# Joint DOA and Array Manifold Estimation for a MIMO Array Using Two Calibrated Antennas

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2 2 II. BACKGROUND Consider a MIMO system with a uniform linear array (ULA) of M antennas used for both transmitting and receiving. For simplicity of notation and without loss of generality, we assume that the first two antennas are perfectly calibrated. The steering vector of the ULA is then given by a(θ) = [1,e j2πdsin(θ)/λ,α 3 e jφ3 e j2π2dsin(θ)/λ,, α N e jφn e j2π(m 1)dsin(θ)/λ ] T (1) where [ ] T denotes the transpose operation, θ is the angle of the pointing direction, d is the inter-element spacing, λ is the signal wavelength, and α i and φ i denote the gain and phase errors, respectively. Assume that K targets are present. The output of the matched filters at the receiver is given by [5] x[n] = K a(θ k ) a(θ k )b k [n]+n[n] = Ab[n]+n[n] (2) where θ k is the DOA of the kth target, is the Kronecker product,b k [n] = β k e j2πf dn, withβ k being the complex-valued reflection coefficient of thekth target andf d being the Doppler frequency, b[n] = [b 1 [n],b 2 [n],,b K [n]] T, A = [a(θ 1 ) a(θ 1 ),, a(θ K ) a(θ K )] (3) is the overall transmit-receive or virtual array manifold, and n[n] is the white noise vector with a power σ 2. Assume that all target-reflected signals and noise are uncorrelated. Then we have R x = E[x[n]x[n] H ] = AR b A H +σ 2 I = U s ΛU H s +σ2 U n U H n (4) where E[ ] and [ ] H denote expectation and Hermitian transpose, respectively, R b = E[b[n]b[n] H ], Λ = diag{λ 1,,λ K } consists of the K principal eigenvalues of R x, U s is the signal subspace, specified by the principal eigenvectors of R x, and the remaining eigenvectors U n is the noise subspace. In practice, R x will be replaced by ˆR x = 1 L L n=1 x[n]x[n]h, where L is the number of snapshots. The MUSIC algorithm for DOA estimation for MIMO radar can be constructed as [18], [19] f(θ) = 1/[a(θ) a(θ)] H U n U H n [a(θ) a(θ)]. (5) The K largest peaks of f(θ) indicate the DOAs of the targets. It requires the spacing between two adjacent antennas to be within a half wavelength to avoid estimation ambiguity. For ESPRIT estimator [20], it is based on the signal subspace U s. Let U s,1 be the subset of U s, which relates to the first to the (M 1)-th transmit antennas, and U s,2 be the subset of U s, which relates to the second to the M-th transmit antennas. We then have the following relationship U s,2 = U s,1 T e Q e T 1 e (6) where T e is an unknown nonsingular matrix and Q e is a diagonal matrix, with its kth main diagonal element being e j2πdsin(θ k)/λ. Thus, the DOAs can be found from the eigenvalues of (U H s,1 U s,1) 1 U H s,1 U s,2. III. PROPOSED METHOD In this section, we first perform an initial DOA estimation using the two sets of received data associated with the first and the second transmit antennas by applying the ESPRIT algorithm, then the gain and phase errors can be estimated using the initial DOA results by applying a MUSIC-based approach. A. Estimating initial DOAs Since the array manifold is unknown, we can not apply the traditional subspace-based methods directly. To solve the problem, define A 1 and A 2 as the first and the second M rows of A, respectively, with A 1 = [a(θ 1 ),, a(θ K )], (7) A 2 = [e j2πdsin(θ1)/λ a(θ 1 ),,e j2πdsin(θk)/λ a(θ K )] = A 1 Q (8) where Q is an M M diagonal matrix, with e j2πdsin(θ k)/λ being its kth main diagonal element. Although there are model errors in both A 1 and A 2, a rotational invariance property between A 1 and A 2 is still maintained, which enables the use of ESPRIT for DOA estimation. A and U s have a relationship determined by a unique nonsingular matrix T as A = U s T. (9) Define U 1 and U 2 as the first and second M rows of U s, respectively. We have Then, A 1 = U 1 T, (10) A 2 = U 2 T = A 1 Q. (11) U 2 = U 1 TQT 1. (12) Now using the traditional ESPRIT technique, the main diagonal elements of Q can be obtained via eigendecomposition of (U H 1 U 1 ) 1 U H 1 U 2. Since the two transmit antennas have been well calibrated, {θ k } K can be obtained easily from Q. Note that the rotational invariance property exploited here depends only on the two calibrated transmit antennas and is not related to the uncalibrated part. Thus, the initial DOAs

