Waveform Optimization for Target Scattering Coefficients Estimation Under Detection and Peak-to-Average Power Ratio Constraints in Cognitive Radar

Abstract

This work investigates the estimation of target scattering coefficients (TSC) in cognitive radar systems with temporally correlated targets. An estimation method based on Kalman filtering (KF) is proposed to exploit the temporal TSC correlation between the pulses in the frequency domain. To minimize the mean square error of the estimated TSC at each KF iteration, unlike existing indirect methods, in this paper the radar waveform is optimized directly under the constraints of transmitted power, peak-to-average power ratio (PAPR) and detection probability. Since the optimization problem regarding the waveform design is non-convex, a novel method is proposed to convert this problem into a convex one. Simulation results demonstrate that the performance of the TSC estimation for the temporally correlated target is significantly improved by radar waveform optimization. Meanwhile, no performance degradation is observed with the introduction of the additional PAPR constraints and the detection constraints for KF estimation with the optimized waveform.

Appendices

Appendix 1: Derivation of the Objective Function

TSC estimation based on the MAP can be written as

$$\begin{aligned} \hat{\mathbf {g}}_k=\arg \max _{\mathbf {g}_k}~p\left( \left. \mathbf {g}_k\right| \mathbf {y}_k\right) , \end{aligned}$$

(60)

where the probability distribution of the TSC $\mathbf {g}_k$ given the echo waveform $\mathbf {y}_k$ can be written as

$$\begin{aligned} p\left( \left. \mathbf {g}_k\right| \mathbf {y}_k\right) = \frac{p\left( \mathbf {g}_k,\mathbf {y}_k\right) }{p\left( \mathbf {y}_k\right) } = \frac{p\left( \left. \mathbf {y}_k\right| \mathbf {g}_k\right) p \left( \mathbf {g}_k\right) }{p\left( \mathbf {y}_k\right) }. \end{aligned}$$

(61)

The probability distribution of the echo waveform $\mathbf {y}_k$ given the TSC $\mathbf {g}_k$ is

$$\begin{aligned} p\left( \left. \mathbf {y}_k\right| \mathbf {g}_k\right) = \frac{1}{\left( 2\pi \right) ^{\frac{M}{2}}\det \left( \mathbf {R}_N\right) ^{\frac{1}{2}}}e^{-\frac{1}{2} \left( \mathbf {y}_k-\mathbf {Z}_k\mathbf {g_k}\right) ^H\mathbf {R}^{-1}_N \left( \mathbf {y}_k-\mathbf {Z}_k\mathbf {g_k}\right) }. \end{aligned}$$

(62)

The probability distribution of the TSC $\mathbf {g}_k$ is

$$\begin{aligned} p\left( \mathbf {g}_k\right) =\frac{1}{\left( 2\pi \right) ^{\frac{M}{2}} \det \left( \mathbf {R}_T\right) ^{\frac{1}{2}}}e^{-\frac{1}{2} \mathbf {g}_k^H\mathbf {R}^{-1}_T\mathbf {g}_k}, \end{aligned}$$

(63)

and the probability distribution of the echo waveform $\mathbf {y}_k$ is

$$\begin{aligned} p\left( \mathbf {y}_k\right) =\frac{1}{\left( 2\pi \right) ^{\frac{M}{2}} \det \left( \mathbf {R}_k\right) ^{\frac{1}{2}}}e^{-\frac{1}{2} \mathbf {y}_k^H\mathbf {R}^{-1}_k\mathbf {y}_k}. \end{aligned}$$

(64)

Then, we substitute (62), (63) and (64) into (61) and derive

$$\begin{aligned} p\left( \left. \mathbf {g}_k\right| \mathbf {y}_k\right)&= \frac{p\left( \left. \mathbf {y}_k\right| \mathbf {g}_k\right) p\left( \mathbf {g}_k\right) }{p\left( \mathbf {y}_k\right) }\\&= \frac{e^{-\frac{1}{2}\left( \mathbf {y}_k-\mathbf {Z}_k\mathbf {g_k}\right) ^H\mathbf {R}^{-1}_N\left( \mathbf {y}_k-\mathbf {Z}_k\mathbf {g_k}\right) } e^{-\frac{1}{2}\mathbf {g}_k^H\mathbf {R}^{-1}_T\mathbf {g}_k}}{\left( 2\pi \right) ^{\frac{M}{2}}\sqrt{\frac{\det \left( \mathbf {R}_T\right) \det \left( \mathbf {R}_N\right) }{\det \left( \mathbf {R}_k\right) }}e^{-\frac{1}{2}\mathbf {y}_k^H\mathbf {R}^{-1}_k\mathbf {y}_k}}\nonumber \\&= \frac{e^{-\frac{1}{2}f\left( \mathbf {g}_k\right) }}{\left( 2\pi \right) ^{\frac{M}{2}}\sqrt{\frac{\det \left( \mathbf {R}_T\right) \det \left( \mathbf {R}_N\right) }{\det \left( \mathbf {R}_k\right) }}},\nonumber \end{aligned}$$

