We investigate the possibility of utilizing the chaotic dynamic system for the measurement matrix design in the CS-MIMO radar system. The CS-MIMO radar achieves better detection performance than conventional MIMO radar with fewer measurements. For exactly recovering from compressed measurements, we should carefully design the measurement matrix to make the sensing matrix satisfy the restricted isometry property (RIP). A Gaussian random measurement matrix (GRMM), typically used in CS problems, is not satisfied for on-line optimization and the low coherence with the basis matrix corresponding to the MIMO radar scenario can not be well guaranteed. An optimized measurement matrix design method applying the two-dimensional spatiotemporal chaos is proposed in this paper. It incorporates the optimization criterion which restricts the coherence of the sensing matrix and singular value decomposition (SVD) for the optimization process. By varying the initial state of the spatiotemporal chaos and optimizing each spatiotemporal chaotic measurement matrix (SCMM), we can finally obtain the optimized measurement matrix. Its simulation results show that the optimized SCMM can highly reduce the coherence of the sensing matrix and improve the DOA estimation accuracy for the CS-MIMO radar.

The application of compressive sensing (CS) to radar systems has received considerable attention in recent years [

Some works have addressed the measurement matrix design problem [

So far many of the CS schemes employ the Gaussian or Bernoulli matrix for their measurement matrix design or optimization. However, as proposed in [

Chaos is a nonlinear dynamical system, which can generate pseudorandom matrix in deterministic approach. It is easy to be implemented in physical electric circuit and only one initial state is necessary to be memorized. Moreover, since a chaotic dynamic system is quite sensitive to its initial state, slight changes will lead to quite different chaotic behaviors. This property can be applied for adjusting the chaotic measurement matrix to match the known basis matrix corresponding to a MIMO radar scenario, thus realizing on-line optimization in CS-MIMO radar systems. Several literatures have proposed the idea of using chaos in CS [

However, all of the proposed approaches have just applied one-dimensional chaotic systems for a simple CS recovery framework. The fact is, all of the methods have to tolerate the unexpected loss of the independence and randomness of the chaotic sequences during reshaping the sequence into a matrix for CS. In this paper, we propose a novel measurement matrix design method using spatiotemporal chaos for CS-MIMO radar. The measurement matrix for CS-MIMO radar is generated directly by the two-dimensional spatiotemporal chaos. Compared to the low-dimensional chaotic system, spatiotemporal chaos possesses higher complexity and randomness [

The scheme of CS-MIMO radar in this paper is demonstrated in Figure

The scheme of using CS in MIMO radar.

The remainder of the paper is organized as follows. In Section

We consider a monostatic MIMO radar system consisting of

Let

Then the echo received by the

By discretizing the angle space as

In the matrix form we have

Using the measurement matrix

In this section, we present the idea of using spatiotemporal chaos for measurement matrix design. Subsequently, an SVD-based optimization is performed on the SCMM with the purpose of further reducing the coherence of column pairs in the sensing matrix.

Spatiotemporal chaos typically shows disorder in both space and time domain and is capable of exhibiting chaotic behavior for certain parameter values [

Compared with the low-dimensional chaotic systems, spatiotemporal chaos has more complex behavior and more abundant characteristics, which makes it an excellent candidate for pseudorandom matrix design.

According to the signal model proposed in Section

The boundary condition of the OCOML model is [

Despite the deterministic definition via ordinary difference expressions, spatiotemporal chaotic dynamical systems exhibit unpredictable behaviour. The detailed proof that chaotic matrix could satisfy RIP with overwhelming probability is presented in [

The goal of the optimization is to further reduce the cross-correlations between the measurement matrices

However, the normalized cross-correlation is difficult for us to design

With the successful case in [

SVD is applied twice during the whole process. The first SVD is performed on

Calculate

Find

Use the LS estimator to calculate the SCMM

In summary, the optimization approach is demonstrated in Figure

The optimization approach for each SCMM.

By optimizing each SCMM, the optimized SCMM is finally chosen with the criterion that it has the minimum normalized cross-correlations with the fixed basis matrix.

In this section, we will carry out computer simulations on three aspects. Firstly the coherence of the sensing matrix will be calculated to show the effectiveness of using spatiotemporal chaos for measurement matrix design. Secondly the examples of DOA estimation will be given to demonstrate the excellent performance of the CS-MIMO radar with the optimized SCMM. Thirdly the Monte Carlo simulation is employed to verify the recovery accuracy versus various system conditions.

We consider the CS-MIMO radar with

Figure

Average and maximum normalized cross-correlations.

Measurement matrix | Average | Maximum |
---|---|---|

GRMM | 0.3624 | 0.9455 |

SCMM | 0.3549 | 0.7075 |

Optimized GRMM | 0.2844 | 0.5702 |

Optimized SCMM | 0.2286 | 0.4383 |

Histogram for the cross-correlations of the sensing matrix using (a) GRMM, (b) SCMM, (c) optimized GRMM, and (d) optimized SCMM.

We assume that several far-field point targets fall on the angle space and the received signal is mixed with zero mean Gaussian noise. At each receive antenna, the measurement matrix compresses the echo signal and acquires the measurements of length

We tested the CS-MIMO radar system with five targets to demonstrate the performance improvement induced by the optimized SCMM. The far-field targets are assumed to have the same complex amplitudes with

DOA estimation results when using (a) GRMM, (b) optimized GRMM, (c) SCMM, and (d) optimized SCMM.

GRMM

Optimized GRMM

SCMM

Optimized SCMM

Monte Carlo simulations are applied here to verify the robustness of the proposed method. The parameters of the CS-MIMO radar system are similar to previous examples. Figure

Recovery error versus the target sparsity

The recovery errors versus the initial value and the coupling constant of the spatiotemporal chaos in the OCOML model are given in Figures

Recovery error versus the initial value of the spatiotemporal chaos.

Recovery error versus coupling constant of the spatiotemporal chaos.

In this paper, a new notion of applying nonlinear dynamic chaotic system for measurement matrix design incorporating an SVD-based optimization method is proposed for CS-MIMO radar systems. Exploiting the statistical properties of the spatiotemporal chaos, a pseudorandom but deterministic matrix is obtained to match the basis matrix constructed by the MIMO radar signal model. The iterative optimization is performed to update the generated chaotic matrix aiming at further reducing the coherence of the sensing matrix. Simulation results have proved that the proposed method outperforms that of the GRMM in exact recovery and DOA estimation. In summary, the proposed method has several advantages over the method using GRMM: it possesses low coherence of the sensing matrix which enables more accurate recovery and DOA estimation results for the MIMO radar system; it is easy to implement in electric circuit and only one initial state is necessary to be memorized; it can realize on-line optimization only by changing the initial states of the chaotic system, which is practical for the real radar system.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work was supported by the National Natural Science Foundation of China under Grants 61071163, 61071164, 61201367, and 61271327; the Natural Science Foundation of Jiangsu Province under Grant BK2012382; the fundamental research funds for central universities, no. NJ20140011; and a project funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions.