Parameter Estimation Based on Fractional Power Spectrum Density in Bistatic MIMO Radar System Under Impulsive Noise Environment

Abstract

This paper takes an Alpha-stable distribution as the noise model to solve the parameter estimation problem of bistatic multiple-input multiple-output (MIMO) radar system in the impulsive noise environment. For a moving target, its echo often contains a time-varying Doppler frequency. Furthermore, the echo signal may be corrupted by a non-Gaussian noise. It causes the conventional algorithms and signal models degenerating severely in this case. Thus, this paper proposes a new signal model and a novel method for parameter estimation in bistatic MIMO radar system in the impulsive noise environment. It combines the fractional lower-order statistics (FLOS) and fractional power spectrum density (FPSD), for suppressing the impulse noise and estimating parameters of the target in fractional Fourier transform domain. Firstly, a new signal array model is constructed based on the $\alpha$-stable distribution model. Secondly, Doppler parameters are jointly estimated by peak searching of the FLOS–FPSD. Furthermore, two modified algorithms are proposed for the estimation of direction-of-departure and direction-of-arrival (DOA), including the fractional power spectrum density based on MUSIC algorithm (FLOS–FPSD–MUSIC) and the fractional lower-order ambiguity function based on ESPRIT algorithm (FLOS–FPSD–ESPRIT). Simulation results are presented to verity the effectiveness of the proposed method.

Appendix a Derivation of the Expression \(MP_{yz,n}^{\left( p \right) } \)

According to (17) and (19), we can define a variable $z_l \left( t \right)$ as

$$\begin{aligned} z_l \left( t \right) =\sigma _l \exp \left( {j2\pi \left( {f_l t+{\mu _l t^{2}}/2} \right) } \right) +w\left( t \right) . \end{aligned}$$

(32)

Supposing the noise $w_n (t)$ of (14) and (20) is independent and zero-mean Gaussian white noise, the fractional correlation functions $\hat{{R}}_{yz,qnl}^\rho \left( \tau \right)$ between $y_{qnl} \left( t \right)$ and $z_l \left( t \right)$ are defined as

$$\begin{aligned} \hat{{R}}_{yz,qnl}^\rho \left( \tau \right)= & {} \frac{1}{N}\sum _{-N/2}^{N/2-1} {y_{qnl} \left( {t+\tau } \right) z_l ^{*}\left( t \right) } \exp \left( {jt\tau \cot \rho } \right) \nonumber \\= & {} \frac{1}{N}\sum _{-N/2}^{N/2-1} {\exp } \left( {j\left( {2\pi \mu _l +\cot \rho } \right) t\tau } \right) \nonumber \\&\exp \left( {j2\pi \left( {f_l \tau + \frac{1}{2}\mu _l \tau ^{2}} \right) } \right) A_q \left( {\varphi _l } \right) {{B}}_n \left( {\theta _l } \right) +\hat{{R}}_{yw}^\rho \left( \tau \right) \end{aligned}$$

(33)

where $\hat{{R}}_{yw}^\rho \left( \tau \right)$ is treated as a random interference.

The FPSD $P_{yz,qnl}^\rho \left( m \right)$ between $y_{qnl} \left( t \right)$ and $z_l \left( t \right)$ can be expressed as

$$\begin{aligned} P_{yz,qnl}^\rho \left( m \right)= & {} A_{-\rho } F^{\rho }\left[ {\hat{{R}}_{yz,qnl}^{\rho _0 } \left( \tau \right) } \right] \left( m \right) \exp \left( {{-jm^{2}\cot \rho }/2} \right) \nonumber \\= & {} \sqrt{\frac{1}{2\pi }}A_{-\rho } A_\rho \int _{-\frac{T}{2}}^{+\frac{T}{2}} \exp \left( j\left( \tau ^{2}\left( {{\cot \rho }/2 +\pi a_2 } \right) \right. \right. \nonumber \\&\left. \left. +\tau \left( {-m\csc \rho +2\pi a_1 } \right) \right) \right) {\text {d}}\tau +P_{yw}^\rho \left( m \right) \end{aligned}$$

(34)

where $P_{yw}^\rho \left( m \right)$ denotes the fractional power spectrum of the noise. The $P_{yz,qnl}^\rho \left( m \right)$ can increase the peak value when both $\rho$ and m meet the conditions shown in (11), and the position of the peak is also located at $\left( {\rho _l ,m_l } \right)$. The peak value of $P_{yz,qnl}^\rho \left( m \right)$ is

$$\begin{aligned} P_{ss}^{\rho _l } \left( {m_l } \right) =\sqrt{\frac{1}{2\pi }}A_{-\rho _l } A_{\rho _l } TA_q \left( {\varphi _l } \right) {{B}}_n \left( {\theta _l } \right) \end{aligned}$$

