Abstract
The mutual coupling among various components of the collocated crossdipole (CCD) vector-sensor is severe, and its application is greatly limited. The spatial spread dipole (SSD) vector-sensor can avoid this problem, but the multiple signal classification (MUSIC) algorithm for the SSD array is rarely developed. In view of this situation, this paper proposed a MUSIC-like algorithm for the SSD array. The biquaternion model was first established, and the biquaternion MUSIC (BQ-MUSIC) algorithm was developed on the basis of this model, for the two-dimensional direction-of-arrival (2D-DOA) estimation. Our proposed algorithm requires low computational complexity by adopting the dimensionality reduction method. Numerical simulations verify the effectiveness of the proposed algorithm.
1. Introduction
The electromagnetic vector-sensor (EMVS) array is also known as polarization sensitive array due to their ability to sense the polarization information of electromagnetic waves. Compared with traditional scalar array, EMVS array can obtain more information in the time domain and frequency domain such as phase and frequency waveforms, with stronger anti-interference ability, higher spatial resolution, and better detection robustness ability [1]. EMVS array also has the following advantages: (1) they can sense electromagnetic waves in a vector mode and provide polarization information of electromagnetic sources; (2) collocated EMVS that appeared in the early stage is suitable for the direction of arrival estimation for near-field or far-field, broad-band, or narrow-band signals; (3) different polarization information can be exploited in polarization domain when polarization electromagnetic wave cannot be distinguished in the spatial domain. Meanwhile, the appropriate application of polarization diversity technology could bring great convenience to the subsequent target recognition and classification; (4) when the spatial aperture of the array is small, the EMVS array still have a better resolution. Therefore, it is suitable for limited physical space such as mobile platform [2]. EMVS array has become a research hotspot in the field of array signal processing and is widely used in sonar radar communications.
Direction-of-arrival (DOA) estimation represents a vital research direction of array signal processing. Based on previous researches on DOA estimation method for scalar array, scholars have transplanted and applied MUSIC [3], estimated signal parameters via rotational invariance techniques (ESPRIT) [4], and maximum likelihood (ML) [5] to vector-sensor array signal processing, and achieved favorable estimation results. Particularly, some studies have used the Poynting vector crossproduct relationship that only the vector-sensor can perceive. However, the time-domain ESPRIT algorithm developed based on this relationship has high requirements on the source, thus limiting the application scenarios [6, 7]. Polarization smoothing is used to realize the DOA estimation of coherent sources by using the method of dephasing intervention processing [8]. On the other hand, this method is suitable for all array structures and does not lose the effective aperture of the array compared with spatial smoothness [9].
In the long vector (LV) model, multiple components of each antenna are directly “stacked” into a column vector, and this representation has the advantages of simplicity and manipulation. However, the orthogonal relationship among the components of each array element is lost at the same time. The hypercomplex number as typified by quaternion is very suitable for the component representation of EMVS because of its strict orthogonality [10]. With the breakthrough of some mathematical problems such as the eigenvalue decomposition (EVD) of the quaternion value matrix [11], the hypercomplex number tool has been introduced into the research of EMVS array signal processing successfully [10–16]. The frequency-domain quaternion MUSIC algorithm [12] and frequency-domain biquaternion MUSIC algorithm [13] are explored, respectively. Even though the two algorithms can simultaneously estimate direction parameters and polarization parameters, they require a four-dimensional (4-D) spectral peak search at the very least, which brings enormous computing burdens. Li et al. have proposed a Q-MUSIC algorithm that can realize dimensionality reduction search on the premise of excluding only a few special parameters in the time domain [14]. Correspondingly, quaternion ESPRIT [15, 16] and augment ESPRIT [17] algorithms have also been developed.
