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Robust Waveform Design for MIMO Radar with Imperfect Prior Knowledge

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Abstract

Waveform optimization for multi-input multi-output radar usually depends on the initial parameter estimates (i.e., some prior information on the target of interest and scenario). However, it is sensitive to estimate errors and uncertainty in the parameters. Robust waveform design attempts to systematically alleviate the sensitivity by explicitly incorporating a parameter uncertainty model into the optimization problem. In this paper, we consider the robust waveform optimization to improve the worst-case performance of parameter estimation over a convex uncertainty model, which is based on the Cramer–Rao bound. An iterative algorithm is proposed to optimize the waveform covariance matrix such that the worst-case performance can be improved. Each iteration step in the proposed algorithm is solved by resorting to convex relaxation that belongs to the semidefinite programming class. Numerical results show that the worst-case performance can be improved considerably by the proposed method compared to that of uncorrelated waveforms and the non-robust method.

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Acknowledgments

The authors would like to thank Dr. M. N. S. Swamy, Dr. Q. Guo, and the anonymous reviewers for their thoughtful and to-the-point comments and suggestions, which greatly improved the manuscript. This work is sponsored in part by NSFC under Grant Nos. 61301258, 61271379, and 61271292.

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Correspondence to Hongyan Wang.

Appendices

Appendix 1

1.1 Constrained Cramero–Rao Bound

Based on the signal model in (3), in the following, we derive the constrained CRB of the unknown target parameters \({\varvec{\uptheta }} =[\theta _1, \theta _2, \ldots , \theta _K ]^{\mathrm{T}}\). According to [30], the constrained CRB can be written as

$$\begin{aligned} \mathbf{C}_{CCRB} =\mathbf{U}(\mathbf{U}^{H}\mathbf{FU})^{-1}\mathbf{U}^{H}, \end{aligned}$$
(30)

where \(\mathbf{U}\) satisfies

$$\begin{aligned} \mathbf{G}(\mathbf{x})\mathbf{U}(\mathbf{x})=0,\quad \mathbf{U}^{H}(\mathbf{x})\mathbf{U}(\mathbf{x})=\mathbf{I}, \end{aligned}$$
(31)

in which \(G(\mathbf{x})=\frac{\partial \mathbf{g}(\mathbf{x})}{\partial \mathbf{x}}\) is assumed to have full row rank, and \(\mathbf{U}\) is the tangent hyperplane of \(\mathbf{g}(\mathbf{x})\) that is equality constraint set on \(\mathbf{x}\). In addition, \(\mathbf{F}\) is the Fisher information matrix (FIM) with respect to \(\mathbf{x}=[{\varvec{\uptheta }} ^{\mathrm{T}},{\varvec{\upbeta }} _R^T, {\varvec{\upbeta }}_I^T ]^{\mathrm{T}}\), where \({\varvec{\upbeta }} _R =[\beta _{R,1}, \beta _{R,2}, \ldots ,\beta _{R,K} ]^{\mathrm{T}}\), \({\varvec{\upbeta }} _I =[\beta _{I,1}, \beta _{I,2}, \ldots ,\beta _{I,K} ]^{\mathrm{T}}\), \(\beta _R =\hbox {Re}(\beta )\), and \({{\beta }} _I =\hbox {Im}(\beta )\).

Following [10] and [27], some prior information can be available in array signal processing, for example, constant modulus constraint on the transmitted waveform and the signal subspace constraints in the estimation of the angle of arrival. Here, we assume that the complex amplitude matrix \({\varvec{\upbeta }} =\hbox {diag}(\beta _{1}, \beta _{2}, \ldots , \beta _K )\) is known as

$$\begin{aligned} g_i (\mathbf{x})= & {} \beta _{R,i} -1=0,\quad i=1,\ldots , K \nonumber \\ g_j (\mathbf{x})= & {} \beta _{I,j} -1=0,\quad j=K+1,\ldots , 2K. \end{aligned}$$
(32)

Therefore, \(\mathbf{G}(\mathbf{x})=\frac{\partial \mathbf{g}(\mathbf{x})}{\partial \mathbf{x}}\) has the form \(\mathbf{G}=[\mathbf{0}_{2K\times K} ,\;\mathbf{I}_{2K\times 2K} ]\) with \(\mathbf{0}_{2K\times K} \) denoting a zero matrix of size \(2K\times K\), and the corresponding null space \(\mathbf{U}\) can be represented as

$$\begin{aligned} \mathbf{U}=[\mathbf{I}_{K\times K} {}\;{}\quad \mathbf{0}_{K\times 2K} ]^{H}. \end{aligned}$$
(33)

In what follows, we will calculate the FIM with respect to \(\mathbf{x}\) (here we only consider one-dimensional targets). According to [31], we can obtain

$$\begin{aligned} \mathbf{F}(x_i, x_j )=2\hbox {Re}\left\{ {\hbox {tr}\left[ {\frac{\partial \left( \sum \nolimits _{k=1}^K {\beta _k \mathbf{H}_k \mathbf{S}} \right) ^{H}}{\partial x_i }\mathbf{Q}^{-1}\frac{\partial \left( \sum \nolimits _{k=1}^K {\beta _k \mathbf{H}_k \mathbf{S}} \right) }{\partial x_j }} \right] } \right\} . \end{aligned}$$
(34)

Then

(35)

Hence, \(\mathbf{F}({\varvec{\uptheta }},{\varvec{\uptheta }} ) =\hbox {2Re}(\mathbf{F}_{11} )\) with \(\mathbf{F}_{11} \) given in (6).

Similar to [18], we have

$$\begin{aligned} \mathbf{F}^{\mathrm{T}}({\varvec{\upbeta }} _R, {\varvec{\uptheta }} )=\mathbf{F}({\varvec{\uptheta }}, {\varvec{\upbeta }} _R )=2\hbox {Re}(\mathbf{F}_{12} ), \end{aligned}$$
(36)

and

$$\begin{aligned} \mathbf{F}^{\mathrm{T}}({\varvec{\upbeta }} _I, {\varvec{\uptheta }} )=\mathbf{F} ({\varvec{\uptheta }}, {\varvec{\upbeta }} _I )=-2\hbox {Im}(\mathbf{F}_{12} ), \end{aligned}$$
(37)

where \(\left[ {\mathbf{F}_{12} } \right] _{ij} =\hbox {tr}[\beta _i^*\dot{\mathbf{H}}_i^H \mathbf{Q}^{-1}\mathbf{H}_j \mathbf{R}_\mathbf{S} ]\).

Also,

$$\begin{aligned} \mathbf{F}({\varvec{\upbeta }} _R, {\varvec{\upbeta }} _R ) =\mathbf{F}({\varvec{\upbeta }} _I, {\varvec{\upbeta }} _I )=2\hbox {Re}(\mathbf{F}_{22} ), \end{aligned}$$
(38)

and

$$\begin{aligned} \mathbf{F}({\varvec{\upbeta }} _R, {\varvec{\upbeta }} _I ) =\mathbf{F}^{\mathrm{T}}({\varvec{\upbeta }} _I, {\varvec{\upbeta }} _R )=-2\hbox {Im}(\mathbf{F}_{22} ), \end{aligned}$$
(39)

where \(\left[ {\mathbf{F}_{22} } \right] _{ij} =\hbox {tr}[\mathbf{H}_i^H \mathbf{Q}^{-1}\mathbf{H}_j \mathbf{R}_\mathbf{S} ]\).

Based on the discussion above, the FIM \(\mathbf{F}\) with respect to \(\mathbf{x}\) can be expressed as

$$\begin{aligned} \mathbf{F}=2\left[ {{\begin{array}{lll} {\hbox {Re}(\mathbf{F}_{11} )}&{}\quad {\hbox {Re}(\mathbf{F}_{12} )}&{}\quad {-\hbox {Im}(\mathbf{F}_{12} )} \\ {\hbox {Re}^{\mathrm{T}}(\mathbf{F}_{12} )}&{}\quad {\hbox {Re}(\mathbf{F}_{22} )}&{}\quad {-\hbox {Im}(\mathbf{F}_{22} )} \\ {-\hbox {Im}^{\mathrm{T}}(\mathbf{F}_{12} )}&{}\quad {-\hbox {Im}^{\mathrm{T}}(\mathbf{F}_{22} )}&{}\quad {\hbox {Re}(\mathbf{F}_{22} )} \\ \end{array} }} \right] . \end{aligned}$$
(40)

With (30), (33), (6), and (36)–(40), (4) can be obtained immediately.

