Parametric matched filter based on interference iteration

Abstract

In this paper, an iteration algorithm based on interference iteration is proposed and referred to as I-PMF. Derived from the parametric matched filter (PMF), I-PMF has a low computational cost and reveals not only an iteration relationship between the autoregressive (AR) coefficient matrices and interferences in mathematics, but also a mechanism for PMF in suppressing interferences. Then the iteration of I-PMF is utilized in an adaptive method, namely I-PAMF. The average minimum output signal-to-interference-and-noise ratio (SINR) of I-PAMF is analyzed with separated interferences and verified by numerical results. It is shown that the computational cost of the proposed iterative algorithm, i.e. I-PMF, is low and the detection performance of the proposed adaptive iterative algorithm, i.e. I-PAMF, outperforms most of its counterparts with small training size. Although there might be a small loss of detection performance for I-PAMF compared with a few of its counterparts in some cases, I-PAMF still has a much lower computational cost than these counterparts.

Introduction

The problem of detecting a multichannel signal under correlated interferences is often encountered in a variety of applications including radar, wireless communications, and other fields. Taking advantage of the spatial information brought by multiple sensors, detection systems can achieve better performance, compared with single-sensor systems. Space-time adaptive processing (STAP) [1] is a widely explored technique in multichannel signal detection problems. Most classic STAP-based methods incur a high computational cost for matrix inversion and estimation, such as the generalized likelihood ratio test (GLRT) [2] and the adaptive matched filter (AMF) [3].

As a classical parametric STAP method, the parametric adaptive matched filter (PAMF) [4] is proposed to reduce both the training data requirement and the computation cost. So far, PAMF has been improved and extended further, such as in [5], [6], [7], [8], [9], [10], [11], [12]. In [5] and [6], PAMF is applied to multiple input multiple output (MIMO) and beam control, respectively. Pu Wang et al. proposed a persymmetric parametric adaptive matched filter (Per-PAMF) to improve the detection performance of PAMF in [7]. In [11], the conjugate gradient algorithm [13] is used in the order selection of autoregressive (AR) model in PAMF, which leads to the CG-PAMF detector. In [12], the bootstrap resampling method is employed in PAMF method, which improves the accuracy of the threshold of false alarm when training data is limited.

As is well known, PAMF is an adaptive version of the parametric matched filter (PMF) [4]. As an original method, PMF stems from the auto-regressive (AR) [14] model, which is the inverse system of linear predictor (LP) [15]. Nevertheless, the relationship between the coefficient matrices and corresponding interferences remains unclear in PMF.

In this paper, a parametric matched filter based on interference iteration (i.e. I-PMF) is proposed, which depends on the recursive superposition of interferences. The goal of the paper is to propose an iteration algorithm which has a low cost and presents a mechanism for the parametric matched filter in suppressing interferences. In I-PMF, the number of interferences determines the number of iterations. Compared with some other conventional approaches (e.g. direct matrix inversion and conjugate gradient method), the I-PMF has a lower computation cost. In addition, combined with other methods (such as the Clean method in [16]), the iteration of the proposed algorithm can be utilized in some space-time adaptive methods. In this paper, the iteration of I-PMF is combined with a Clean method, which leads to I-PAMF algorithm. Then the average minimum output signal-to-interference-and-noise ratio of I-PAMF is analyzed in a special case. In the simulation part later, it is shown that I-PAMF has its advantages in the detection performance with small training size, and a low computation cost. As a classical tool, the matrix inversion lemma plays an important role in the deduction of I-PMF. In the process for obtaining coefficient matrices of PMF, the matrix inversion lemma is utilized to separate each of interference components (one by one) from the inversion of the covariance matrix in PMF, which is crucial for the recursive superposition of interferences and leads to I-PMF in the end.

The rest of the paper is organized as follows. The data model and a detailed overview of PMF are presented in Section 2. In Section 3, we present a parametric matched filter based on interference iteration, i.e. I-PMF, which shows the relationship between coefficient matrices and interferences in PMF. Then an application of I-PMF in STAP, i.e. I-PAMF, and a detailed overview of the clean method in [16] are presented in Section 4. Besides, the average minimum output signal-to-interference-and-noise ratio of I-PAMF is analyzed with separated interferences in Section 4. Section 5 presents some numerical results of the detection performance of I-PAMF and the low computational cost brought by I-PMF (compared with direct matrix inversion and the conjugate gradient method). Moreover, the numerical analysis of the average minimum output SINR of I-PAMF is validated in Section 5. Finally, conclusion is shown in Section 6.

Throughout this paper, boldface uppercase letters denote matrices, and boldface lowercase letters denote vectors. ⊗ and ${(\cdot )}^{\mathrm{T}}$ denote the Kronecker product and transpose, respectively. Complex conjugate transpose and complex conjugate are denoted by ${(\cdot )}^{\mathrm{H}}$ and ${(\cdot )}^{⁎}$, respectively. $\mathbb{C}$ represents the complex number field.