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Parameter estimation and network sorting for noncooperative wideband frequency-hopping (FH) signals have been essential and challenging tasks, especially in the case with little or even no prior information at all. In this paper, we propose a nearly blind estimation approach to estimate signal parameters based on sparse Bayesian reconstruction. Taking the sparsity in the spatial frequency domain of multiple FH signals into account, we propose a sparse Bayesian algorithm to estimate the spatial frequency parameters. As a result, the frequency and direction of arrival (DOA) parameters can be obtained. In order to improve the accuracy of the estimation parameters, we employ morphological filter methods to further clean the data poisoned by the noise. Moreover, our method is applicable to the wideband signal models with little prior information. We also conduct extensive numerical simulations to verify the performance of our method. Notably, the proposed method works well even in low signal-to-noise ratio (SNR) environment.

Frequency-hopping (FH) communication is widely used in the military communications, due to its high confidentiality, antijamming capacity, low susceptibility, and strong networking capabilities [

Among various methods of parameter estimation, estimation methods based on maximum likelihood (ML) are practically intractable with few or no prior information, and thus several ML based methods have been proposed [

In order to achieve accurate estimation and network sorting results simultaneously, direction of arrival (DOA) information and blind separation (BS) is utilized with the assumption of multiple array elements. In many cases, FH signals are actually wideband, while many existing methods of DOA estimation consider the FH signals to be narrow-band signals [

With the assumption of wideband multiple FH signals, we propose a new method based on sparse reconstruction and morphological filtering to achieve better parameter estimation and network sorting results. We exploit the sparsity of the spatial frequencies of multiple FH signals to build redundant sparse dictionaries, based on which spatial frequencies are estimated for the reconstruction of FH signals and the DOA estimation can also be achieved. To improve the estimation accuracy in low SNR conditions, morphological filtering is used to achieve more accurate time-frequency and time-space spectrum. Therefore, parameters of FH signals can be estimated and then networks can be sorted with DOA. Moreover, the proposed method can be used for different multiple FH signals, different parameters, and different SNR

The rest of the paper is organized as follows. In signal model, the FH signal model and the sparse signal model are formulated. In proposed method, parameter estimation and network sorting algorithms are formulated mainly including sparse Bayesian reconstruction method and morphological filtering. In Algorithm Procedure, the procedure of the algorithm is summarized step by step. Several simulations and conclusions are given in Simulation and Analysis and Conclusion, respectively.

Assume that

The model after discrete sampling can be expressed as follows:

Assume elements of the array are isotropic, ignoring the effect of mismatch and mutual of antenna. Assume the incidence angle of the

Both frequency and incidence angle can be deduced from (

Therefore, the signal vector at the

In this paper, the interelement spacing meet

By the optimization above, the sparse solution

The proposed method based on sparse Bayesian reconstruction is presented as the following steps. Firstly, signal frequency and DOA are roughly estimated by spatial frequencies obtained from reconstruction processing. Then, the estimated frequency and DOA are processed by morphological filtering, which produces precise parameter estimation and network sorting.

According to sparse Bayesian theory and (

To facilitate notation, intermediate variable

From (

To obtain

Taking logarithm of (

In practice, it is difficult to optimize directly on (

To improve the speed of iteration and avoid exception caused by many zeros in

The estimated power

Based on the rough estimation above, the spatial frequency estimations

The reconstruction in Section

By the algorithm in Section

The target function for signal power and spatial frequency is

It can be inferred from

Taking (

The precise estimation of spatial frequency is as follows:

In practical applications, the peak position can be obtained by searching in the smaller range

The spatial frequencies of

Therefore, the incidence angle can be estimated by

The incidence angle and frequency of

In the frequency estimation of FH signals, one frame data may include information about two FH frequencies of the same signal; therefore, the estimation results are often in low credulity, since they suffered from great fluctuation in low SNR condition.

Many image filtering algorithms have been proposed to exclude these data points. Among these methods, median filter, wiener filter, and morphological filter are conventional methods to deal with binary image. Median filter is often used to suppress salt-and-pepper noise and it is a nonlinear filter. But the image we are processing has the feature that there is only one nonzero value in every time index; the median filter method is inefficient with the image of this type. Wiener filter is also a nonlinear filter. Under the criterion of minimum mean-square error, wiener filter is the best filter. Since the wiener filter needs iteration, it has a high computational complexity. morphological filtering method is used to improve the time-frequency spectrum since it has lower computational complexity and it is easier to be realized.

Consider

The obtained binary image is going through erosion and dilation operation. The scatter points of estimates can be eliminated by erosion, while the boundary can be restricted. The void in the signal can be filled by dilation, but the boundary would also be expanded. The erosion and dilation operations using structural element

There is only one nonzero value in each column of the image, and the probability of presence of consecutive points with greater fluctuation is small, so that the structural element is as

The modified time-frequency spectrum is turned into binary form

Four connected components and eight connected components.

Four connected components

Eight connected components

The sets of points of each hop are

Therefore, the frequency estimate of

The DOA estimate of

The procedure of joint parameter estimation and network sorting of multiple FH signals is summarized as follows.

Initialize

Update

Update

Repeat Steps

Conduct precise estimation of spatial frequency by (

Estimate the instantaneous frequency and DOA by (

Update frame for operation and repeat (1), (2), and Step

The time-frequency spectrum generated by (

Detect the connected components of modified binary spectrum and extract the time-frequency point set

The computational complexity of the proposed method mainly depends on the heuristic and search algorithm. From the procedure of the algorithm, we can find that the computational complexity of one iteration mainly focuses on Steps

In this section, the proposed method is analyzed by simulation to verify factors contributing to its performance.

Consider an ULA with eight elements and interelement spacing of 2.5 m. The bandwidth of receiver is

The simulated time-frequency spectrum and space-time pattern with SNR = 0 dB are depicted in Figures

Estimate of DOA and period.

DOA 1/° | DOA 2/° | Period 1/us | Period 2/us | |
---|---|---|---|---|

Real value | 38 | −44 | 100 | 50 |

Estimate | 38.93 | −43.01 | 101 | 50.6 |

Estimate of frequency sets.

Frequency set 1/MHz | Frequency set 2/MHz | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Real value | 42 | 35 | 53.5 | 48.7 | 47 | 53 | 34.3 | 42.5 | 49 | 56 | 39 |

Estimate | 42.02 | 35.03 | 53.51 | 48.67 | 47.02 | 52.96 | 34.33 | 42.46 | 49.04 | 55.97 | 38.95 |

Time-frequency spectrum and space-time spectrum after reconstruction.

Time-frequency spectrum and space-time spectrum after morphological filtering.

As seen in Figure

As seen from Tables

Consider an ULA with eight elements and interelement spacing of 2.5 m. The bandwidth of receiver is

Figures

Bias of frequency estimation in different SNRs.

Bias of DOA estimation in different SNRs.

Bias of period estimation in different SNRs.

From Figures

The reconnaissance and processing of multiple FH signals are challenging tasks in modern warfare. By sparse Bayesian reconstruction method, the parameter estimation and network sorting of multiple wideband FH signals are successfully achieved and precise estimation of parameters in low SNR conditions is conducted by the proposed method. The procedure of the algorithm has been deduced in theoretical analysis. And the excellent performance of the algorithm is verified by simulation.

The authors declare that they have no conflicts of interest.

This research was supported by the National Natural Science Foundation of China (no. 61601500).