diff --git a/parts/B-mathBackground.tex b/parts/B-mathBackground.tex index 70bf811..fd3e501 100644 --- a/parts/B-mathBackground.tex +++ b/parts/B-mathBackground.tex @@ -117,7 +117,7 @@ $\mathbf{Z} \sim \mathcal{CN} \left( \bm{\mu}, \mathbf{R} \right)$ 的 \gls{pdf} \mathrm{var} ( \hat\theta ) \geqslant \frac{1}{-\mathbb{E}_X \left[ - \dfrac{ + \frac{ \partial^2 \ln f_X \left( x; \theta \right) }{ \partial \theta^2 diff --git a/parts/D-singleArrayCrb.tex b/parts/D-singleArrayCrb.tex index 20aa3c1..f47a75a 100644 --- a/parts/D-singleArrayCrb.tex +++ b/parts/D-singleArrayCrb.tex @@ -25,15 +25,15 @@ \begin{equation} \label{eq:wirtinger_derivative} \begin{cases} - \dfrac{\partial}{\partial z} = \dfrac{1}{2}\left(\dfrac{\partial}{\partial x} - j\dfrac{\partial}{\partial y} \right) + \frac{\partial}{\partial z} = \frac{1}{2}\left(\frac{\partial}{\partial x} - j\frac{\partial}{\partial y} \right) \\ - \dfrac{\partial}{\partial z^*} = \dfrac{1}{2}\left(\dfrac{\partial}{\partial x} + j\dfrac{\partial}{\partial y} \right) + \frac{\partial}{\partial z^*} = \frac{1}{2}\left(\frac{\partial}{\partial x} + j\frac{\partial}{\partial y} \right) \end{cases} \quad\Rightarrow\quad \begin{cases} - \dfrac{\partial}{\partial x} = \dfrac{\partial}{\partial z} + \dfrac{\partial}{\partial z^*} + \frac{\partial}{\partial x} = \frac{\partial}{\partial z} + \frac{\partial}{\partial z^*} \\ - \dfrac{\partial}{\partial y} = j\left(\dfrac{\partial}{\partial z} - \dfrac{\partial}{\partial z^*} \right) + \frac{\partial}{\partial y} = j\left(\frac{\partial}{\partial z} - \frac{\partial}{\partial z^*} \right) \end{cases}, \end{equation} % @@ -41,13 +41,13 @@ \begin{subequations} \begin{equation} \frac{\partial z^*}{\partial z} - = \dfrac{1}{2}\left(\dfrac{\partial}{\partial x} - j\dfrac{\partial}{\partial y} \right) (x - jy) + = \frac{1}{2}\left(\frac{\partial}{\partial x} - j\frac{\partial}{\partial y} \right) (x - jy) = \frac{1}{2}\left(\frac{\partial x}{\partial x} - j\frac{\partial (-jy)}{\partial y} \right) = 0, \end{equation} % \begin{equation} \frac{\partial z}{\partial z^*} - = \dfrac{1}{2}\left(\dfrac{\partial}{\partial x} + j\dfrac{\partial}{\partial y} \right) (x + jy) + = \frac{1}{2}\left(\frac{\partial}{\partial x} + j\frac{\partial}{\partial y} \right) (x + jy) = \frac{1}{2}\left(\frac{\partial x}{\partial x} + j\frac{\partial (jy)}{\partial y} \right) = 0, \end{equation} \end{subequations} @@ -110,11 +110,11 @@ \begin{equation} \frac{\partial\ln \mathcal{L}}{\partial \bm{\theta}} = \begin{bmatrix} - \dfrac{\partial}{\partial \theta_1} + \frac{\partial}{\partial \theta_1} \\ \vdots \\ - \dfrac{\partial}{\partial \theta_K} + \frac{\partial}{\partial \theta_K} \end{bmatrix} \ln \mathcal{L} = \frac{2}{\sigma} \sum_{t=1}^T \Re\left[ \mathbf{X}^\Herm _t \mathbf{D}^\Herm \mathbf{n}_t \right]. \end{equation} @@ -394,9 +394,9 @@ % \begin{align} \mathrm{CRB}^{-1}(\bm{\theta}) &= \mathcal{I}_{\theta\theta} - \sum_{t=1}^T \Re\{\bm{\Delta}_t^\Herm \mathbf{G}\bm{\Delta}_t\} \nonumber - = \dfrac{2}{\sigma} \sum_{t=1}^T \Re\left\{ \mathbf{X}_t^\Herm \mathbf{D}^\Herm \mathbf{D} \mathbf{X}_t - \mathbf{X}_t^\Herm \mathbf{D}^\Herm \mathbf{A}(\mathbf{A}^\Herm \mathbf{A})^{-1}\mathbf{A}^\Herm \mathbf{D} \mathbf{X}_t \right\} \nonumber + = \frac{2}{\sigma} \sum_{t=1}^T \Re\left\{ \mathbf{X}_t^\Herm \mathbf{D}^\Herm \mathbf{D} \mathbf{X}_t - \mathbf{X}_t^\Herm \mathbf{D}^\Herm \mathbf{A}(\mathbf{A}^\Herm \mathbf{A})^{-1}\mathbf{A}^\Herm \mathbf{D} \mathbf{X}_t \right\} \nonumber \\ - &= \dfrac{2}{\sigma} \sum_{t=1}^T \Re\left\{ \mathbf{X}_t^\Herm \mathbf{D}^\Herm \left[ \mathbf{I} - \mathbf{A}(\mathbf{A}^\Herm \mathbf{A})^{-1}\mathbf{A}^\Herm \right] \mathbf{D} \mathbf{X}_t \right\}. + &= \frac{2}{\sigma} \sum_{t=1}^T \Re\left\{ \mathbf{X}_t^\Herm \mathbf{D}^\Herm \left[ \mathbf{I} - \mathbf{A}(\mathbf{A}^\Herm \mathbf{A})^{-1}\mathbf{A}^\Herm \right] \mathbf{D} \mathbf{X}_t \right\}. \end{align} 参考 \cite{stoicaMusicMaximumLikelihood1989} 进一步定义阵列导向矩阵的列空间的投影矩阵 $\mathcal{P}_{\mathbf{A}} = \mathbf{A}(\mathbf{A}^\Herm \mathbf{A})^{-1}\mathbf{A}^\Herm$，及其正交补空间的投影矩阵 $\mathcal{P}_{\mathbf{A}}^\perp = \mathbf{I} -\mathcal{P}_{\mathbf{A}}$，投影矩阵下的等效信息矩阵 $\mathcal{H} = \mathbf{D}^\Herm \mathcal{P}_{\mathbf{A}}^\perp \mathbf{D}$，确定信号的经验协方差 $\mathbf R_{x,\mathrm{det}} = \dfrac{1}{T} \displaystyle\sum_{t=1}^T \mathbf{x}_t \mathbf{x}_t^\Herm$，则关于角度的 \gls{efim} 可以表示为 diff --git a/parts/E-multiSatStaticCrb.tex b/parts/E-multiSatStaticCrb.tex index 3726783..4e4ebf0 100644 --- a/parts/E-multiSatStaticCrb.tex +++ b/parts/E-multiSatStaticCrb.tex @@ -193,7 +193,7 @@ $\zeta_\ell\in\mathbb C$ 为第 $\ell$ 条链路的未知复干扰系数，$\sig \frac{\partial \nu_\ell}{\partial \boldsymbol\eta^\Tran} = \begin{bmatrix} - \dfrac{\partial \Delta\nu_\ell(\boldsymbol\xi)}{\partial \boldsymbol\xi^\Tran} + \frac{\partial \Delta\nu_\ell(\boldsymbol\xi)}{\partial \boldsymbol\xi^\Tran} & 1 \end{bmatrix} \in\mathbb R^{1\times3}, @@ -242,15 +242,15 @@ $\zeta_\ell\in\mathbb C$ 为第 $\ell$ 条链路的未知复干扰系数，$\sig \frac{\partial \boldsymbol\xi_\ell^{\mathrm{loc}}}{\partial \boldsymbol\xi^\Tran} = \begin{bmatrix} - \dfrac{ - u_{\ell,1}^{\mathrm{loc}}\,\dfrac{\partial u_{\ell,2}^{\mathrm{loc}}}{\partial \boldsymbol\xi^\Tran} + \frac{ + u_{\ell,1}^{\mathrm{loc}}\,\frac{\partial u_{\ell,2}^{\mathrm{loc}}}{\partial \boldsymbol\xi^\Tran} - - u_{\ell,2}^{\mathrm{loc}}\,\dfrac{\partial u_{\ell,1}^{\mathrm{loc}}}{\partial \boldsymbol\xi^\Tran} + u_{\ell,2}^{\mathrm{loc}}\,\frac{\partial u_{\ell,1}^{\mathrm{loc}}}{\partial \boldsymbol\xi^\Tran} }{ \big(u_{\ell,1}^{\mathrm{loc}}\big)^2 + \big(u_{\ell,2}^{\mathrm{loc}}\big)^2 } \\[1.2em] - \dfrac{\partial u_{\ell,3}^{\mathrm{loc}}/\partial \boldsymbol\xi^\Tran}{\sqrt{1-\big(u_{\ell,3}^{\mathrm{loc}}\big)^2}} + \frac{\partial u_{\ell,3}^{\mathrm{loc}}/\partial \boldsymbol\xi^\Tran}{\sqrt{1-\big(u_{\ell,3}^{\mathrm{loc}}\big)^2}} \end{bmatrix} \in\mathbb R^{2\times2}, \label{eq:A6_local_angle_chain} @@ -264,9 +264,9 @@ $\zeta_\ell\in\mathbb C$ 为第 $\ell$ 条链路的未知复干扰系数，$\sig \frac{\partial \mathbf a_\ell}{\partial (\boldsymbol\xi_\ell^{\mathrm{loc}})^\Tran} = \begin{bmatrix} - \dfrac{\partial \mathbf a_\ell}{\partial \alpha_\ell^{\mathrm{loc}}} + \frac{\partial \mathbf a_\ell}{\partial \alpha_\ell^{\mathrm{loc}}} & - \dfrac{\partial \mathbf a_\ell}{\partial \beta_\ell^{\mathrm{loc}}} + \frac{\partial \mathbf a_\ell}{\partial \beta_\ell^{\mathrm{loc}}} \end{bmatrix} \in\mathbb C^{M_\ell\times2}. \label{eq:A6_a_local_jac} @@ -325,11 +325,11 @@ $\zeta_\ell\in\mathbb C$ 为第 $\ell$ 条链路的未知复干扰系数，$\sig \frac{\partial \mathbf g_\ell}{\partial \boldsymbol\eta^\Tran} = \begin{bmatrix} - \mathbf q_\ell\otimes\dfrac{\partial \mathbf a_\ell}{\partial \boldsymbol\xi^\Tran} + \mathbf q_\ell\otimes\frac{\partial \mathbf a_\ell}{\partial \boldsymbol\xi^\Tran} + - \dfrac{\partial \mathbf q_\ell}{\partial \boldsymbol\xi^\Tran}\otimes\mathbf a_\ell + \frac{\partial \mathbf q_\ell}{\partial \boldsymbol\xi^\Tran}\otimes\mathbf a_\ell & - \dfrac{\partial \mathbf q_\ell}{\partial \nu_\ell}\otimes\mathbf a_\ell + \frac{\partial \mathbf q_\ell}{\partial \nu_\ell}\otimes\mathbf a_\ell \end{bmatrix} \in \mathbb C^{M_\ell T \times 3}, \label{eq:A6_Dl} @@ -403,9 +403,9 @@ $\zeta_\ell\in\mathbb C$ 为第 $\ell$ 条链路的未知复干扰系数，$\sig \mathcal I_{\eta\psi} = \begin{bmatrix} - \dfrac{2}{\sigma_0}\Re\left\{\mathbf J_{\eta,0}^{\Herm}\mathbf J_{\psi,0}\right\} + \frac{2}{\sigma_0}\Re\left\{\mathbf J_{\eta,0}^{\Herm}\mathbf J_{\psi,0}\right\} & \cdots & - \dfrac{2}{\sigma_{L-1}}\Re\left\{\mathbf J_{\eta,L-1}^{\Herm}\mathbf J_{\psi,L-1}\right\} + \frac{2}{\sigma_{L-1}}\Re\left\{\mathbf J_{\eta,L-1}^{\Herm}\mathbf J_{\psi,L-1}\right\} \end{bmatrix}, \label{eq:A6_Itb} \end{equation} diff --git a/parts/sec5_multiFrameCp.tex b/parts/sec5_multiFrameCp.tex index af28eed..a5b3b2c 100644 --- a/parts/sec5_multiFrameCp.tex +++ b/parts/sec5_multiFrameCp.tex @@ -282,7 +282,7 @@ $\boldsymbol\xi$ 的增益主要来自空间视线与阵列流形几何多样性 该 comb 是全局 profile likelihood 的周期窗口效应，与 \gls{efim} 局部曲率并不矛盾——\gls{efim} 只刻画某一个 tooth 内的二阶曲率，并不保证全局唯一 tooth。 相邻 tooth 是否容易区分由两层因素共同决定：单帧 pilot block 的长度 $T$（控制块内 Doppler 相干带宽）与非周期 subset 的窗口形状（控制 comb 旁瓣压制）。 -当 $T\ll\Delta\tau_{\mathrm{f}}$ 时，相邻 tooth 的块内相干损失约为 $\frac{\pi^2}{3}(T/\Delta\tau_{\mathrm{f}})^2$，因此增大 $T$ 可增强相邻 tooth 的分离度。 +当 $T\ll\Delta\tau_{\mathrm{f}}$ 时，相邻 tooth 的块内相干损失约为 $\dfrac{\pi^2}{3}(T/\Delta\tau_{\mathrm{f}})^2$，因此增大 $T$ 可增强相邻 tooth 的分离度。 非周期 subset 打破 comb 的严格周期性，使本来等高的假 tooth 退化为受压制的旁瓣、真峰凸显，因而更适合 tooth acquisition；完整周期数据在 tooth 选对之后由于保留更多相干能量，更适合同 tooth 内 refine。 若定义 tooth 可分离性为不同 $k$ 下 profile optimum 之间的目标差 %