Modified UCA-ESPRIT for Estimating DOA of Coherent Signals Using One Snapshot Kareem Al Jabr
Hyuck M. Kwon
Nizar Tayem
Dept. of ECE Wichita State University 1845 N. Fairmount Wichita, KS 67260, USA
[email protected]
Dept. of ECE Wichita State University 1845 N. Fairmount Wichita, KS 67260, USA
[email protected]
Dept. of EE ITT Technical Institute Owings Mills, MD 21117 USA
[email protected]
covariance matrices for all subarrays. This approach results in a high computational load and is timeconsuming. The spatial smoothing technique uses a large number of snapshots, which, in turn, increases the computational burden. Moreover, coherency between the signals is unknown in practice, so the number of spatial smoothings is uncertain. The proposed method uses only a single snapshot of the received signals to estimate the DOA for incident sources. This reduces the computational load drastically and makes the proposed method a good candidate for real-time implementation, whereas the methods in [6], [7] require a large number of snapshots (e.g., 100 to 200) in order to produce a good estimate.
ABSTRACT - This paper introduces a modified UCAESPRIT algorithm to estimate the direction of arrival (DOA) of coherent signals using one single snapshot. First, the mode excitation method is used to transform the uniform circular array (UCA) in element space into a virtual uniform linear array (VULA) in mode space. Then, the Hermitian Toeplitz matrix can be reconstructed from the observed data vector as a decorrelated algorithm. The purpose for using the Hermitian Toeplitz matrix is to rearrange the data in order to extend the dimensionality of the noise space. Consequently, signal and noise spaces can be estimated more accurately. The proposed method does not need forward/backward spatial smoothing of the covariance matrix, which would result in reduced computational complexity. Simulation results verify that the proposed algorithm with one snapshot has a close performance to the conventional mode of spatial smoothing with 100 snapshots.
In radar and other signal processing applications, the Hermitian Toeplitz matrix algorithm has been shown to provide accurate estimates with a single snapshot [8] and an extremely low computation burden for the ULA case. However, it has not been applied to the UCA case. This paper is the first to introduce the estimation of DOA of coherent signals using one single snapshot for the UCA. The key idea in the proposed algorithm is a technique that maps a ( 2h + 1) × 1 data vector into a
I. INTRODUCTION Estimating the direction of arrival (DOA) of propagating plane waves is an interesting problem found in many applications, including radar, sonar and mobile communications. By using smart antennas, the communications capacity can be greatly increased. This allows the system to manipulate received signals in the spatial domain. The uniform circular array (UCA) is a special planner array with many effective properties over the uniform linear array (ULA), since it is able to provide 360 degrees of azimuthal coverage and a certain degree of source elevation information. Coherent signals appear in the realistic mobile environment as a result of multipath propagation or intentional unfriendly jamming. The ESPRIT algorithm [1] does not work for coherent signals. Considering that this algorithm cannot be directly applied to the UCA, spatial smoothing techniques [2-4] and subspace smoothing [5] have been developed. A mode spatial smoothing algorithm [6], [7] is developed, which is based on partitioning the total number of antennas in the antenna array into a number of subarrays, and finding the covariance matrix for each before averaging the output covariance matrices. This method requires that the covariance matrix be obtained for each subarray using a large number of snapshots and then averaging the
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( h + 1) × ( h + 1) symmetric Toeplitz matrix whose rank is related to the DOA of the incoming signals, even though they may be highly correlated with each other. This allows for proper estimation of signal and noise spaces. These advantages improve performance significantly, thus allowing the proposed method to work efficiently at a low signal-to-noise ratio (SNR) and for a single snapshot.
