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Spatial polarimetric timefrequency distribution based DOA estimation: combining ESPRIT with MUSIC
EURASIP Journal on Wireless Communications and Networking volume 2018, Article number: 51 (2018)
Abstract
Spatial spectrum estimation technology has always been the focused research project in an array signal processing of High Frequency Surface Wave Radar (HFSWR). This paper takes the polarization sensitive array for the model and combines with timefrequency analysis method for DOA estimation of HFSWR echo signals, which can make full use of the spatial, timefrequency, and polarized domain information of signals. The timefrequency MUSIC and ESPRIT algorithms of integrated polarization information are studied, and the improved algorithm is proposed based on the advantages and disadvantages of both algorithms. The improved algorithm first estimates the azimuth angles of signals roughly with the polarized timefrequency ESPRIT algorithm and then determines a small scope centered on each azimuth. Within the scope, a more accurate DOAestimated value is got by spectral peak search with MUSIC algorithm, so it can save computation time on the basis of accuracy of the DOA estimation, and their performance is analyzed by computer simulations.
Introduction
In recent years, as the coastal area of space gets smaller to make HFSWR array antenna aperture get smaller too, using a signal processing technology to improve the precision of the direction of arrival (DOA) estimation algorithm has also become a research hotspot. The timefrequency analysis method uses the joint function of time and frequency to describe the energy density of the processed signal and has its unique advantage in dealing with nonstationary signals. In 1998, A. Belouchrani and M. Amin first observed that the WignerVille distribution of the array output and the correlation matrix of the array output had a similar structure and applied it to the DOA estimation [1, 2]. Yimin Zhang [3, 4] found the intrinsic relation between the spatial timefrequency distribution matrix and the covariance matrix of the array output. For the first time, it has been proved theoretically that the structure of the spatial timefrequency distribution matrix can be used to estimate the DOA of the signals, which provides theoretical support to study the timefrequency signal processing for later scholars. Polarization information is the inherent attribute of the electromagnetic wave signal, and it is also the information of the echo signals. The electromagneticvectorsensor (EMVS) has received extensive attention in an array signal processing due to the fact that it cannot only provide DOA of the signal, but can also give the polarization information. An electromagnetic vectorsensor usually consists of three orthogonally oriented dipoles to measure the electric field, plus three orthogonally oriented loops to measure the magnetic field of the source [5]. A unique DOA estimation algorithm for the EMVS, the vectorcross product algorithm, has been proposed in [5] and been advanced in [6, 7] along with various eigenstructurebased directionfinding schemes [8,9,10]. Dr. Ferrara and Parks of Stanford University in the USA used a polarized sensitive array composed of crossed dipole and combined with classical super resolution algorithms to study DOA [11]. Dr. Jian Li of the USA studied the polarization state parameters and the estimation of the DOA of electromagnetic waves in the case of polarized sensitive arrays based on uniform linear array [12]. In the literature [13], the polarization state parameters and DOA estimation in the case of polarized sensitive arrays based on rectangular arrays are studied. In [14], the joint estimation of the DOA and the polarization state in the case of the vector sensors in the absence of regularity is studied. In literature [15], the DOA and polarization parameters of the source signals are estimated firstly and then the variation of the estimated performance with the SNR is analyzed.
This paper intends to use the dualpolarized receiving antenna array from the compact highfrequency ground wave radar of the LFM system; make full use of the spatial, timefrequency, and polarized domain information of signals; and combined with the modern spatial spectrum estimation technologies. The superresolution algorithm of timefrequency spectrum estimation of the integrated polarization information is studied.
Signal model
Polarization sensitive array
Polarization sensitive array refers to polarized sensitive elements arranged in a particular way in space, and the array can receive spatial and polarized domain information of the source signals. The spatial information can be expressed by the phase delay between different elements, but the polarization information needs to analyze the structure of the polarized sensitive elements. Polarization sensitive elements can receive six electromagnetic components most at the same time, but there will be some redundancy. By the relationship between the electric and magnetic field, a part of the components can be selected to form the element, but the polarization information of the electromagnetic wave signals can be obtained at least two components [16]. Therefore, it is usual to form a uniform linear array with orthogonal cross dipoles, as shown in Fig. 1. This structure is relatively simple and easy to achieve.
The polarization information of the electromagnetic wave signals can be expressed by the amplitude ratio and the phase difference of two mutually orthogonal electric fields, i.e.,
where\( \tan \gamma =\frac{A_y}{A_x} \)represents the ratio of electric field amplitudes between the ydirection and the xdirection, in addition,η = ϕ_{ y } − ϕ_{ x } represents the phase difference of electric field between the ydirection and the xdirection. Regardless of the energy information of the electromagnetic wave, the polarization parameter (γ, η) can represent the polarization information of any state.
When there are k signal sources inciding N elements array antenna and the noise is independent and stationary, the received signal can be expressed as
where
represents the polarizedspatial domain joint steering vector of received signals, S_{p} is polarized domain steering vector, and S_{s} is spatial domain steering vector, symbol ⊗ represents kernel product of S_{p} and S_{s}. The polarized domain steering vector can be expressed as
and the spatial domainoriented vector can be expressed as
Spatial polarimetric timefrequency distributions
The Cohen’s class of STFD of a data vector x(t) is expressed as
where φ(t, τ) is the timefrequency kernel function and it uniquely be used to define timefrequency distributions.
From the knowledge of timefrequency transform and the polarization sensitive array model introduced abovementioned, we can see that for the kth dual polarized orthogonal dipole, the selfterm and crossterm timefrequency distributions can respectively be expressed as
where the superscript i and j both represent the vertical component v or the horizontal component h of the array element.
Therefore, the timefrequency distribution of the kth dual polarized orthogonal dipole is a 2 × 2 matrix, which consists of the vertical and horizontal oscillator’s selfterm and crossterm timefrequency distributions. Considering a uniform linear array consists of n dual polarized orthogonal dipole, the received data vector can be expressed as
The timefrequency distributions of its selfterm and crossterm can be expressed as
Spreading Formula (9) in accordance with the horizontal and vertical component, we can get the following formula
It is clear that the dual polarized orthogonal dipole element has more available information than the unipolar element. After theoretical analysis, we can combine the spatial, polarized, and timefrequency domain information of the signal received by the polarization sensitive array. Thus, the spatial polarization timefrequency distribution (SPTFD) of the received data x(t) can be expressed as
Spatial polarimetric timefrequency DOA estimation algorithm
Polarimetric timefrequency MUSIC
In [3], it has been proved that the structure of spatial timefrequency distribution matrix is similar to that of the traditional array covariance matrix. Therefore, the STFD matrix can be applied to the subspace class algorithm instead of the covariance matrix in the MUSIC algorithm, called timefrequency MUSIC algorithm.
Similar to the timefrequency MUSIC algorithm, the polarimetric timefrequency MUSIC algorithm, which integrated polarization information, takes the polarization sensitive array for the model and then do timefrequency transform of the received signal shown in Eq. (2) which can obtain the SPTFD matrix. The SPTFD matrix of the corresponding signal can be obtained by sampling each signal along its timefrequency ridge, and the corresponding noise subspace can be obtained by characteristic decomposition.
The following summarizes specific steps of the polarimetric timefrequency MUSIC (PTFMUSIC) algorithm:

