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 nearfield parameter estimation problem. In this scheme, a parallel factor analysis model is firstly constructed by computing five timefrequency distribution matrices of the properly chosen observation data. In addition, the uniqueness of the constructed model is proved, and both the twodimensional (2D) directionofarrival (DOA) and range can be jointly obtained via trilinear alternating least squares regression (TALS). The investigated algorithm is well suitable for nearfield nonstationary source localization and does not require parameterpairing 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 highresolution 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 nearfield sources localization, the twodimensional (2D) MUSIC method, the highorder ESPRIT method, and the pathfollowing method were, respectively, proposed in [
While quadratic timefrequency 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 timefrequency matrices to construct a parallel factor (PARAFAC) analysis model. Then, we obtain threedimensional (3D) nearfield 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 nearfield scenario of
Sensorsource configuration for the nearfield 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 zeromean 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 threedimensional tensor
The discrete form of Cohen’s class of timefrequency distribution of a signal
Substituting (
Under the assumptions
Using a rectangular window of old length
Assume that the thirdorder 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 timefrequency 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 frequencymodulated 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 fourthorder cumulant based method versus SNRs.
Elevation
Azimuth
Range
In the second example, the proposed method is used to deal with the situation that two nearfield 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 farfield and nearfield 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 fourthorder cumulant based method versus SNRs.
Elevation
Azimuth
Range
We have developed a spatial timefrequency distribution based algorithm for 3D nearfield 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 timefrequency distribution and parallel factor together significantly solves the problem of the joint estimation of elevation, azimuth, and range of nonstationary signals. However, the spatial timefrequency 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).