DMCE-FrPSD-Based Parameter Estimation Method for Bistatic MIMO Radar Under Alpha-Stable Distributed Noise

Abstract

Parameter estimation is an essential step for various MIMO radar applications and attracts the attention of many scholars. However, conventional methods typically model moving objects with a constant Doppler frequency shift under Gaussian distributed noise, which may fail in a real radar electromagnetic environment. To make up for the lack of research in this field, a multi-parameter estimation scheme for bistatic MIMO radar in the presence of time-varying Doppler frequency and impulsive noise is proposed. First, a novel operator which combines the deviation from the median-based correntropy (DMCE) and fractional power spectral density (FrPSD), termed DMCE-FrPSD, is defined, and boundedness proof is given to ensure the effectiveness of the proposed methods. Afterwards, Doppler parameters are estimated by peak searching of the DMCE-FrPSD spectrogram. Furthermore, two algorithms based on DMCE-FrPSD are proposed to estimate direction-of-arrival and direction-of-departure. Compared with the existing algorithms, the proposed scheme can estimate multiple parameters with higher accuracy and lower error even under intense impulsive noise. Comprehensive Monte Carlo simulations are carried out to verify the proposed scheme’s superiority under various impulsive noise conditions.

Appendix A. Boundedness Proof of DMCE Under SαS Noise

The method proposed in Sect. 3 is based on DMCE. Since the second-order statistics will diverge under α-stable distributed noise, conventional second-order statistics-based methods will deteriorate seriously. Therefore, we need to prove the boundedness of DMCE to ensure the effectiveness of the proposed method.

Theorem 1

Assuming that X and Y are two i.i.d SαS random variables, the DMCE-FrPSD between X and Y is bounded.

Proof

Based on (21), we can obtain the DMCE between X and Y as

$$R^{{{\text{DMCE}}}} = E\left[ {\exp \left( { - \frac{{\left| {\left| X \right| - m_{X} } \right| + \left| {\left| Y \right| - m_{Y} } \right|}}{\sigma }} \right)XY} \right]$$

(49)

where $m_{X} = {\text{ med}}\left( X \right)$ and $m_{Y} = {\text{ med}}\left( Y \right)$.

It is apparent that $XY \le \left[ {\max \left( {X,Y} \right)} \right]^{2}$. Assuming $\left| X \right| \ge \left| Y \right|$, we can get

$$\begin{aligned} \left| {R^{{{\text{DMCE}}}} } \right| & \le E\left[ {\exp \left( { - \frac{{\left| {\left| X \right| - m_{X} } \right| + \left| {\left| Y \right| - m_{Y} } \right|}}{\sigma }} \right)\left| X \right|^{2} } \right] \\ & \le E\left[ {\exp \left( { - \frac{{\left| {\left| X \right| - m_{X} } \right|}}{\sigma }} \right)\left| X \right|^{2} } \right] \\ \end{aligned}$$

(50)

Substituting the characteristic function (11) into (48), we can get

$$\begin{aligned} \left| {R^{{{\text{DMCE}}}} } \right| & \le E\left[ {\exp \left( { - \frac{{\left| {\left| X \right| - m_{X} } \right|}}{\sigma }} \right)\left| X \right|^{2} } \right] \\ & = \frac{1}{2\pi }\int\limits_{ - \infty }^{ + \infty } {\int\limits_{ - \infty }^{ + \infty } {{\text{e}}^{{ - \frac{{\left| {\left| X \right| - m_{X} } \right|}}{\sigma }}} } \left| X \right|^{2} {\text{e}}^{{j\eta \omega - \gamma \left| \omega \right|^{\alpha } }} {\text{e}}^{ - j\omega X} {\text{d}}X{\text{d}}} \omega \\ & \le \frac{1}{\pi }\int\limits_{0}^{ + \infty } {\int\limits_{ - \infty }^{ + \infty } {{\text{e}}^{{ - \frac{{\left| {X - m_{X} } \right|}}{\sigma }}} } X^{2} {\text{e}}^{{ - \gamma \left| \omega \right|^{\alpha } }} {\text{d}}\omega {\text{d}}X} \\ & = \frac{1}{\pi }\int\limits_{0}^{ + \infty } {{\text{e}}^{{ - \frac{{\left| {X - m_{X} } \right|}}{\sigma }}} } X^{2} {\text{d}}X\int\limits_{ - \infty }^{ + \infty } {{\text{e}}^{{ - \gamma \left| \omega \right|^{\alpha } }} } {\text{d}}\omega \\ \end{aligned}$$

