cdot \\ \;\;\;\;\;\;\;{{\rm{e}}^{ - \left[ {{\gamma _{\rm{B}}}{\gamma _{\rm{S}}}{v_{\rm{B}}}{\alpha _{\rm{B}}}2{\tau _{\rm{B}}} + {\gamma _{\rm{S}}}{v_{\rm{S}}}{\alpha _{\rm{S}}}\left( {{\tau _{\rm{S}}} + {\tau _{\rm{B}}}} \right) + {\gamma _{\rm{A}}}{v_{\rm{A}}}{\alpha _{\rm{A}}}\left( {{\tau _{\rm{A}}} + {\tau _{\rm{B}}}} \right)} \right]N}} \end{array}$ （10）

 $\Delta T = \left\{ {\begin{array}{*{20}{l}} {{T_1}}&{\;\;\;\;\;{P_{\rm{u}}}}\\ {{T_1} + {T_2}}&{\left( {1 - {P_{\rm{u}}}} \right){P_{\rm{u}}}}\\ {\; \vdots }&{}\\ {\sum\limits_{i = 1}^I {{T_i}} }&{{{\left( {1 - {P_{\rm{u}}}} \right)}^{I - 1}}{P_{\rm{u}}}}\\ {\; \vdots }&{} \end{array}} \right.$ （11）

 ${P_{\rm{f}}}\left( H \right) = 1 - {\left( {1 - {P_{\rm{f}}}} \right)^H} \approx H{P_{\rm{f}}}$ （12）

 ${P_{\rm{d}}} = {\left( {1 - {P_{{\rm{e}},{\rm{d}}}}} \right)^n} \approx 1 - n{P_{{\rm{e}},{\rm{d}}}}$ （13）

 $\Delta {T_{{\rm{In}}}} = \left\{ \begin{array}{l} \begin{array}{*{20}{l}} {{T_1}}&{\;\;\;\;\;\;\;\;\;\;\;\;\;\;{P_{\rm{u}}}\left( {1 - H \cdot {P_{\rm{f}}}} \right){P_{\rm{d}}}}\\ {{T_1} + {T_2}}&{{P_{\rm{u}}}\left( {1 - H \cdot {P_{\rm{f}}}} \right){P_{\rm{d}}} \cdot \left[ {1 - {P_{\rm{u}}}\left( {1 - H \cdot {P_{\rm{f}}}} \right){P_{\rm{d}}}} \right]}\\ {{T_1} + {T_2} + {T_3}}&{{P_{\rm{u}}}\left( {1 - H \cdot {P_{\rm{f}}}} \right){P_{\rm{d}}} \cdot {{\left[ {1 - {P_{\rm{u}}}\left( {1 - H \cdot {P_{\rm{f}}}} \right){P_{\rm{d}}}} \right]}^2}}\\ {\; \vdots }&{}\\ {\sum\limits_{i = 1}^I {{T_i}} }&{{P_{\rm{u}}}\left( {1 - H \cdot {P_{\rm{f}}}} \right){P_{\rm{d}}} \cdot {{\left[ {1 - {P_{\rm{u}}}\left( {1 - H \cdot {P_{\rm{f}}}} \right){P_{\rm{d}}}} \right]}^{I - 1}}} \end{array}\\ \; \vdots \end{array} \right.$ （14）

 $\begin{array}{*{20}{c}} {E\left[ {\Delta {T_{{\rm{In}}}}|{T_i},i = 1,2, \cdots } \right] = {P_{\rm{u}}}\left( {1 - H{P_{\rm{f}}}} \right){P_{\rm{d}}} \cdot }\\ {\sum\limits_{i = 1}^\infty {\left\{ {{{\left[ {1 - {P_{\rm{u}}}\left( {1 - H{P_{\rm{f}}}} \right){P_{\rm{d}}}} \right]}^{i - 1}}\sum\limits_{j = 1}^i {{T_j}} } \right\}} } \end{array}$ （15）

