基于OFDM的通感一体化信号的定位CRLB推导

标签：采样 OFDM 载波 CRLB 通感信号无人机

本文的CRLB推导结果来源于文章Cooperative Trajectory Planning and Resource Allocation for UAV-Enabled Integrated Sensing and Communication Systems，引用格式：

{Yu Pan, Ruoguang Li*, Xinyu Da, Hang Hu, Miao Zhang, Dong Zhai, Kanapathippillai Cuman, and Octavia A. Dobre, "Cooperative Trajectory Planning and Resource Allocation for UAV-Enabled Integrated Sensing and Communication Systems." IEEE Transactions on Vehicular Technology, vol.73, no. 5, pp. 6502-6516, May 2024. doi: 10.1109/TVT.2023.3337106. }

首先简单介绍一下单个OFDM符号传输的原理：

假设原连续信号为N个载波下信号的叠加，每个载波传输的码元为 $X_k$ ，则该信号为：

${s_l}(t) = \sum\limits_{k = 0}^{N - 1} {{X_k}{e^{\frac{{j2\pi kt}}{T}}}}$

在每个时刻 ${t_n} = \frac{{nT}}{N}$ 抽样，则此时得到 $N$ 个离散的时间信号：

${s_n} = {s_l}(\frac{{nT}}{N}) = \sum\limits_{k = 0}^{N - 1} {{X_k}{e^{\frac{{j2\pi kn}}{N}}}}$

这意味着我们可以直接通过生成 $s_n$ 得到原连续信号的采样信号，这必然可以在一定条件下恢复出原信号。在解调端，为恢复传输码元，有

${X_k} = \frac{1}{N}\sum\limits_{n = 0}^{N - 1} {{s_n}{e^{ - \frac{{j2\pi kn}}{N}}}}$

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接下来考虑如下场景， $K$ 架无人机利用OFDM信号基于时延对目标进行测距，并通过多点法利用多架无人机的测距结果定位目标，第 $k$ 架无人机的发射信号为

$S_k^i(t) = {e^{{\rm{j}}2\pi {f_{c,k}}t}}\sum\limits_{n = 0}^{{N_k} - 1} {\sum\limits_{m = 0}^{M - 1} {a_{kn}^i} } C_{k,nm}^i{e^{{\rm{j}}2\pi n\Delta f(t - mT)}}{\mathop{\rm rect}\nolimits} \left[ {\frac{{(t - mT)}}{T}} \right]$ （1）

该式表示第 $k$ 架无人机的发射的时域信号，在其每个符号间隔 $T$ 内采样 $N_k$ 次（与载波数相等），则第 $m$ 个符号间隔内的第 $l$ 个采样信号为：（忽略载波）

（OFDM调制中1个连续时间码元由 $N_k$ 个载波携带的码元生成，生成 $N_k$ 个采样点）

$\tilde S_k^i(l) = S_k^i(mT + \frac{{lT}}{{{N_k}}}) = \sum\limits_{n = 0}^{{N_k} - 1} {a_{kn}^iC_{k,nm}^i{e^{\frac{{{\rm{j}}2\pi nl}}{{{N_k}}}}}}$ （2）

其中 $l$ =1,…, $N_k$ 。也就是说如果需要传输 $M*N_k$ (这里的 $N_k$ 表示载波数， $M$ 表示传输的符号数)，就需要在串并转换后对每一列的 $N_k$ 个符号进行FFT变换，得到 $N_k$ 个离散时间采样值，同样的操作需进行 $M$ 次。

由发射信号我们可以得到第 $k$ 架无人机的回波信号为：

$r_k^i(t) = \sum\limits_{n = 0}^{{N_{\rm{c}}} - 1} {\sum\limits_{m = 0}^{M - 1} {\zeta _k^i} } a_{kn}^iC_{k,nm}^i{e^{{\rm{j}}2\pi n\Delta f\left( {t - mT - \tau _k^i} \right)}}{e^{{\rm{j}}2\pi {f_{{\rm{c,k}}}}\left( {t - \tau _k^i} \right)}}{e^{{\rm{j}}2\pi f_{D,k}^it}} + {\mathop{\rm rect}\nolimits} \left[ {\frac{{t - mT - \tau _k^i}}{T}} \right] + {\phi _k}(t)$ （3）