(1)
Performing pseudoWignerVille timefrequency transform to the polarization data received on the first receive channel and estimating the instantaneous frequencies and the frequency modulation slopes of the incoming wave signals in their timefrequency domain.

(2)
Using the estimated signal parameters to select points in the timefrequency domain, the different frequency parameters of the signal were selected on their respective timefrequency ridge points to construct their own spatial polarimetric timefrequency distribution matrix.

(3)
The eigenvalue decomposition, construction of the noise subspace, and construction of the spatial spectrum are carried out on the constructed spatial polarimetric timefrequency distribution matrix, respectively.
Polarimetric timefrequency ESPRIT
Based on the previous study of MUSIC algorithm and ESPRIT algorithm, we know that they both belong to subspace class algorithm, so similar to MUSIC algorithm, ESPRIT algorithm can also be combined with timefrequency analysis integrated polarization information, which is polarimetric timefrequency ESPRIT (PTFESPRIT) algorithm. The biggest advantage of the MUSIC algorithm is the high accuracy of estimation, while the ESPRIT algorithm has the biggest advantage of fast calculation. Therefore, the PTFESPRIT algorithm should have a faster calculation speed than PTFMUSIC algorithm in theory. The TLSESPRIT algorithm is used in this section.
The principle of PTFESPRIT algorithm is similar to the PTFMUSIC algorithm. Its algorithm flow is as follows:

(1)
Performing pseudoWignerVille timefrequency transform to the polarization data received on the first receive channel and estimating the instantaneous frequencies and the frequency modulation slopes of the incoming wave signals in their timefrequency domain.

