Abstract
In estimating the two-dimensional (2D) direction-of-arrival (DOA) using a coprime planar array, there are problems of the limited degree of freedom (DOF) and high complexity caused by the spectral peak search. We utilize the time-domain characteristics of signals and present a high DOF algorithm with low complexity based on the noncircular signals. The paper first analyzes the covariance matrix and ellipse covariance matrix of the received signals, vectorizes these matrices, and then constructs the received data of a virtual uniform rectangular array (URA). 2D spatial smoothing processing is applied to calculate the covariance of the virtual URA. Finally, the paper presents an algorithm using 2D multiple signal classification and an improved algorithm using unitary estimating signal parameters via rotational invariance techniques, where the latter solves the closed-form solutions of DOAs replacing the spectral peak search to reduce the complexity. The simulation experiments demonstrate that the proposed algorithms obtain the high DOF and enable to estimate the underdetermined signals. Furthermore, both two proposed algorithms can acquire the high accuracy.
1. Introduction
Direction-of-arrival (DOA) is a significant research field in many applications, such as radar [1], underwater acoustics [2, 3], and indoor navigation. At present, the uniform and nonsparse arrays are widely applied, including uniform line arrays (ULAs) [4], uniform rectangular arrays (URAs) [5], and uniform L-shaped arrays [6]. The array spacing of those is set to no more than half the wavelength of the impinging signals. We then can obtain the accurate parameters using the high-resolution algorithms, such as multiple signal classification (MUSIC) [7], root-MUSIC [8], estimating signal parameters via rotational invariance techniques (ESPRIT) [9], and propagator method (PM) [10]. With the development of technology, the nonsparse array generally needs more sensors to expand array aperture to meet the requirement for more precise location determination. However, this processing makes the systems more complicated and enhances the antenna mutual coupling interference, which increases the estimation errors.
To acquire both high array degree of freedom (DOF) and precision, researchers have begun to examine the signal features. A method proposed in [11] constructs a virtual array in the frequency domain combined with spacing features based on orthogonal frequency division multiplexing systems. Thus, that method not only improves the accuracy of estimated parameters, but also estimates underdetermined signals (no less than the number of sensors). Furthermore, the sparse and nonuniform array structure has become a focus. The nested array in [12] consists of a uniform and sparse array and a uniform and nonsparse array, of which DOF and precision are high. However, the nested array cannot avoid the mutual coupling interference among the sensors of the nonsparse array. Thus, a one-dimensional coprime array consisting of two uniform sparse line arrays is introduced in [13]. This weakens the mutual coupling interference and simultaneously constructs a larger aperture with fewer sensors when compared with uniform and nonsparse arrays. Moreover, the two-dimensional (2D) coprime planar arrays are proposed in [14], which can estimate both azimuth and elevation angles. A partial spectral search (PSS) method is introduced to reduce the complexity caused by 2D-MUSIC mentioned in [14], but PSS is still based on peak search, which does not significantly reduce the complexity. Moreover, the DOF is limited by the number of subarray sensors. We have proposed a method in [15], which uses the covariance matrix of coprime planar array to estimate a new covariance matrix with matrix completion. This processing has much improved DOF bigger than the number of sensors. However, matrix completion can introduce the additional errors so that its precision is limited, and it costs much complexity. All the methods mentioned above do not take some signal features into consideration.
There are many kinds of noncircular signals in modern communication systems, with, such as, BPSK, ASK, and AM modulation. The methods, which construct extended virtual arrays and improve the accuracy of estimated parameters based on noncircular signals, are proposed in [16, 17]. Considering the problems that existing methods for estimating DOAs of general uncorrelated signals using coprime planar arrays have huge computational complexity and low DOF, we present an algorithm to estimate 2D DOAs of underdetermined noncircular signals with low complexity. The algorithm vectorizes the covariance matrix and ellipse covariance matrix of received signals to construct the new received data of a virtual URA. Combining 2D spatial smoothing processing and Unitary-ESPRIT, we then realize the fast estimation of 2D DOAs. Through constructing a virtual URA with more number of sensors than that of coprime planar array, we improve the DOF. Moreover, Unitary-ESPRIT can solve the closed-form solutions of DOAs replacing the spectral peak search to reduce the complexity.
