第四章：波束稳健性分析

4.1 最佳波束形成器稳健性影响因素

{{w}_{opt}}=\alpha R_{x}^{-1}{{\bar{p}}_{s}}

4.2 导向向量失配对波束性能的影响

{{G}_{MVDR}}({{\bar{p}}_{s}}:{{\tilde{p}}_{s}})=\frac{{{\left| \bar{p}_{s}^{H}R_{x}^{-1}{{{\tilde{p}}}_{s}} \right|}^{2}}}{\bar{p}_{s}^{H}R_{x}^{-1}{{\rho }_{n}}R_{x}^{-1}{{{\bar{p}}}_{s}}}

{{G}_{MVDR}}=\frac{\bar{p}_{s}^{H}\rho _{n}^{-1}{{{\tilde{p}}}_{s}}{{\cos }^{2}}({{{\bar{p}}}_{s}},{{{\tilde{p}}}_{s}},\rho _{n}^{-1})}{1+{2(\sigma _{s}^{2}/\sigma _{n}^{2})\tilde{p}_{s}^{H}\rho _{n}^{-1}{{{\tilde{p}}}_{s}}+{{[(\sigma _{s}^{2}/\sigma _{n}^{2})\tilde{p}_{s}^{H}\rho _{n}^{-1}{{{\tilde{p}}}_{s}}]}^{2}}}{{\sin }^{2}}({{{\bar{p}}}_{s}},{{{\tilde{p}}}_{s}},\rho _{n}^{-1})}

4.3 协方差矩阵失配对波束性能的影响

4.3.1 样本协方差矩阵求逆波束形成

{{w}_{MVDR}}=\alpha {{R}^{-1}}{{\bar{p}}_{s}}

\alpha \text{=}{{(\bar{p}_{s}^{H}{{R}^{-1}}{{\bar{p}}_{s}})}^{-1}}

\hat{R}\text{=}\frac{1}{N}\sum\limits_{n=1}^{N}{[x(n){{x}^{H}}(n)]}

{{w}_{MVDR}}=\alpha {{\hat{R}}^{-1}}{{\bar{p}}_{s}}

O({{M}^{3}})

4.3.2 样本协方差矩阵求逆法波束性能

{{\rho }_{0}}\text{=}{{\left. \frac{SINR({{{\hat{w}}}_{SMI}})}{SIN{{R}_{opt}}} \right|}_{\beta =0}}

E({{\rho }_{0}})=\frac{N-M+2}{N+1}

N\ge 2M-3\approx 2M

E(SLL)\text{=}\frac{1}{N+1}

N\ge SIN{{R}_{opt}}\centerdot (M-1)\gg M

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