其中 $\tau _k^i = 2d_k^i/{c_0},{\rm{ }}f_{D,k}^i = 2v_k^i{f_c}/{c_0}$ （4）

对该接收信号进行下变频

$y_k^i(t) = {e^{{\rm{ - j}}2\pi {f_{{\rm{c,k}}}}\tau _k^i}}\sum\limits_{n = 0}^{{N_{\rm{c}}} - 1} {\sum\limits_{m = 0}^{M - 1} {\zeta _k^i} } a_{kn}^iC_{k,nm}^i{e^{{\rm{j}}2\pi n\Delta f\left( {t - mT - \tau _k^i} \right)}}{e^{{\rm{j}}2\pi f_{D,k}^it}}{\mathop{\rm rect}\nolimits} \left[ {\frac{{t - mT - \tau _k^i}}{T}} \right] + n(t)$ （5）

对其取样，则第 $m$ 个符号间隔内的采样信号为（其中 ${T_s} = \frac{T}{{{N_k}}}$ ）：

$\begin{array}{l} \tilde y_k^i(l,m) = \tilde y_k^i(mT + {\frac{{lT}}{{{N_k}}}}) = {e^{{\rm{ - j}}2\pi {f_{{\rm{c,k}}}}\tau _k^i}}\sum\limits_{n = 0}^{{N_{\rm{c}}} - 1} {\zeta _k^ia_{kn}^iC_{k,nm}^i{e^{{\rm{j}}2\pi n\Delta f\left( {\frac{{lT}}{{{N_k}}} - \tau _k^i} \right)}}{e^{{\rm{j}}2\pi f_{D,k}^i\left( {mT + \frac{{lT}}{{{N_k}}}} \right)}} + n(t)} \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} = \sum\limits_{n = 0}^{{N_{\rm{c}}} - 1} {\zeta _k^ia_{kn}^iC_{k,nm}^i{e^{{\rm{ - j}}2\pi {f_{{\rm{c,k}}}}\tau _k^i}}{e^{{\rm{ - j}}2\pi n\Delta f\tau _k^i}}{e^{{\rm{j}}2\pi f_{D,k}^i\left( {mT + \frac{{lT}}{{{N_k}}}} \right)}}{e^{\frac{{{\rm{j}}2\pi nl}}{{{N_k}}}}} + n(t)} \end{array}$

（6）

然后对采样得到的 $N_k$ 个时间序列进行FFT（OFDM解调），即

$\begin{array}{l} {\bf{Y}}_k^i(n,m) = \frac{1}{{{N_k}}}\sum\limits_{l = 0}^{{N_k} - 1} {\tilde y_k^i(l,m){e^{ - \frac{{{\rm{j}}2\pi nl}}{{{N_k}}}}}} \\ = \sum\limits_{l = 0}^{{N_k} - 1} {\sum\limits_{n = 0}^{{N_{\rm{c}}} - 1} {\zeta _k^ia_{kn}^iC_{k,nm}^i{e^{{\rm{ - j}}2\pi {f_{{\rm{c,k}}}}\tau _k^i}}{e^{{\rm{ - j}}2\pi n\Delta f\tau _k^i}}{e^{{\rm{j}}2\pi f_{D,k}^i\left( {mT + \frac{{lT}}{{{N_k}}}} \right)}}{e^{\frac{{{\rm{j}}2\pi nl}}{{{N_k}}}}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} } {e^{ - \frac{{{\rm{j}}2\pi nl}}{{{N_k}}}}}} \\ = \zeta _k^ia_{kn}^iC_{k,nm}^i{e^{{\rm{ - j}}2\pi {f_{c,kn}}\tau _k^i}}{e^{{\rm{ - j}}2\pi n\Delta f\tau _k^i}}{e^{{\rm{j}}2\pi f_{D,k}^imT}} + {\bf{N}}(n,m) \end{array}$ （7）

（忽略了 $\frac{{lT}}{{{N_k}}}$ 的频偏）

由于每个无人机对自己的发射矩阵 $C_{k,nm}^i$ 已知，所以可以写成

${\bf{Y}}_k^i(n,m) = a_{kn}^i\zeta _k^i{e^{{\rm{ - j}}2\pi {f_{{\rm{c,kn}}}}\tau _k^i}}{e^{{\rm{j}}2\pi f_{D,k}^imT}} + {\bf{N}}(n,m),$ （8）