(2)
Using the estimated signal parameters to select points in the timefrequency domain, for the different frequency parameters of the signal were selected on their respective timefrequency ridge points to construct their own SPTFD matrix.

(3)
The covariance matrices R_{ 11 } and R_{ 22 } are obtained by the SPTFD matrix.

(4)
The covariance matrices are decomposed and two signal subspaces E_{ 1 } and E_{ 2 } are obtained, then combine them into a new matrix\( \overline{\mathbf{E}}=\left[{\mathbf{E}}_{\mathbf{1}}\kern0.5em {\mathbf{E}}_{\mathbf{2}}\right] \).

(5)
Construct the matrix \( {\overline{\mathbf{E}}}^H\overline{\mathbf{E}} \) and characterize it, we can get the feature matrixE, E can be divided into four P × P dimensional matrix.

(6)
Calculate \( {\boldsymbol{\Psi}}_{\mathbf{TLS}}={\mathbf{E}}_{\mathbf{12}}{\mathbf{E}}_{\mathbf{22}}^{1} \) and characterize it, we can get P eigenvalues. And the wave direction information of signals can be obtained according to the formulaϕ_{ k } = 2πd sin θ_{ k }/λ.
Improved algorithm
We know that the estimation accuracy of the PTFMUSIC algorithm is very high, but because of its need for a larger angle range for peak search, coupled with the complexity of the timefrequency transform, it takes a long time to get the DOA estimation of the incoming wave signals. The PTFESPRIT algorithm does not require spectral peak search, so it takes much less time than the PTFMUSIC algorithm, but its estimation accuracy is not so good. Therefore, an improved algorithm is proposed in this paper, and the advantages of the two algorithms are comprehensively used to estimate the DOA.
The principle of the improved algorithm is as follows. Firstly, the pseudoWignerVille timefrequency transform of the first channel polarization data received by the array is used to obtain its polarization timefrequency distribution figure and then analyze and extract the SPTFD matrix on the timefrequency ridge of each signal, and use it to replace the traditional covariance matrix; the DOA of signal will be estimated by TLSESPRIT algorithm roughly. The arrival angles of the different signals are locked in a range. In this range, the MUSIC algorithm is used to estimate the DOA accurately. This will not only save time but also improve the estimation accuracy; the algorithm flow is shown in Fig. 2.
As shown in Fig. 2, the specific process of the improved algorithm is as follows:

(1)
Firstly, the pseudoWignerVille timefrequency transform is performed on the polarization data received by the first channel of the array, and the instantaneous frequency and frequency modulation slope of the incoming wave signal are estimated in the timefrequency domain.

(2)
With the estimated signal parameters to select points in the timefrequency domain, for signals of different frequency parameters, the points on respective timefrequency ridge were selected to construct their own SPTFD matrix.

(3)
The DOA estimation of the timefrequency ESPRIT algorithm is used to obtain a general range of DOAs for each signal.

