Cognitive Design of Radar Waveform and the Receive Filter for Multitarget Parameter Estimation

Abstract

This research work considers waveform design for an adaptive radar system. The aim is to achieve enhanced feature extraction performance for multiple extended targets. There are two scenarios to consider: multiple extended targets separated in range and multiple extended targets close in range. We propose a waveform optimization scheme based on Kalman filtering by minimizing the mean square error of separated target scattering coefficient estimation and a waveform optimization approach by minimizing the mean square error of closed power spectrum density estimation. A convex cost function is established, and the optimal solution can be obtained using the existing convex programming algorithm. With subsequent iterations of the algorithm, the simulation results demonstrate an improvement in the estimation of target parameters from the dynamic scene, such as target scattering coefficient and power spectrum density, while maintaining relatively lower computational complexity.

Appendices

Appendix A: Derivation of the Probability Constraints in (12)

Since the true TSC is unknown, the estimate of TSC based on Kalman filtering is used to replace the true TSC to design the radar waveform. We assume that $H_{1}$ and $H_{0}$ are the presence and absence of a target, respectively. Then, the distribution of backscattered signals can be expressed as

$$ \begin{aligned} & \left. {{\mathbf{Y}}_{i} } \right|H_{0} \sim{\mathbf{N}}\left\{ {0,{\mathbf{C}}_{N} } \right\}, \\ & \left. {{\mathbf{Y}}_{i} } \right|H_{1} \sim{\mathbf{N}}\left\{ {{\mathbf{\rm Z}}_{i} {\hat{\mathbf{G}}}_{i} ,{\mathbf{C}}_{N} } \right\}, \\ \end{aligned} $$

(31)

where ${\hat{\mathbf{G}}}_{i}$ is the estimate of TSC. The likelihood estimation is

$$ \begin{aligned} & l\left( {{\mathbf{Y}}_{i}^{{}} } \right) = \frac{{{\mathbf{N}}\left\{ {{\mathbf{\rm Z}}_{i} {\hat{\mathbf{G}}}_{i} ,{\mathbf{C}}_{N} } \right\}}}{{{\mathbf{N}}\left\{ {0,{\mathbf{C}}_{N} } \right\}}}\mathop { \lessgtr }\limits_{{H_{2} }}^{{H_{1} }} T \\ & \Leftrightarrow {\mathbf{Y}}_{i}^{H} {\mathbf{C}}_{N}^{ - 1} {\mathbf{\rm Z}}_{i} {\hat{\mathbf{G}}}_{i} \mathop { \lessgtr }\limits_{{H_{2} }}^{{H_{1} }} T, \\ \end{aligned} $$

(32)

where $T$ is the detection threshold. The false alarm probability of CFAR detection is

$$ \begin{aligned} P_{fa} & = P\left( {{\mathbf{W}}_{{}}^{H} {\mathbf{C}}_{N}^{ - 1} {\mathbf{\rm Z}}_{i} {\hat{\mathbf{G}}}_{i} \ge T} \right) \\ & = Q\left( {T/\sqrt {\left( {{\mathbf{\rm Z}}_{i} {\hat{\mathbf{G}}}_{i} } \right)^{H} {\mathbf{C}}_{N}^{ - 1} {\mathbf{\rm Z}}_{i} {\hat{\mathbf{G}}}_{i} } } \right). \\ \end{aligned} $$

(33)

${\text{Q}}\left( . \right)$ is the Q-function. The detection threshold is $T = Q^{ - 1} \left( {P_{fa} } \right)\sqrt {\left( {{\mathbf{\rm Z}}_{i} {\hat{\mathbf{G}}}_{i} } \right)^{H} {\mathbf{C}}_{N}^{ - 1} {\mathbf{\rm Z}}_{i} {\hat{\mathbf{G}}}_{i} }$. The probability of detection can be rewritten as

$$ \begin{aligned} P_{d} & = P\left( {\left( {{\mathbf{\rm Z}}_{i} {\hat{\mathbf{G}}}_{i} + {\mathbf{W}}} \right)^{H} {\mathbf{C}}_{N}^{ - 1} {\mathbf{\rm Z}}_{i} {\hat{\mathbf{G}}}_{i} \ge T} \right) \\ & = Q\left( {Q^{ - 1} \left( {P_{fa} } \right) - \sqrt {\left( {{\mathbf{\rm Z}}_{i} {\hat{\mathbf{G}}}_{i} } \right)^{H} {\mathbf{C}}_{N}^{ - 1} {\mathbf{\rm Z}}_{i} {\hat{\mathbf{G}}}_{i} } } \right). \\ \end{aligned} $$

(34)