由此写出似然函数

${\rm{ - In}}f({{\bf{Y}}^i}\left| {{\theta ^i}} \right.) = - \sum\limits_{k \in {\cal K}} {\sum\limits_{n \in {\cal N}_k^i} {\sum\limits_{m = 0}^{M - 1} {{\rm{In}}\frac{1}{{\sqrt {2\pi \sigma _{r,k}^2} }}} } } - \frac{1}{{2\sigma _{r,k}^2}}{\left| {{\bf{Y}}_k^i(n,m) - a_{kn}^i\zeta _k^i{e^{{\rm{ - j}}2\pi {f_{{\rm{c,kn}}}}\tau _k^i}}{e^{{\rm{j}}2\pi f_{D,k}^imT}}} \right|^2}$

$= - \sum\limits_{k \in {\cal K}} {\sum\limits_{n \in {\cal N}_k^i} {\sum\limits_{m = 0}^{M - 1} {{\rm{In}}\frac{1}{{\sqrt {2\pi \sigma _{r,k}^2} }}} } } - \frac{{{{\left| {{\bf{Y}}_k^i(n,m)} \right|}^2} + {{\left| {a_{kn}^i\zeta _k^i{e^{{\rm{ - j}}2\pi {f_{{\rm{c,kn}}}}\tau _k^i}}{e^{{\rm{j}}2\pi f_{D,k}^imT}}} \right|}^2}}}{{2\sigma _{r,k}^2}}+ \frac{{{\mathop{\rm Re}\nolimits} \left[ {{\bf{Y}}_k^i{{(n,m)}^*}a_{kn}^i\zeta _k^i{e^{{\rm{ - j}}2\pi {f_{{\rm{c,kn}}}}\tau _k^i}}{e^{{\rm{j}}2\pi f_{D,k}^imT}}} \right]}}{{\sigma _{r,k}^2}}$

${\kern 1pt} = - \sum\limits_{k \in {\cal K}} {\sum\limits_{n \in {\cal N}_k^i} {\sum\limits_{m = 0}^{M - 1} {{\rm{constant}}} } } + \frac{{{\mathop{\rm Re}\nolimits} \left[ {{\bf{Y}}_k^i{{(n,m)}^*}a_{kn}^i\zeta _k^i{e^{{\rm{ - j}}2\pi {f_{{\rm{c,kn}}}}\tau _k^i}}{e^{{\rm{j}}2\pi f_{D,k}^imT}}} \right]}}{{\sigma _{r,k}^2}}$ （9）

常数项求导时可约去，即令

$\begin{array}{l} {\rm{ - In}}f({{\bf{Y}}^i}\left| {{\theta ^i}} \right.) = - \sum\limits_{k \in {\cal K}} {\sum\limits_{n \in {\cal N}_k^i} {\sum\limits_{m = 0}^{M - 1} {\frac{{{\mathop{\rm Re}\nolimits} \left[ {{\bf{Y}}_k^i{{(n,m)}^*}a_{kn}^i\zeta _k^i{e^{{\rm{ - j}}2\pi {f_{{\rm{c,kn}}}}\tau _k^i}}{e^{{\rm{j}}2\pi f_{D,k}^imT}}} \right]}}{{\sigma _{r,k}^2}}} } } \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} = - \sum\limits_{k \in {\cal K}} {\sum\limits_{n \in {\cal N}_k^i} {\sum\limits_{m = 0}^{M - 1} {\frac{{{{\left| {a_{kn}^i\zeta _k^i} \right|}^2}\cos \left[ {{\rm{ - }}2\pi {f_{{\rm{c,kn}}}}\left( {\tau _k^i - \tilde \tau _k^i} \right) + 2\pi mT\left( {f_{D,k}^i - \tilde f_{D,k}^i} \right)} \right]}}{{\sigma _{r,k}^2}}} } } \end{array}$ （10）

