ℓp-Based complex approximate message passing with application to sparse stepped frequency radar
Introduction
Compressed sensing (CS) theory suggests that a high-dimensional unknown sparse signal can be recovered exactly from a small number of measurements with high probability by solving a sparsity-regularized optimization problem [17], [31], [33], [34]. It can significantly reduce the sampling cost and is of great interest in a wide range of applications, such as magnetic resonance imaging (MRI) [11], [29], wireless communications [26] and radar signal processing [37], [35], [25], [13].
Compressed sensing for complex-valued signal is required in many applications such as radar and MRI [19], [21]. The last decade has witnessed a surge of reconstruction algorithms for complex signal [27], [17], [21]. One algorithm that has attracted significant attention is the complex approximate message passing (CAMP) algorithm [21]. The following properties of CAMP have made it appealing [1]: (1) the algorithm converges very fast, and hence is suitable for real-time processing. (2) Its mean-squared-error (MSE) can be accurately predicted at every iteration, which enables a dynamic tuning of its parameters on the fly. (3) It is straightforward to control the false alarm rate, which is important for applications such as radar.
Despite the appealing features of CAMP, some evidence inspires us to research better reconstruction algorithms. It is shown in CS that the ℓp-norm constraint with can lead to more accurate solutions than the ℓ1-norm constraint [8], [23], which encourages researchers to use the other algorithms such as SL0 and OMP despite all the nice properties of CAMP. The main objective of this paper is to develop an algorithm, called adaptive ℓp-CAMP. As an application, we apply our ℓp-CAMP algorithm to reconstruct the range profiles in sparse stepped frequency waveform (SFW) radar, which has the following properties: (1) It is guaranteed to outperform the original CAMP on any recovery problem. (2) All the parameters are tuned dynamically and efficiently (to achieve minimum MSE) in the algorithm, and hence it does not require any parameter tuning from the user. (3) Its MSE performance can be analyzed theoretically. In applications such as radar, there is a trade-off between the SNR of reconstruction and the data rate, so the third property helps in designing the parameter in sensing to improve the overall system performance.
The remainder of the paper is organized as follows. We start with the problem formulation and the signal model of the sparse SFW radar in Section 2. The adaptive ℓp-CAMP is developed in Section 3. The performance of ℓp-CAMP is theoretically analyzed by characterizing the statistical properties of the recovered signal in Section 4. In Section 5, the performance of the proposed algorithm is verified by using the real data collected by an experimental radar system. Finally we conclude the paper in Section 6.
Section snippets
Motivating example: sparse stepped frequency radar
To motivate this work by an application, we introduce the signal model of sparse SFW radar and the problem formulation. Conventional full-band SFW radars transmit a series of narrowband pulses, which are stepped in frequency from pulse to pulse and form a burst1 that covers a wide bandwidth. For high resolution radar, the target is usually composed of several scatterers which take up a small fraction of the
Solving c-LPLS with CAMP
In this section, we develop an algorithm for solving (5), (6) under the framework of AMP, along with an analysis framework reviewed in Section 4, which enables the evaluation and automatic optimization of the parameters.
State evolution
In this section, we first explain the state evolution (SE) framework that characterizes the performance of the -CAMP and adaptive -CAMP. Then the performance of the adaptive -CAMP is compared with that of the adaptive -CAMP. In Theorem 1 only, we write the vectors and matrices as , , , , to emphasize the dimensions of . In [36], the SE for -AMP has been given. The result can be extended to the complex-valued signal case and we obtain the following theorem. Theorem 1 Let
Data description
In this section, we verify the performance of the proposed algorithm in the sparse SFW radar application. The adaptive ℓp-CAMP algorithm is run on the data collected by an experimental full-band SFW radar system and its performance is compared with that of the other existing methods. The experimental radar is an S-band synthetic wideband pulse doppler radar reported in [14]. The waveform parameter of the radar in this experiment is listed in Table 1. The radar transmits 256 pulses in each burst
Conclusions
In this paper, we have proposed the adaptive ℓp-CAMP algorithm and showed its application to the sparse SFW radar signal processing. The experiments on real data show that the proposed algorithm provides better MSE performance compared to ℓ1-CAMP and some other existing compressed sensing reconstruction algorithms. It is also shown that the performance of the proposed algorithm can be accurately predicted by a theoretical framework, known as state evolution (SE). SE allows the system designer
Acknowledgements
The work of Quanhua Liu is supported by the National Natural Science Foundation of China (Grant No. 61301189), and 111 Project of China (Grant No. B14010).
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