(65)

where

$$\begin{aligned} f\left( \mathbf {g}_k\right)&\triangleq \left( \mathbf {y}_k-\mathbf {Z}_k\mathbf {g}_k\right) ^H\mathbf {R}^{-1}_N\left( \mathbf {y}_k-\mathbf {Z}_k\mathbf {g_k}\right) +\mathbf {g}_k^H\mathbf {R}^{-1}_T\mathbf {g}_k-\mathbf {y}_k^H\mathbf {R}^{-1}_k\mathbf {y}_k\\&= \mathbf {y}^H_k\mathbf {R}^{-1}_N\mathbf {y}_k-\mathbf {g}^H_k\mathbf {Z}^H_k\mathbf {R}^{-1}_N\mathbf {y}_k-\mathbf {y}^H_k\mathbf {R}^{-1}_N\mathbf {Z}_k\mathbf {g_k}+\mathbf {g}^H_k\mathbf {Z}^H_k\mathbf {R}^{-1}_N\mathbf {Z}_k\mathbf {g_k}\nonumber \\&\quad +\mathbf {g}_k^H\mathbf {R}^{-1}_T\mathbf {g}_k-\mathbf {y}_k^H\mathbf {R}^{-1}_k\mathbf {y}_k.\nonumber \end{aligned}$$

(66)

Therefore, the TSC $\mathbf {g}_k$ that can maximize the posteriori probability $p\left( \left. \mathbf {g}_k\right| \mathbf {y}_k\right)$ can also minimize $f\left( \mathbf {g}_k\right)$. (60) can then be simplified as

$$\begin{aligned} \hat{\mathbf {g}}_k&=\arg \max _{\mathbf {g}_k}~p\left( \left. \mathbf {g}_k\right| \mathbf {y}_k\right) \\&= \arg \min _{\mathbf {g}_k}~f\left( \mathbf {g}_k\right) \nonumber \\&= \arg \min _{\mathbf {g}_k}~\mathbf {y}^H_k\mathbf {R}^{-1}_N\mathbf {y}_k-\mathbf {g}^H_k\mathbf {Z}^H_k\mathbf {R}^{-1}_N\mathbf {y}_k-\mathbf {y}^H_k\mathbf {R}^{-1}_N\mathbf {Z}_k\mathbf {g_k}\nonumber \\&\quad +\mathbf {g}^H_k\mathbf {Z}^H_k\mathbf {R}^{-1}_N\mathbf {Z}_k\mathbf {g_k}+\mathbf {g}_k^H\mathbf {R}^{-1}_T\mathbf {g}_k-\mathbf {y}_k^H\mathbf {R}^{-1}_k\mathbf {y}_k\nonumber \\&= \arg \min _{\mathbf {g}_k}~\mathbf {g}^H_k\left( \mathbf {Z}_k^H\mathbf {R}^{-1}_N\mathbf {Z}_k+\mathbf {R}^{-1}_T\right) \mathbf {g}_{k}-\mathbf {y}^H_k\mathbf {R}^{-1}_N\mathbf {Z}_k\mathbf {g}_k-\mathbf {g}^H_k\mathbf {Z}_k^H\mathbf {R}^{-1}_N\mathbf {y}_k,\nonumber \end{aligned}$$

(67)

which is (7).

Appendix 2: Simplification of the Optimization Problem

First, we present the derivation of (38). In (31), the objective function is

$$\begin{aligned} \mathbf {z}^*_k&=\arg \min _{\mathbf {z}_k}~{\text {Tr}}\left\{ \left( \mathbf {P}^{-1}_{\left. k\right| k}+\mathbf {V}_k\circ \mathbf {R}^{-1}_N\right) ^{-1}\right\} \left( \text {where } \mathbf {V}_k\triangleq \left( \mathbf {z}_k\mathbf {z}_k^H\right) ^T \right) \\&= \arg \min _{\mathbf {z}_k}~{\text {Tr}}\left\{ \left( \mathbf {P}^{-1}_{\left. k\right| k}+\left( \mathbf {z}_k\mathbf {z}_k^H\right) ^T\circ \mathbf {R}^{-1}_N\right) ^{-1}\right\} .\nonumber \end{aligned}$$

