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Parameter estimation of air maneuvering target for multi-antenna system via reconstructing time samples and signal

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Abstract

A novel algorithm for estimating the motion parameters of air maneuvering target by means of reconstructing time samples and signal is proposed in this paper. Firstly, the received data of multiple antennas are spliced together to reconstruct time samples of a single antenna by compensating a proper phase. This reconstruction is equivalent to increasing time samples within a single coherent processing interval for a single antenna. After that, an ideal signal whose time sample number is equal to the length of the reconstructed time samples is constructed. At last, the estimation results of initial velocity and acceleration of the air maneuvering target are obtained by applying the nonlinear least squares method to compare the similarity between the reconstructed time samples and signal. The proposed algorithm can achieve accurate parameter estimation with limited pulses. The effectiveness of this algorithm is verified and its performance is very close to the Cramer–Rao bound as shown by our simulation results.

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Acknowledgements

This work was supported by National Nature Science Foundation of China (NSFC) under Grant 61471365, U1633106, 61231017, National University’s Basic Research Foundation of China under Grant No. 3122017007. The work is also supported by the Foundation for Sky Young Scholars of Civil Aviation University of China.

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Correspondence to Hai Li.

Appendix 1: The proof of cost function

Appendix 1: The proof of cost function

Without losing generality, assume that the amplitude of signal is 1. Considering an N elements system, the number of samples of each antenna is K. Without considering the space phase, the signal after reconstructing time samples is a \(NK \times 1\) vector, given by

$$\begin{aligned} \mathbf{{x}}_{rec} (v,a,v_0 ,a_0 )= & {} \left[ {\begin{array}{l@{\quad }l@{\quad }l@{\quad }l} {\varvec{x}_{proj1}^T (v_0 ,a_0 )} &{} {\varvec{x}_{proj2}^T (v_0 ,a_0 )} &{} \cdots &{} {\varvec{x}_{projN}^T (v_0 ,a_0 )} \\ \end{array}} \right] _{NK \times 1}^T \nonumber \\&\odot \left[ {\begin{array}{l@{\quad }l@{\quad }l@{\quad }l} {\left( {e^{\Delta \varvec{\varphi } _1 (v,a)} } \right) ^T } &{} {\left( {e^{\Delta \varvec{\varphi } _2 (v,a)} } \right) ^T } &{} \cdots &{} {\left( {e^{\Delta \varvec{\varphi } _N (v,a)} } \right) ^T } \\ \end{array}} \right] _{NK \times 1}^T \end{aligned}$$
(21)

where

$$\begin{aligned} \mathbf{{x}}_{proj1}^T (v_0 ,a_0 )= & {} \mathbf{{x}}_{proj2}^T (v_0 ,a_0 ) = \cdots \mathbf{{x}}_{projN}^T (v_0 ,a_0 ) \nonumber \\= & {} \left[ {\begin{array}{l@{\quad }l@{\quad }l@{\quad }l} 1 &{} { e^{j2\pi \frac{{2v_0 }}{{\lambda f_r }} + j\pi \frac{{2a_0 }}{{\lambda f_r^2 }}} } &{} \cdots &{} {e^{j2\pi \frac{{2v_0 }}{{\lambda f_r }}(K - 1) + j\pi \frac{{2a_0 }}{{\lambda f_r^2 }}(K - 1)^2 } } \\ \end{array}} \right] _{K \times 1}^T \end{aligned}$$
(22)

\(v_0\) and \(a_0\) are the actual motion parameters. Thus the reconstructed signal is

$$\begin{aligned} \mathbf{{x}}_z (v,a)= & {} \left[ {\begin{array}{l@{\quad }l@{\quad }l@{\quad }l} 1 &{} {e^{j2\pi \frac{{2v}}{{\lambda f_r }} + j\pi \frac{{2a}}{{\lambda f_r^2 }}} } &{} \cdots &{} {e^{j2\pi \frac{{2v}}{{\lambda f_r }}(NK - 1) + j\pi \frac{{2a}}{{\lambda f_r^2 }}(NK - 1)^2 } } \nonumber \\ \end{array}} \right] _{NK \times 1}^T \\= & {} \left[ {\begin{array}{l@{\quad }l@{\quad }l@{\quad }l} {\mathbf{{x}}_{z1}^T (v,a)} &{} {\mathbf{{x}}_{z2}^T (v,a)} &{} \cdots &{} {\mathbf{{x}}_{zN}^T (v,a)} \\ \end{array}} \right] _{NK \times 1}^T \nonumber \\&\odot \left[ {\begin{array}{l@{\quad }l@{\quad }l@{\quad }l} {\left( {e^{\Delta \varvec{\varphi } _1 (v,a)} } \right) ^T } &{} {\left( {e^{\Delta \varvec{\varphi } _2 (v,a)} } \right) ^T } \cdots {\left( {e^{\Delta \varvec{\varphi } _N (v,a)} } \right) ^T } \\ \end{array}} \right] _{NK \times 1}^T \end{aligned}$$
(23)

where

$$\begin{aligned} \mathbf{{x}}_{z1}^T (v,a)= & {} \mathbf{{x}}_{z2}^T (v,a) = \cdots \mathbf{{x}}_{zN}^T (v,a) \nonumber \\= & {} \left[ {\begin{array}{l@{\quad }l@{\quad }l@{\quad }l} 1 &{} {e^{j2\pi \frac{{2v}}{{\lambda f_r }} + j\pi \frac{{2a}}{{\lambda f_r^2 }}} } &{} \cdots &{} {e^{j2\pi \frac{{2v}}{{\lambda f_r }}(K - 1) + j\pi \frac{{2a}}{{\lambda f_r^2 }}(K - 1)^2 } } \\ \end{array}} \right] _{K \times 1}^T \end{aligned}$$
(24)

v and a are the unknown parameters. Thus, the cost function can be expressed as

$$\begin{aligned} f(v,a)= & {} \left\| {\mathbf{{x}}_{rec} (v,a,v_0 ,a_0 ) - \mathbf{{x}}_z (v,a)} \right\| _2 \nonumber \\= & {} \left\| {\left[ {\begin{array}{ccc} {\left( {\mathbf{{x}}_{proj1}^T (v_0 ,a_0 ) - \mathbf{{x}}_1^T (v,a)} \right) } &{} \cdots &{} {\left( {\mathbf{{x}}_{projN}^T (v_0 ,a_0 ) - \mathbf{{x}}_N^T (v,a)} \right) } \\ \end{array}} \right] } \right. _{NK \times 1}^T \nonumber \\&\left. { \odot \left[ {\begin{array}{ccc} {(e^{\Delta \varvec{\varphi } _1 (v,a)} )^T } &{} \cdots &{} {(e^{\Delta \varvec{\varphi } _N (v,a)} )^T } \\ \end{array}} \right] _{NK \times 1}^T } \right\| _2 \nonumber \\= & {} \left\| {\left[ {\begin{array}{ccc} {\left( {\mathbf{{x}}_{proj1}^T (v_0 ,a_0 ) - \mathbf{{x}}_1^T (v,a)} \right) } &{} \cdots &{} {\left( {\mathbf{{x}}_{projN}^T (v_0 ,a_0 ) - \mathbf{{x}}_N^T (v,a)} \right) } \\ \end{array}} \right] _{NK \times 1}^T } \right\| _2 \cdot 1 \nonumber \\ \end{aligned}$$
(25)

Equation (25) can be simplified as

$$\begin{aligned} f(v,a)= & {} \left\| {\mathbf{{x}}_{rec} (v,a,v_0 ,a_0 ) - \mathbf{{x}}_z (v,a)} \right\| _2 \nonumber \\= & {} \left\| {\left[ {\begin{array}{ccc} {\left( {\mathbf{{x}}_{proj1}^T (v_0 ,a_0 ) - \mathbf{{x}}_1^T (v,a)} \right) } &{} \cdots &{} {\left( {\mathbf{{x}}_{projN}^T (v_0 ,a_0 ) - \mathbf{{x}}_N^T (v,a)} \right) } \\ \end{array}} \right] _{NK \times 1}^T } \right\| _2 \cdot 1 \nonumber \\= & {} \sqrt{N\sum \limits _{i = 0}^{K - 1} {\left| {e^{j2\pi \frac{{2v_0 }}{{\lambda f_r }}i + j\pi \frac{{2a_0 }}{{\lambda f_r^2 }}i^2 } - e^{j2\pi \frac{{2v}}{{\lambda f_r }}i + j\pi \frac{{2a}}{{\lambda f_r^2 }}i^2 } } \right| ^2 } } \nonumber \\= & {} \sqrt{N\sum \limits _{i = 0}^{K - 1} {2\left| {1 - \cos \left[ {2\pi \frac{{(v_0 - v)}}{{\lambda f_r }}i + \pi \frac{{(a_0 - a)}}{{\lambda f_r^2 }}i^2 } \right] } \right| ^2 } } \end{aligned}$$
(26)