3 3 can be estimated accurately without any knowledge of array model errors. Additionally, in this initial DOA estimation, the proposed ESPRIT-based method imposes less constraints on the spacing of the uncalibrated part, which can be arranged to be much larger than a half-wavelength for a high-resolution DOA estimation. B. Estimating array manifold From (5), with exactly known R x, the DOAs can also be found by solving the following equation [13]: [a(θ) a(θ)] H U n U H n [a(θ) a(θ)] = 0. (13) The actual steering vector can also be expressed as a(θ) = Γā(θ) (14) where Γ = diag[1,1,α 3 e jφ3,,α M e jφm ] and ā(θ) = [1,e j2πdsin(θ)/λ,,e j2π(m 1)dsin(θ)/λ ] T. Therefore, the estimate of antenna gains and phases can be obtained using the initially estimated DOAs as follows: K [( min Γā(ˆθk ) ) ( Γā(ˆθ k ) )] H Un U H [( n Γā(ˆθk ) ) ( Γā(ˆθ k ) )] K = min [V k δ] H U n U H n [V kδ] δ subject to δ H e 1 = 1, δ H e 2 = 1 (15) where δ is the M 2 1 gain and phase vector, with its elements being the diagonal elements of [Γ Γ], V k = diag[ā(ˆθ k ) ā(ˆθ k )], with ˆθ k being the initial DOA estimate of thekth target, e 1 = [1,0,,0] T and e 2 = [0,1,0,,0] T. It should be noted that both the (M +1)-th and the (M +2)-th elements of δ should also be equal to 1; however, we find that the above two constraints are able to give a satisfactory result. The problem in (15) can be rewritten as min δ δ H Zδ subject to δ H e = f T (16) where Z = K VH k U nu H n V k, e = [e 1, e 2 ], and f = [1,1] T. Its solution is given by δ = Z 1 e[e H Z 1 e] 1 f T. (17) Using the estimates (17), the DOAs can be estimated from the K highest peaks of the following function: 1 f(θ) = [ ] HUn diag[δ][ā(θ) ā(θ)] U H [ ]. n diag[δ][ā(θ) ā(θ)] (18) Since a set of initial DOA estimates has already been obtained, we can search for each DOA estimate over a small DOA region corresponding to each initial DOA estimate. Thus, the interelement spacing of the uncalibrated array does not have to be smaller than half wavelength to avoid estimation ambiguity. Actually, we can increase the inter-element spacing of the uncalibrated array to improve the accuracy of estimation. The proposed joint DOA and array manifold estimation scheme is summarized as follows: 1) Estimate the initial DOAs using the ESPRIT algorithm. 2) Estimate the array manifold using (17). 3) Use the results in Step 2 to find updated DOAs by local searching through (18). 4) Repeat Steps 2 and 3 until some convergence criterion is satisfied. One such a criterion could be the difference between the estimation results of the last round and the current one. When this difference is smaller than a pre-set threshold value, we can then stop the iteration. Note that we have assumed implicitly that the antenna positions have been calibrated, and we consider the fixed uncalibrated gain and phase errors only. This is because the calibration of array position is more convenient than the calibration of gain and phase which may vary due to environmental changes. On the other hand, the position error can be transformed into phase errors. However, the phase errors caused by position errors are not fixed for the targets because the targets have different DOAs. In such a case, a simple way is to obtain the gain and phase errors corresponding to each target, i.e. we should estimate the gain and phase errors when obtaining one target s DOA other than all the DOAs. C. Complexity analysis To estimate the sample covariance matrix, a computational complexity of O(M 4 L) is needed. The eigendecomposition operation needs a computational