(68)

Then, by using the Fourier transform $\mathbf {z}_k=\mathbf {Fs}_k$, this objective function can be rewritten in time domain as:

$$\begin{aligned} \mathbf {s}^*_k&= \arg \min _{\mathbf {s}_k}~{\text {Tr}}\left\{ \left( \mathbf {P}^{-1}_{\left. k\right| k}+\left( \mathbf {Fs}_k\mathbf {s}^H_k\mathbf {F}^H\right) ^T\circ \mathbf {R}^{-1}_N\right) ^{-1}\right\} . \end{aligned}$$

(69)

In (38), we define $\mathbf {W}_k\triangleq \mathbf {s}_k\mathbf {s}_k^H$. Then the objective function to attain the optimal waveform matrix $\mathbf {W}_k$ is

$$\begin{aligned} \mathbf {W}_k^*=&\arg \min _{\mathbf {W}_k}~{\text {Tr}}\left\{ \left[ \mathbf {P}^{-1}_{\left. k\right| k-1}+\left( \mathbf {F}\mathbf {W}_k\mathbf {F}^H\right) ^T\circ \mathbf {R}^{-1}_N\right] ^{-1}\right\} . \end{aligned}$$

(70)

The constraints in (38) can be represented by the waveform matrix $\mathbf {W}_k$. The first power constraint is equivalent to

$$\begin{aligned} {\text {Tr}}\left\{ \mathbf {W}_k\right\} = \mathbf {s}_k^H\mathbf {s}_k=E_s. \end{aligned}$$

(71)

The second target detection constraint is

$$\begin{aligned} p\left( \mathbf {z}_k\right)&=\mathbf {z}_k^H \hat{\mathbf {G}}^H_k\mathbf {R}^{-1}_N\hat{\mathbf {G}}_k\mathbf {z}_k'\\&= {\text {Tr}}\left\{ \mathbf {z}_k^H\hat{\mathbf {G}}^H_k\mathbf {R}^{-1}_N\hat{\mathbf {G}}_k\mathbf {z}_k\right\} \nonumber \\&={\text {Tr}}\left\{ \hat{\mathbf {G}}^H_k\mathbf {R}^{-1}_N\hat{\mathbf {G}}_k\mathbf {F}\mathbf {W}_k\mathbf {F}^H\right\} \ge \epsilon '.\nonumber \end{aligned}$$

(72)

And the third PAPR constraint can be written as

$$\begin{aligned} {\text {diag}}\left\{ \mathbf {W}_k\right\} \le \zeta 'E_s. \end{aligned}$$

(73)

Therefore, by combining the objective function (70) with the three constraints (71), (72) and (73), we derive the optimization problem and obtain the optimized waveform matrix $\mathbf {W}_k^*$ as

$$\begin{aligned} \mathbf {W}_k^*=&\arg \min _{\mathbf {W}_k}~{\text {Tr}}\left\{ \left[ \mathbf {P}^{-1}_{\left. k\right| k-1}+\left( \mathbf {F}\mathbf {W}_k\mathbf {F}^H\right) ^T\circ \mathbf {R}^{-1}_N\right] ^{-1}\right\} \nonumber \\ \text {s.t. }&{\text {Tr}}\left\{ \mathbf {W}_k\right\} =E_s\\&{\text {Tr}}\left\{ \hat{\mathbf {G}}^H_k\mathbf {R}^{-1}_N\hat{\mathbf {G}}_k\mathbf {F}\mathbf {W}_k\mathbf {F}^H\right\} \ge \epsilon '\nonumber \\&{\text {diag}}\left\{ \mathbf {W}_k\right\} \le \zeta 'E_s,\nonumber \end{aligned}$$

(74)

which is (38).

Second, if we only consider the transmitted power constraint, (74) reduces to

$$\begin{aligned} \mathbf {W}^*_k=&\arg \min _{\mathbf {W}}~{\text {Tr}}\left\{ \left[ \mathbf {P}^{-1}_{\left. k\right| k-1}+\left( \mathbf {F}\mathbf {W}\mathbf {F}^H\right) ^T\circ \mathbf {R}^{-1}_N\right] ^{-1}\right\} \nonumber \\ \text {s.t. }&{\text {Tr}}\left\{ \mathbf {W}\right\} =E_s, \end{aligned}$$

(75)

which is (44). If we add the target detection constraint, we can obtain the optimization problem in (50).