Abstract
This paper proposes a multi-invariance ESPRIT-based method for estimation of 2D direction (MIMED) of multiple non-Gaussian monochromatic signals using cumulants. In the MIMED, we consider an array geometry containing sparse -shaped diversely polarized vector sensors plus an arbitrarily-placed single polarized scalar sensor. Firstly, we define a set of cumulant matrices to construct two matrix blocks with multi-invariance property. Then, we develop a multi-invariance ESPRIT-based algorithm with aperture extension using the defined matrix blocks to estimate two-dimensional directions of the signals. The MIMED can provide highly accurate and unambiguous direction estimates by extending the array element spacing beyond a half-wavelength. Finally, we present several simulation results to demonstrate the superiority of the MIMED.
1. Introduction
Estimation of direction-of-arrival (DOA) and polarization parameters using an array of diversely polarized electromagnetic vector sensors is very important in many applications, such as radar and wireless communications. In the past two decades, many electromagnetic vector sensor direction finding algorithms have been proposed [1–8]. Among them, ESPRIT-based algorithms [2–4] are highly popular due to their low computations and high estimation accuracy. Generally, the ESPRIT-based algorithm exploits the spatial invariance between two identical subarrays to offer direction estimation.
Multi-invariance ESPRIT (MI-ESPRIT), first proposed in [9], is an extension of conventional ESPRIT by exploiting the inherent multiple invariances embedded in the array structure. With exploitation of the special multi-invariance structure, MI-ESPRIT can offer better estimation performance than original version of ESPRIT [9]. In recent years, MI-ESPRIT algorithm has been applied to solve direction finding problems, where the impinging signals can be MC-CDMA [10], Noncircular [11], and narrowband chirp [12]. However, all the MI-ESPRIT-based algorithms mentioned above are developed using scalar arrays, with array intersensor spacing being not beyond a half-wavelength to guarantee unique and unambiguous direction estimates.
In this paper, we apply the idea of MI-ESPRIT to sparse electromagnetic vector sensor arrays to achieve aperture extension for highly accurate direction estimation. We present a multi-invariance ESPRIT-based method for estimation of two-dimensional direction (MIMED) of multiple non-Gaussian signals using cumulants. In the proposed MIMED, we consider an array geometry that contains sparse -shaped diversely polarized vector sensors plus an arbitrarily-placed single polarized scalar sensor (dipole). Firstly, a set of cumulant matrices is defined to form two matrix blocks that possess the multi-invariance property. Then, a multi-invariance ESPRIT-based algorithm with aperture extension using the defined matrix blocks to estimate two-dimensional directions of the signals is developed. The MIMED can provide highly accurate and unambiguous angle estimates by exploiting the fact that the spatial phase factor is larger than a half-wavelength.
Note also that the MIMED algorithm is related to the algorithm presented in [13] in the sense that both algorithms define the cumulant matrices to estimate the directions of signals. However, since the sensors used in the MIMED algorithm are different from those in [13], the algorithmic derivation and performance bounds are different from those in [13]. For example, (i) [13] is derived based on a scalar sensor array, whereas the MEMID algorithm is derived using a vector sensor array; (ii) the algorithm in [13] defines cumulant matrices to formulate the tensor model, whereas the MIMED algorithm defines cumulant matrices to form the multi-invariance ESPRIT model.
Notation. Throughout the paper, superscripts , , , and , respectively, represent the transpose, conjugate transpose, complex conjugate, and pseudoinverse, and denotes the Kronecker-product operator. denotes the smallest integer not less than , represents the largest integer not greater than , and signifies the principal argument of the complex number between and .
2. Signal Model
Consider narrowband completely polarized source signals, parameterized by , , where , , , and , respectively, denote the elevation angle, azimuth angle, orientation angle, and ellipticity angle of the th source, impinging upon an array, which is composed of sparse -shaped diversely polarized vector sensors plus an arbitrarily-placed single polarized scalar sensor, as shown in Figure 1. The location of the th sensor lied on the -axis is , and the location of the th sensor lied on the -axis is , where and are the intersensor spacing. Each electromagnetic vector sensor consists of six spatially orthogonal but colocated components: three electric dipoles plus three magnetic loops. The electromagnetic vector sensor response for the th source signal produces the following steering vector [2]:where the first three entries and the last three entries represent the electric and the magnetic field vectors, respectively. Note that the array manifold of electromagnetic vector sensors does not contain the angle-related phase factor. This fact is pivotal to the MIMED in improving the estimation accuracy by extending the array aperture beyond that limited by the spatial Nyquist sampling theorem. Also, note that the Frobenius-norm of the normalized Poynting vector is independent of the parameters of the signal and is equal to unity. That is, the electric field vector and the magnetic field vector are orthogonal to each other and to the source signal’s normalized Poynting vector , i.e., [2]where , , and denote the direction cosines of the -axis, -axis, and -axis.