Equation (26) reaches the minimum when the values of the cosine function family in it are all 1, given by

$$\begin{aligned} \textit{COS}(a,v,i) = \cos \left[ {2\pi \frac{{(v_0 - v)}}{{\lambda f_r }}i + \pi \frac{{(a_0 - a)}}{{\lambda f_r^2 }}i^2 } \right] = 1 , (i = 0,1, \ldots ,K - 1) \end{aligned}$$
(27)

Thus, the results of the cosine function family can be obtain by the linear function family as follow:

$$\begin{aligned} 2\pi \frac{{(v_0 - v)}}{{\lambda f_r }}i + \pi \frac{{(a_0 - a)}}{{\lambda f_r^2 }}i^2 = 2p\pi , (p = 0, \pm 1, \pm 2 \cdots ;i = 0,1, \ldots ,K - 1) \end{aligned}$$
(28)

where \(p=0\) is the unambiguous situation and \(p \ne 0\) is the ambiguous situation caused by the periodicity of trigonometric function. The explicit solutions to the Eq. (28) is

$$\begin{aligned} \left\{ \begin{array}{l} v = \frac{i}{{2f_r }}(a - a_0 ) - \frac{{f_r }}{i}2\lambda p + v_0 \\ a = \frac{{f_r }}{i}(v - v_0 ) - \left( {\frac{{f_r }}{i}} \right) ^2 2\lambda p + a_0 \\ \end{array} \right. , (p = 0, \pm 1, \pm 2 \cdots ;i = 0,1, \ldots ,K - 1) \end{aligned}$$
(29)

It is obvious that the solution of the linear function family is \(v=v_0\), \(a=a_0\), \(p=0\), as shown in Fig. 8a. When \(p=0\), each line represents the solution of the equations in Eq. (29). The whole linear function family converges to only one common solution, which is the true value of the estimation results. When \(p \ne 0\), the whole linear function family has no common solution as shown in Fig. 8b.

Fig. 8
figure 8

The solution of the cost function. a \(p=0\), b \(p \ne 0\)

With the above discussion, the cost function reaches the minimum when and only when \(v=v_0\) and \(a=a_0\). Through the proof of cost function, we can draw a conclusion that the parameter estimation results can be acquired successfully if Eq. (15) is minimized. Meanwhile, the right term of Eq. (15) can be rewritten as

$$\begin{aligned} f(\hat{v},\hat{a}) = \left\| {\mathbf{{x}}_{rec} (v,a,v_0 ,a_0 ) - \mathbf{{x}}_z (v,a)} \right\| _2 = \left\| {\mathbf{{x}}_{rec} (v,a,v_0 ,a_0 ) - \hat{A}{{\hat{\mathbf{a}}}}(\omega _t )} \right\| _2 \end{aligned}$$
(30)

If \({{{\hat{\mathbf{a}}}}(\omega _t )}\) is a full column rank vector, \(\left[ {{{\hat{\mathbf{a}}}}^H (\omega _t ){{\hat{\mathbf{a}}}}(\omega _t )} \right] ^{ - 1}\) could be easily acquired. Equation (30) could be rewritten as