 $\begin{array}{l} \Delta {{\bar T}_{{\rm{ln}}}} = {E_{{T_i}}}\left\{ {E\left[ {\Delta {T_{{\rm{ln}}}}|{T_i},i = 1,2, \cdots } \right]} \right\} = \\ \;\;\;\;\;{P_{\rm{u}}}\left( {1 - H{P_{\rm{f}}}} \right){P_{\rm{d}}} \cdot \\ \;\;\;\;\;\sum\limits_{i = 1}^\infty {\left\{ {{{\left[ {1 - {P_{\rm{u}}}\left( {1 - H{P_{\rm{f}}}} \right){P_{\rm{d}}}} \right]}^{i - 1}}i{T_{{\rm{pos}}}}} \right\}} \end{array}$ （16）

 $\Delta {{\bar T}_{{\rm{In}}}} = \frac{{{T_{{\rm{pos}}}}}}{{{P_{\rm{u}}}\left( {1 - H{P_{\rm{f}}}} \right){P_{\rm{d}}}}}$ （17）

 ${{\bar \lambda }_{{\rm{In}}}} = \frac{1}{{\Delta {{\bar T}_{{\rm{In}}}}}} = \frac{{{P_{\rm{u}}}\left( {1 - H{P_{\rm{f}}}} \right){P_{\rm{d}}}}}{{{T_{{\rm{pos}}}}}}$ （18）

 $f(t) = {{\bar \lambda }_{{\rm{ln}}}}{{\rm{e}}^{ - {{\bar \lambda }_{{\rm{ln}}}t}}}\;\;\;\;t > 0$ （19）

 $\Delta {T_{{\rm{Output }}}} = \Delta {T_{{\rm{In}}}}$ （20）

2.6 95%位置报告的更新间隔

 $P\left( {\Delta {T_{{\rm{Output }}}} \le \Delta {T_{95\% }}} \right) = 95\%$ （21）

 $J = \frac{{\ln 0.05}}{{\ln \left( {1 - {P_{\rm{u}}}{P_{\rm{d}}}} \right)}}$ （22）

 $\Delta {T_{95\% }} = E\left[ {\sum\limits_{j = 1}^J {{T_j}} } \right] = JE\left[ {{T_i}} \right] = J{T_{{\rm{pos}}}}$ （23）

 $\Delta {T_{95\% }} = \frac{{\ln \;0.05}}{{\ln \left[ {1 - {{\rm{e}}^{ - \left[ {{\gamma _{\rm{B}}}{\gamma _{\rm{S}}}{v_{\rm{B}}}{\alpha _{\rm{B}}}2{\tau _{\rm{B}}} + {\gamma _{\rm{S}}}{v_{\rm{S}}}{\alpha _{\rm{S}}}\left( {{\tau _{\rm{S}}} + {\tau _{\rm{B}}}} \right) + {\gamma _{\rm{A}}}{v_{\rm{A}}}{\alpha _{\rm{A}}}\left( {{\tau _{\rm{A}}} + {\tau _{\rm{B}}}} \right)} \right]N}}{{\left( {1 - {P_{{\rm{e}},{\rm{u}}}}} \right)}^n}{{\left( {1 - {P_{{\rm{e}},{\rm{d}}}}} \right)}^n}} \right]}} \cdot {T_{{\rm{pos}}}}$ （24）

 $\begin{array}{l} \Delta {T_{95\% }} \approx \\ \frac{{\ln \;0.05}}{{\ln \left[ {1 - {{\rm{e}}^{ - 697.68 \times {{10}^{ - 6}} \times N}}\left( {1 - 112{P_{{\rm{e}},{\rm{u}}}}} \right)\left( {1 - 112{P_{{\rm{e}},{\rm{d}}}}} \right)} \right]}} \end{array}$ （25）

 $\Delta {T_{95\% }} \approx - \frac{{\ln 0.05}}{{{P_{\rm{u}}}{P_{\rm{d}}}}}{T_{{\rm{pos}}}}$ （26）