则

${{\bf{J}}_z} = \left[ \begin{array}{l} {J_{{\tau _1}{\tau _1}}}{\rm{ }}{\kern 12pt}{J_{{\tau _1}{\tau _2}}}{\rm{ }} {\kern 12pt} \ldots {\rm{ }}{\kern 9pt}{J_{{\tau _1}{f_{D,K}}}}\\ {J_{{\tau _2}{\tau _1}}}{\rm{ }}{\kern 12pt}{J_{{\tau _2}{\tau _2}}}{\rm{ }} {\kern 12pt}\ldots {\rm{ }}{\kern 9pt}{J_{{\tau _2}{f_{D,K}}}}\\ {\rm{ }} {\kern 12pt}\vdots {\rm{ }} {\kern 30pt}\vdots {\rm{ }} {\kern 25pt}\ddots {\rm{ }} {\kern 15pt}\vdots \\ {J_{{f_{D,K}}{\tau _1}}}{\rm{ }}{J_{{f_{D,K}}{\tau _2}}}{\rm{ }} \cdots {\rm{ }}{J_{{f_{D,K}}{f_{D,K}}}} \end{array} \right]$ （11）

$- E\left[ {\frac{{{\partial ^2}{\rm{In}}f({{\bf{Y}}^i}\left| {{\theta ^i}} \right.)}}{{\partial \tau _k^{{i^2}}}}} \right] = {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} E\left\{ {\sum\limits_{n \in {\cal N}_k^i} {\sum\limits_{m = 0}^{M - 1} {\frac{{{{\left| {a_{kn}^i\zeta _k^i2\pi {f_{{\rm{c,kn}}}}} \right|}^2}\cos \left[ {{\rm{ - }}2\pi {f_{{\rm{c,kn}}}}\left( {\tau _k^i - \tilde \tau _k^i} \right) + 2\pi mT\left( {f_{D,k}^i - \tilde f_{D,k}^i} \right)} \right]}}{{\sigma _{r,k}^2}}} } } \right\}$

$= E\left\{ {\sum\limits_{n \in {\cal N}_k^i} {\sum\limits_{m = 0}^{M - 1} {\frac{{{{\left| {a_k^i\zeta _k^i2\pi {f_{{\rm{c,kn}}}}} \right|}^2}{\mathop{\rm Re}\nolimits} \left( {{e^{{\rm{j}}2\pi {f_{{\rm{c,kn}}}}\tilde \tau _k^i}}{e^{{\rm{ - j}}2\pi \tilde f_{D,k}^imT}}*{e^{{\rm{ - j}}2\pi {f_{{\rm{c,kn}}}}\tau _k^i}}{e^{{\rm{j}}2\pi f_{D,k}^imT}}} \right)}}{{\sigma _{r,k}^2}}} } } \right\}$

${\kern 1pt} = \sum\limits_{n \in {\cal N}_k^i} {\sum\limits_{m = 0}^{M - 1} {\frac{{{{\left| {a_{kn}^i\zeta _k^i2\pi {f_{{\rm{c,kn}}}}} \right|}^2}}}{{\sigma _{r,k}^2}}} } = \frac{{Mc_0^2{\sigma _{{\rm{RCS}}}}}}{{16\pi \sigma _{r,k}^2}}\frac{{P_k^i}}{{d{{_k^i}^4}}}$ （12）

同理可得

$- E\left[ {\frac{{{\partial ^2}{\rm{In}}f({{\bf{Y}}^i}\left| {{\theta ^i}} \right.)}}{{\partial f_{D,k}^{{i^2}}}}} \right] = \sum\limits_{n \in {\cal N}_k^i} {\sum\limits_{m = 0}^{M - 1} {\frac{{{{\left| {a_{kn}^i\zeta _k^i2\pi mT} \right|}^2}}}{{\sigma _{r,k}^2}}} } \approx \frac{{M(M - 1)(2M - 1){T^2}c_0^2{\sigma _{{\rm{RCS}}}}}}{{96\pi \sigma _{r,k}^2}}\frac{{P_k^i}}{{d{{_k^i}^4}{f_{c,k}}^2}}$ （13）

