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The frequency diverse multiple-input-multiple-output (FD-MIMO) radar synthesizes a wideband waveform by transmitting and receiving multiple frequency signals simultaneously. For FD-MIMO radar imaging, conventional imaging methods based on Matched Filter (MF) cannot enjoy good imaging performance owing to the few and incomplete wavenumber-domain coverage. Higher resolution and better imaging performance can be obtained by exploiting the sparsity of the target. However, good sparse recovery performance is based on the assumption that the scatterers of the target are positioned at the pre-discretized grid locations; otherwise, the performance would significantly degrade. Here, we propose a novel approach of sparse adaptive calibration recovery via iterative maximum

The multiple-input-multiple-output (MIMO) radar system has attracted much attention recently, due to the additional degrees of freedom and the higher spatial resolution [

For a restricted number of transmitters and receivers, the wavenumber-domain coverage is incomplete, so the traditional imaging methods based on matched filter (MF) fail to achieve good performance. Nonetheless, in most radar imaging applications, the scatterers of the target are often distributed sparsely,

However, most existing sparse recovery techniques require the scatterers to be located exactly on the pre-discretized grid. When off-grid scatterers exist, their performance would be severely affected. In MIMO radar imaging [

For linear measurement equation

In this paper, we propose an approach of sparse adaptive calibration recovery via iterative maximum

The outline of this paper is as follows. In Section 2, the sparse recovery of the off-grid target for the FD-MIMO imaging problem is formulated. In Section 3, a novel algorithm, named SACR-iMAP, for the off-grid sparse recovery problem is proposed. In Section 4, extensive numerical simulations are presented to verify the proposed algorithm. Finally, in Section 5, the conclusions are drawn.

Notations used in this paper are as follows. Bold-case letters are reserved for vectors and matrices. ‖_{p}_{p} norm of a vector _{2} is the spectral norm of the matrix ^{T}^{T}

Consider the scenario of a MIMO radar system composed by M transmitters and N receivers, as illustrated in

Assuming that all of the _{m}_{m}_{m}_{c}

Suppose the imaging scene contains _{k}_{k}_{k}_{1}(_{2}(_{M}^{T}, the echoes of _{1}(_{2}(_{N}^{T} can be written in the following form:
_{k}_{k}_{k}_{k}_{c}_{ti}_{ri}

After orthogonal separation, we can obtain the received signal of the _{m}_{k}_{k}_{n}_{k}_{k}_{mn}

Supposing that the reference range is _{0}, for LFM signal, we have:
_{0}/_{k}_{k}_{0}. After dechirping and removing the residual video phase in

Here, we consider uniform linear arrays (ULA) for the transmitters and receivers with _{t}_{r}_{t}_{r}_{tm}_{rn}

Define _{mn}(_{mn}(

Moreover, we let:

Defining _{1},…,_{K}

The available observations in the wavenumber domain point are (2(_{m} + _{q}

In most cases of radar imaging, the scatterers are distributed in a continuous scene. The sparsity is more of a kind of quantitative description. However, if we use the sparsity prior directly into _{k}_{0}, _{k}_{0}) is the actual location of the _{k}_{k}_{k}_{k}

Substituting

Since the off-grid errors are restricted in the region of one radial range bin or angle bin, we can assume that the off-grid errors are significantly small; then, we can make the following approximations by the Taylor expansion:

Since {_{k}_{k}_{k}_{k}_{k}_{k}_{1}_{k}_{k}_{q}_{2}_{k}_{k}_{tm} + _{rn})/_{k}_{k}

Then, we can get the matrix-vector form corresponding to

Thus, our pursuit is to find out the optimized

Inheriting the Bayesian idea, we can estimate

Further, suppose that _{r}_{θ}

Taking the negative logarithm of

The minimization of

SACR-iMAP includes mainly three steps. Defining ^{(}^{l}^{+1)}, the off-grid errors ^{(}^{l}^{+1)} and the parameter ^{(}^{l}^{+1)} in the (

Assuming that ^{(}^{l}^{)} and ^{(}^{l}^{)} are obtained, we seek for the optimal ^{(}^{l}^{+1)} to minimize the following equivalent cost function:

Letting ∂_{1}/∂^{(}^{l}^{+1)} = 0, we have:

From ^{(}^{l}^{)} and ^{(}^{l}^{)}/2, we can get that:
^{(}^{l}^{+1)}[^{(}^{l,s}^{=0)} = ^{(}^{l}^{)}).

Moreover, considering ^{(}^{l}^{+1,s)} converges, yielding ^{(}^{l}^{+1)}.

In this stage, we use ^{(}^{l}^{+1)} and ^{(}^{l}^{)} to estimate {_{k}^{(}^{l}^{+1)}, _{k}^{(}^{l}^{+1)}}, for

Observe that if _{k}^{(}^{l}^{+1)} = 0, then Δ_{k}_{k}_{k}^{(}^{l}^{+1)}, _{k}^{(}^{l}^{+1)}} that correspond to _{k}^{(}^{l}^{+1)} ≠ 0, while setting other errors to zero. Let _{1},...,_{1},...,^{T}, and:
_{k̃}^{(}^{l}^{+1)}. _{Ω} and _{Ω} correspond to the _{Ω}^{(}^{l}^{+1)} and _{Ω}^{(}^{l}^{+1)} correspond to the ^{(}^{l}^{+1)} and ^{(}^{l}^{+1)}, respectively, where ^{(}^{l}^{+1)} can be obtained.

To alleviate the effect of parameter estimation on the performance of SACR-iMAP, here, we embed a dynamically and adaptively parameter update process in SACR-iMAP. Similarly, setting ∂^{(}^{l}^{+1)} = 0 leads to:

Then, set

Remark (1) The Initialization of SACR-iMAP

Here, we initialize ^{(0)} = 0, then, we get ^{(0)} by matched filter (MF). The ^{(0)} satisfies:
_{j}

Then, we have the initial estimation of the noise power as:

Remark(2) The Convergence of SACR-iMAP

We can conclude that the cost function

Remark(3) The Applicability and Limitation of SACR-iMAP

Though the algorithm of SACR-iMAP is deduced in the case of the FD-MIMO radar imaging scene, it is also applicable for the other imaging scenarios that can be described by

Based on the above idea, SACR-iMAP can be described, as shown in

In this section, we present several numerical simulation results to illustrate the performance of the proposed algorithm. The SACR-iMAP is implemented in Matlab, and

The simulation conditions are given in _{r} = 1 m, _{θ}

The original scatterer distribution of the target is shown in

In the following simulations, besides SACR-iMAP, other algorithms will be involved: MF, OMP, FOCUSS, S-TLS [

From the Bayesian perspective, S-TLS is equivalent to searching for the MAP solutions by assuming that the added noise is white Gaussian,

Similarly, TLS-FOCUSS is equivalent to searching for the MAP solutions by assuming that the added noise is white Gaussian, _{p}-term forced and

In this subsection, we study the imaging error with respect to the SNR level. The target recovery errors, as well as the off-grid recovery errors, are averaged over 30 trials. In each trial, the off-grid errors are uniformly distributed in the region of one radial range bin or angle bin. We use the same data as Section 4.1, and expect that the SNR varies from 0 dB to 40 dB with interval 5 dB.

Assuming that the off-grid errors are known

In this subsection, we assume that the size of the imaging area and the actual scatterer distribution of the target are invariant, then we change the discretized grid interval and, consequently, the number of the discretized radial range bins and angle bins. We assign SNR = 20 dB, and the target recovery errors, as well as the off-grid recovery errors, are averaged over 30 trials.

_{r}_{θ}_{r}_{θ}_{r}_{θ}

This paper presents the SACR-iMAP method to realize high resolution imaging of sparse, but off-grid, targets for the FD-MIMO radar system. Unlike traditional sparse recovery methods, SACR-iMAP adaptively adjusts the off-grid errors, meanwhile seeking the optimal target reconstruction result. Through iterative MAP, it turns the non-convex optimization problem of off-grid sparse recovery to three main stages: sparse recovery, off-grid errors calibration and parameter update. Benefited from adaptively adjusting and updating, SACR-iMAP has some merits, e.g., there is no need for accurate initialization and improved robustness to noise. The derivations and numerical simulations illustrate the effectiveness of the new method, which shows the potential for the method to be applied in practical systems.

In this paper, we only consider the case of FD-MIMO radar imaging; however, the framework in this paper can be extended to other imaging radar systems, such as generalized MIMO radar imaging and passive radar imaging, so SACR-iMAP will have wider applications. Here, we only consider the first order Taylor approximation of the off-grid errors. Higher order approximations can be adopted in the future to further reduce the modeling error and, hence, to achieve higher precision. Moreover, in the off-grid problem formulation, the scatterers and off-grid errors are jointly sparse. Inspired by the recent works on block and structured sparsity [

This work was supported by the Hi-Tech Research and Development Program of China under Grant Project No. 2011AA120103.

^{(}^{l}^{)} and ^{(}^{l}^{)} are kept unchanged. We write _{1} + _{1}, where _{1} is independent of _{1}(^{(}^{l}^{+1)}) and _{1}(^{(}^{l}^{)}) totally reflects the change of _{1}(^{(}^{l}^{,}^{s}^{+1)}) in the (_{1}(^{(}^{l}^{,}^{s}^{+1)}) and _{1}(^{(}^{l}^{,}^{s}^{)}). It is showed in [_{1}(^{(}^{l}^{,}^{s}^{+1)}) is smaller than _{1}(^{(}^{l}^{,}^{s}^{)}) based on the fact that ^{q} (^{(}^{l}^{+1)}. So, there is no doubt that _{1}(^{(}^{l}^{+1)}) < _{1}(^{(}^{l}^{)}) as ^{(}^{l}^{)} and ^{(}^{l}^{+1)} represent the initial value and the converged value of the internal iteration, respectively. Therefore, we prove that _{1},

Secondly, in the off-grid errors calibration stage, similarly, we rewrite _{2} + _{2}, where _{2} is independent of _{2}(^{(}^{l}^{+1)}) = _{2}(^{(}^{l}^{+1)}) and _{2}(^{(}^{l}^{)}) = _{2}(^{(}^{l}^{)}). Furthermore, we can deduce that:
_{2},

Finally, since the parameter ^{(}^{l}^{+1)} is computed according to ∂^{(}^{l}^{+1)} = 0, it would definitely result in the decrease of the cost function

Based on all the analysis above, we know that the cost function

Imaging scenario for multiple-input-multiple-output with frequency diversity (FD-MIMO) radar.

The original target.

Imaging results of off-grid target by (

Cost function

Normalized mean square error (NMSE)

NMSE

The main steps of sparse adaptive calibration recovery via iterative maximum

Input: |

Initialization: |

Iteration (denote estimate calculate update end: The loops are terminated when the norm of reflection coefficients changes moderately between the adjacent loop. |

Output: ^{(}^{l}^{+1)}, ^{(}^{l}^{+1)}. |

Simulation conditions.

Number of transmitters |
10 |

Number of receivers |
10 |

Bandwidth of each transmitted signal |
15 MHz |

Waveform pulse duration |
2 μs |

Number of snapshots |
10 |

Carrier frequency of the first transmitter _{c} |
10 GHz |

Inter-element spacing of the transmitters |
0.3 m |

Inter-element spacing of the receivers |
0.03 m |