 $\begin{array}{l} N = \\ \frac{{\ln \left[ {{{\left( {1 - {P_{{\rm{e}},{\rm{u}}}}} \right)}^n}{{\left( {1 - {P_{{\rm{e}},{\rm{d}}}}} \right)}^n}} \right] - \ln \left( {1 - {{0.05}^{\frac{{{T_{{\rm{pos}}}}}}{{\Delta {T_{95\% }}}}}}} \right)}}{{{\gamma _{\rm{B}}}{\gamma _{\rm{S}}}{v_{\rm{B}}}{\alpha _{\rm{B}}}2{\tau _{\rm{B}}} + {\gamma _{\rm{S}}}{v_{\rm{S}}}{\alpha _{\rm{S}}}\left( {{\tau _{\rm{S}}} + {\tau _{\rm{B}}}} \right) + {\gamma _{\rm{A}}}{v_{\rm{A}}}{\alpha _{\rm{A}}}\left( {{\tau _{\rm{A}}} + {\tau _{\rm{B}}}} \right)}} \end{array}$ （27）

 $\begin{array}{l} {C_{\rm{s}}} = \\ \frac{{\ln \left[ {{{\left( {1 - {P_{{\rm{e}},{\rm{u}}}}} \right)}^n}{{\left( {1 - {P_{{\rm{e}},{\rm{d}}}}} \right)}^n}} \right] - \ln \left( {1 - {{0.05}^{\frac{{{T_{{\rm{pos}}}}}}{{\Delta {T_{{\rm{req}}}}}}}}} \right)}}{{{\gamma _{\rm{B}}}{\gamma _{\rm{S}}}{v_{\rm{B}}}{\alpha _{\rm{B}}}2{\tau _{\rm{B}}} + {\gamma _{\rm{S}}}{v_{\rm{S}}}{\alpha _{\rm{S}}}\left( {{\tau _{\rm{S}}} + {\tau _{\rm{B}}}} \right) + {\gamma _{\rm{A}}}{v_{\rm{A}}}{\alpha _{\rm{A}}}\left( {{\tau _{\rm{A}}} + {\tau _{\rm{B}}}} \right)}} \end{array}$ （28）

 $\begin{array}{l} {C_{\rm{s}}} \approx 1433 \times \\ \;\;\;\;\;\left\{ {\ln \left[ {\left( {1 - 112{P_{{\rm{e}},{\rm{u}}}}} \right)\left( {1 - 112{P_{{\rm{e}},{\rm{d}}}}} \right)} \right]} \right. - \\ \;\;\;\;\;\left. {\ln \left( {1 - {{0.05}^{1/\Delta {T_{{\rm{req}}}}}}} \right)} \right\} \end{array}$ （29）

 $C = {M_{\rm{s}}}{C_{\rm{s}}}$ （30）

3 仿真结果 3.1 仿真环境及仿真参数设置

 参数名称 取值 γA 0.1 γS 0.9 γB 0.3 αA 0 αS 0.5 αB 0.5 vA/(消息数·s-1) 60 vS/(消息数·s-1) 6 vB/(消息数·s-1) 6.2 τA/μs 21 τS/μs 64 τB/μs 120 n/bit 112 Pe, u 10-4~5×10-3 Pf 10-3 H 3 Pe, d 10-6~10-4
3.2 消息正确接收概率

 图 2 消息正确接收概率与航空器数量的关系曲线(Pe, d=1×10-4) Fig. 2 Correct reception probability of message versus aircraft counts(Pe, d=1×10-4)

3.3 95%位置报告更新间隔

 图 3 位置报告更新间隔的概率密度函数及累积分布函数曲线(Pe, u=1×10-3，Pe, d=1×10-4，N=2 500) Fig. 3 PDF and CDF of position report update interval(Pe, u=1×10-3, Pe, d=1×10-4, N=2 500)