$- E\left[ {\frac{{{\partial ^2}{\rm{In}}f({{\bf{Y}}^i}\left| {{\theta ^i}} \right.)}}{{\partial f_{D,k}^i\partial \tau _k^i}}} \right] = \sum\limits_{n \in {\cal N}_k^i} {\sum\limits_{m = 0}^{M - 1} {\frac{{{{\left| {a_{kn}^i\zeta _k^i2\pi } \right|}^2}{f_{{\rm{c,kn}}}}mT}}{{\sigma _{r,k}^2}}} } = \frac{{M(M - 1)Tc_0^2{\sigma _{{\rm{RCS}}}}}}{{32\pi \sigma _{r,k}^2}}\frac{{P_k^i}}{{d{{_k^i}^4}{f_{c,k}}}}$ （14）

因此由文献[Posterior_Cramer-Rao_bounds_for_discrete-time_nonlinear_filtering]有

${{\bf{J}}_e}\left( {{{\bf{\theta }}^{\bf{i}}}_1} \right) = {\bf{J}}\left( {{\bf{\theta }}_1^i,{{\bf{\theta }}^{\bf{i}}}_1} \right) - {\bf{J}}\left( {{\bf{\theta }}_1^i,{\bf{\theta }}_2^i} \right){\bf{J}}{\left( {{\bf{\theta }}_2^i,{\bf{\theta }}_2^i} \right)^{ - 1}}{\bf{J}}{\left( {{\bf{\theta }}_1^i,{\bf{\theta }}_2^i} \right)^{\rm{T}}}$ （15）

${{\bf{J}}_e}{\left( {{{\bf{\theta }}^{\bf{i}}}_1} \right)_{k,k}} = \frac{{M(M + 1)c_0^2{\sigma _{{\rm{RCS}}}}}}{{32\pi (2M - 1)}}\frac{{P_k^i}}{{{{\left( {d_k^i} \right)}^4}\sigma _{r,k}^2}}$ （16）

${\rm{CRLB}}\left( {\tau _k^i} \right) = \frac{{32\pi (2M - 1)}}{{M(M + 1)c_0^2{\sigma _{{\rm{RCS}}}}}}\frac{{{{\left( {d_k^i} \right)}^4}\sigma _{r,k}^2}}{{P_k^i}}$ （17）

${{\bf{J}}_e}{\left( {{{\bf{\theta }}^{\bf{i}}}_2} \right)_{k,k}} = \frac{{{T^2}M({M^2} - 1)c_0^2{\sigma _{{\rm{RCS}}}}}}{{192\pi }}\frac{{P_k^i}}{{{f_{c,k}}^2{{\left( {d_k^i} \right)}^4}\sigma _{r,k}^2}}$ （18）

${\rm{CRLB}}\left( {f_{D,k}^i} \right) = \frac{{192\pi }}{{{T^2}M({M^2} - 1)c_0^2{\sigma _{{\rm{RCS}}}}}}\frac{{{f_{c,k}}^2{{\left( {d_k^i} \right)}^4}\sigma _{r,k}^2}}{{P_k^i}}$ （19）

此时有 $\tau _k^i = \frac{2}{{{c_0}}}d_k^i = \frac{2}{{{c_0}}}{\left[ {{{\left( {x_0^i - x_k^i} \right)}^2} + {{\left( {y_0^i - y_k^i} \right)}^2}} \right]^{1/2}}$ ，将 ${{\bf{\theta }}^{\bf{i}}}_1$ 与 ${\omega ^i}$ 的关系线性化，即

${{\bf{\theta }}^{\bf{i}}}_1 = {f_1}({\omega ^i}) = {f_1}({\bar \omega ^i}) + {\Psi _i}({\omega ^i} - {\bar \omega ^i})$ （20）

其中 ${{\bf{\Psi }}_i}$ 为Jacbian矩阵

${{\bf{\Psi }}_i} = \frac{{\partial {{\bf{\theta }}^i}_1}}{{\partial {{\bf{\omega }}^i}}} = \frac{2}{{{c_0}}}\left[ {\begin{array}{*{20}{l}} {\frac{{x_0^i - x_1^i}}{{d_1^i}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \frac{{y_0^i - y_1^i}}{{d_1^i}}}\\ \begin{array}{l} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \vdots {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \vdots \\ \frac{{x_0^i - x_K^i}}{{d_K^i}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \frac{{y_0^i - y_K^i}}{{d_K^i}} \end{array} \end{array}} \right]$ （21）