Appendix 2

1.1 Derivation of (16)

To simplify notation, we define

$$\begin{aligned} \dot{\mathbf{H}}_{k,x}= & {} \hbox {Re}(\dot{\mathbf{H}}_k ),\;\dot{\mathbf{H}}_{k,y} =\hbox {Im}(\dot{\mathbf{H}}_k ),\;\dot{\tilde{\mathbf{H}}}_{k,x} =\hbox {Re}(\dot{\tilde{\mathbf{H}}}_k ),\;\dot{\tilde{\mathbf{H}}}_{k,y} =\hbox {Im}(\dot{\tilde{\mathbf{H}}}_k ),\;\dot{\varvec{\updelta } }_{k,x} =\hbox {Re}(\dot{\varvec{\updelta } }_k ) \nonumber \\ \dot{\varvec{\updelta } }_{k,y}= & {} \hbox {Im}(\dot{\varvec{\updelta } }_k ),\;\mathbf{R}_{\mathbf{S},x} =\hbox {Re}(\mathbf{R}_\mathbf{S} ),\;\mathbf{R}_{\mathbf{S},{y}} =\hbox {Im}(\mathbf{R}_\mathbf{S} ),\;\mathbf{Z}_x =\hbox {Re}(\mathbf{Z}),\;\mathbf{Z}_y =\hbox {Im}(\mathbf{Z}), \end{aligned}$$
(41)

where \(\mathbf{Z}=\mathbf{Q}^{-1}\).

Then

$$\begin{aligned}&\dot{\tilde{\mathbf{H}}}_k^H \mathbf{Q}^{-1}\dot{\tilde{\mathbf{H}}}_k \mathbf{R}_\mathbf{S}\nonumber \\&\quad =(\dot{\tilde{\mathbf{H}}}_{k,x} +j\dot{\tilde{\mathbf{H}}}_{k,y} )^{H}(\mathbf{Z}_x +j\mathbf{Z}_y )(\dot{\tilde{\mathbf{H}}}_{k,x} +j\dot{\tilde{\mathbf{H}}}_{k,y} )(\mathbf{R}_{\mathbf{S},x} +j\mathbf{R}_{\mathbf{S},\mathrm{y}} )\nonumber \\&\quad =(\dot{\tilde{\mathbf{H}}}_{k,x}^T \mathbf{Z}_x \dot{\tilde{\mathbf{H}}}_{k,x} +j\dot{\tilde{\mathbf{H}}}_{k,x}^T \mathbf{Z}_y \dot{\tilde{\mathbf{H}}}_{k,x} -j\dot{\tilde{\mathbf{H}}}_{k,y}^T \mathbf{Z}_x \dot{\tilde{\mathbf{H}}}_{k,x} +\dot{\tilde{\mathbf{H}}}_{k,y}^T \mathbf{Z}_y \dot{\tilde{\mathbf{H}}}_{k,x} +j\dot{\tilde{\mathbf{H}}}_{k,x}^T \mathbf{Z}_x \dot{\tilde{\mathbf{H}}}_{k,y} \nonumber \\&\quad \quad -\dot{\tilde{\mathbf{H}}}_{k,x}^T \mathbf{Z}_y \dot{\tilde{\mathbf{H}}}_{k,y} +\dot{\tilde{\mathbf{H}}}_{k,y}^T \mathbf{Z}_x \dot{\tilde{\mathbf{H}}}_{k,y} +j\dot{\tilde{\mathbf{H}}}_{k,y}^T \mathbf{Z}_y \dot{\tilde{\mathbf{H}}}_{k,y} )(\mathbf{R}_{\mathbf{S},x} +j\mathbf{R}_{\mathbf{S},\mathrm{y}} ) \nonumber \\&\quad =\dot{\tilde{\mathbf{H}}}_{k,x}^T \mathbf{Z}_x \dot{\tilde{\mathbf{H}}}_{k,x} \mathbf{R}_{\mathbf{S},x} -\dot{\tilde{\mathbf{H}}}_{k,x}^T \mathbf{Z}_y \dot{\tilde{\mathbf{H}}}_{k,y} \mathbf{R}_{\mathbf{S},x} +\dot{\tilde{\mathbf{H}}}_{k,y}^T \mathbf{Z}_x \dot{\tilde{\mathbf{H}}}_{k,y} \mathbf{R}_{\mathbf{S},x} +\dot{\tilde{\mathbf{H}}}_{k,y}^T \mathbf{Z}_y \dot{\tilde{\mathbf{H}}}_{k,x} \mathbf{R}_{\mathbf{S},x}\nonumber \\&\qquad -\dot{\tilde{\mathbf{H}}}_{k,x}^T \mathbf{Z}_y \dot{\tilde{\mathbf{H}}}_{k,x} \mathbf{R}_{\mathbf{S},{y}} +\dot{\tilde{\mathbf{H}}}_{k,y}^T \mathbf{Z}_x \dot{\tilde{\mathbf{H}}}_{k,x} \mathbf{R}_{\mathbf{S},{y}} -\dot{\tilde{\mathbf{H}}}_{k,x}^T \mathbf{Z}_x \dot{\tilde{\mathbf{H}}}_{k,y} \mathbf{R}_{\mathbf{S},{y}} -\dot{\tilde{\mathbf{H}}}_{k,y}^T \mathbf{Z}_y \dot{\tilde{\mathbf{H}}}_{k,y} \mathbf{R}_{\mathbf{S},{y}}.\nonumber \\ \end{aligned}$$
(42)