The remainder of this paper is arranged as follows. Section 2 introduces the model of coprime planar array, and Section 3 describes the steps of the algorithm. Sections 4 and 5 analyze the computational complexity and performance of the model, respectively, to demonstrate the validity of this algorithm. Section 6 gives the conclusion to this paper.
The notations used in this paper are as follows: , , and , respectively, represent the transposition, conjugation, and conjugate transposition; denotes the mathematical expectation; expresses the transformation of a vector to a diagonal matrix; , , and denote the Kronecker product, Khatri-Rao product, and pseudoinverse operator, respectively.
2. System Model
Considering the coprime planar array model, the array geometry is shown in Figure 1. The coprime planar array consists of two URAs. Subarray 1 has sensors and subarray 2 has sensors, where and are the coprime integers (generally assuming ) and denote the sensor numbers on the axis. Correspondingly, the distance between the two adjacent sensors is and , respectively, where the represents the wavelength of the impinging signals. The subarrays coincide at the origin, so the total number of sensors is . Suppose there are uncorrelated narrowband far-field noncircular signals impinging on the array with power . The th signal is located at elevation angle , which is downward from the z-axis, and azimuth angle , which is counterclockwise from the x-axis.
We define and as the location set of the subarrays. and , where and . Hence, the location set of the coprime planar array is expressed as , and we have . The received signals at the array can be represented as
The array manifold iswhere
The noncircular signal data vector iswhere is the sampling time, and is the number of snapshots. And the noise vector iswhere the elements are usually Gaussian random variables with zero means and variance .
Considering (1), the covariance matrix of the received signals is defined aswhere . Vectorize the covariance matrix in (7) aswhere , , andDefine the new set as . Thus, the any element of can be expressed as . can be denoted as the received data of a virtual array containing one snapshot, whose manifold matrix is defined as . And and represent the location set of real array and virtual array, respectively. As shown in Figure 2, the virtual array is not a completed URA because there are holes. We can use the consecutive uniform and nonsparse virtual array to resolve DOAs, which avoids the ambiguous results and enhances the DOF, but this method loses much array aperture and has not significant improvement on DOF.
3. DOA Estimation for Noncircular Signals
3.1. Construct Extended Virtual Array
In our paper, we assume the received signals as noncircular signals of maximum noncircular rate [16]. The ellipse covariance matrix of noncircular signals is nonzeros, where we can obtain the additional information besides the covariance matrix. Hence, we calculate the ellipse covariance matrices of received signals in (1) given bywhere the ellipse covariance of noise is 0 because noise is circular, , and . And the noncircular signals , so . Vectorize the ellipse covariance matrix aswhere and . Thus, and can be denoted as the received data of virtual arrays whose manifold matrices are defined as and , respectively. Furthermore, combining the received data of virtual arrays, the received data of an extended virtual array is expressed aswhere and is the noise vector. Define another two sets and . Hence, the location of the extended virtual array is denoted as , and the values are shown in Figure 3. The extended virtual array has the biggest URA size as . As shown in Table 1, using noncircular signals has increased the array aperture of virtual URA. The received data of the virtual URA after removing the repeated rows is given bywhere is the manifold matrix of size and is a vector of all zeros except a corresponding to the virtual sensor at . In practice, because of the limited number of snapshots, the covariance matrix is usually estimated asSimilarly, the ellipse covariance matrices are and . Hence, the real received data of virtual URA is denoted aswhere and is the real impinging signals and noise of virtual array.
3.2. 2D Spatial Smoothing Processing
We apply the 2D spatial smoothing processing to received data of the virtual URA. The detailed smoothing scheme is presented in Figure 4.
We first assume the smoothing subarray of size . The received data of the sensors in the upper right corner of the virtual URA can be expressed aswhere is the subarray manifold matrix of size and given byDefine the set . Thus, the any element of can be expressed as . There are overlapping smoothing subarray from upper right corner to upper left corner. The received data of the sensors in the upper left corner, can be given bywhere . Similarly, there are overlapping smoothing subarray from upper right corner to lower right corner. The received data of the sensors in the lower right corner, , can be given bywhere . Thus, as for a virtual URA, there are total overlapping subarrays. Hence, the smoothing covariance matrix can be calculated by averaging over the corresponding covariance matrix of the subarrays to obtainGiven (20) and (21), (22) can be rewritten aswhere and . Considering the proof in [18], when , , which satisfies the requirement of using subspace algorithms.