Most of the studies mentioned above [3–6, 8–16] mainly focus on collocated vector-sensor array. In actual production, this collocated EMVS with the same geometric center requires high degree of electromagnetic isolation. As the frequency increases, the cost of implementation will go up. Therefore, some scholars have studied the noncollocated EMVS. According to the modified vector crossproduct method based on the ingenious array arrangement, Wong et al. [18] have obtained a 2-D DOA estimation of the spatially spread vector-sensors. Spatial ESPIRT [19] was developed by Zheng based on Ref. [7]. Then, by combining the coarse and fine estimations, the high-accuracy 2-D DOA estimation can be obtained. Gong et al. have put forward a spatially spread quint of dipoles or loops that are symmetrical in space. With this feature, an efficient DOA and polarization estimator is realized using the vector crossproduct method [20].
However, the introduction of the spatial phase shift factor inside the sensor makes LV-MUSIC dimensionality reduction difficult to be realized, and the 4-D search poses unacceptable calculation burdens. Therefore, most DOA estimation algorithms for SSD array are based on ESPRIT-like. In this paper, we first introduce the biquaternion operation and EVD of its Hermitian matrix and then create a high-dimensional algebraic model of the SSD-EMVS array that has a very compact form. The characteristics of the approximation of the covariance matrix of CCD array and SSD array are illustrated. The resulting BQ-MUSIC algorithm is developed to easily achieve dimensionality reduction, with promising results. Finally, we use simulation experiments to verify the effectiveness of the proposed algorithm.
2. Biquaternion Operation and EVD of Hermitian Matrix
Notation: , , , and represent real numbers, complex numbers, quaternion numbers, and biquaternion numbers, respectively. marks the complex numbers with the imaginary part of and . “” stands for quaternion or biquaternion conjugate transpose; “” refers to complex conjugate transpose; “” is the complex conjugate.
Quaternion, as a kind of high-dimensional complex number, has one real component and three imaginary components. By extending the real number coefficients of each component in quaternions to the complex number, the biquaternions (also known as complexified quaternions) are obtained. And a biquaternion is defined as where . The multiplications among the three imaginary units of a biquaternion characterizing the orthogonality relation are not commutative, but the multiplication between the complex imaginary unit and the quaternion imaginary units satisfied the commutative law [10]. The following standard relations among all imaginary units (quaternion and complex) hold
Biquaternion, as an extension of quaternion, also can be written in the form of the sum of real and imaginary parts, but the coefficients of each part are all complex values, shown as follows: where
The conjugate of a , noted , is given by
And the norm of , noted , is given by
It should be noted that , , and .
A biquaternion value matrix can be rewritten as , where . Usually, EVD operation on the high-dimensional algebraic matrix is realized by its isomorphic adjoint matrix. The adjoint matrix of is recorded as , and
Particularly, if is a Hermitian matrix, then is a Hermitian matrix. Then, has the following quaternion value matrix EVD: , where and . According to the “isomorphism” property, the eigenvalues of and are the same (both results are in ), and contains the eigenvectors of on its columns, as shown in where , , and are the unit matrix. Finally, we obtain the EVD of a biquaternion valued matrix:
3. Array Signal Model
A collocated crossed dipole electromagnetic vector-sensor (CCD EMVS) is shown in Figure 1 and consists of three electric dipoles, namely, . The output of each component is , respectively, and the value are [1] where and are the elevation-azimuth 2D-DOA (labeled in Figure 1) and polarization of an electromagnetic wave, respectively, and , , , and . The output of the EMVS, as a whole, is expressed as biquaternion scalar ,
Since its real part coefficient is zero, it is also called a pure quaternion. Compared with LV-models, this pure biquaternion representation method is not only more compact in structure but also preserves orthogonality among its components.
The spatial spread dipole electromagnetic vector-sensor (SSD-EMVS) is physically isolated by translating them to a certain distance along its respective coordinate axes. In particular, SDD-EMVS is uniform when . The spatial phase shift factor should be introduced subsequently with . In the SSD-EMVS, as shown in Figure 2, the spatial phase shift factors on the three components along the -axis are , , and in order [18]. Thus, the polarization response of this vector-sensor is expressed by applying phase shift factor and a biquaternion scalar as
where , and is the wavelength of electromagnetic waves.