Let

$$\begin{aligned} \dot{\mathbf{H}}_{R,k}= & {} \left[ {{\begin{array}{ll} {\dot{\mathbf{H}}_{k,x} }&{}\quad {-\dot{\mathbf{H}}_{k,y} } \\ {\dot{\mathbf{H}}_{k,y} }&{}\quad {\dot{\mathbf{H}}_{k,x} } \\ \end{array} }} \right] ,\;\dot{\tilde{\mathbf{H}}}_{R,k} =\left[ {{\begin{array}{ll} {\dot{\tilde{\mathbf{H}}}_{k,x} }&{}\quad {-\dot{\tilde{\mathbf{H}}}_{k,y} } \\ {\dot{\tilde{\mathbf{H}}}_{k,y} }&{}\quad {\dot{\tilde{\mathbf{H}}}_{k,x} } \\ \end{array} }} \right] ,\;\dot{\varvec{\updelta } }_{R,k} =\left[ {{\begin{array}{ll} {\dot{\varvec{\updelta } }_{k,x} }&{}\quad {-\dot{\varvec{\updelta } }_{k,y} } \\ {\dot{\varvec{\updelta } }_{k,y} }&{}\quad {\dot{\varvec{\updelta } }_{k,x} } \\ \end{array} }} \right] \nonumber \\ \mathbf{R}_{R,\mathbf{S}}= & {} \left[ {{\begin{array}{ll} {\mathbf{R}_{\mathbf{S},x} }&{}\quad {-\mathbf{R}_{\mathbf{S},{y}} } \\ {\mathbf{R}_{\mathbf{S},{y}} }&{}\quad {\mathbf{R}_{\mathbf{S},x} } \\ \end{array} }} \right] ,\;\mathbf{Z}_R =\left[ {{\begin{array}{ll} {\mathbf{Z}_x }&{}\quad {-\mathbf{Z}_y } \\ {\mathbf{Z}_y }&{}\quad {\mathbf{Z}_x } \\ \end{array} }} \right] . \end{aligned}$$
(43)