Then, the data vectors measured by the first to the th vector sensors at time can be expressed aswhere denotes the phasor representation of the th signal, denotes the signal vector, , with , and represent the additive noise vector. Likewise, the output vector measured by the th to th vector sensors can be expressed aswhere , with , and represent the additive noise vector.
Next, we assume the single polarized scalar sensor is placed arbitrarily at the location . Then, the data collected by the single polarized scalar sensor can be represented aswhere and is additive noise.
With a total of snapshots taken at , the problem is to determine the azimuth-elevation directions from these snapshots. For beamforming purposes, it may be also useful to subsequently estimate the corresponding polarization parameters . We will present the MIMED algorithm to solve the above-mentioned problems in Section 3, under the following assumptions: (i) the parameters are distinct with each other, and the array steering vector matrix is of full column rank; (ii) the impinging signals are zero-mean and stationary, mutually independent, and non-Gaussian, having nonzero fourth-order cumulants; (iii) the noise is zero-mean, complex Gaussian, and possibly spatially correlated.
3. Algorithm Development
3.1. Formulation of the Cumulant Matrices
In this subsection, we define cumulant matrices that can be linked to multi-invariance model for direction estimation. In forming the cumulant matrices, the single polarized scalar sensor is used as a reference sensor. Let , , and be the data measured by the th, th, and th () vector sensor element from the -shaped vector sensors, and let the fourth-order cumulants of , , , and be . Using the assumptions made in Section 2 and the cumulant properties in [14], we have where and with being the location of the th vector sensor, andThen, denoting the vector , we define the cumulant matrices aswhere is the th row of . After some mathematical computations, we can obtainwherewith .
Next, we construct two data blocks using the defined cumulant matrices asWith the foregoing definitions, we will show in next subsection that and are of multi-invariance properties, which can be exploited to estimate the directions of signals using MI-ESPRIT algorithm.
3.2. MI-ESPRIT Algorithm for Direction Estimation
It is easily to verify that the block is of the formThe first rows of can be considered as measurement of signal by physical vector sensors, while the th rows ( to rows) of can be viewed as measurement of signal by virtual vector sensors, which are virtually placed by shifting the physical sensors along -axis with distance . Then, for all , altogether different virtual sensor groups can be formed. Each virtual sensor group has its own spatial invariance, so that distinct spatial invariances can be provided [9]. Therefore, the matrix (17) is of multi-invariance characteristic [9] which can be exploited for use of MI-ESPRIT for estimating directions. The signal subspace matrix , which contains the eigenvectors associated with largest eigenvalues of , can be represented aswhere is an full rank matrix.
Let and , respectively, be the first rows and the last rows of ; we haveFrom (19), we can infer that, with the estimation of and , the diagonal elements of can be estimated from eigenvalues of and the matrix can be estimated from the eigenvectors of , i.e.,where . Since , a set of ambiguous direction cosine estimates that satisfy (20) can be obtained. These estimates are expressed as [15]
Similarly, we can obtain a set of cyclically ambiguous direction cosine estimates from the data block . These estimates are expressed as [15]where .
The above low-variance but cyclically ambiguous direction cosine estimates can be disambiguated by using the set of high-variance but unambiguous direction cosine estimates extracted from the estimation of electromagnetic vector sensor array manifolds. This extraction may be accomplished by decoupling the matrix aswhere denotes the th to th elements of . According to (18), can be estimated as the first rows of . Denoting the th column of as , the high-variance but unambiguous direction cosine estimates of the th signal can be directly calculated by the vector cross product between the normalized electric field vector and the magnetic field vector, i.e.,where and , respectively, correspond to the first three and the last three columns of .
Finally, the high-variance but unambiguous direction cosine estimates would serve as reference direction cosine estimates to resolve the cyclically ambiguous direction cosines estimates . The disambiguated -axis and -axis direction cosines, and , are found from and when the value of and are minimized. Mathematically, the disambiguated -axis direction cosine estimates are given bywhere is estimated asAnalogously, the disambiguated -axis direction cosine estimates arewhere is estimated asNote that since is paired with , the estimated and are then automatically paired without any additional processing.