$$\begin{aligned} f(\hat{v},\hat{a})= & {} \left[ {\mathbf{{x}}_{rec} - \hat{A} \cdot {{\hat{\mathbf{a}}}}\mathrm{{(}}\omega _t \mathrm{{)}}} \right] ^H \cdot \left[ {\mathbf{{x}}_{rec} - \hat{A} \cdot {{\hat{\mathbf{a}}}}\mathrm{{(}}\omega _t \mathrm{{)}}} \right] \nonumber \\= & {} \left\{ {\hat{A} - } {\left[ {{{\hat{\mathbf{a}}}}\mathrm{{(}}\omega _t \mathrm{{)}}^H \cdot {{\hat{\mathbf{a}}}}\mathrm{{(}}\omega _t \mathrm{{)}}} \right] ^{ - 1} \cdot {{\hat{\mathbf{a}}}}\mathrm{{(}} \omega _t \mathrm{{)}}^H \cdot \mathbf{{x}}_{rec} } \right\} ^H \cdot \left[ {{{\hat{\mathbf{a}}}}\mathrm{{(}}\omega _t \mathrm{{)}}^H \cdot {{\hat{\mathbf{a}}}}\mathrm{{(}}\omega _t \mathrm{{)}}} \right] \nonumber \\&\cdot \left\{ {\hat{A} - } {\left[ {{{\hat{\mathbf{a}}}}\mathrm{{(}}\omega _t \mathrm{{)}}^H \cdot {{\hat{\mathbf{a}}}}\mathrm{{(}}\omega _t \mathrm{{)}}} \right] ^{ - 1} \cdot {{\hat{\mathbf{a}}}}\mathrm{{(}}\omega _t \mathrm{{)}}^H \cdot \mathbf{{x}}_{rec} } \right\} + \mathbf{{x}}_{rec}^H \cdot \mathbf{{x}}_{rec} \nonumber \\&- \mathbf{{x}}_{rec}^H \cdot {{\hat{\mathbf{a}}}}\mathrm{{(}}\omega _t \mathrm{{)}} \cdot \left[ {{{\hat{\mathbf{a}}}}\mathrm{{(}}\omega _t \mathrm{{)}}^H \cdot {{\hat{\mathbf{a}}}}\mathrm{{(}}\omega _t \mathrm{{)}}} \right] ^{ - 1} \cdot {{\hat{\mathbf{a}}}}\mathrm{{(}}\omega _t \mathrm{{)}}^H \cdot \mathbf{{x}}_{rec} \end{aligned}$$
(31)

Then, the estimated values of v and a are also obtained by this cost function

$$\begin{aligned} (\hat{v}_0,\hat{a}_0) = \arg \mathop {\max }\limits _{(v,a)} \left\{ {\left. {\mathbf{{x}}_{rec}^H \cdot {{\hat{\mathbf{a}}}}\mathrm{{(}}\omega _t \mathrm{{)}} \cdot \left[ {{{\hat{\mathbf{a}}}}\mathrm{{(}}\omega _t \mathrm{{)}}^H \cdot {{\hat{\mathbf{a}}}}\mathrm{{(}}\omega _t \mathrm{{)}}} \right] ^{ - 1} \cdot {{\hat{\mathbf{a}}}}\mathrm{{(}}\omega _t \mathrm{{)}}^H \cdot \mathbf{{x}}_{rec} } \right\} } \right. \end{aligned}$$
(32)

Actually, Eqs. (15) and (32) are equivalent in estimating these parameters. The parameters are estimated by Eq. (32) for convenience in engineering.

1.1 Cramer–Rao bound of parameter estimation for air maneuvering target

It is assumed that the unknown parameters are initial velocity, acceleration, amplitude and phase. The covariance matrix of clutter and noise could be obtained from

$$\begin{aligned} \mathbf{{R}} = E\{ (\mathbf{{x}}_c + \mathbf{{x}}_n )(\mathbf{{x}}_c + \mathbf{{x}}_n )^H \} \end{aligned}$$
(33)

The parameters v and a are decided by the space–time steering vector \(\tilde{A}\mathbf{{a}}(\omega _s ,\omega _t )\). Since the snapshot is assumed to obey a multivariate Gaussian distribution, the probability density function is

$$\begin{aligned} p(\mathbf{{x}}|a,v,\tilde{b}_t ) = \frac{1}{{\pi ^{NK} \det \mathbf{{R}}}}e^{ - (\mathbf{{x}} - \tilde{A}\mathbf{{a}}\mathrm{{(}}\omega _s \mathrm{{,}}\omega _t \mathrm{{)}})^H \mathbf{{R}}^{ - 1} (\mathbf{{x}} - \tilde{A}\mathbf{{a}}\mathrm{{(}}\omega _s \mathrm{{,}}\omega _t \mathrm{{)}})} \end{aligned}$$
(34)