3.3. 2D DOA Estimation
According to [19], we apply 2D-MUSIC to the smoothing covariance matrix to estimate 2D angle. Take the eigenvalue decomposition of and obtain the noise subspace . The spatial spectrum function is given by
Using 2D-MUSIC can estimate the angles but the spectral peak search costs much complexity. Considering this, we present an improved algorithm which uses Unitary-ESPRIT [20, 21] to replace 2D-MUSIC. Unitary-ESPRIT transforms the complex matrix into real matrix and solves the closed-form solutions of DOAs to reduce the complexity.
First, define the inverse matrix , whose counter-diagonal values are 1 while the others are 0, as . The unitary matrix is defined to satisfyMoreover, the matrix can be expressed asWe then define the selection matrix as and and havewhere and denote as the real part and imaginary part of the matrix or vector, respectively. Therefore, we defineNext, we transform to givewhere is a real matrix. Taking eigenvalue decomposition of this real matrix can cost less complexity than and obtain the noise subspace . At last, we combine the last two step and haveDefine and calculate its eigenvalue vector . We can acquire the estimationwhere is the estimated value, , and . Equations (37) and (38) give the closed-form solutions of DOAs, thus avoiding the spectral peak search.
3.4. Algorithm Step Conclusion
The main steps of the proposed algorithm can be summarized as follows:
Step 1. Calculate the received signals’ covariance matrix and ellipse covariance matrices , .
Step 2. Vectorize those covariance matrices and construct the received data of the virtual URA.
Step 3. Apply 2D spatial smoothing processing and calculate the smoothing covariance matrix via (22).
Step 4. Use Unitary-ESPRIT to estimate .
4. DOF and Computational Complexity Analysis
4.1. DOF Analysis
The DOF determines the maximum number of signals that we can estimate directly. Compared with the coprime planar array, the URA with the same number of sensors has much smaller array aperture and DOF is no more than the , but the algorithm in [22] enhances the DOF to based on the noncircular signals. The DOF of PSS presented in [14] is depended on . As for two proposed algorithms, the smoothing covariance matrix enables us to resolve DOA estimation of maximum signals, when . The detailed values are presented in Figure 5. The DOF of PSS is limited by the number of subarray sensors, so it is the lowest. The proposed algorithm has much improved the DOF more than the number of sensors, which means it can estimate underdetermined signals.
4.2. Complexity Analysis
We analyze the computational complexity of the proposed algorithm using 2D-MUSIC and improved proposed algorithm using Unitary-ESPRIT, comparing them with that of PSS in [14].
The complexity of the proposed algorithm is made up of four parts: covariance matrix estimation, calculating smoothing covariance matrix, eigenvalue decomposition, and 2D-MUSIC. The complexities of these parts are , , , and , respectively, where and represent the number of spectral points. The complexity of improved proposed algorithm also has four parts, which is only different in fourth part, Unitary-ESPRIT with . Therefore, the computational complexity of proposed algorithm is and that of improved proposed algorithm is . Moreover, the complexity of PSS is . For the sake of clarity, the computational complexities of all these methods are summarized in Table 2. We also compare the complexity of algorithm mentioned above versus snapshots (), the number of sensors (), and the searching step (, where , ) in Figures 6(a)–6(c), respectively, under the condition that .
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As shown in Figure 6, the improved proposed algorithm using Unitary-ESPRIT has the smallest complexity. The proposed algorithm using 2D-MUSIC and PSS has the larger complexity due to the 2D spectral peak search. The complexity is huge when the searching step is small. Therefore, we introduce the Unitary-ESPRIT to solve the closed-form solutions of DOAs, where the searching step does not affect the complexity. Compared with the searching step, the number of snapshots and sensors has the weaker impact on complexity. As a result, the improved proposed algorithm can efficiently reduce the complexity.
