Reduced-Dimension Noncircular-Capon Algorithm for DOA Estimation of Noncircular Signals

The problem of the direction of arrival (DOA) estimation for the noncircular (NC) signals, which have been widely used in communications, is investigated. A reduced-dimension NC-Capon algorithm is proposed hereby for the DOA estimation of noncircular signals. The proposed algorithm, which only requires one-dimensional search, can avoid the high computational cost within the two-dimensional NC-Capon algorithm. The angle estimation performance of the proposed algorithm is much better than that of the conventional Capon algorithm and very close to that of the two-dimensional NC-Capon algorithm, which has a much higher complexity than the proposed algorithm. Furthermore, the proposed algorithm can be applied to arbitrary arrays and works well without estimating the noncircular phases. The simulation results verify the effectiveness and improvement of the proposed algorithm.


Introduction
Direction of arrival (DOA) estimation is a hot topic in the array signal processing field and has been widely used in communication, radar, sonar, and medical image [1][2][3][4].Classical DOA estimation algorithms include multiple signal classification (MUSIC) [5], estimation of signal parameters via rotational invariance technique (ESPRIT) [6][7][8], propagator method [9], and the Capon [10].Besides, compressive sensing (CS) [11] and Bayesian compressive sensing (BCS) [12] have recently been used to solve the problem of DOA estimation, and they have an advantage of not requiring knowledge of the number of impinging signals.
To improve the DOA estimation performance, the noncircular property of incoming signals has been considered in [13][14][15][16][17][18][19][20][21][22].In wireless communications, the noncircular signals have been extensively used, for example, the binary phase shift keying, amplitude modulation, and unbalanced quadrature phase shift keying [22].If {()} = 0, {()  ()} ̸ = 0, and {()  ()} ̸ = 0, then () is a noncircular signal [13][14][15].This statistics redundancy can be properly exploited to enhance the DOA estimation performance.In general, we use the array output and its conjugation to extend the data model and array aperture.A noncircular MUSIC (NC-MUSIC) algorithm was proposed in [14] for the DOA estimation of the noncircular signals.In order to avoid the peak search in NC-MUSIC, a polynomial rooting NC-MUSIC (NC-Root-MUSIC) was presented in [15].NC-ESPRIT algorithms were proposed in [16,17] for DOA estimation without spectrum search.Real-valued implementation of unitary ESPRIT (NC-Unitary-ESPRIT) for noncircular sources was presented in [18], and it has a low complexity.Besides, a noncircular propagator method (NC-PM) for direction estimation of noncircular signals was proposed in [19], which has better angle estimation performance than PM in [9].Based on the parallel factor (PARAFAC) technique, a noncircular PARAFAC (NC-PARAFAC) algorithm was proposed in [20] to obtain the two-dimensional (2D) DOA estimation of the noncircular signals for arbitrarily spaced acoustic vectorsensor array.Moreover, a two-dimensional direction-finding for noncircular signals using two parallel linear arrays via the extended rank reduction algorithm was presented in [21].

DOA Estimation Algorithm
In this section, we will propose the RD-NC-Capon algorithm for the DOA estimation of noncircular signals.In order to double the array aperture, we will first give the extended data model by exploiting the noncircular property, which is a common method of expansion for noncircular signals and has been widely used in many noncircular DOA estimation algorithms [13][14][15][16][17][18][19][20][21][22].Then we will talk about the conventional 2D-NC-Capon algorithm in Section 3.2, whereas the RD-NC-Capon algorithm will be proposed in Section 3.3.

Data Construction.
When the noncircular signals impinge on the array, we use the array output and its conjugation to extend the data model [13][14][15]: International Journal of Antennas and Propagation 3 where , and it can be written as ] . ( The covariance matrix of the extended data model can be expressed as R = [y()y  ()].

2D-NC-Capon Algorithm.
The following 2D-NC-Capon function can be utilized to estimate the DOAs of the noncircular signals [27,28] where The 2D-NC-Capon algorithm can obtain  local peak values of ( 6) by 2D search of  and .Since the 2D-NC-Capon algorithm requires an exhaustive 2D search, this approach is normally inefficient due to its high computational cost.A reduced-dimension (RD) Capon algorithm was proposed in [29] for the angle estimation of bistatic multiple-input multiple-output radar.In the following subsections, we make reference to the RD idea from [29] and present a RD-NC-Capon algorithm for the DOA estimation of noncircular signals just through one-dimensional (1D) search.
There are some differences between the work in [29] and our work.Firstly, [29] used the RD-Capon method for angle estimation in bistatic multiple-input multiple-output radar, while our work is to extend the RD-Capon idea and propose RD-NC-Capon for angle estimation of noncircular signals.Secondly, the noncircular property is not considered in [29], and we use the noncircular property of the sources to double the array aperture and enhance the angle estimation performance.Thirdly, the modeling in each work is different.The received signal in [26] can be used directly for RD processing, while a data model extension is required in our work.
We define q() = e 0 ()  = [ 1  2 ] and Q() = P  ()R −1 P().Equation ( 13) can be rewritten as As ( 14) is a quadratic optimization, we consider eliminating trivial solution q() = 0 2×1 and add a constraint of e  q() = 1, where e = [1, 0]  .Then the optimization problem in ( 14) is reconstructed as follows: min We construct the following cost function using Lagrange multiplier: where  is a constant.If the derivative of ( 16) is set to zero, that is,  q ()  (, ) = 2Q () q () + e = 0, then q() = −0.5Q−1 ()e.We define  = −0.5;then we get As e  q() = 1 and  = 1/e  Q() −1 e, we have From ( 14) and ( 19), we have As Q() = P()  R −1 P(), a new 1D search cost function is used for DOA estimation: In the finite sample case, the covariance matrix can be estimated as where  denotes the number of snapshots.Then, the DOAs   ( = 1, 2, . . ., ) can be obtained through the following search: 2D-NC-Capon algorithm obtains DOA estimation by 2D search of  and , whereas the proposed algorithm gets DOA estimation via 1D search according to the function in (23).The proposed RD-NC-Capon algorithm can work well without estimation of noncircular phases.
The main steps of RD-NC-Capon algorithm are shown as follows.
Step 1. Construct the extended data model from the array output via (4).
Step 2. Compute the covariance matrix R of the extended data model via (22).
Step 3. Use the 1D spectrum search function to estimate DOAs via (23).