Since the Q-function describes a monotonically decreasing function, the expression $P_{d} \ge \varepsilon$ can be rewritten as

$$ {\mathbf{\rm Z}}_{i}^{H} {\hat{\mathbf{G}}}_{i}^{H} {\mathbf{C}}_{N}^{ - 1} {\hat{\mathbf{G}}}_{i} {\mathbf{\rm Z}}_{i}^{{}} \ge \varepsilon^{'} . $$

(35)

Appendix B: Derivation of (26)

The estimation error of $P_{{q_{j,i} }} \left( {f_{p} } \right)$ can be expressed as

$$ \begin{aligned} e\left( {f_{p} } \right) & = P_{{q_{j,i} }} \left( {f_{p} } \right) - \hat{P}_{{q_{j,i} }} \left( {f_{p} } \right) \\ & = \frac{{G_{C} \left( {f_{p} } \right) - \hat{P}_{C} \left( {f_{p} } \right) - 2\text{Re} \left( {\sqrt {\eta_{j} } Q_{j,i} \left( {f_{p} } \right)F_{i} \left( {f_{p} } \right)C\left( {f_{p} } \right)} \right)}}{{P_{{f_{i} }} \left( {f_{p} } \right)}}. \\ \end{aligned} $$

(36)

The MSE of the jth target PSD estimation at time $i$ can be expressed by

$$ \begin{aligned} E\left( {\left\| {e\left( f \right)} \right\|_{2}^{2} } \right) & = \sum\limits_{p = 1}^{P} {E\left( {P_{{q_{j,i} }} \left( {f_{p} } \right) - \hat{P}_{{q_{j,i} }} \left( {f_{p} } \right)} \right)}^{2} \\ & = \sum\limits_{p = 1}^{P} {\frac{1}{{P_{{f_{i} }}^{2} \left( {f_{p} } \right)}}} E\left\{ {G_{C} \left( {f_{p} } \right) - \hat{P}_{C} \left( {f_{p} } \right) - 2\text{Re} \left( {\sqrt {\eta_{j} } Q_{j,i} \left( {f_{p} } \right)F_{i} \left( {f_{p} } \right)C\left( {f_{p} } \right)} \right)} \right\}^{2} \\ & { = }\sum\limits_{p = 1}^{P} {\frac{1}{{P_{{f_{i} }}^{2} \left( {f_{p} } \right)}}} E\left( \begin{aligned} \left({G_{C} \left( {f_{p} } \right) - \hat{P}_{C} \left( {f_{p} } \right)} \right)^{2} + 4\eta_{j} \left( {\text{Re} \left( {Q_{j,i} \left( {f_{p} } \right)F_{i} \left( {f_{p} } \right)C\left( {f_{p} } \right)} \right)} \right)^{2} \hfill \\ - 4\text{Re} \left( {\sqrt {\eta_{j} } Q_{j,i} \left( {f_{p} } \right)F_{i} \left( {f_{p} } \right)C\left( {f_{p} } \right)} \right)\left( {G_{C} \left( {f_{p} } \right) - \hat{P}_{C} \left( {f_{p} } \right)} \right) \hfill \\ \end{aligned} \right). \\ \end{aligned} $$

(37)

In (37), we have

$$ E\left( {\left( {G_{C} \left( {f_{p} } \right) - \hat{P}_{C} \left( {f_{p} } \right)} \right)^{2} } \right) = G_{C}^{2} \left( {f_{p} } \right). $$

(38)

$$ E\left( {\text{Re} \left( {Q_{j,i} \left( {f_{p} } \right)F_{i} \left( {f_{p} } \right)C\left( {f_{p} } \right)} \right)\left( {G_{C} \left( {f_{p} } \right) - \hat{P}_{C} \left( {f_{p} } \right)} \right)} \right) = 0. $$

(39)

$$ E\left( {4\eta_{j} \text{Re} \left( {Q_{j,i} \left( {f_{p} } \right)F_{i} \left( {f_{p} } \right)C\left( {f_{p} } \right)} \right)^{2} } \right) = 2\eta_{j} G_{C}^{{}} \left( {f_{p} } \right)P_{{q_{j,i} }}^{{}} \left( {f_{p} } \right)P_{{f_{i} }}^{{}} \left( {f_{p} } \right). $$

(40)

Substituting (38)–(40) into (37), we have

$$ E\left( {\left\| {e\left( f \right)} \right\|_{2}^{2} } \right) = \sum\limits_{p = 1}^{P} {\left[ {\frac{{G_{C}^{2} \left( {f_{p} } \right)}}{{P_{{f_{i} }}^{2} \left( {f_{p} } \right)}} + \frac{{2\eta_{j} G_{C} \left( {f_{p} } \right)P_{{q_{j,i} }}^{{}} \left( {f_{p} } \right)}}{{P_{{f_{i} }}^{{}} \left( {f_{p} } \right)}}} \right]} . $$

(41)