(35)

According to (17), (35) can be expressed as

$$\begin{aligned} P_{yz,qnl}^{\rho _l } \left( {m_l } \right) =A_q \left( {\varphi _l } \right) {{B}}_n \left( {\theta _l } \right) P_{yy,qnl}^{\rho _l } \left( {m_l } \right) \end{aligned}$$

(36)

When the noise in (14) and (20) are sequences of i.i.d isotropic complex $S\alpha S$ random variables with $1<\alpha \le 2$, we can obtain the FLOS–FPSD $P_{yy,qnl}^{\left( p \right) } \left( {\rho _l ,m_l } \right)$ and $P_{yz,qnl}^{\left( p \right) } \left( {\rho _l ,m_l } \right)$. According to (36), we have

$$\begin{aligned} P_{yz,qnl}^{\left( p \right) } \left( {\rho _l ,m_l } \right) =A_q \left( {\varphi _l } \right) {{B}}_n \left( {\theta _l } \right) P_{yy,qnl}^{\left( p \right) } \left( {\rho _l ,m_l } \right) \end{aligned}$$

(37)

At $(\rho _l ,m_l )$, the FLOS–FPSD $MP_{yz,qnl}^{\left( p \right) } \left( {\rho _l ,m_l } \right)$ between (2) and (20) can be expressed as

$$\begin{aligned} MP_{yz,qnl}^{\left( p \right) } \left( {\rho _l ,m_l } \right) =P_{yz,qnl}^{\left( p \right) } \left( {\rho _l ,m_l } \right) +\sum _{\kappa \ne l}^L {P_{yz,l\kappa }^{\left( p \right) } \left( {\rho _l ,m_l } \right) } \end{aligned}$$

(38)

Because the amplitudes of FLOS–FPSD of different targets are very low at $(\rho _l ,m_l )$, these signals are not considered as random interfering. Therefore, (38) can be rewritten as

$$\begin{aligned} MP_{yz,qnl}^{\left( p \right) } \left( {\rho _l ,m_l } \right) =A_q \left( {\varphi _l } \right) B_n \left( {\theta _l } \right) P_{yy,qnl}^{\left( p \right) } \left( {\rho _l ,m_l } \right) \end{aligned}$$

(39)

Selecting the data of L peak points $MP_{yz,qnl}^{\left( p \right) } \left( {m_l } \right)$ for $l=1,\ldots ,L$ as observed data at the receiver, the output of the nth receive antenna in the FRFT domain can be expressed as

$$\begin{aligned} {\varvec{MP}}_{yz,n}^{\left( p \right) } = \left[ \begin{array}{cccc} {MP_{yz,n1}^{\left( p \right) } \left( {\rho _1 ,m_1 } \right) }&{MP_{yz,n2}^{\left( p \right) } \left( {\rho _2 ,m_2 } \right) }&{\ldots }&{MP_{yz,nL}^{\left( p \right) } \left( {\rho _L ,m_L } \right) } \end{array} \right] \end{aligned}$$

(40)

The vector of the receiver outputs can be modeled as

$$\begin{aligned} {\varvec{R}}={\varvec{GAB}}+{\varvec{N}} \end{aligned}$$

(41)

where ${\varvec{R}}=\left[ \begin{array}{cccc} {{\varvec{MP}}_{yz,1}^{\left( p \right) } }&{{\varvec{MP}}_{yz,2}^{\left( p \right) } }&{\ldots }&{{\varvec{MP}}_{yz,N}^{\left( p \right) } } \end{array} \right] ^{T}$, ${\varvec{G}}=\hbox {diag} \left\{ {P_{yy,qn1}^{\left( p \right) } \left( {\rho _1 ,m_1 } \right) ,} {\ldots } {,P_{yy,qnL}^{\left( p \right) } \left( {\rho _L ,m_L } \right) } \right\}$, ${\varvec{A}}=\hbox {diag}\left\{ \begin{array}{lll} {A_q \left( {\varphi _1 } \right) ,}&{\ldots }&{,A_q \left( {\varphi _L } \right) } \end{array} \right\}$ and ${\varvec{B}}=\left[ \begin{array}{cccc} {{\varvec{B}}_1 }&{{\varvec{B}}_2 }&{\ldots }&{{\varvec{B}}_L } \end{array} \right]$, where ${\varvec{B}}_l = \left[ \begin{array}{ccc} {{{B}}_1 \left( {\varphi _l } \right) }&{\ldots }&{{{B}}_N \left( {\varphi _l } \right) } \end{array} \right] ^{T}$, $\left( \right) ^{T}$ and $\hbox {diag}\left( \bullet \right)$ denote transpose and diagonal matrix, respectively.