 图 4 95%位置报告更新间隔与航空器数量的关系曲线(Pe, d=1×10-4) Fig. 4 95% update interval of position report versus aircraft counts(Pe, d=1×10-4)

 图 5 95%位置报告更新间隔ΔT95%与N和Pe, d的关系图(Pe, u=1×10-3) Fig. 5 95% update interval of position report versus aircraft counts and SER of satellite-earth link (Pe, u=1×10-3)

 图 6 95%位置报告更新间隔ΔT95%与N和Pe, u的关系图(Pe, d=1×10-4) Fig. 6 95% update interval of position report versus aircraft counts and SER of aircraft-satellite link (Pe, d=1×10-4)

 图 7 星基ADS-B接收机监视容量与位置报告更新间隔的关系曲线(Pe, d=1×10-4) Fig. 7 Surveillance capacity of satellite receiver versus update interval of position report (Pe, d=1× 10-4)

 图 8 监视容量C与Pe, u和Pe, d的关系图(ΔTreq=8 s) Fig. 8 Surveillance capacity versus SER of aircraft-satellite link and SER of satellite-earth link (ΔTreq=8 s)

 误码率(Pe, u) 5×10-3 3×10-3 1×10-3 1×10-4 监视容量/架 847 1 169 1 491 1 635

4 结论

 $P\left( {A|{B_0}} \right) = {\left( {1 - {P_{{\rm{e}},{\rm{u}}}}} \right)^n}$ （A1）

 $P\left( {{B_{01}}} \right) = {P_{\rm{B}}}\left( {0,2{\tau _{\rm{B}}}} \right) = {{\rm{e}}^{ - {\lambda _{\rm{B}}}2{\tau _{\rm{B}}}}}$ （A2）

 $P\left( {{B_{02}}} \right) = {P_{\rm{S}}}\left( {0,{\tau _{\rm{S}}} + {\tau _{\rm{B}}}} \right) = {{\rm{e}}^{ - {\lambda _{\rm{S}}}\left( {{\tau _{\rm{S}}} + {\tau _{\rm{B}}}} \right)}}$ （A3）

 $P\left( {{B_{03}}} \right) = {P_{\rm{A}}}\left( {0,{\tau _{\rm{A}}} + {\tau _{\rm{B}}}} \right) = {{\rm{e}}^{ - {\lambda _{\rm{A}}}\left( {{\tau _{\rm{A}}} + {\tau _{\rm{B}}}} \right)}}$ （A4）

 $\begin{array}{l} P\left( {{B_0}} \right) = P\left( {{B_{01}} \cap {B_{02}} \cap {B_{03}}} \right) = \\ \;\;\;\;\;\;\;P\left( {{B_{01}}} \right)P\left( {{B_{02}}} \right)P\left( {{B_{03}}} \right) = \\ \;\;\;\;\;\;\;{{\rm{e}}^{ - \left[ {{\lambda _{\rm{B}}}2{\tau _{\rm{B}}} + {\lambda _{\rm{S}}}\left( {{\tau _{\rm{S}}} + {\tau _{\rm{B}}}} \right) + {\lambda _{\rm{A}}}\left( {{\tau _{\rm{A}}} + {\tau _{\rm{B}}}} \right)} \right]}} \end{array}$ （A5）

 $\begin{array}{l} P\left( {A|{B_0}} \right)P\left( {{B_0}} \right) = {\left( {1 - {P_{{\rm{e}},{\rm{u}}}}} \right)^n} \cdot \\ \;\;\;\;\;\;{{\rm{e}}^{ - \left[ {{\lambda _{\rm{B}}}2{\tau _{\rm{B}}} + {\lambda _{\rm{S}}}\left( {{\tau _{\rm{S}}} + {\tau _{\rm{B}}}} \right) + {\lambda _{\rm{A}}}\left( {{\tau _{\rm{A}}} + {\tau _{\rm{B}}}} \right)} \right]}} \end{array}$ （A6）