3.3. Implementation of the MIMED Algorithm
The implementation of the proposed MIMED algorithm is summarized as follows:(S1)Estimate the cumulant matrices , , from the data samples.(S2)Form the matrices and using (16); then perform the eigenvalue decomposition to and to construct the signal subspaces and , whose columns correspond to the eigenvectors associated with largest eigenvalues of and .(S3)Let and , respectively, be the first rows and the last rows of , , and , respectively, be the first rows and the last rows of . Perform eigenvalue decomposition of and .(S4)Estimate the ambiguous direction cosine estimates and from (21) to (23) and (24) to (26).(S5)Estimate using (27).(S6)Compute reference direction cosine estimates and using (28).(S7)Obtain the disambiguated direction cosine estimates and using (29) and (31).
4. Simulation Results
Simulation results are presented to demonstrate the efficacy of the MIMED. The array configuration in Figure 1 with elements is used. Two narrowband uncorrelated monochromatic signals with identical power impinge on the array. The signal direction cosines , , , and are simulated. The first signal is left-circularly polarized and the second right-circularly polarized. The snapshots used are snapshots. independent Monte Carlo trials are performed. Further the additive white noise is assumed to be complex Gaussian. In all the simulations, the performance metric used is the root mean squared error (RMSE) of the first signal.
We first assess the performance of the MIMED for various values of , the intersensor spacing. Figure 2 shows, on a log-log scale, the RMSEs of the reference direction estimates and the disambiguated direction estimates as a function of , varying from to , being the wavelength. The SNR for each of the signals is set to dB. It is seen that the RMSE of the disambiguated direction estimates decrease linearly as the intersensor spacing increases from up to . The performance of the reference direction estimates keeps almost unchanged as the sensor spacing increases from to . For , the disambiguated direction estimates exhibit almost the same statistical errors as the reference direction estimates. This behavior is similar to that in [4] and can be explained as follows. Referring to (29) and (31), the estimates and suffer ambiguities of some unknown integer multiples of the grid size . As the intersensor spacing increases, the grid sizes shrink relative to the variances of the reference estimates and . Therefore, it becomes increasingly probable that and would identify the wrong grid point. As the intersensor spacing continues to increase, the grid misidentification will become the dominant error, and the disambiguated estimates and eventually have the same error statistics as the reference estimates and . These imply that and would have the same error statistics as and . Also note that the performance of the MIMED algorithm is very close to the CRB.
In the second example, we compare the RMSEs of the MIMED with the cumulant-based ESPRIT algorithm proposed by Liu and Mendel [16] and the second-order statistics based ESPRIT-based algorithm proposed by Zoltowski and Wong [4]. For the algorithm in [16], we assume -element arbitrarily-spaced array configuration that contains three presets guiding sensors located at , and . For the algorithm in [4], we use a -element square-shaped electromagnetic vector sensor array. For these two algorithms, the estimated sets of directions are assumed to have been correctly paired. For the MIMED and the one in [4], we set . Figure 3 shows the RMSEs of the three algorithms as a function of the SNR, varying from dB to dB. It is seen from the figure that the MIMED has a performance that is better than those of the other two algorithms for a wide range of the SNR (SNR dB).
In the third example, we compare the RMSEs of the algorithms as a function of the number of snapshots. The simulation conditions are similar to those in the second example, except that the SNR is set at dB, and the number of snapshots is varying from to . The RMSEs of the three algorithms are shown in Figure 4, where the CRBs are also plotted for comparison. It is seen from the figure that the RMSEs of the MIMED algorithm are lower than those of the other two algorithms and are close to the CRBs.
5. Conclusions
We have presented a MIMED method for finding directions of non-Gaussian signals using cumulants. By using an array containing sparse -shaped diversely polarized vector sensors plus an arbitrarily-placed single polarized scalar sensor, we have defined a set of cumulant matrices to construct two matrix blocks with multi-invariance property. We have then developed a MI-ESPRIT-based algorithm using the defined matrix blocks to estimate two-dimensional directions of the signals. The MIMED extends the array aperture by being able to space the sensors much farther apart than a half-wavelength.
Data Availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This work was supported in part by the National Natural Science Foundation of China under Grant 81601568, in part by the National Key Technology Support Program 2015BAI02B04, in part by the Natural Science Foundation of Jiangsu Province under Grant SBK2014043201, and in part by the Fundamental Research Funds for the Central Universities under Grant 30917011316.