Passive localization of nonstationary sources in the spherical coordinates (azimuth, elevation, and range) is considered, and a parallel factor analysis based method is addressed for the near-field parameter estimation problem. In this scheme, a parallel factor analysis model is firstly constructed by computing five time-frequency distribution matrices of the properly chosen observation data. In addition, the uniqueness of the constructed model is proved, and both the two-dimensional (2D) direction-of-arrival (DOA) and range can be jointly obtained via trilinear alternating least squares regression (TALS). The investigated algorithm is well suitable for near-field nonstationary source localization and does not require parameter-pairing or multidimensional search. Several simulation examples confirm the effectiveness of the proposed algorithm.
Bearing estimation has been a strong interest in radar and sonar as well as communication. In the last three decades, various high-resolution algorithms for direction finding of multiple narrowband sources assume that the propagating waves are considered to be plane waves at the sensor array. However, when the sources are located in the Fresnel region [
By applying the Fresnel approximation to the near-field sources localization, the two-dimensional (2D) MUSIC method, the high-order ESPRIT method, and the path-following method were, respectively, proposed in [
While quadratic time-frequency distribution [
In this paper, by exploiting favorable characteristics of a uniform cross array, we present a joint 2D DOA and range estimation algorithm. We first compute five time-frequency matrices to construct a parallel factor (PARAFAC) analysis model. Then, we obtain three-dimensional (3D) near-field parameters via trilinear alternating least squares regression (TALS). Compared with the other methods, the main contribution for the proposed method can be summarized as follows:
The rest of this paper is organized as follows. Section
We consider a near-field scenario of
Sensor-source configuration for the near-field problem.
The
The main problem addressed in this paper is to jointly estimate the sets of parameters The source signal The additive noise is spatially white Gaussian with zero-mean and independent from the source signals. For unique estimation, we require
We need to introduce the following notation that will be used in the sequel.
Let
For a matrix
Consider a three-dimensional tensor
The discrete form of Cohen’s class of time-frequency distribution of a signal
Substituting (
Under the assumptions
Using a rectangular window of old length
Assume that the third-order derivative of the phase can be negligible over the rectangular window length
We construct matrix
On the other hand, following the same process described above, we can easily obtain
Considering the situation of limited samples, we build a parallel factor analysis model that uses the spatial time-frequency distribution as
Letting
Similarly, (
As it stands,
Then using these estimates, we can get each pair
Finally, the sources parameters can be estimated as
In this section, we explicit several simulation results to evaluate the performance of proposed method. For all examples, a symmetrical cross array with a number of 17 elements and interelements spacing of
In the first example, we examine the performance of the elevation, azimuth, and range estimations accuracy versus the SNR. The snapshot number is set at 512. Two linear frequency-modulated signals arrival at the sensor array with start and end frequencies
The RMSEs of the elevation, azimuth, and range estimation using the proposed method and the fourth-order cumulant based method versus SNRs.
Elevation
Azimuth
Range
In the second example, the proposed method is used to deal with the situation that two near-field FM signals are impinging on the sensor array shown in Figure
The mean and variance of the proposed method for the second example.
True | Mean | Variance | ||
---|---|---|---|---|
Source 1 | Elevation (°) | 35 | 34.1036 | 0.0854 |
Azimuth (°) | 40 | 38.8652 | 0.0514 | |
Range | 1/6 |
0.1956 |
|
|
|
||||
Source 2 | Elevation (°) | 20 | 19.8274 | 0.0963 |
Azimuth (°) | 60 | 59.3610 | 0.1827 | |
Range | 0.4 |
0.4449 |
|
In the last example, we consider the situation when far-field and near-field nonstationary sources are incoming on the sensor array mentioned above, and they are located at
The RMSEs of the elevation, azimuth, and range estimation using the proposed method and the fourth-order cumulant based method versus SNRs.
Elevation
Azimuth
Range
We have developed a spatial time-frequency distribution based algorithm for 3D near-field nonstationary source localization problems. Additionally, with parallel factor analysis technique, there is no parameter pairing or multidimensional searching. Finally, the computer simulation results indicate that using spatial time-frequency distribution and parallel factor together significantly solves the problem of the joint estimation of elevation, azimuth, and range of nonstationary signals. However, the spatial time-frequency averaging methods may lead to the additional computation load.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work is supported by the National Natural Science Foundation of China (61171137) and Specialized Research Fund for the Doctoral Program of Higher Education (20090061120042).