High-resolution synthetic aperture radar (SAR) is an active remote sensing modality for real-time information acquisition. It plays a significant role in the field of civil exploration and military reconnaissance owing to its capability of all-weather, all-time, and high-resolution imaging. A SAR system usually operates at a wide range of microwave frequencies and it is inevitably subject to various kinds of electromagnetic interference. These kinds of interference with the characteristics of high power and narrowband may seriously degrade the quality of SAR images and cause trouble for the subsequent interpretation.
Multi-channel technology and signal processing are two typical methods of interference suppression for SAR. A multi-channel processing method [1
] uses the space information and extracts the signal of interest (SOI) from the contaminated echo by zeroing the interference direction, and this method outperforms that of a single channel. However, this special multiplex architecture increases the complexity of the radar system and cannot be directly applied to existing devices.
From the perspective of pure signal processing, narrowband interference suppression can be mainly divided into parametric, non-parametric, and semi-parametric methods. The parametric methods such as high-order ambiguity function [3
] and complex empirical mode decomposition [4
] are based on interference modeling with multi-order terms. However, it is heavily dependent on model accuracy and has a large amount of calculation in the process of parameter searching. A non-parametric method such as notched filtering (NF) [5
], least mean square (LMS) filtering [7
], eigen-subspace filtering (ESF) [8
], independent component analysis (ICA) [9
], independent subspace analysis (ISA) [10
], and robust principal component analysis (RPCA) [12
] can suppress the interference from raw data without any prior knowledge or parametric model. Notched filtering and LMS filtering are actually equivalent to adding a band-stop filter where the interference is located, regardless of whether there is a signal component in this frequency range. The basic idea of ESF, ICA, ISA, and RPCA is the singular value decomposition (SVD) of the data matrix, and the signal or interference is reconstructed by inverse transform after extracting the dominant components. The main problem of the non-parametric method is the signal distortion, since the SOI is also suppressed when the interference is eliminated.
Sparse recovery, as a typical semi-parametric method for interference suppression, is state-of-the-art, especially in terms of reducing signal distortion. It can be considered as an optimization problem of reconstructing few coefficients with a given dictionary. The sparsity-based method is mainly used for suppressing RFI that appears in the form of spikes in a large frequency range. Considering the sparse property in the range-frequency domain and the low-rank property in the azimuth, in References [13
], RFI was extracted and suppressed based on a sparse and low-rank model. In Reference [15
], the matrix factorization technique was introduced into the sparse and low-rank model to avoid large residuals after SVD and further reduce the computational complexity at the same time. In our previous work [16
], we proposed an RFI suppression method for SAR based on morphological component analysis (MCA), in which a stepwise reconstruction algorithm was adopted to the reconstruction. Given that the steps of interference reconstruction and cancellation may limit the suppression performance and increase the system complexity, the alternating direction multiplier method (ADMM) [17
] was adopted to reconstruct the signal and the interference simultaneously in Reference [18
]. The premise of this method is that both the SOI and interference are sparse in their respective domains.
The observed scene in most SAR images is not sparse and it is difficult to find a proper dictionary to represent the echo signal with few nonzero coefficients. Moreover, the narrowband interference (NBI) of a noise-modulated type with a certain bandwidth is not sparse either in the frequency domain. Classical recovery algorithms [19
] such as basis pursuit (BP), matching pursuit (MP), and orthogonal matching pursuit (OMP) fail to recover the signal accurately. The block MP (BMP) and block OMP (BOMP) algorithm proposed in Reference [21
] can improve the reconstruction probability with a slight requirement for sparsity by exploiting the block sparse structure. Still, with the increase in scene complexity and interference bandwidth, the reconstruction probability decreases, since the block sparse feature gradually weakens. The global minimum of the above algorithm is not really the sparsest solution, unless strict conditions are satisfied. Hence, sparse Bayesian learning (SBL) [22
], which considers all unknown parameters as random variables and adds appropriate prior distributions according to the sparse structure, is no doubt a better choice. Derived from the SBL framework, block sparse Bayesian learning (BSBL) [23
] is a robust recovery algorithm for both sparse and non-sparse signals from a low-dimensional space by exploiting the temporal correlation of intra-block data. In Reference [25
], the BSBL framework is first used and modified for RFI suppression where the target or observed scene is not strictly sparse but block sparse, and the S-BSBL and A-BSBL algorithms are, respectively, proposed to improve reconstruction performance and reduce the amount of computation. Judging from the results of interference suppression, the BSBL-based approach is indeed superior to other advanced ones and can be used more widely.
Nevertheless, there remain several problems to be solved. As is known, radar signals are complex-valued in most processing steps, so the BSBL algorithm cannot be directly applied. A widely accepted trade-off approach is to concatenate the real and imaginary parts of the signal into a new vector. There are two main shortcomings in this scheme. One is that the length of the new real-valued vector is twice as long as the original complex-valued vector and the corresponding sensing matrix will expand in square with the signal length increasing, which will result in a huge amount of computational burden. The other is that the reconstruction performance may be degraded due to the loss of structural information, since the real and imaginary part of the signal are processed separately. In addition, while the BSBL framework is robust to the interatomic coherence in the sensing matrix for the reconstruction of a clean signal, the block coherence of sub-dictionaries corresponding to different components in the contaminated echo has a great impact on the separated reconstruction performance, since the diagonal block of the covariance matrix cannot be effectively distinguished.
To solve these problems above, our goal is to reduce the amount of calculation with a modified BSBL algorithm, which can be applied to the complex-valued signal directly and further improve the performance of NBI separation by optimizing the cascaded sensing matrix.
The main contents of this paper are divided into three parts. In Section 2
, the problem of separated reconstruction for SOI and NBI based on complex-valued block sparse Bayesian learning framework is formulated. In Section 3
, the optimal sensing matrix is designed by minimizing the newly defined block coherence measure, and the SMO-BSBL algorithm for NBI separation, which is embedded in the entire procedure of SAR imaging, is presented. In Section 4
, numerical experiments with simulated data are carried out, and results of the proposed algorithm in this paper are compared with existing BSBL-based algorithms.