- Research
- Open Access
RZA-NLMF algorithm-based adaptive sparse sensing for realizing compressive sensing
- Guan Gui^{1}Email author,
- Li Xu^{1} and
- Fumiyuki Adachi^{2}
https://doi.org/10.1186/1687-6180-2014-125
© Gui et al.; licensee Springer. 2014
- Received: 1 March 2014
- Accepted: 31 July 2014
- Published: 9 August 2014
Abstract
Nonlinear sparse sensing (NSS) techniques have been adopted for realizing compressive sensing in many applications such as radar imaging. Unlike the NSS, in this paper, we propose an adaptive sparse sensing (ASS) approach using the reweighted zero-attracting normalized least mean fourth (RZA-NLMF) algorithm which depends on several given parameters, i.e., reweighted factor, regularization parameter, and initial step size. First, based on the independent assumption, Cramer-Rao lower bound (CRLB) is derived as for the performance comparisons. In addition, reweighted factor selection method is proposed for achieving robust estimation performance. Finally, to verify the algorithm, Monte Carlo-based computer simulations are given to show that the ASS achieves much better mean square error (MSE) performance than the NSS.
Keywords
- Nonlinear sparse sensing (NSS)
- Adaptive sparse sensing (ASS)
- Normalized least mean fourth (NLMF)
- Reweighted zero-attracting NLMF (RZA-NLMF)
- Sparse constraint
- Compressive sensing
1 Introduction
It is well known that the CS provides a robust framework that can reduce the number of measurements required to estimate a sparse signal. Many nonlinear sparse sensing (NSS) algorithms and their variants have been proposed to deal with CS problems. They mainly fall into two basic categories: convex relaxation (basis pursuit de-noise (BPDN) [6]) and greedy pursuit (orthogonal matching pursuit (OMP) [7]). The above NSS-based CS methods have either high complexity or low performance, especially in the case of low signal-to-noise ratio (SNR) regime. Indeed, it was very hard to adapt trade-off between high complexity and good performance.
In this paper, we propose an adaptive sparse sensing (ASS) method using the reweighted zero-attracting normalized mean fourth error algorithm (RZA-NLMF) [8] to solve the CS problems. Different from NSS methods, each observation and corresponding sensing signal vector will be implemented by the RZA-NLMF algorithm to reconstruct the sparse signal during the process of adaptive filtering. According to the concrete requirements, the complexity of the proposed ASS method could be adaptively reduced without sacrificing much recovery performance. The effectiveness of our proposed method is confirmed via computer simulation when comparing with NSS.
The remainder of the paper is organized as follows. The basic CS problem is introduced and the typical NSS method is presented in Section 2. In Section 3, ASS using the RZA-NLMF algorithm is proposed for solving CS problems and its derivation process is highlighted. Computer simulations are given in Section 4 in order to evaluate and compare performances of the proposed ASS method. Finally, our contributions are summarized in Section 5.
2 Nonlinear sparse sensing
3 Adaptive sparse sensing
which is a variable step size (VSS) that is adaptive to change as square sensing error e_{ m }^{2}(n); a smaller error incurs a smaller step size to ensure the stability of the gradient descent while a larger error yields a larger step size to accelerate the convergence speed of this algorithm [12]. According to the update equation in (9), our proposed ASS method can be concluded in Algorithm 1.
4 Computer simulations
Simulation parameters
Parameters | Values |
---|---|
Signal length | N = 40 |
Measurement length | M = 20 |
Sensing matrix | Random Gaussian distribution |
Number of nonzero coefficients | K ∈ {2, 6, 10} |
Distribution of nonzero coefficients | Random Gaussian |
Signal-to-noise ratio (SNR) | (0 dB, 12 dB) |
Initial step size: μ_{iss} | 1.5 |
Regularization parameter: λ | 5 × 10^{-8} |
Reweighted factor: ϵ | 2,000 |
where h and $\tilde{\mathbf{h}}\left(n\right)$ are the actual channel vector and its n th iterative adaptive channel estimator, respectively. According to our previous work [8], the regularization parameter for RZA-NLMF is set as λ = 5 × 10^{-8} so that it can exploit signal sparsity robustly. Since the RZA-NLMF-based ASS method depends highly on the reweighted factor ϵ, hence, we first select the reasonable factor ϵ by virtue of the Monte Carlo method. Later, we compare the proposed method with two typical NSS ones, i.e., BPDN [6] and OMP [7].
4.1 Reweighted factor selection
4.2 Performance comparisons with NSS
5 Conclusions
In this paper, we proposed an ASS method using the RZA-NLMF algorithm for dealing with CS problems. First, we selected the reweighted factor and regularization parameter for the proposed algorithm by virtue of the Monte Carlo method. Later, based on the update equation of RZA-NLMF, the CRLB of ASS was also derived based on random independent assumptions. Finally, several representative simulations have been given to show that the proposed method achieves much better MSE performance than NSS with respect to different signal sparsities, especially in the case of low SNR regime.
Since the empirical reweighted factor was selected for RZA-NLMF in the noise environment, in the future work, we will develop the learning reweighted factor for RZA-NLMF in the case of a noiseless environment. It is expected that RZA-NLMF using learning reweighted factor can achieve much better recovery performance without sacrificing much computational complexity.
Declarations
Acknowledgements
The authors would like to appreciate the editor and the anonymous reviewers for their help comments and suggestion to improve the quality of this paper.
Authors’ Affiliations
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