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SCIENCE CHINA Information Sciences, Volume 64 , Issue 9 : 192305(2021) https://doi.org/10.1007/s11432-019-2773-1

Localization deception performance of FDA signals under passive bi-satellite reconnaissance

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  • ReceivedOct 22, 2019
  • AcceptedJan 20, 2020
  • PublishedAug 18, 2021

Abstract


Supplement

Appendix

To prove (17), note that the elements of the Fisher information matrix ${\boldsymbol~I}\left(~\text{~}\!\!\phi\!\!\text{~}~\right)$ for the complex Gaussian scenario PDF in (15) is a standard result given by [32] \begin{equation}{\boldsymbol{I}}{\left( \phi \right)_{ij}} = 2{\rm{Re}}\left\{ {{{\left[ {\frac{{\partial {\boldsymbol s}\left( \phi \right)}}{{\partial {\phi _i}}}} \right]}^{\rm H}}{{\boldsymbol C}^{ - 1}}\left( \phi \right)\left[ {\frac{{\partial {\boldsymbol s}\left( \phi \right)}}{{\partial {\phi _j}}}} \right]} \right\} + {\rm tr}\left[ {{{\boldsymbol C}_\phi }^{ - 1}\frac{{\partial {{\boldsymbol C}_\phi }}}{{\partial {\phi _i}}}{{\boldsymbol C}_\phi }^{ - 1}\frac{{\partial {{\boldsymbol C}_\phi }}}{{\partial {\phi _j}}}} \right], \tag{30}\end{equation} \begin{equation}{\boldsymbol C}\left( \phi \right) = \left[ {\begin{array}{*{20}{c}} {\sigma _1}&0 \\ 0&{\sigma _2} \end{array}} \right]. \tag{31}\end{equation} The covariance matrices is not depend on interest parameters which means (A1) can be abbreviated as \begin{equation}{\boldsymbol{I}}{\left( \phi \right)_{ij}} = 2{\rm{Re}}\left\{ {{{\left[ {\frac{{\partial {\boldsymbol s}\left( \phi \right)}}{{\partial {\phi _i}}}} \right]}^{\rm H}}{{\boldsymbol C}^{ - 1}}\left( \phi \right)\left[ {\frac{{\partial {\boldsymbol s}\left( \phi \right)}}{{\partial {\phi _j}}}} \right]} \right\}, \tag{32}\end{equation} formula $\frac{{\partial~{\boldsymbol~s}\left(~\phi~\right)}}{{\partial~\phi~}}$ can be considered by items. We get $\frac{{\partial~{\boldsymbol~s}\left(~\phi~\right)}}{{\partial~\tau~}}$ in (A4), where ${\boldsymbol~s}'_{i}$, $i=1,2$ is the time-derivative of the received FDA signal by BPS. Similarly, we can also get $\frac{{\partial~{\boldsymbol~s}\left(~\phi~\right)}}{{\partial~f}}$ shown in (A5). \begin{equation}\frac{{\partial {{\boldsymbol s}_\phi }}}{{\partial \tau }} = {\left[ {\begin{array}{*{20}{c}} \displaystyle {\frac{{\partial {{\boldsymbol s}_1}}}{{\partial \tau }}}&\displaystyle {\frac{{\partial {{\boldsymbol s}_2}}}{{\partial \tau }}} \end{array}} \right]^{\rm T}} = {\left[ {\begin{array}{*{20}{c}} \displaystyle {\frac{{\left( {\frac{{\partial {{\boldsymbol s}_1}}}{{\partial {\tau _1}}}} \right)}}{{\left( {\frac{{\partial \tau }}{{\partial {\tau _1}}}} \right)}}} &\displaystyle {\frac{{\left( {\frac{{\partial {{\boldsymbol