 $P\left( {A|{B_1}} \right)P\left( {{B_1}} \right) = P\left( {{B_1}|A} \right)P(A)$ （A7）

 $\begin{array}{l} P\left( {A|{B_1}} \right)P\left( {{B_1}} \right) = \\ \;\;\;\;\;\;P\left( {{B_{11}} \cup {B_{12}} \cup {B_{13}}|A} \right)P(A) \end{array}$ （A8）

 $\begin{array}{l} P\left( {A|{B_1}} \right)P\left( {{B_1}} \right) = \left[ {P\left( {{B_{11}}|A} \right) + } \right.\\ \;\;\;\;\;\;\;\left. {P\left( {{B_{12}}|A} \right) + P\left( {{B_{13}}|A} \right)} \right]P(A) = \\ \;\;\;\;\;\;\;P\left( {A|{B_{11}}} \right)P\left( {{B_{11}}} \right) + P\left( {A|{B_{12}}} \right)P\left( {{B_{12}}} \right) + \\ \;\;\;\;\;\;\;P\left( {A|{B_{13}}} \right)P\left( {{B_{13}}} \right) \end{array}$ （A9）

 $\begin{array}{l} P\left( {{B_{13}}} \right) = {P_{\rm{A}}}\left( {1,{\tau _{\rm{A}}} + {\tau _{\rm{B}}}} \right){P_{\rm{B}}}\left( {0,2{\tau _{\rm{B}}}} \right) \cdot \\ \;\;\;\;\;\;{P_{\rm{S}}}\left( {0,{\tau _{\rm{S}}} + {\tau _{\rm{B}}}} \right) = {\lambda _{\rm{A}}}\left( {{\tau _{\rm{A}}} + {\tau _{\rm{B}}}} \right) \cdot \\ \;\;\;\;\;\;{{\rm{e}}^{ - \left[ {{\lambda _{\rm{B}}}2{\tau _{\rm{B}}} + {\lambda _{\rm{S}}}\left( {{\tau _{\rm{S}}} + {\tau _{\rm{B}}}} \right) + {\lambda _{\rm{A}}}\left( {{\tau _{\rm{A}}} + {\tau _{\rm{B}}}} \right)} \right]}} \end{array}$ （A10）

 $\begin{array}{*{20}{c}} {P\left( {A|{B_1}} \right)P\left( {{B_1}} \right) = 0.89{{\left( {1 - {P_{{\rm{e}},{\rm{u}}}}} \right)}^n}{\lambda _{\rm{A}}} \cdot }\\ {\left( {{\tau _{\rm{A}}} + {\tau _{\rm{B}}}} \right){{\rm{e}}^{ - \left[ {{\lambda _{\rm{B}}}2{\tau _{\rm{B}}} + {\lambda _{\rm{S}}}\left( {{\tau _{\rm{S}}} + {\tau _{\rm{B}}}} \right) + {\lambda _{\rm{A}}}\left( {{\tau _{\rm{A}}} + {\tau _{\rm{B}}}} \right)} \right]}}} \end{array}$ （A11）

 $\begin{array}{l} P\left( {A|{B_2}} \right)P\left( {{B_2}} \right) = \\ \;\;\;\;\;0.64{\left( {1 - {P_{{\rm{e}}.{\rm{u}}}}} \right)^n}\frac{{{{\left[ {{\lambda _{\rm{A}}}\left( {{\tau _{\rm{A}}} + {\tau _{\rm{B}}}} \right)} \right]}^2}}}{2}\\ \;\;\;\;\;{{\rm{e}}^{ - \left[ {{\lambda _{\rm{B}}}2{\tau _{\rm{B}}} + {\lambda _{\rm{S}}}\left( {{\tau _{\rm{S}}} + {\tau _{\rm{B}}}} \right) + {\lambda _{\rm{A}}}\left( {{\tau _{\rm{A}}} + {\tau _{\rm{B}}}} \right)} \right]}} \end{array}$ （A12）