complexity of O(M 6 ). The proposed ESPRIT requires a computational complexity of O(M 3 ). In the estimation of array manifold, the computational complexity of O(M 6 n) is needed, where n is the iteration number. Therefore, the proposed scheme has at least a complexity of O(M 6 n+m 6 +M 4 L+M 3 ). D. Cramér-Rao Bound for Uncalibrated Array In this section, we derive the stochastic CRB for uncalibrated array by extending the results of [11], [21]. Define h i = α i e jφi, i=3,, M, as the gain and phase error that corresponds to the ith sensor and the (2M 4 + K) 1 vector η = [θ T,ξ T,ζ T ] T containing the unknown parameters, where θ = [θ 1,,θ K ] T (19) ξ = [Re{h 3 },,Re{h M }] T (20) ζ = [Im{h 3 },,Im{h M }] T. (21)

4 4 The snapshots are assumed to satisfy the stochastic model x[n] = N{0, R x } (22) where N{, } is the complex Gaussian distribution. The unknown parameters include the elements of η, the noise variance σ 2, and the parameters of the source covariance matrix {[R b ] ii } K i=1 and {Re{[R b] ij },Im{[R b ] ij };j > i} K i,j=1. Considering the problem with respect to the parameters of the source covariance matrix and the noise variance, the(2m 4 + K) (2M 4 + K) Fisher information matrix can be written as [11], [21] [F(η)] i,j = 2L { σ 2Re trace (W AH η j P A A )} η i (23) where P A = I A(A H A) 1 A H is the M M orthogonal projection matrix and the K K matrix W = R b (A H AR b + σ 2 I) 1 A H AR b. Then the CRB matrix is CRB = F 1. IV. SIMULATIONS Simulations are carried out to investigate the performance of the proposed method compared with the traditional ESPRIT estimator in [20] and the MUSIC estimator. We consider a MIMO array with M = 10 antennas and half-wavelength spacing. The first two antennas are perfectly calibrated.k = 3 targets are located at 10, 20, and 30, respectively. Results from 100 simulation runs are averaged to give the root mean square error (RMSE) of the estimates. For all simulations, the number of snapshots L = 100 is used. We first study MUSIC and ESPRIT algorithms. However, the proposed one is quite robust and has a much better performance. In this figure, we also showed the result of our proposed method with 5 iterations, and a clear improvement can be observed compared to the initial estimation Fig. 2. Iteration number RMSEs of DOA estimation versus iteration number. Proposed MUSIC based In the second example, the effect of the iteration number on the performance of the proposed method is demonstrated. The input SNR is set to 20 db and the antenna gain and phase errors are set as (the diagonal elements of Γ) [1,1,1.13e j0.020,0.89e j0.180,1.1e j0.130,1.05e j0.038, 0.98e j0.101,0.90e j0.057,1.15e j0.187,0.88e j0.247 ]. (24) The RMSE for DOA estimation versus the iteration number is shown in Fig. 2 and the result for unknown parameters estimation is shown in Fig. 3. Clearly the first or two iterations have already led to an accurate enough result Proposed ESPRIT based Proposed MUSIC based with 5 iterations Traditional ESPRIT Traditional MUSIC CRB (Calibrated Array) SNR (db) RMSE RMSE of real part RMSE of imaginary part CRB of real part CRB of imaginary part Fig. 1. RMSEs of DOA estimation versus input SNR. the performance of the proposed ESPRIT-based algorithm for initial DOA estimation. The antenna gain and phase errors are assumed to have a uniform distribution: α k [0.8,1.2] and φ k [ π/10,π/10].α k and φ k change from run to run while remaining constant for all snapshots. Fig. 1 shows the RMSE results versus input SNR. We see that the gain and phase errors have significantly degraded the performance of the traditional Fig. 3. Iteration number RMSEs of gain and phase estimation versus iteration number. Now we study the effect of antenna spacing on the performance of the proposed method with 5 iterations. The spacing between the two calibrated antennas is 0.5λ, while the spacing

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