II. System Model The M antenna elements are assumed to be omnidirectional, identical, and uniformly distributed over the circumference of a circle with radius r in the X-Y plane. The p equipowered sources arrive at the center of the circular array of radius r λ from the far field, with azimuth angle φk and elevation angle θ k . The φk and θ k are shown in Figure 1. We assume that one sample is observed by the array, because we are using only one snapshot. The signal vector X(t) impinging on the array at time t is defined as
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A (φ ) = TA(φ ) , N (t ) = TN (t ) ,
X (t ) = A(θ )S(t ) + N(t) = [ X0 (t), X1(t),..., XM−1(t)]T (1) where A(θ ) = [ a (θ1 ), a (θ 2 ),..., a (θ p )] is the matrix
(7)
(8) the transforming matrix T is defined as (9) T = JF , F is the (2h+1)×M submatrix of the spatial discrete Fourier transform (DFT)
consisting of steering vectors, is the signal vector S ( t ) = [ s1 ( t ), s 2 ( t ),..., s p ( t )]T consisting of p different source signals, and N ( t ) = [ N 0 ( t ), N1 (t ),..., N M −1 (t )]T is the white noise vector generated at each array element with zero mean and variance ı2.
§1 w− h ¨ ¨# # ¨ 1 w −1 1 ¨ F= ¨1 1 M¨ 1 w1 ¨ ¨# # ¨1 wh ©
w −2 h " w − ( M −1) h · ¸ # " # ¸ w −2 " w − ( M −1) ¸ ¸, 1 " 1 ¸ w 2 " w ( M −1) ¸ ¸ # " # ¸ w 2 h " w ( M −1) h ¸¹
w = e j 2π / M ,
(10)
(11)
and J is the (2h+1) × (2h+1) diagonal matrix
1 ° °½ J = diag ® ¾ m = − h," , 0," , h . m ¯° M j J m ( kr ) ¿° Figure 1.
The J matrix is useful for preserving the angle-dependent phase. The maximum mode number that UCA can intrigue is h ≈ k0 r . The m-order first kind of the Bessel function
The antenna array output in matrix notation is (2) X = AS + N where X is an M×1 (one sample) element space data vector, A is an M×p element space array manifold, S is the p×1 (one sample) complex signal envelopes at the array center, and N is the matrix of noise complex envelopes. The element space array manifold matrix A is
A = ª¬a1 (ς ,ϕ ) , a2 (ς ,ϕ ) , a3 (ς ,ϕ ) ,..., aP (ς ,ϕ )º¼ . The columns of matrix A are modeled as a(θ) = a(ς,ϕ) =[ejς cos(φ−γ0) ,ejς cos(φ−γ1) ,...,ejς cos(φ−γM−1) ]
is J m ( kr ) . If 2h 1 b M , then the manifold matrix of the VULA will be
§ e − jhφ1 ! e − jhφ p · ¨ ¸ % # ¸ A (φ ) = ¨ # jhφ ¸ ¨ jhφ1 " e p ¹ ©e
(3)
arrival
directions.
Here,
III. Modified UCA-ESPRIT
ς k0 r sin θ ,
As we know, forward/backward (FB) averaging is widely used in the ULA for the centro-Hermition property of the ULA manifold. On the other hand, FB averaging cannot directly be used in the UCA for decorrelating. In [9], the modified spatial smoothing method, or alternative forward/backward (AFB) method, is introduced. Reference [10] introduces the propagator method (PM)ESPRIT for UCA, which uses spatial smoothing as its decorrelating algorithm. The disadvantage of using these two methods is the large number of snapshots that are required.
k 0 2π / λ is the wave number, r is the radius, and γ n 2π n / M , (n=1, 2, …, M) is the sensor location. The manifold matrix A does not agree with the Vandermonde structure [6]. Therefore, decorrelated algorithms cannot be applied directly to the UCA, hence, the reason for applying the mode excitation method to transform the UCA in element space into virtual uniform linear array (VULA) in phase mode space. We shall transform this array to a virtual array of size 2h+1, where 2h 1 b M , which is amenable to decorrelated algorithms.
Reconstructing a Hermitian Toeplitz matrix from the observed signal vector is applied as follows:
The mode excitation method can be applied by using the transformation matrix T. Then the observed signal vector of X (t ) can be obtained by
X (t ) = A (φ ) S (t ) + N (t )
The observed signal vector X (t ) of (2h+1) rows is
(5)
where
X (t ) = TX ( t ) ,
(13)
where A (φ ) has a similar form as ULA and a size of (2h+1)×p.