则由求导链式法则可以得到 ${\bf{J}}\left( {{{\bf{\omega }}^i}} \right) \approx {\bf{Q}}_i^{\rm{T}}{{\bf{J}}_e}\left( {{{\bf{\theta }}^i}_1} \right){{\bf{Q}}_i}$ ，已知

则

${\bf{J}}\left( {{{\bf{\omega }}^i}} \right) = \frac{{M(M + 1){\sigma _{{\rm{RCS}}}}}}{{8\pi (2M - 1)}}\left[ {\begin{array}{*{20}{l}} {\sum\limits_{k \in {\cal K}} {\frac{{P_k^i{{\left( {x_0^i - x_k^i} \right)}^2}}}{{d{{_k^i}^6}\sigma _{r,k}^2}}} }&{\sum\limits_{k \in {\cal K}} {\frac{{P_k^i\left( {x_0^i - x_k^i} \right)\left( {y_0^i - y_k^i} \right)}}{{d{{_k^i}^6}\sigma _{r,k}^2}}} }\\ {\sum\limits_{k \in {\cal K}} {\frac{{P_k^i\left( {x_0^i - x_k^i} \right)\left( {y_0^i - y_k^i} \right)}}{{d{{_k^i}^6}\sigma _{r,k}^2}}} }&{\sum\limits_{k \in {\cal K}} {\frac{{P_k^i{{\left( {y_0^i - y_k^i} \right)}^2}}}{{d{{_k^i}^6}\sigma _{r,k}^2}}} } \end{array}} \right]$ （23）

即

${\bf{J}}\left( {{{\bf{\omega }}^i}} \right) = {A_1}\sum\limits_{k \in {\cal K}} {\left[ {\begin{array}{*{20}{l}} {\frac{{P_k^i{{\left( {x_0^i - x_k^i} \right)}^2}}}{{d{{_k^i}^6}\sigma _{r,k}^2}}}&{\frac{{P_k^i\left( {x_0^i - x_k^i} \right)\left( {y_0^i - y_k^i} \right)}}{{d{{_k^i}^6}\sigma _{r,k}^2}}}\\ {\frac{{P_k^i\left( {x_0^i - x_k^i} \right)\left( {y_0^i - y_k^i} \right)}}{{d{{_k^i}^6}\sigma _{r,k}^2}}}&{\frac{{P_k^i{{\left( {y_0^i - y_k^i} \right)}^2}}}{{d{{_k^i}^6}\sigma _{r,k}^2}}} \end{array}} \right]}$ （24）

其中 ${A_1} = \frac{{M(M + 1){\sigma _{{\rm{RCS}}}}}}{{8\pi (2M - 1)}}$ ，关于目标位置估计的CRLB矩阵为 ${\bf{J}}{\left( {{{\bf{\omega }}^i}} \right)^{ - 1}}$ 。

即：位置估计的克拉美罗下界为

${\left( {\sigma _l^i} \right)^2} \triangleq E\left[ {{{\left( {\hat x_T^i - x_T^i} \right)}^2} + {{\left( {\hat y_T^i - y_T^i} \right)}^2}}\right] \geqslant {\left[ {{\mathbf{J}}{{\left( {{{\mathbf{\omega }}^i}} \right)}^{ - 1}}} \right]_{11}} + {\left[ {{\mathbf{J}}{{\left( {{{\mathbf{\omega }}^i}} \right)}^{ - 1}}} \right]_{22}}$ （25）

再考虑多普勒频移的估计误差，同时又有

$\begin{gathered} f_{D,k}^i = \frac{{2{f_{c,k}}}}{{{c_0}d_k^i}}\left[ {\left( {v_{x,k}^i - v_{x,0}^i} \right)\left( {x_0^i - x_k^i} \right) + \left( {v_{y,k}^i - v_{y,0}^i} \right)\left( {y_0^i - y_k^i} \right)} \right] \hfill \\ = \frac{{2{f_{c,k}}}}{{{c_0}d_k^i}}\left[ {\left( {v_{x,0}^i - v_{x,k}^i} \right)\left( {x_k^i - x_0^i} \right) + \left( {v_{y,0}^i - v_{y,k}^i} \right)\left( {y_k^i - y_0^i} \right)} \right] \hfill \\ \end{gathered}$