The unknown parameters are arranged to form a \(4 \times 1\) vector

$$\begin{aligned} \varvec{\theta } = \left[ {\begin{array}{l@{\quad }l@{\quad }l@{\quad }l} {\theta _1 } &{} {\theta _2 } &{} {\theta _3 } &{} {\theta _4 } \\ \end{array}} \right] ^T = \left[ {\begin{array}{l@{\quad }l@{\quad }l@{\quad }l} a &{} v &{} \rho &{} \varphi \\ \end{array}} \right] ^T \end{aligned}$$
(35)

where \(\rho \) and \(\varphi \) are the amplitude and phase, i.e., \(\tilde{A} = \rho e^{j\varphi } \). There are four parameters needed to be estimated, namely, the amplitude, phase, initial velocity and acceleration. The log-likelihood function is

$$\begin{aligned} l\left( {\varvec{\theta }} \right) = \ln p(\mathbf{{x}}|{\varvec{\theta }}) = - \ln (\pi ^{NK} \det \mathbf{{R}}) - (\mathbf{{x}} - \tilde{A}\mathbf{{a}}\mathrm{{(}}\omega _s \mathrm{{,}}\omega _t \mathrm{{)}})^H \mathbf{{R}}^{ - 1} (\mathbf{{x}} - \tilde{A}\mathbf{{a}}\mathrm{{(}}\omega _s \mathrm{{,}}\omega _t \mathrm{{)}}) \end{aligned}$$
(36)

The Cramer–Rao bound for the error covariance matrix of an unbiased estimator \({{\hat{\varvec{\theta }}}}\) is given by (Stoica and Moses 2005)

$$\begin{aligned} E\{ ({{\hat{{\varvec{\theta }}}}} - {{\varvec{\theta }}})({{\hat{{\varvec{\theta }}}}} - {{ \varvec{\theta }}})^T \} \ge \mathbf{{J}}^{ - 1} \end{aligned}$$
(37)

where \(\mathbf{{J}}\) is the Fisher information matrix. It is convenient to compute \(\mathbf{{J}}\) as

$$\begin{aligned} \mathbf{{J}} = - E\{ \mathbf{{L}}_{\theta \theta } \} = \frac{{\partial ^2 l}}{{\partial \theta _k \theta _l }} \end{aligned}$$
(38)

The Fisher information matrix \(\mathbf{{J}}\) is

$$\begin{aligned} \mathbf{{J}} = - E\{ \mathbf{{L}}_{\theta \theta } \} = - E\left[ {\begin{array}{l@{\quad }l@{\quad }l@{\quad }l} {\frac{{\partial ^2 l}}{{\partial a^2 }}} &{} {\frac{{\partial ^2 l}}{{\partial a\partial v}}} &{} {\frac{{\partial ^2 l}}{{\partial a\partial \rho }}} &{} {\frac{{\partial ^2 l}}{{\partial a\partial \varphi }}} \\ {\frac{{\partial ^2 l}}{{\partial v\partial a}}} &{} {\frac{{\partial ^2 l}}{{\partial v^2 }}} &{} {\frac{{\partial ^2 l}}{{\partial v\partial \rho }}} &{} {\frac{{\partial ^2 l}}{{\partial v\partial \varphi }}} \\ {\frac{{\partial ^2 l}}{{\partial \rho \partial a}}} &{} {\frac{{\partial ^2 l}}{{\partial \rho \partial v}}} &{} {\frac{{\partial ^2 l}}{{\partial \rho ^2 }}} &{} {\frac{{\partial ^2 l}}{{\partial \rho \partial \varphi }}} \\ {\frac{{\partial ^2 l}}{{\partial \varphi \partial a}}} &{} {\frac{{\partial ^2 l}}{{\partial \varphi \partial v}}} &{} {\frac{{\partial ^2 l}}{{\partial \varphi \partial \rho }}} &{} {\frac{{\partial ^2 l}}{{\partial \varphi ^2 }}} \\ \end{array}} \right] \end{aligned}$$
(39)

Let us define the derivative vectors

$$\begin{aligned} \mathbf{{d}}_a= & {} \frac{{\partial \mathbf{{a}}}}{{\partial a}} = \frac{{\partial \mathbf{{a}}(\omega _t )}}{{\partial a}} \otimes \mathbf{{a}}(\omega _s ) \end{aligned}$$
(40)
$$\begin{aligned} \mathbf{{d}}_v= & {} \frac{{\partial \mathbf{{a}}}}{{\partial v}} = \frac{{\partial \mathbf{{a}}(\omega _t )}}{{\partial v}} \otimes \mathbf{{a}}(\omega _s ) \end{aligned}$$
(41)