5. Simulation Results
This section performs the results of simulation experiments comparing the proposed algorithm using Unitary-ESPRIT with that using 2D-MUSIC and PSS proposed in [14]. To measure the accuracy of the algorithms, define the root mean square error (RMSE) aswhere represents the number of simulations; and are denoted as the real values and the th estimated values, respectively. We assume that the impinging signals are BPSK modulation, which are a kind of noncircular signals of maximum noncircular rate. Moreover, we set .
Simulation 1 (performance of estimating underdetermined signals). Through the analysis of DOF, the proposed algorithm and improved proposed algorithm can estimate underdetermined signals. To verify the ability estimating underdetermined signals, we consider the case of with a signal-of-noise (SNR) of dB and . We set the coprime planar array as and . The distribution of the estimated values from 20 simulations using 2D-MUSIC and Unitary-ESPRIT is presented in Figures 7(a) and 7(b), respectively. The figure proves that both algorithms can estimate DOAs of signals, which is more than the number of sensors.
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Simulation 2. RMSE comparison of the proposed algorithm, improved proposed algorithm, PSS, and the URA with the same number of sensors [22] under different SNRs.
Simulations are conducted with , , , , and SNRs from dB to dB at dB intervals. And we set , and . The RMSE results are shown in Figures 8 and 9, respectively.
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The RMSEs of the two proposed algorithms are lower than that of URA with the same number of sensors, because the virtual URA has a bigger array aperture. The figures show that there are gaps between the RMSE of two proposed algorithm and that of PSS. The virtual URA has the same size as the coprime planar array, but the PSS uses more information. Nevertheless, the gap is not big when their array apertures are same. Compared with proposed algorithm, the improved proposed algorithm obtains the similarly precise estimations with much less complexity. Moreover, when applied to multiple signals (), both proposed algorithms maintain their accuracy, but the PSS fails.
Simulation 3. RMSE comparison of the proposed algorithm, improved proposed algorithm, PSS, and the URA with the same number of sensors [22] under different snapshots.
In Simulation 3, we set the SNR to dB, , , , , and and vary the number of snapshots to . The results are presented in Figure 10. The RMSE decreases as the number of snapshots increase, although the decline is negligible once .
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Simulation 4. RMSE comparison under different SNRs with a bigger array aperture than Simulation 2.
We conduct this simulation as Simulation 2 with the condition that , , and . The results are shown in Figure 11. There are two differences in Figure 11 compared with Figure 8. One is that the gaps between the RMSE of URA and those of two proposed algorithms are larger, because the array aperture gap between the virtual URA and URA is bigger with the more number of sensors. The other is that the gaps between the RMSE of PSS and those of two proposed algorithms are also larger. Given the Table 1, the virtual smoothing URA has the array aperture as while that of the coprime planar array is . When , the array aperture of virtual URA is smaller. Hence, the RMSEs of two proposed algorithms are higher. Moreover, this means that the proposed algorithms improve the DOF with the loss of array aperture and accuracy. And the loss can get bigger with the increase of , and this is still true when . How to deal with the balance between the DOF enhancement and accuracy improvement is worthy our further research.
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6. Conclusions
The paper has proposed a 2D DOA estimation algorithm using a coprime planar array for noncircular signals, which has much improved the DOF and reduced the complexity. The paper has described the model and the associated algorithms and analyzed the DOF and computational complexity of the proposed algorithms in comparison with that of existing algorithms. Through the theoretical analysis and simulation experiments, we conclude that the proposed algorithms can obtain the higher accuracy than the URA with the same number of sensors, where the proposed algorithms construct a more bigger virtual URA aperture based on the coprime planar array. Compared with DOF of PSS as , the proposed algorithms has improved DOF bigger than , which has an ability to estimate underdetermined signals. Furthermore, both two proposed algorithms have the close RMSEs, but the improved algorithm applying Unitary-ESPRIT has greatly reduced the complexity compared with the algorithm using 2D-MUSIC.
Data Availability
No data were used to support this study.
Disclosure
The authors claim that the data used in this article are provided by their simulations and this is developed without using any data in a published article to support their results.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The work is supported by the National Natural Science Foundation of China (Grant no. 61401513).