Then, we have

$$\begin{aligned}&\dot{\tilde{\mathbf{H}}}_{R,k}^T \mathbf{Z}_R \dot{\tilde{\mathbf{H}}}_{R,k} \mathbf{R}_{R,\mathbf{S}} =\left[ {{\begin{array}{ll} {\dot{\tilde{\mathbf{H}}}_{k,x} }&{}\quad {-\dot{\tilde{\mathbf{H}}}_{k,y} } \\ {\dot{\tilde{\mathbf{H}}}_{k,y} }&{}\quad {\dot{\tilde{\mathbf{H}}}_{k,x} } \\ \end{array} }} \right] ^{\mathrm{T}}\left[ {{\begin{array}{ll} {\mathbf{Z}_x }&{}\quad {-\mathbf{Z}_y } \\ {\mathbf{Z}_y }&{}\quad {\mathbf{Z}_x } \\ \end{array} }} \right] \left[ {{\begin{array}{ll} {\dot{\tilde{\mathbf{H}}}_{k,x} }&{}\quad {-\dot{\tilde{\mathbf{H}}}_{k,y} } \\ {\dot{\tilde{\mathbf{H}}}_{k,y} }&{}\quad {\dot{\tilde{\mathbf{H}}}_{k,x} } \\ \end{array} }} \right] \nonumber \\&\qquad \left[ {{\begin{array}{ll} {\mathbf{R}_{\mathbf{S},x} }&{}\quad {-\mathbf{R}_{\mathbf{S},{y}} } \\ {\mathbf{R}_{\mathbf{S},{y}} }&{}\quad {\mathbf{R}_{\mathbf{S},x} } \\ \end{array} }} \right] \nonumber \\&\quad =2(\dot{\tilde{\mathbf{H}}}_{k,x}^T \mathbf{Z}_x \dot{\tilde{\mathbf{H}}}_{k,x} \mathbf{R}_{\mathbf{S},x} -\dot{\tilde{\mathbf{H}}}_{k,x}^T \mathbf{Z}_y \dot{\tilde{\mathbf{H}}}_{k,y} \mathbf{R}_{\mathbf{S},x} +\dot{\tilde{\mathbf{H}}}_{k,y}^T \mathbf{Z}_x \dot{\tilde{\mathbf{H}}}_{k,y} \mathbf{R}_{\mathbf{S},x} +\dot{\tilde{\mathbf{H}}}_{k,y}^T \mathbf{Z}_y \dot{\tilde{\mathbf{H}}}_{k,x} \mathbf{R}_{\mathbf{S},x}\nonumber \\&\qquad -\dot{\tilde{\mathbf{H}}}_{k,x}^T \mathbf{Z}_y \dot{\tilde{\mathbf{H}}}_{k,x} \mathbf{R}_{\mathbf{S},{y}} +\dot{\tilde{\mathbf{H}}}_{k,y}^T \mathbf{Z}_x \dot{\tilde{\mathbf{H}}}_{k,x} \mathbf{R}_{\mathbf{S},{y}} -\dot{\tilde{\mathbf{H}}}_{k,x}^T \mathbf{Z}_x \dot{\tilde{\mathbf{H}}}_{k,y} \mathbf{R}_{\mathbf{S},{y}} -\dot{\tilde{\mathbf{H}}}_{k,y}^T \mathbf{Z}_y \dot{\tilde{\mathbf{H}}}_{k,y} \mathbf{R}_{\mathbf{S},{y}} ).\nonumber \\ \end{aligned}$$
(44)

Therefore, (16) holds immediately.

Besides, because

$$\begin{aligned}&\dot{\tilde{\mathbf{H}}}_{R,k} =\dot{\mathbf{H}}_{R,k} +\dot{\varvec{\updelta } }_{R,k}, \end{aligned}$$
(45)
$$\begin{aligned}&\dot{\varvec{\updelta } }_{R,k} =\left[ {{\begin{array}{ll} {\dot{\varvec{\updelta } }_{k,x} }&{}\quad {-\dot{\varvec{\updelta } }_{k,y} } \\ {\dot{\varvec{\updelta } }_{k,y} }&{}\quad {\dot{\varvec{\updelta } }_{k,x} } \\ \end{array} }} \right] , \end{aligned}$$
(46)

and

$$\begin{aligned} \left\| {\dot{\varvec{\updelta } }_k } \right\| _{\mathrm{F}} \le \sigma _k , \end{aligned}$$
(47)

we can obtain

$$\begin{aligned} \left\| {\dot{\varvec{\updelta } }_{R,k} } \right\| _{\mathrm{F}} \le \gamma _k, \end{aligned}$$
(48)

where

$$\begin{aligned} \gamma _k =\sqrt{2}\sigma _k . \end{aligned}$$
(49)

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Wang, H., Pei, B. & Li, J. Robust Waveform Design for MIMO Radar with Imperfect Prior Knowledge. Circuits Syst Signal Process 35, 1239–1255 (2016). https://doi.org/10.1007/s00034-015-0116-3

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