(4)
where the elevation θ dependence through parameter ς and the vector θ (ς ,ϕ ) are used to represent the source
(12)
(6)
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§ x − h (t ) · ¨ x ( t ) ¸ ¨ − h+1 ¸ ¨ # ¸ ¨ ¸ X (t ) = ¨ x 0 (t ) ¸ ¨ # ¸ ¸ ¨ ¨ xh −1 (t ) ¸ ¨ x ( t ) ¸ © h ¹
where (1:h,:) is the collection of rows from 1 to h, and (2:h+1,:) is the collection of rows from 2 to h+1. Then, a matrix R of Q1 and Q2 is formed as R = pinv(Q1 ) × Q2 (24) where pinv denotes the pseudo-inverse.
(14)
By taking the angle of the largest eigenvalues of R,, we estimate ª¬φ1 ,",φ p º¼ .
where X (t ) has a form similar to the ULA. Reconstructing the Hermitian Toeplitz data matrix from the observed signal vector X (t ) becomes
X Toep
x1 (t ) " x h ( t ) · § x0 ( t ) ¨ ¸ x−1 (t ) x0 (t ) " x h−1 (t ) ¸ =¨ ¨ # # # # ¸ ¨ ¸ © x − h (t ) x − h+1 ( t ) " x0 (t ) ¹
IV. Simulation Results To demonstrate the performance of the proposed method, we performed several simulated experiments, whereby we compared the DOA errors obtained by applying the UCAESPRIT algorithm [3] using the modified spatial smoothing method and our proposed method or modified UCA-ESPRIT.
(15)
where X Toep has a size of (h+1)×(h+1).
The omnidirectional sensors are M=14, and their diameter is r λ . The virtual array is computed using (6) with h = 6, hence consisting of 13 elements as in (14). And the highest modes, m=±6, are computed using 2h 1 b M , and two equipower coherent sources are located at 100º and 120º. The number of snapshots used is 128 for the UCA-ESPRIT using spatial smoothing. The proposed method uses the same number of omnidirectional sensors and two equipower coherent sources located at 100º and 120º. The same virtual array size is used with only one snapshot. Figure 2 shows the root mean square (RMS) DOA error as a function of SNR after 100 trials.
By mapping the data vector into such a Hermitian Toeplitz form, the proposed method can estimate DOAs from up to h coherent sources because the incident sources become decorrelated, and the rank of X Toep can be higher than that of the covariance matrix of the received vector. The Hermitian Toeplitz matrix in a novel algorithm [11] that is applied to the MUSIC algorithm is constructed from the covariance matrix of the observed data matrix. But in this paper, the Hermitian Toeplitz matrix is constructed from the observed data vector, thus finding the covariance matrix from the constructed Toeplitz matrix as H (16) º¼ . X Toep _ cov = E ª¬ X Toep X Toep To separate the signal space from noise space, the total least squares method is used, which begins by computing the singular value decomposition of X Toep _ cov as
X Toep _ cov = U ¦V H .
(17)
Now, the number of source signals p is estimated to be the number of dominant singular values. In Matlab notation, this becomes U ′ = U (:,1: p ) , (18)
V ′ = V (:,1: p ) , ¦′ = ¦(1: p,1: p ) .
(19) (20)
Then, a matrix Q of size (h+1)×(h+1), which corresponds to the p largest eigenvalues of X Toep _ cov , is generated
Q = U ′ ¦′V ′H .
Figure 2.
(21)
Now, the data matrix X Toep is partitioned into two submatrices Q1 and Q2 as
Q1 = Q(1: h,:) Q2 = Q(2 : h + 1,:)
As shown in Figure 2, our proposed method using one single snapshot has a very close performance to the spatial smoothing using 100 snapshots.