 $\begin{array}{l} P\left( {A|{B_3}} \right)P\left( {{B_3}} \right) = \\ \;\;\;\;\;\;0.52{\left( {1 - {P_{{\rm{e}},{\rm{u}}}}} \right)^n} \cdot \frac{{{{\left[ {{\lambda _{\rm{A}}}\left( {{\tau _{\rm{A}}} + {\tau _{\rm{B}}}} \right)} \right]}^3}}}{6} \cdot \\ \;\;\;\;\;\;{{\rm{e}}^{ - \left[ {{\lambda _{\rm{B}}}2{\tau _{\rm{B}}} + {\lambda _{\rm{S}}}\left( {{\tau _{\rm{S}}} + {\tau _{\rm{B}}}} \right) + {\lambda _{\rm{A}}}\left( {{\tau _{\rm{A}}} + {\tau _{\rm{B}}}} \right)} \right]}} \end{array}$ （A13）

 $P\left( {A|{B_4}} \right)P\left( {{B_4}} \right) = 0$ （A14）

 $\begin{array}{l} \Delta {{\bar T}_{{\rm{ln}}}} = {P_{\rm{u}}}\left( {1 - H{P_{\rm{f}}}} \right){P_{\rm{d}}} \cdot \\ \;\;\;\;\;\;\sum\limits_{i = 1}^\infty {\left\{ {{{\left[ {1 - {P_{\rm{u}}}\left( {1 - H{P_{\rm{f}}}} \right){P_{\rm{d}}}} \right]}^{i - 1}}i{T_{{\rm{pos}}}}} \right\}} = \\ \;\;\;\;\;\;{P_{\rm{u}}}\left( {1 - H{P_{\rm{f}}}} \right){P_{\rm{d}}} \cdot \\ \;\;\;\;\;\;\sum\limits_{i = 1}^I {\left\{ {{{\left[ {1 - {P_u}\left( {1 - H{P_{\rm{f}}}} \right){P_{\rm{d}}}} \right]}^{i - 1}}i{T_{{\rm{pos}}}}} \right\}} + \\ \;\;\;\;\;\;{P_{\rm{u}}}\left( {1 - H{P_{\rm{f}}}} \right){P_{\rm{d}}} \cdot \\ \;\;\;\;\;\sum\limits_{i = I + 1}^\infty {\left\{ {{{\left[ {1 - {P_{\rm{u}}}\left( {1 - H{P_{\rm{f}}}} \right){P_{\rm{d}}}} \right]}^{i - 1}}i{T_{{\rm{pos}}}}} \right\}} \end{array}$ （B1）

I充分大时，Pu(1-HPf)Pd·$\sum\limits_{i = I + 1}^\infty {\left\{ {{{\left[ {1 - {P_{\rm{u}}}\left( {1 - H{P_{\rm{f}}}} \right){P_{\rm{d}}}} \right]}^{i - 1}}i{T_{{\rm{pos}}}}} \right\}}$近似约等于0，因此式(B1)化简为

 $\begin{array}{l} \Delta {{\bar T}_{{\rm{ln}}}} \approx {P_{\rm{u}}}\left( {1 - H{P_{\rm{f}}}} \right){P_{\rm{d}}} \cdot \\ \;\;\;\;\;\;\sum\limits_{i = 1}^I {\left\{ {{{\left[ {1 - {P_{\rm{u}}}\left( {1 - H{P_{\rm{f}}}} \right){P_{\rm{d}}}} \right]}^{i - 1}}i{T_{{\rm{pos}}}}} \right\}} = \\ \;\;\;\;\;{P_{\rm{u}}}\left( {1 - H{P_{\rm{f}}}} \right){P_{\rm{d}}}{T_{{\rm{pos}}}} + \\ \;\;\;\;\;{P_{\rm{u}}}\left( {1 - H{P_{\rm{f}}}} \right){P_{\rm{d}}} \cdot \\ \;\;\;\;\;\left[ {1 - {P_{\rm{u}}}\left( {1 - H{P_{\rm{f}}}} \right){P_{\rm{d}}}} \right] \cdot 2{T_{{\rm{pos}}}} + \cdots + \\ \;\;\;\;\;{P_{\rm{u}}}\left( {1 - H{P_{\rm{f}}}} \right){P_{\rm{d}}} \cdot \\ \;\;\;\;\;{\left[ {1 - {P_{\rm{u}}}\left( {1 - H{P_{\rm{f}}}} \right){P_{\rm{d}}}} \right]^{{\rm{I}} - 1}}I{T_{{\rm{pos}}}} \end{array}$ （B2）