(51)

where $\int\limits_{ - \infty }^{ + \infty } {{\text{e}}^{{ - \gamma \left| \omega \right|^{\alpha } }} } {\text{d}}\omega = 2\int\limits_{0}^{ + \infty } {{\text{e}}^{{ - \gamma \omega^{\alpha } }} } {\text{d}}\omega = \frac{2}{\alpha }\gamma^{{ - \frac{1}{\alpha }}} \Gamma \left( {\frac{1}{\alpha }} \right) = h$ and $\Gamma \left( x \right) = \int\limits_{0}^{ + \infty } {t^{x - 1} {\text{e}}^{ - t} {\text{d}}t}$ denotes the gamma function.

Then, (51) can be simplified as

$$\left| {R^{{{\text{DMCE}}}} } \right| \le \frac{h}{\pi }\int\limits_{0}^{ + \infty } {{\text{e}}^{{ - \frac{{\left| {X - m_{X} } \right|}}{\sigma }}} } X^{2} {\text{d}}X$$

(52)

If $m_{X} \le 0$, (52) can further be expressed as

$$\begin{aligned} \left| {R^{{{\text{DMCE}}}} } \right| & \le \frac{h}{\pi }\int\limits_{0}^{ + \infty } {{\text{e}}^{{ - \frac{{X - m_{X} }}{\sigma }}} } X^{2} {\text{d}}X \\ & = \frac{h}{\pi }{\text{e}}^{{\frac{{m_{X} }}{\sigma }}} \int\limits_{0}^{ + \infty } {{\text{e}}^{{ - \frac{X}{\sigma }}} } X^{2} \,{\text{d}}X \\ & = \frac{{2h\sigma^{3} }}{\pi }{\text{e}}^{{\frac{{m_{X} }}{\sigma }}} < + \infty \\ \end{aligned}$$

(53)

If $m_{X} > 0$, (52) can further be expressed as

$$\begin{aligned} \left| {R^{{{\text{DMCE}}}} } \right| & \le \frac{h}{\pi }\int\limits_{0}^{ + \infty } {{\text{e}}^{{ - \frac{{\left| {X - m_{X} } \right|}}{\sigma }}} } X^{2} {\text{d}}X \\ & = \frac{h}{\pi }e^{{ - \frac{{m_{X} }}{\sigma }}} \int\limits_{0}^{{m_{X} }} {{\text{e}}^{{\frac{X}{\sigma }}} } X^{2} {\text{d}}X + \frac{h}{\pi }{\text{e}}^{{\frac{{m_{X} }}{\sigma }}} \int\limits_{{m_{X} }}^{ + \infty } {{\text{e}}^{{ - \frac{X}{\sigma }}} } X^{2} {\text{d}}X \\ & < \frac{h}{\pi }e^{{ - \frac{{m_{X} }}{\sigma }}} \int\limits_{0}^{{m_{X} }} {{\text{e}}^{{\frac{X}{\sigma }}} } X^{2} {\text{d}}X + \frac{{2h\sigma^{3} }}{\pi }{\text{e}}^{{\frac{{m_{X} }}{\sigma }}} < + \infty \\ \end{aligned}$$

(54)

According to (53) and (54), we can infer that the DMCE between X and Y is bounded. This is the end of the proof.