将 ${{\mathbf{\theta }}^{\mathbf{i}}}_2$ 与 $v_0^i$ 的关系线性化，即

${{\mathbf{\theta }}^i}_2 = {f_2}(v_0^i) = {f_2}(\bar v_0^i) + {\Phi _i}(v_0^i - \bar v_0^i)$ （26）

其中 ${Q_i}$ 为Jacbian矩阵

${\Phi _i} = \frac{{\partial {{\mathbf{\theta }}^i}_2}}{{\partial v_0^i}} = \frac{2}{{{c_0}}}\left[ {\begin{array}{*{20}{l}} {\frac{{{f_{c,1}}\left( {x_1^i - x_0^i} \right)}}{{d_1^i}}{\kern 44pt} \frac{{{f_{c,1}}\left( {y_1^i - y_0^i} \right)}}{{d_1^i}}} \\ \begin{gathered} {\kern 14pt} \vdots {\kern 94pt} \vdots \hfill \\ \frac{{{f_{c,K}}\left( {x_K^i - x_0^i} \right)}}{{d_K^i}}{\kern 14pt} \frac{{{f_{c,K}}\left( {y_K^i - y_0^i} \right)}}{{d_K^i}} \hfill \\ \end{gathered} \end{array}} \right]$ （27）

则由求导链式法则可以得到 ${\mathbf{J}}\left( {v_0^i} \right) \approx {\Phi _i}^{\text{T}}{{\mathbf{J}}_e}\left( {{{\mathbf{\theta }}^i}_2} \right){\Phi _i}$ ，已知

${{\mathbf{J}}_e}{\left( {{{\mathbf{\theta }}^{\mathbf{i}}}_2} \right)_{k,k}} = \frac{{{T^2}M({M^2} - 1)c_0^2{\sigma _{{\text{RCS}}}}}}{{192\pi }}\frac{{P_k^i}}{{{f_{c,k}}^2{{\left( {d_k^i} \right)}^4}\sigma _{r,k}^2}}$ （28）

则

${\mathbf{J}}\left( {v_0^i} \right) = {A_2}\sum\limits_{k \in \mathcal{K}} {\left[ {\begin{array}{*{20}{l}} {\frac{{P_k^i{{\left( {x_k^i - x_0^i} \right)}^2}}}{{d{{_k^i}^6}\sigma _{r,k}^2}}}&{\frac{{P_k^i\left( {x_0^i - x_k^i} \right)\left( {y_0^i - y_k^i} \right)}}{{d{{_k^i}^6}\sigma _{r,k}^2}}} \\ {\frac{{P_k^i\left( {x_0^i - x_k^i} \right)\left( {y_0^i - y_k^i} \right)}}{{d{{_k^i}^6}\sigma _{r,k}^2}}}&{\frac{{P_k^i{{\left( {y_0^i - y_k^i} \right)}^2}}}{{d{{_k^i}^6}\sigma _{r,k}^2}}} \end{array}} \right]}$ （29）

其中 ${A_2} = \frac{{{T^2}M({M^2} - 1){\sigma _{{\text{RCS}}}}}}{{48\pi }}$ ，关于目标速度估计的CRLB矩阵为 ${\mathbf{J}}{\left( {v_0^i} \right)^{ - 1}}$ 。

即：速度估计的克拉美罗下界为

${\left( {\sigma _v^i} \right)^2} \triangleq E\left[ {{{\left( {\hat v_{x,T}^i - v_{x,T}^i} \right)}^2} + {{\left( {\hat v_{y,T}^i - v_{y,T}^i} \right)}^2}} \right]\geq {\left[ {{\mathbf{J}}{{\left( {v_T^i} \right)}^{ - 1}}} \right]_{11}} + {\left[ {{\mathbf{J}}{{\left( {v_T^i} \right)}^{ - 1}}} \right]_{22}}$

由此即得到多架无人机对目标进行测距后，所进行的目标位置估计和速度估计的克拉美罗界，针对该性能指标可以对通感一体化网络进行进一步优化。

标签：采样,OFDM,载波,CRLB,通感,信号,无人机
From： https://blog.csdn.net/panyu_kgd/article/details/141319656

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