Performance Analysis
In this section, we first analyze the computational complexity of the proposed algorithm and then derive the Cramér-Rao bound (CRB) of DOA estimation.

Complexity Analysis.
For the proposed algorithm, computing the covariance matrix requires (4 2 ), inversion operation of the covariance matrix needs (8 3 ), and a spectrum searching wants (8( 2 +)).The main computational complexity of the proposed RD-NC-Capon algorithm is (4 2  + 8 3 + 8( 2 + )), where  is the number of searches, whereas the 2D-NC-Capon algorithm needs (4 2  + 8 3 + (4 2 + 2) 2 ).Thus, the RD-NC-Capon algorithm has a much lower computational complexity than the 2D-NC-Capon algorithm.Figure 2 shows the complexity comparison versus the snapshots , where  = 3,  = 6000, and  = 8 are considered.Figure 3 presents the running time comparison, where we compare the proposed algorithm against the 2D-NC-Capon algorithm.From Figures 2 and  3, we find that the proposed algorithm has a much lower complexity than the 2D-NC-Capon algorithm.

Cramér-Rao Bound.
In the finite sample case, the data model can be rewritten as where Y = [y( 1 ), . . ., y(  )] and S 0 = [s 0 ( 1 ), . . ., s 0 (  )].B ∈ C 2× is the same as that shown in (5).We assume that the signal is deterministic, and then the estimation parameter vector is expressed as where s  0 (  ) is the th column of S 0 . 2 is the noise power.According to (24), we have where vec(⋅) is to convert a matrix into a vector.The expected value  and the covariance matrix Γ of y are . . .
where According to [22,30,31], we know that the (, ) element of the CRB matrix (P cr ) can be expressed as where Γ   , Γ   and    ,    are the derivatives of Γ and  on the th or th element of , respectively.tr[⋅] is to obtain the trace of a matrix.Since the covariance matrix is just related to  2 , the first part of ( 29) is zero.Then We have . . .
where  0 () is the th element of s 0 (), d  = b  /  , and d  = b  /  with b  being the th column of B.
We define As  = GS, Now we have According to (34), where where Q  and Q  are the real and imaginary parts of Q, respectively.
It can be demonstrated that where Then we have International Journal of Antennas and Propagation According to (37), we know that Hence, combining (37)-( 38), F  JF can be rewritten as It is obvious that J −1 = F(F  JF) −1 F  .Then J −1 is written as Furthermore, where  denotes the part we are not concerned about.Till now, we can give the CRB matrix [30,31]: After further simplification, we rewrite the CRB matrix as where  2 is noise power, R = (1/) ∑  =1 s 0 (  )s 0  (  ), and We also estimate DOAs of the general noncircular signals in a similar way.
Remark 2. The proposed algorithm works well for the uniform circular array (UCA).We assume a(, ) is the direction vector of the UCA with DOA being (, ).3D-NC-Capon function can be utilized to estimate the 2D-DOA of the noncircular signals for UCA.2D-DOA estimation function of the proposed algorithm is  NC-RD-Capon (, ) = e  (P (, )  R−1 P (, )) where

Simulation Results
This section uses Monte Carlo simulations to assess the DOA estimation performance of the proposed algorithm.We define the root mean square error (RMSE) of DOA as Figure 4 shows the spectrum search result of the proposed algorithm for the nonuniform linear array with d = [0, 0.45, 0.9, 1.3, 1.78, 2.2, 2.64, 3.1], where  is the wavelength.In this simulation, signal-to-noise ratio (SNR) = 20 dB,  = 8, and  = 200 are considered.From Figure 4, we find that the proposed algorithm can work well.5, we find that proposed algorithm works well even when there are more sources than sensors, while the conventional Capon algorithm fails to work in this condition.The Capon algorithm with  elements can identify  − 1 impinging signals.The proposed algorithm uses the noncircular property to double the number of resolvable sources.
We compare the proposed algorithm against other DOA estimation algorithms.Figure 6  the invariance shift property.Therefore, ESPRIT and NC-ESPRIT are not considered in Figure 6.estimation performance to the 2D-NC-Capon algorithm, which has a much higher complexity.
Figure 8 illustrates the DOA estimation performance of the proposed algorithm with different number of antennas ().It is clearly shown that the angle estimation performance of the proposed algorithm is gradually improved with the number of antennas increasing.Multiple antennas improve the DOA estimation performance because of antenna diversity gain.
with   being the accurate angle of the th signal and θ, being the estimate of   of the th Monte Carlo trial.For Figures4 and 6-9, we assume that there are 3 ( = 3) noncircular signals impinging on the array with angles being [10 ∘ , 20 ∘ , 30 ∘ ] and the noncircular phases being [10 ∘ , 30 ∘ , 50 ∘ ].