(4)
The PTFMUSIC algorithm is carried out in the DOA range obtained in the previous step, and the spatial spectrum is constructed to obtain the accurate DOA estimation value.
Simulation results
Polarimetric timefrequency MUSIC
Experiment 1: TF MUSIC vs PTF MUSIC
Assuming a 4element polarized sensitive array consists of dual polarized orthogonal dipoles and three linear frequency modulation signals incident on it, the normalized frequencies of the three signals are 0.2 to 0.4, 0.22 to 0.42, and 0.1; DOAs are − 3°, 3°, and 9°; polarized angles are 45°, 45°, and 20°; and polarized phase angles are 0°, 180°, and 0°, respectively. Take the data length of 256, the signal to noise ratio of 13 dB, assuming that the number of signals and their polarized parameters are known. In this experiment, DOA estimation of signal 1 and signal 2 are carried out by timefrequency MUSIC and polarized timefrequency MUSIC algorithm. Each algorithm performs three independent experiments, analyzes its timefrequency transform, and draws spatial spectrum. First of all, the WVD timefrequency transform of the received data is shown in Fig. 3.
By the polarized state of the signals, we can see that s_{1} (source 1) and s_{2} (source 2) satisfies the conditions of electromagnetic vector quadrature. So crossterm between the two signals is suppressed when the timefrequency distribution of the two directional components is taken into account. This helps us select the timefrequency points of signals more accurately.
Figure 4 is the spatial spectrum obtained by timefrequency MUSIC algorithm and polarized timefrequency MUSIC algorithm to signal 1 and signal 2, each algorithm has been done three independent experiments.
From Fig. 4, we can see that signal 1 and signal 2 cannot be distinguished correctly by the MUSIC spatial spectrum obtained by the timefrequency information of the unipolar component because their timefrequency points are too near and there are very serious crossterm, resulting in fuzzy of their timefrequency information. However, the PTF MUSIC algorithm can distinguish the two signals very easily because the two signals are orthogonal to the electromagnetic information, so that their timefrequency information is very clear and their timefrequency points can be easily extracted, and the resulting spectral peak is very sharp.
Experiment 2: estimated accuracy of TF MUSIC vs PTF MUSIC
Assuming a 4element polarized sensitive array consists of dual polarized orthogonal dipoles and two LFM signal incidents on it, the normalized frequencies of the two signals are 0.2 to 0.4 and 0.1, DOAs are − 3° and 9°, polarized angles are 45° and 20°, and polarized phase angles are 0° and 180°, respectively. Taking the data length of 256, assuming the number of signals and their polarized parameters are known, the SNR increases from − 10 to 10 dB with an increase of 2 dB. The Monte Carlo experiments were carried out 200 times to calculate the RMSE of each DOA estimation by using the TF MUSIC algorithm and PTF MUSIC algorithm, respectively; the results are shown in Fig. 5.
As can be seen from Fig. 5, the RMSE of the two DOA estimates both decreases with the increase of the SNR between two algorithms, which means that the accuracy of the DOA estimation is getting higher and higher. Overall, the accuracy of the DOA estimate is improved after taking full advantage of the polarized information of the electromagnetic signals, especially in the case of low SNR.
Polarimetric timefrequency ESPRIT
Experiment 3: TF ESPRIT vs PTF ESPRIT
Assuming a 4element polarized sensitive array consists of dual polarized orthogonal dipoles and three LFM signal incidents on it, the signal parameters are the same as in experiment 1. Take the data length of 256, and the SNR of 13 dB, assuming that the number of signals and their polarized parameters are known. In this experiment, the TF ESPRIT and PTF ESPRIT algorithm are used to estimate DOAs of signal 1 and signal 2 and each algorithm performs 100 independent Monte Carlo experiments. The average values of DOAs are calculated respectively, and the results are shown in Table 1.
From Table 1, we know that since the timefrequency information of signal 1 and signal 2 is very similar and there are cross terms, the TF ESPRIT algorithm cannot distinguish them easily. But because the signals have different polarized information, the PTF ESPRIT algorithm can distinguish the two signals correctly.
Experiment 4: PTF ESPRIT vs PTF MUSIC
Assuming a 4element polarized sensitive array consists of dual polarized orthogonal dipoles and two LFM signal incidents on it, the signal parameters are the same as in experiment 2. Take the data length of 256, and the noise is Gaussian white noise uniformly distributed, assuming that the number of signals and their polarized parameters are known. The SNR increases from − 10 to 10 dB with an increase of 2 dB, the Monte Carlo experiments were carried out 100 times to calculate the RMSE of each DOA estimation by using the PTF MUSIC and PTF ESPRIT algorithm, respectively; the results are shown in Fig. 6. In addition, the time to calculate the timefrequency transformation and the two algorithms are shown in Table 2.
It can be seen from Fig. 6 that the RMSE of the DOA estimates of the two signals estimated by the PTF MUSIC and PTF ESPRIT algorithm both decreases with the increase of the SNR, which means that the accuracy of DOA estimation is also getting higher and higher. Comparing the two algorithms, it can be seen that the estimated accuracy of the PTF MUSIC algorithm is higher than PTF ESPRIT algorithm under the same conditions, especially in the case of low SNR.
Table 2 describes the computer consumed times of two algorithms, the computer configuration are as follows: CPU is Intel Pentium E5200 (Dual core), memory is 4G, and OS is Window10 platform. It can be seen from Table 2 that the TF transformations occupy a considerable amount of time; in addition, the PTF ESPRIT algorithm only needs 18.95 s, while the PTF MUSIC algorithm needs 105.7865 s, about 6 times of the PTF ESPRIT algorithm. In this experiment, the spectral peak search is only between − 20° to 20°, if the DOA of signal incident is from a wide range of angles, the time consuming of PTF MUSIC algorithm will be as several dozen times as PTF ESPRIT algorithm.
Improved algorithm
Assuming a 4element polarized sensitive array consists of dual polarized orthogonal dipoles and two LFM signal incidents on it, the snapshot numbers are 256 and the normalized start and end frequencies of the two signals are 0.2~0.4 and 0.1~0.1. The SNR increases from 0 to 10 dB with an increase of 2 dB, use the PTF MUSIC, PTF ESPRIT and improved algorithm to estimate the DOAs of incident signals, respectively. Each algorithm performs 100 independent experiments to calculate the RMSE, the results are shown in Fig. 7.
It can be seen from Fig. 7 that the improved algorithm and the PTF MUSIC algorithm have similar estimated accuracy, which are higher than that of the PTF ESPRIT algorithm.
Experiment 6: the run time of the improved algorithm
The experimental condition is similar to those of experiment 5, except that the SNR is fixed at 10 dB. This experiment calculates the run time required for 100 times of Monte Carlo experiments for each algorithm, as shown in Table 3.
As can be seen from Table 3, the time spent of PTF MUSIC and improved PTF DOA estimation algorithms are 138.7203 and 48.8823 s, respectively. Yet timefrequency transform needs to be carried out in each experiment and the time of 100 times of timefrequency transform is 43.8828 s, which means that the PTF MUSIC algorithm and improved PTF DOA estimation algorithm spend 94.8195 and 4.9995 s, respectively. Meanwhile, the PTF ESPRIT algorithm spends about 3.2668 s, so the time required for PTF MUSIC algorithm is about 19 times of the time for the improved PTF DOA estimation algorithm, and is about 29 times of the time used for the PTF ESPRIT algorithm under the same circumstances. This experiment shows that the improved PTF DOA estimation algorithm optimizes the timeconsuming problem to a large extent under the condition of guaranteeing estimation accuracy.
Conclusions
This paper focuses on the principle of polarized timefrequency MUSIC and ESPRIT algorithms, and the improved algorithm is proposed based on the advantages and disadvantages of both of them. The improved algorithm makes full use of the advantages of the small calculation of ESPRIT algorithm and the high accuracy of MUSIC algorithm, which makes the algorithm more practical and the computer simulation also proves its performance. However, there are still some places need to be improved, such as the pseudoWV distribution is suitable for the processing of LFM signals, but its kernel function cannot adapt to different forms of signals, and it also has cross terms to a certain extent. Therefore, it is worth considering how to design a kernel function that changes adaptively with the signal forms and does not have cross terms.
Abbreviations
 DOA:

