High resolution range-reflectivity estimation of radar targets via compressive sampling and Memetic Algorithm

Abstract

Recent results of Compressive Sampling (CS) have demonstrated its feasibility in high-resolution radar targets estimation and imaging [2], [10], [14], [15], [17], [19], [23], [29], [30], [32], [33], [34]. However, the signal recovery is reduced to seeking a sparse solution to an underdetermined linear system of equations. It is potentially very difficult because even finding a solution that approximates the true minimum is NP-hard. In this paper, we introduce Memetic Algorithm (MA) to solve this non-convex l₀-norm minimization problem, and design a compressive receiver for high-resolution range-reflectivity estimation of multiple radar targets. A double-population MA is proposed, where the position population is used to evaluate the ranges, and the coefficient population is used to realize a local search of target reflectivities. By combining the global search with a local searching operation to exploit the available knowledge in the recovery, the proposed MA outperforms the general purpose optimization algorithms in terms of the quality of solution. Some experiments are taken to investigate the performance of this compressive receiver at different sampling rates, and the results show the superiority to its counterparts in both noiseless environment and noisy, cluttered environment.

Introduction

High resolution about the range and azimuth angular is desirable in radar systems for the increasing demand of radar in military, civilian, and biomedical applications. However, the range resolution of radar systems is limited by the time–frequency uncertainty principle. According to this principle, the time-bandwidth product of signals is constant, that is, signals of short duration have wide spectral width and vice versa. Consequently, wideband or ultra-wideband radar systems that transmit signals across a wider frequency than conventional narrow-band radar systems, are used to obtain higher range resolution. Unfortunately, high-bandwidth signals require very high sampling rate that is indicated by Shannon–Nyquist theorem, which imposes demanding challenges on the acquisition hardware and on the subsequent storage and Digital Signal Processing (DSP) processors [18], [24].

The recently developed Compressive Sampling (or Compressive Sensing, CS) theory indicates that it is possible to recover sparse signals from their projection onto a small number of random measurements [6], [11]. Some preliminary results on CS have demonstrated its feasibility in radar system because radar data has proven to be compressible with no significant losses for most of the applications where it is used [17], [23], [32], [33]. Several authors have considered the CS idea for radar target estimation [14], [15], imaging [2], [10], [29], [30], [34] and space-time adaptive processing (STAP) [31]. The sparsity of targets is explored to enhance the range, azimuth and spatial resolution, and the signal recovery is reduced to seeking a sparse solution to an underdetermined linear system of equations. However, it is potentially very difficult because even finding a solution that approximates the true minimum is Non-deterministic Polynomial-time hard (NP-hard). For the NP-hardness of the problem, exhaustive search cannot be computed in reasonable time for its exponential complexity. Currently several general purpose algorithms have been proposed in paper [2], [10], [14], [15], [17], [19], [23], [29], [30], [31], [32], [33], [34] to avoid the exhaustive search, which can be divided into three catalogs: greedy methods [12], [21], [22], [25], convex methods [5], [8], [13] and relax methods [7], as shown in Fig. 1.

The greedy methods abandon exhaustive search in favor of a series of locally optimal single-term updates. Their basic idea is to represent a signal as a weighted sum of atoms taken from a dictionary, such as Matching Pursuit (MP) [21], Orthogonal Matching Pursuit (OMP) [25] and their variants including Stagewise orthogonal Matching Pursuit (StoMP) [12], and Regularized orthogonal Matching Pursuit (RoMP) [22]. They can correctly pick up atoms in the case of existing sparse solution and the selection rule is simple to understand. However, they are characteristics of heavy computation, slow convergence, and can only work well in the noiseless case. Convex methods convert the non-convex l₀-norm to the convex l₁-norm, such as Basis Pursuit (BP) algorithm [8] that reduces the optimization problem to a linear programming structure. It is solvable comparing to the l₀-norm minimization and is easy to be integrated into other variational models. However, its optimization is difficult and the parameter tuning is not straightforward. Gradient based methods are discussed in paper [13], [5] to solve this problem. Relax methods use the l_p-norm (0 < p < 1) or weak l_p-norm to serve as a candidate function for l₀-norm [7]. Although it is still non-convex, it is almost equivalent to l₀-norm and can be represented as a weighted l₁-norm form by the Iterative-Reweighed-Least-Squares (IRLS) method. However, the algorithm is very sensitive to the initialization of solutions. Moreover, it is guaranteed to converge to a fixed-point that is not necessarily the optimal one. Fig. 1 compares the objective function, optimization type and the sparsity of solutions of the three catalogs.

Theoretical studies seek for various guarantees for the success of those approximate algorithms. Some metrics such as the Restricted Isometry Constant (RIC) and Spark are used to analyze the success of these algorithms in addressing the original l₀-norm objective [3], [9], [26]. However, it is not feasible to validate the exact equivalence of those approximated algorithms to l₀-norm, because the calculation of RIC and Spark are themselves NP-hard. Moreover, these guarantees succeed only in the noiseless environment and fail when the compressive sampled data are noisy. In the range-reflectivity estimation of multiple radar targets, the received echo signal is polluted by the system noise and clutters, where the available optimization algorithms cannot efficiently deal with.

Evolutionary Algorithms (EAs) simulate natural evolution over the populations of candidate solution to explore better solution [10]. They prove to provide a general and global searching approach for solving complex optimization problems. In this study, we address the original l₀-norm objective using EAs, to realize an accurate and robust angle-reflectivity estimation of multiple radar targets. An Analog-to-Information Converter (AIC) [20] with low-rate sampling is used in the receiver, and a double-population Memetic Algorithm (MA) is proposed to recover the signal from a small number of data sampled far less than that of Nyquist sampling. The position population is used to evaluate the ranges, and the coefficient population is used to realize a local search of target reflectivities. By combining the global search with a local searching operation to exploit the available knowledge in the recovery, the proposed MA outperforms the general purpose optimization algorithms in terms of the quality of solution. In summary, the contributions of the study are as follows:

• A compressive receiver is designed, with a sub-Nyquist sampling strategy of signals followed by an MA based optimizer, to enhance the resolution of range-reflectivity estimation of multiple radar targets.

• In order to overcome the disadvantage of the general purpose optimization algorithms, we firstly introduce EAs to estimate the target-parameters vector of multiple radar targets. MA optimizer is proposed to exploit the available knowledge to find a sparse solution.

• Double-population: position population and coefficient population are designed in MA, to represent the range and reflectivity of targets respectively. Both the global search and local search are explored to solve an NP-hard l₀-norm optimization problem.

• Some experiments are taken on a monostatic, single-pulse, far-field radar system with static multiple targets to investigate the performance of our proposed scheme, and the results show that the proposed method outperforms its counterparts in both estimation accuracy and robustness.

The remainder of this paper is organized as follows. Section 2 introduces the compressive receiver based on sub-Nyquist sampling and MA based optimizer. In Section 3, some experiments are conducted and their results together with some relevant discussions are reported. The conclusions are finally summarized in Section 4.