Through deduction, the Fisher information matrix is shown to have the compact form

$$\begin{aligned} \mathbf{{J}} = 2\left[ {\begin{array}{l@{\quad }l@{\quad }l@{\quad }l} {\rho ^2 \delta _a } &{} {\rho ^2 \beta _d } &{} {\rho \beta _a } &{} {\rho ^2 \gamma _a } \\ {\rho ^2 \beta _d } &{} {\rho ^2 \delta _v } &{} {\rho \beta _v } &{} {\rho ^2 \gamma _v } \\ {\rho \beta _a } &{} {\rho \beta _v } &{} \xi &{} 0 \\ {\rho ^2 \gamma _a } &{} {\rho ^2 \gamma _v } &{} 0 &{} {\rho ^2 \xi } \\ \end{array}} \right] \end{aligned}$$
(42)

The vectors are defined as

$$\begin{aligned} \delta _a= & {} \mathbf{{d}}_a ^H \mathbf{{R}}^{ - 1} \mathbf{{d}}_a \end{aligned}$$
(43)
$$\begin{aligned} \delta _v= & {} \mathbf{{d}}_v ^H \mathbf{{R}}^{ - 1} \mathbf{{d}}_v \end{aligned}$$
(44)
$$\begin{aligned} \xi= & {} \mathbf {a}^H \mathbf{{R}}^{ - 1} \mathbf {a} \end{aligned}$$
(45)
$$\begin{aligned} \beta _d= & {} \frac{1}{2}(\mathbf{{d}}_a^H \mathbf{{R}}^{ - 1} \mathbf{{d}}_v + \mathbf{{d}}_v^H \mathbf{{R}}^{ - 1} \mathbf{{d}}_a ) \end{aligned}$$
(46)
$$\begin{aligned} \beta _a= & {} \frac{1}{2}(\mathbf{{d}}_a^H \mathbf{{R}}^{ - 1} \mathbf {a} + \mathbf {a}^H \mathbf{{R}}^{ - 1} \mathbf{{d}}_a ) \end{aligned}$$
(47)
$$\begin{aligned} \gamma _a= & {} \frac{1}{2}j(\mathbf{{d}}_a^H \mathbf{{R}}^{ - 1} \mathbf {a} - \mathbf {a}^H \mathbf{{R}}^{ - 1} \mathbf{{d}}_a ) \end{aligned}$$
(48)
$$\begin{aligned} \beta _v= & {} \frac{1}{2}(\mathbf{{d}}_v^H \mathbf{{R}}^{ - 1} \mathbf {a} + \mathbf {a}^H \mathbf{{R}}^{ - 1} \mathbf{{d}}_v ) \end{aligned}$$
(49)
$$\begin{aligned} \gamma _v= & {} \frac{1}{2}j(\mathbf{{d}}_v^H \mathbf{{R}}^{ - 1} \mathbf {a} - \mathbf {a}^H \mathbf{{R}}^{ - 1} \mathbf{{d}}_v ) \end{aligned}$$
(50)

The CRB of \({{\hat{\varvec{\theta }}}}\) is obtained from the diagonal elements of \(\mathbf{{J}}^{ - 1}\) , that is

$$\begin{aligned} \textit{CRB}({{\hat{\varvec{\theta }}}}_{ii} ) = (\mathbf{{J}}^{ - 1} )_{ii} \end{aligned}$$
(51)

\(\mathbf{{J}}^{ - 1}\) obtained with the inverse lemma of partitioned matrix. As initial velocity and acceleration errors are the results we need, the upper left block of \(\mathbf{{J}}^{ - 1}\) is focused. The inverse partitioned matrix is

$$\begin{aligned} \left[ {\begin{array}{l@{\quad }l} \mathbf{{A}} &{} \mathbf{{U}} \\ \mathbf{{V}} &{} \mathbf{{D}} \\ \end{array}} \right] ^{ - 1} = \left[ {\begin{array}{l@{\quad }l} {(\mathbf{{A - UD}}^{\mathbf{{ - 1}}} \mathbf{{V}})^{ - 1} } &{} { - (\mathbf{{A - UD}}^{\mathbf{{ - 1}}} \mathbf{{V}})^{ - 1} \mathbf{{UD}}^{ - 1} } \\ { - (\mathbf{{D - VA}}^{\mathbf{{ - 1}}} \mathbf{{U}})^{ - 1} \mathbf{{VA}}^{\mathbf{{ - 1}}} } &{} {(\mathbf{{D - VA}}^{\mathbf{{ - 1}}} \mathbf{{U}})^{ - 1} } \\ \end{array}} \right] \end{aligned}$$
(52)