Direction of Arrival
 ESPRIT:

Estimating Signal Parameters Viarotational Invariance Techniques
 HFSWR:

High Frequency Surface Wave Radar
 LFM:

Linear Frequency Modulation
 MUSIC:

Multiple Signal Classification
 SNR:

SignaltoNoise Ratio
 SPTFD:

Spatial Polarization TimeFrequency Distribution
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Acknowledgements
This project is sponsored by the National Marine Technology Program for Public Welfare (no. 201505002), the National Natural Science Foundation of China (nos. 61571157, 61401118, and 61371100), and the Subject Guide Fund of Harbin Institute of Technology at Weihai (no. WH20150111).
Funding
The National Marine Technology Program for Public Welfare (no. 201505002) is supporting the data acquisition devices and materials, the National Natural Science Foundations of China (nos. 61571157, 61401118, and 61371100) are supporting the simulations, and the Subject Guide Fund of Harbin Institute of Technology at Weihai (no. WH20150111) is supporting the data analyses.
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AL and BL conceived and designed the experiments, FL performed the experiments, QL and XS contributed simulation tools, and AL and BL wrote the paper. All authors have read and approved the final manuscript.
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Correspondence to Bo Li.
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Keywords
 DOA estimation
 Timefrequency
 Polarization sensitive array