s}_2}}}{{\partial {\tau _2}}}} \right)}}{{\left( {\frac{{\partial \tau }}{{\partial {\tau _2}}}} \right)}}} \end{array}} \right]^{\rm T}} = {\left[ {\begin{array}{*{20}{c}} \displaystyle {\frac{{\partial {{\boldsymbol s}_1}}}{{\partial {\tau _1}}}} &\displaystyle { - \frac{{\partial {{\boldsymbol s}_2}}}{{\partial {\tau _2}}}} \end{array}} \right]^{\rm T}} = {\left[ {{ - {\boldsymbol s}}}'_1 {{{\boldsymbol s}}'_2} \right]^{\rm T}}, \tag{33}\end{equation} \begin{equation}\frac{{\partial {{\boldsymbol s}_\phi }}}{{\partial f}} = {\left[ {\begin{array}{*{20}{c}} \displaystyle {\frac{{\partial {{\boldsymbol s}_1}}}{{\partial f}}} &\displaystyle {\frac{{\partial {{\boldsymbol s}_2}}}{{\partial f}}} \end{array}} \right]^{\rm T}} = {\left[ {\begin{array}{*{20}{c}} \displaystyle {\frac{{\left( {\frac{{\partial {{\boldsymbol s}_1}}}{{\partial {\xi_1}}}} \right)}}{{\left( {\frac{{\partial f}}{{\partial {\xi_1}}}} \right)}}} &\displaystyle {\frac{{\left( {\frac{{\partial {{\boldsymbol s}_2}}}{{\partial {\xi_2}}}} \right)}}{{\left( {\frac{{\partial \tau }}{{\partial {\xi_2}}}} \right)}}} \end{array}} \right]^{\rm T}}{\left[ {\begin{array}{*{20}{c}} \displaystyle {\frac{{\partial {{\boldsymbol s}_1}}}{{\partial {\xi_1}}}} &\displaystyle { - \frac{{\partial {{\boldsymbol s}_2}}}{{\partial {\xi_2}}}} \end{array}} \right]^{\rm T}} = {\left[ - {\hat{\boldsymbol s}}'_1 {{\hat{\boldsymbol s}}'_2} \right]^{\rm T}}, \tag{34}\end{equation} where ${\hat{\boldsymbol~s}}'_i$, $i=1,2$ is the frequency-derivative of the received FDA signal by BPS. After tedious calculations, Fisher information matrix ${\boldsymbol~I}\left(~\phi~\right)$ is calculated as (A6). For the convenience of verification, discrete expressions are given here, and the continuous proof method is similar. \begin{equation}{\boldsymbol I}\left( \phi \right) = \left[ {\begin{array}{*{20}{c}} {{I_{11}}\left( \phi \right)}&{{I_{12}}\left( \phi \right)} \\ {{I_{21}}\left( \phi \right)}&{I{}_{22}\left( \phi \right)} \end{array}} \right], \tag{35}\end{equation} \begin{equation}{I_{11}} = 2{\mathop{\rm Re}\nolimits} \left\{ {{{\left[ \begin{array}{c} {{ - {\boldsymbol s}}}'_1 \\ {\boldsymbol s}'_2 \end{array} \right]}^{\rm H}}{{\boldsymbol C}^{ - 1}}\left[ \begin{array}{c} {{ - {\boldsymbol s}}}'_1 \\ {\boldsymbol s}'_2 \end{array} \right]} \right\} = \frac{2}{{\sigma _1^2}}\sum\limits_{n = 0}^{N - 1} {{{\left| {{\boldsymbol s}'\left( {{\theta _i},{r_i},n{T_s} - {\tau _1}} \right)} \right|}^2}} + \frac{2}{{\sigma _2^2}}\sum\limits_{n = 0}^{N - 1} {{{\left| {{\boldsymbol s}'\left( {{\theta _i},{r_i},n{T_s} - {\tau _2}} \right)} \right|}^2}}, \tag{36}\end{equation} \begin{equation}{I_{22}} = 2{\mathop{\rm Re}\nolimits} \left\{ {{{\left[ \begin{array}{c} {\hat {\boldsymbol s}}'_1 \\ {- {\hat {\boldsymbol s}}}'_2 \end{array} \right]}^{\rm H}}{{\boldsymbol C}^{ - 1}}\left[ \begin{array}{c} {\hat {\boldsymbol s}}'_1 \\ {- {\hat {\boldsymbol s}}}'_2 \end{array} \right]} \right\} =\frac{{2T_s^2}}{{\sigma _1^2}}\sum\limits_{n = 0}^{N - 1} {{n^2}{{\left| {{\boldsymbol s}\left( {{\theta _i},{r_i},n{T_s} - {\tau _1}} \right)} \right|}^2}} + \frac{{2T_s^2}}{{\sigma _2^2}}\sum\limits_{n = 0}^{N - 1} {{n^2}{{\left| {{\boldsymbol s}\left( {{\theta _i},{r_i},n{T_s} - {\tau _2}} \right)} \right|}^2}}, \tag{37}\end{equation} \begin{align}{I_{12}} = {I_{21}} =&\,2{\mathop{\rm Re}\nolimits} \left\{ {{{\left[ \begin{array}{c} {{ - {\boldsymbol s}}}'_1 \\ {\boldsymbol s}'_2 \end{array} \right]}^{\rm H}}{{\boldsymbol C}^{ - 1}}\left[ \begin{array}{c} {\hat {\boldsymbol s}}'_1 \\ {- {\hat {\boldsymbol s}}}'_2 \end{array} \right]} \right\} = 2{\mathop{\rm Re}\nolimits} \left\{ {{{ - {\boldsymbol s}}'_1}^{\rm H} {\boldsymbol C}_1^{ - 1}{\boldsymbol s}'_1- {\hat {\boldsymbol s}}'_2{}^{\rm H} {\boldsymbol C}_2^{ - 1}{\hat {\boldsymbol s}}'_2} \right\} \\ =&\,2{\mathop{\rm Re}\nolimits} \left\{ {\frac{1}{{\sigma _1^2}}\sum\limits_{n = 0}^{N - 1} { - {\rm j}n{T_s}{{\boldsymbol s}^*}\left( {{\theta _i},{r_i},n{T_s} - {\tau _1}} \right){{\boldsymbol s}^\prime }\left( {{\theta _i},{r_i},n{T_s} - {\tau _1}} \right)} } \right\} \\ & +2{\mathop{\rm Re}\nolimits} \left\{ {\frac{1}{{\sigma _2^2}}\sum\limits_{n = 0}^{N - 1} { - {\rm j}n{T_s}{{\boldsymbol s}^*}\left( {{\theta _i},{r_i},n{T_s} - {\tau _2}} \right){{\boldsymbol s}^\prime }\left( {{\theta _i},{r_i},n{T_s} - {\tau _2}} \right)} } \right\}, \tag{38} \end{align} \begin{align}{\boldsymbol s}\left( {{\theta _i},{r_i},n{T_s} - {\tau _i}} \right) \approx&\, \frac{1}{{{r_i}}}\exp \left[ { - {\rm j}2\pi {f_0}\left( {n{T_s}{\rm{ - }}{\tau _i}} \right)} \right] \\ &\times \sum\limits_{{{m}} = 0}^{M{\rm{ - }}1} {\exp } \left\{ { - {\rm j}2\pi \left[ {\Delta {f_m}\left( {n{T_s} - {\tau _i}} \right) - \frac{{{f_0}md\sin \left( {{\theta _i}} \right)}}{\rm c} - \frac{{m\Delta {f_m}d\sin \left( {{\theta _i}} \right)}}{\rm c}} \right]} \right\}, i = 1,2, \tag{39} \end{align} \begin{align}{\boldsymbol s}'\left( {{\theta _i},{r_i},n{T_s} - {\tau _i}} \right) \approx &\, \frac{{{\rm j}2\pi {f_0}}}{{{r_i}}}{\boldsymbol s}\left( {{\theta _i},{r_i},n{T_s} - {\tau _i}} \right) + \frac{{{\rm j}2\pi }}{{{r_i}}}\exp \left\{ { - {\rm j}2\pi {f_0}\left( {n{T_s}{\rm{ - }}{\tau _i}} \right)} \right\} \\ &\times \sum\limits_{m = 0}^{M - 1} {\Delta {f_m}\exp } \left\{ { - {\rm j}2\pi \left[ {\Delta {f_m}\left( {n{T_s} - {\tau _i}} \right) - \frac{{{f_0}md\sin \left( {{\theta _i}} \right)}}{\rm c} - \frac{{m\Delta {f_m}d\sin \left( {{\theta _i}} \right)}}{\rm c}} \right]} \right\}, i = 1,2. \tag{40} \end{align}