 $\begin{array}{l} \left[ {1 - {P_{\rm{u}}}\left( {1 - H{P_{\rm{f}}}} \right){P_{\rm{d}}}} \right]\Delta {{\bar T}_{{\rm{In}}}} = \\ \;\;\;\;\;{P_{\rm{u}}}\left( {1 - H{P_{\rm{f}}}} \right){P_{\rm{d}}} \cdot \\ \;\;\;\;\;\left[ {1 - {P_{\rm{u}}}\left( {1 - H{P_{\rm{f}}}} \right){P_{\rm{d}}}} \right]{T_{{\rm{pos}}}} + \\ \;\;\;\;\;{P_{\rm{u}}}\left( {1 - H{P_{\rm{f}}}} \right){P_{\rm{d}}} \cdot \\ \;\;\;\;\;{\left[ {1 - {P_{\rm{u}}}\left( {1 - H{P_{\rm{f}}}} \right){P_{\rm{d}}}} \right]^2}2{T_{{\rm{pos}}}} + \cdots + \\ \;\;\;\;\;{P_{\rm{u}}}\left( {1 - H{P_{\rm{f}}}} \right){P_{\rm{d}}} \cdot \\ \;\;\;\;\;{\left[ {1 - {P_{\rm{u}}}\left( {1 - H{P_{\rm{f}}}} \right){P_{\rm{d}}}} \right]^{I - 1}}(I - 1){T_{{\rm{pos}}}} + \\ \;\;\;\;\;{P_{\rm{u}}}\left( {1 - H{P_{\rm{f}}}} \right){P_{\rm{d}}} \cdot \\ \;\;\;\;\;{\left[ {1 - {P_{\rm{u}}}\left( {1 - H{P_{\rm{f}}}} \right){P_{\rm{d}}}} \right]^I}I{T_{{\rm{pos}}}} \end{array}$ （B3）

 $\begin{array}{l} {P_{\rm{u}}}\left( {1 - H{P_{\rm{f}}}} \right){P_{\rm{d}}}\Delta {{\bar T}_{{\rm{In}}}} = \\ \;\;\;\;\;{P_{\rm{u}}}\left( {1 - H{P_{\rm{f}}}} \right){P_{\rm{d}}}{T_{{\rm{pos}}}} + \\ \;\;\;\;\;{P_{\rm{u}}}\left( {1 - H{P_{\rm{f}}}} \right){P_{\rm{d}}} \cdot \\ \;\;\;\;\;\left[ {1 - {P_{\rm{u}}}\left( {1 - H{P_{\rm{f}}}} \right){P_{\rm{d}}}} \right]{T_{{\rm{pos}}}} + \cdots + \\ \;\;\;\;\;{P_{\rm{u}}}\left( {1 - H{P_{\rm{f}}}} \right){P_{\rm{d}}} \cdot \\ \;\;\;\;\;{\left[ {1 - {P_{\rm{u}}}\left( {1 - H{P_{\rm{f}}}} \right){P_{\rm{d}}}} \right]^{I - 1}}{T_{{\rm{pos}}}} - \\ \;\;\;\;\;{P_{\rm{u}}}\left( {1 - H{P_{\rm{f}}}} \right){P_{\rm{d}}} \cdot \\ \;\;\;\;\;{\left[ {1 - {P_{\rm{u}}}\left( {1 - H{P_{\rm{f}}}} \right){P_{\rm{d}}}} \right]^I}I{T_{{\rm{pos}}}} \end{array}$ （B4）