Consider a planar array consisting of SSD-EMVS with the first sensor position located at the origin of the coordinates, as displayed in Figure 3. array elements are placed along the -axis, and the spacing between two adjacent array elements is . elements are placed along the -axis, and the spacing between two adjacent array elements is .The single-column steering vector along the -axis in this planar array is , and
The single-row steering vector along the -axis in this planar array is , and
Hence, the spatial polarization steering vector of this polarization sensitive array is where , , and , and “” is the Kronecker product.
Assuming that there are incoherent far-field fully polarized narrow-band plane waves impinge on this array, then the output of such an array is : where is a diagonal matrix composed of and . is the spatial steering vectors matrix of this array, and . is a signal vector composed of signals, and . The manifold of this polarization sensitive array is , and .
The noise data received by the vector-sensor of the row is expressed by a pure biquaternion , as follows: where , , and are the noise data received by the . Correspondingly, is the noise vector formed by the noise data received by array elements in the biquaternion model.
4. Biquaternion-MUSIC for SSD Array
Under the above signal model, the data covariance matrix is calculated by
Since the signal sources are assumed to be incoherent, the signal covariance matrix is a diagonal matrix. And the multiplications between two diagonal matrices can be exchanged with each other. can be rewritten as
Considering a quaternion model for the collocated double electric dipoles array, its polarization response scalar is, and where . For spatial spread vector-sensor of the same type, the product of its polarization response scalar is , and
Its norm of all internal spatial phase shift factors is 1; so, . Finally, we have a conclusion that in the quaternion model established for the double electric dipole array, the covariance matrix is the same regardless of whether it is collocated or spatial spread (the process analogy from Eq. (18) to Eq. (20)).
Unfortunately, equation does not hold the biquaternion operation anymore. In fact, it can be inferred from Eq. (6) that
For the CCD vector-sensor, there is
Since and , it is clearly that , which directly leads to the of SDD-EMVS that is different from its collocated counterpart one, unlike the quaternion model.
However, thanks to the exponential form of the phase shift factor and the feature that the real part of the pure biquaternion is 0, the difference between the and can be quantified by ,
We perform 1000 independent experiments under different and draw the different curves of and in Figure 4. As can be seen in Figure 4, is always around 0.05 when goes from to 1, and then decreases as increases. The small difference between and eventually leads to that the of SSD-EMVS array, and of collocated vector-sensor array is exactly the same in real part and only slightly different in the imaginary parts.
The EVD of by Eqs. (7)–(9) has where is the biquaternion-value signal subspace matrix corresponding to large eigenvalues matrix , and is the biquaternion-value noise subspace corresponding to small eigenvalues matrix . According to the subspace theory, the signal subspace and the noise subspace are orthogonal to each other as in complex-value matrix and quaternion-valued matrix. The array manifold matrix is orthogonal to as and span the same subspace. On the other hand, there is a more efficient noise subspace projector in the biquaternion model, and projector is that is built with the eigenvectors of . Therefore, the subspace orthogonality over biquaternion can be express as
In practice, this equation does not strictly hold due to the finite number of snapshots and the presence of noise. We can construct the spectrum function of BQ-MUSIC as
As , always holds. is full rank under the assumption of . Based on rank reduction principle, the spectrum function can be simplified from to , namely,
And the final estimated value is
The direction parameters are obtained from through 2-D peak searching avoiding 4-D peak searching successfully. Our proposed BQ-MUSIC is summarized in Table 1.
5. Computational Analysis
Compared with the LV-model, the biquaternion model reduces the memory requirements of the covariance spectrum matrix by 4/9. And it is more valuable that the algorithm proposed in this paper simplifies the unapplicable 4-D peak search for to a 2-D search for , which greatly reduces the calculation burden.