Define

$$\begin{aligned} \mathbf{{J = }}\left[ {\begin{array}{l@{\quad }l} \mathbf{{A}} &{} \mathbf{{U}} \\ \mathbf{{V}} &{} \mathbf{{D}} \\ \end{array}} \right] \end{aligned}$$
(53)

where \(\mathbf{{A}},\mathbf{{U}},\mathbf{{V}},\mathbf{{D}}\) are \(2 \times 2\) matrix.

$$\begin{aligned} \mathbf{{A - UD}}^{\mathbf{{ - 1}}} \mathbf{{V}} = 2\rho ^2 \left[ {\begin{array}{l@{\quad }l} {\delta _a \sin ^2 \eta _a } &{} \varepsilon \\ \varepsilon &{} {\delta _v \sin ^2 \eta _v } \\ \end{array}} \right] \end{aligned}$$
(54)

where

$$\begin{aligned} \cos ^2 \eta _a= & {} \frac{{\beta _a^2 + \gamma _a^2 }}{{\xi \delta _a }} \end{aligned}$$
(55)
$$\begin{aligned} \cos ^2 \eta _v= & {} \frac{{\beta _v^2 + \gamma _v^2 }}{{\xi \delta _v }} \end{aligned}$$
(56)
$$\begin{aligned} \varepsilon= & {} \beta _d - \frac{1}{\xi }(\beta _a \beta _v + \gamma _a \gamma _v ) \end{aligned}$$
(57)

According to Eq. (52), the upper left \(2 \times 2\) matrix is

$$\begin{aligned} (\mathbf{{A - UD}}^{\mathbf{{ - 1}}} \mathbf{{V}})^{ - 1}= & {} \frac{1}{{2\rho ^2 }}\left[ {\begin{array}{l@{\quad }l} {\delta _a \sin ^2 \eta _a } &{} \varepsilon \\ \varepsilon &{} {\delta _v \sin ^2 \eta _v } \\ \end{array}} \right] ^{ - 1} \nonumber \\= & {} \left[ {\begin{array}{l@{\quad }l} {\frac{{\delta _v \sin ^2 \eta _v }}{{2\rho ^2 (\delta _a \delta _v \sin ^2 \eta _a \sin ^2 \eta _v - \varepsilon ^2 )}}} &{} {\frac{{ - \varepsilon }}{{2\rho ^2 (\delta _a \delta _v \sin ^2 \eta _a \sin ^2 \eta _v - \varepsilon ^2 )}}} \\ {\frac{{ - \varepsilon }}{{2\rho ^2 (\delta _a \delta _v \sin ^2 \eta _a \sin ^2 \eta _v - \varepsilon ^2 )}}} &{} {\frac{{\delta _a \sin ^2 \eta _a }}{{2\rho ^2 (\delta _a \delta _v \sin ^2 \eta _a \sin ^2 \eta _v - \varepsilon ^2 )}}} \\ \end{array}} \right] \end{aligned}$$
(58)

Hence, the CRB of initial velocity is obtained as

$$\begin{aligned} \sigma _a^2 = \frac{{\delta _v \sin ^2 \eta _v }}{{2\rho ^2 (\delta _a \delta _v \sin ^2 \eta _a \sin ^2 \eta _v - \varepsilon ^2 )}} \end{aligned}$$
(59)

and the CRB of acceleration is

$$\begin{aligned} \sigma _v^2 = \frac{{\delta _a \sin ^2 \eta _a }}{{2\rho ^2 (\delta _a \delta _v \sin ^2 \eta _a \sin ^2 \eta _v - \varepsilon ^2 )}} \end{aligned}$$
(60)

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Li, H., Zhou, M., Wu, R. et al. Parameter estimation of air maneuvering target for multi-antenna system via reconstructing time samples and signal. Multidim Syst Sign Process 29, 621–641 (2018). https://doi.org/10.1007/s11045-017-0495-7

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