(22) (23)
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For the spatial smoothing case, X (t ) has a size of (2h+1)×N (Snapshots) = 13×100 seven subarrays. Then, the covariance is calculated for each subarray. After that, the summation of all covariance subarrays is computed. Finally, the signal space is separated from the noise space by applying the least squares method as in (17).
snapshot has a close performance (e.g., within a decimal degree difference), to the conventional mode of spatial smoothing with 100 snapshots. Therefore, the proposed modified UCA-ESPRIT algorithm can be employed even under significantly reduced complexity constraint, where the conventional spatial smoothing techniques may not perform.
In the modified UCA-ESPRIT, X (t ) is a row vector of
VI. References
(2h+1)=13 rows. A Toeplitz matrix X Toep is constructed
[1] Roy, R., and Kailath,T., “ESPRIT-Estimation of Signal Parameters via Rotational Invariance Technique,” [J]. IEEE Trans. on ASSP, 1989, 37(7):984-995. [2] Lai, W. K., and Ching, P. C. “A New Approach for Coherent Direction of Arrival Estimation,” [J]. ISAC’ 98. Proceedings of the 1998 IEEE International Symposium on Circuits and Systems, 1998, 5:9-12. [3] Haber, F., and Zoltowski, M. “Spatial Spectrum Estimation in a Coherent Signal Environment Using an Array in Motion,” [J]. IEEE Trans. on AP, 1986, 3:301-310. [4] Krim, H., and Viberg, M. “Two Decades of Array Signal Processing,” [J]. IEEE SP, 1996, 13(4):67-94. [5] HPLin, S., and Okamoto, G. et al. “Multi-Path Direction Finding with Subspace Smoothing,” [A]. IEEE. ICASSP [C]. Munich: IEEE, 1997 pp. 34853488. [6] Wax, M., and Sheinvald, J. “Direction Finding of Coherent Signals via Spatial Smoothing for Uniform Circular Arrays,” [J]. IEEE Trans. on AP, 1994, 42(5):613-619. [7] Eiges, R., and Griffitths, H. D. “Mode Space Spatial Spectral Estimation for Circular Arrays,” Proc. Inst. Elect. Eng. F, Radar, Sonar, Navigat. 1994, 14:300306. [8] Tayem, N., and Naraghi-Pour, M. “Fast Algorithms for Direction of Arrival Estimation in Multipath Environment,” Proc. of SPIE, Wireless Sensing and Processing II, vol. 6577, 2007. [9] Minghao, H., Yixin, Y., and Xiamda, Z. “UCAESPRIT Algorithm for 2-D Angle Estimation,” IEEE ICSP, 2000, pp. 437-440. [10] Lian, X., and Zhou, J. “2-D Estimation for Uniform Circular Arrays with PM,” Antennas, Propagation & EM Theory, IEEE ISAPE. 7th International Symposium, 2006. [11] Shuyan, G., Yongliang, W., and Cangzhen, M. “A Novel Algorithm for Estimating DOA of Coherent Signals on Uniform Circular Array,” IEEE, 2006.
from X (t ) . X Toep has a size of (h+1)×(h+1) = 7×7. Afterwards, a covariance matrix of X Toep is calculated. Next, the signal space is separated from the noise space using equation (17) and continues until equation (24). By taking the angle of the largest eigenvalues of R, we estimate φ1 and φ2.
Figure 3. Figure 3 shows our proposed method with four sources: the first with an angle of 70 º, the second with an angle of 80 º, the third with an angle of 100 º, and the fourth with an angle of 120º. Our proposed method can estimate DOA up to h sources because the incident sources become decorrelated. The proposed algorithm can estimate the coherent sources from only a single snapshot. This makes the proposed algorithm suitable for high-speed wireless communication, where latency is an important issue.
V. Conclusion An UCA-ESPRIT algorithm was modified to estimate the direction of arrival of coherent signals efficiently using only one single snapshot. The mode excitation method and the Hermitian Toeplitz matrix were the keys for the proposed algorithm to function correctly. Simulation results verified that the proposed algorithm with one
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