The multiplication of two biquaternions implies 16 complex multiplications, and the addition of two biquaternions implies 4 complex additions. For the LV-MUSIC in Ref. [13], the received data matrix is , where is the number of snapshots. The number of complex multiplications required to calculate the covariance matrix is . The dimension of the covariance matrix is , and calculating the EVD requires complex multiplications. The number of complex multiplications required to calculate the value of a spectrum function is . It is assumed that the search angle interval is , and the LV-model requires searches. Finally, the total amount of complex multiplications required for LV-MUSIC is
For the BQ-MUSIC, the number of complex multiplications that requires to calculate the dimensional data matrix is since the multiplication of two pure biquaternions implies 9 complex multiplications. The EVD of biquaternion value matrix with the dimension of is replaced by quaternion adjoint matrix with the dimension of . Then the quaternion adjoint matrix is replaced by complex-value matrix with the dimension of , and thus the complex multiplication of EVD is . The complex multiplications required to create the projector matrix is . Calculation of the value of a spatial spectrum function requires complex multiplications. The 2-D DOA search requires about searches. The total amount of complex multiplications required for BQ-MUSIC is
Since one complex multiplication operation contains four real multiplications and three real additions, the final computation and memory requirement space are compared in Table 2. A comparison of the computational complexity of the proposed algorithm and the LV-MUSIC algorithm is exhibited in Figure 5, where we set . Let us evaluate the total number of arithmetic operations using complex multiplication as an indicator. The huge gap between the two is mainly reflected in 2-D search and 4-D search, because BQ-MUSIC can easily achieve dimensionality reduction. Moreover, as the search interval decreases, the difference in computational complexity between the two will further increase.
6. Simulation Results
In this section, we use numerical simulation experiments to verify the effectiveness of the proposed algorithm. In the following simulation, the size of the array is , with a total of 16 SSD-EMVS antennas, the distance between adjacent antennas, and . It is supposed that there are 2 independent targets , and their parameters are , , , and , respectively. Snapshots and signal-to-noise ratio (SNR) are set as with the Gaussian white noise environment.
Simulation experiment: as shown in Figures 6 and 7, respectively, the spatial spectrum for CCD-EMVS array and SSD-EMVS array is obtained by BQ-MUSIC. It is obvious that both of them can accurately obtain the direction parameters. Furthermore, the spatial spectrum of SSD-EMVS has shaper peaks.
In Section 4, we conclude that the covariance matrix of the SSD-EMVS array is identical to the real part of the covariance matrix of the collocated type, with only a small difference in the imaginary part. This difference is actually caused by the introduction of the spatial phase shift factor, because the spatial phase shift factor contains directional information. It ultimately leads to a better estimation of the SSD-EMVS array than the CCD-EMVS array by comparing Figures 6 and 7.
Performance analysis: we keep the above conditions unchanged, only change the SNR to perform 100 Monte-Carlo experiments, and define root mean squared error (RMSE) to measure the estimated performance, as where is 2-D parameter estimation, and is the true value of parameter. As shown in Figure 8, the performance of the CCD-EMVS array is not as good as the SSD-EMVS array, which is in line with the above experimental results. Based on coarse-fine estimation, LV-ESPRIT [19] for SSD-EMVS array has poor performance with minimal computational complexity. The MUSIC algorithm is highly dependent on the matrix dimension, and the biquaternion model compresses the size of the vector to one-third of the long vector, which is the reason why the BQ-MUSIC algorithm is not as superior as the LV-MUSIC.
We assume that , the number of snapshots is various, and other simulation conditions are constant. By plotting the RMSE variation curve with the number of snapshots, a similar conclusion to the previous simulation can be obtained from Figure 9.
7. Conclusion
A new MUSIC algorithm for SSD-EMVS array is proposed based on a biquaternion model. By illustrating the approximation of the CCD-EMVS array and the SSD-EMVS array covariance matrix, it is clear that the use of the subspace orthogonality property enables the SSD-EMVS array to obtain good DOA estimation performance. The dimensionality reduction algorithm of BQ-MUSIC makes it more practical than LV-MUSIC.
Data Availability
There is no available data for this study.
Conflicts of Interest
The authors declared that they have no conflicts of interests.
Acknowledgments
This work was supported by the Natural Science Foundation of Shaanxi Province under grant 2019-JM115.