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  • Figure 1

    (Color online) Received signal model by BPS. (a) FDA; (b) PA.

  • Figure 2

    (Color online) Sketch map of BPS localization scenario.

  • Figure 3

    (Color online) Comparison of $\text{A}(~\tau~,f~)$ in different transmitted baseband complex waveforms. (a) FDA sinusoid signal; (b) FDA chirp signal; (c) FDA QPSK signal; (d) PA sinusoid signal; (e) PA chirp signal; (f) PA QPSK signal.

  • Figure 4

    (Color online) GDOP comparisons. (a) FDA emitter; (b) PA emitter.

  • Figure 5

    (Color online) CRBs in FDA and PA under different sampling starting points. (a) CRB of FDOA; (b) CRB of TDOA.

  • Figure 6

    (Color online) CRBs in FDA and PA under different $\Delta~f$. (a) CRB of FDOA; (b) CRB of TDOA.

  • Figure 7

    (Color online) CRBs in FDA and PA under different number of elements. (a) CRB of FDOA; (b) CRB of TDOA.

  • Figure 8

    (Color online) CRBs in FDA and PA under different non-standard FDAs. (a) CRB of FDOA; (b) CRB of TDOA.

  • Figure 9

    (Color online) GDOP in central symmetrical FDA and log-FDA. (a) Central symmetrical FDA; (b) log-FDA.

  • Table 1  

    Table 1Simulation parameters I

    Simulation parameters $X$ $Y$ $Z$
    Sat-1 position (km) 3909.0 4499.43452.7
    Sat-2 position (km) 3892.3 4527.83454.5
    Sat-1 velocity (km/s) $-$1.667 $-$3.5536.517
    Sat-2 velocity (km/s) $-$1.706 $-$3.5106.520
    Emitter position (km) 3362.1 4006.7 3637.8
  • Table 2  

    Table 2Simulation parameters II

    Simulation parameters Value
    Carrier frequency ${f_0}$ 1 GHz
    Frequency offset $\Delta~f$ 100 Hz
    Numbers of elements $M$ 16
    Integration time ${\tau~_{s}}$ 10 ms
    Intermediate frequency ${f_{{b}}}$ 20 MHz
    Sampling frequency ${f_{s}}$ 80 MHz
    Baseband complex waveform ${\boldsymbol~\phi}~(~t~)$ 1
  • Table 3  

    Table 3Comparison of RMSE between FDA and PA parameters estimation

    Signal type SNR (dB) $-$10 $-$5 0 5 10
    Sinusoid signal$\Delta~{Q_v}$ (Hz) 279 151 51 43 12
    $\Delta~{Q_\tau~}$ (ns) 21 9 4 $~\approx~0$ $~\approx~0$
    Chirp signal$\Delta~{Q_v}$ (Hz) 252 176 87 63 17
    $\Delta~{Q_\tau~}$ (ns) 18 7 2 $~\approx~0$ $~\approx~0$
    QPSK signal$\Delta~{Q_v}$ (Hz) 393 152 58 26 3
    $\Delta~{Q_\tau~}$ (ns) 17 4 $~\approx~0$ $~\approx~0$ $~\approx~0$
  • Table 4  

    Table 4RMSE comparison under standard and non-standard FDAs

    Signal type SNR (dB) $-$10 $-$5 0 5 10
    ULA FDA$\Delta~{Q_v}$ (Hz) 341 176 126 73 53
    $\Delta~{Q_\tau~}$ (ns) 18 4 $~\approx~0$ $~\approx~0$ $~\approx~0$
    Log-FDA$\Delta~{Q_v}$ (Hz) 182 121 55 42 36
    $\Delta~{Q_\tau~}$ (ns)25 9 $~\approx~0$ $~\approx~0$ $~\approx~0$
    Central symmetrical FDA$\Delta~{Q_v}$ (Hz) 162 79 70 51 3
    $\Delta~{Q_\tau~}$ (ns) 12 3 $~\approx~0$ $~\approx~0$ $~\approx~0$
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