 $\begin{array}{l} \Delta {{\bar T}_{{\rm{ln}}}} = \frac{{1 - {\Delta _1}\Delta _2^I}}{{{P_{\rm{u}}}\left( {1 - H{P_{\rm{f}}}} \right){P_{\rm{d}}}}} \cdot {T_{{\rm{pos}}}} = \\ \;\;\;\;\;\;\;\;\;\frac{{{T_{{\rm{pos}}}}}}{{{P_{\rm{u}}}\left( {1 - H{P_{\rm{f}}}} \right){P_{\rm{d}}}}} - \\ \;\;\;\;\;\;\;\;\;\frac{{{\Delta _1}\Delta _2^I}}{{{P_{\rm{u}}}\left( {1 - H{P_{\rm{f}}}} \right){P_{\rm{d}}}}} \cdot {T_{{\rm{pos}}}} \end{array}$ （B5）

I→∞时，输入位置消息更新间隔的均值简化为

 $\Delta {{\bar T}_{{\rm{ln}}}} = \frac{{{T_{{\rm{pos}}}}}}{{{P_{\rm{u}}}\left( {1 - H{P_{\rm{f}}}} \right){P_{\rm{d}}}}}$ （B6）

 $\begin{array}{l} P\left( {\Delta {T_{{\rm{output}}}} \le \Delta {T_{95\% }}} \right) = P\left( {\Delta {T_{{\rm{ln}}}} \le \Delta {T_{95\% }}} \right) = \\ \;\;\;\;\;\;P\left( {\Delta {T_{{\rm{In}}}} \le \sum\limits_{j = 1}^J {{T_j}} } \right) = {P_{\rm{u}}}\left( {1 - H{P_{\rm{f}}}} \right){P_{\rm{d}}} + \\ \;\;\;\;\;\; \cdots + {P_{\rm{u}}}\left( {1 - H{P_{\rm{f}}}} \right){P_{\rm{d}}} \cdot \\ \;\;\;\;\;\;{\left[ {1 - {P_{\rm{u}}}\left( {1 - H{P_{\rm{f}}}} \right){P_{\rm{d}}}} \right]^{J - 1}} = \\ \;\;\;\;\;\;1 - {\left[ {1 - {P_{\rm{u}}}\left( {1 - H{P_{\rm{f}}}} \right){P_{\rm{d}}}} \right]^J} = \\ \;\;\;\;\;\;1 - {\left( {1 - {P_{\rm{u}}}{P_{\rm{d}}} + {P_{\rm{u}}}H{P_{\rm{f}}}{P_{\rm{d}}}} \right)^j} \end{array}$ （C1）

 $P\left( {\Delta {T_{{\rm{ln}}}} \le \Delta {T_{95\% }}} \right) \approx 1 - {\left( {1 - {P_{\rm{u}}}{P_{\rm{d}}}} \right)^j}$ （C2）

 $P\left( {\Delta {T_{{\rm{Output }}}} \le \Delta {T_{95\% }}} \right) = 0.95$ （C3）

 $J = \frac{{\ln 0.05}}{{\ln \left( {1 - {P_{\rm{u}}}{P_{\rm{d}}}} \right)}}$ （C4）

http://dx.doi.org/10.7527/S1000-6893.2019.23292

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#### 文章信息

LIU Haitao, LI Shaoyang, QIN Dingben, LI Dongxia

Surveillance performance of satellite-based ADS-B system in co-channel interference environment

Acta Aeronautica et Astronautica Sinica, 2019, 40(12): 323292.
http://dx.doi.org/10.7527/S1000-6893.2019.23292