Vol.23 No.5
JOURNAL OF ELECTRONICS (CHINA)
September 2006 1
A SUBARRAY-SYNTHESIS BASED 2D DOA ESTIMATION METHOD Xu Wenlong
Jiang Wei
Li Zengfu
Shang Yong
Xiang Haige
(Satellite Communications Lab, Department of Electronics, Peking University, Beijing 100871, China) Abstract In some satellite communications, we need to perform Direction Of Arrival (DOA) angle estimation under the restriction that the number of receivers is less than that of the array elements in an array antenna. To solve the conundrum, a method named subarray-synthesis-based Two-Dimensional DOA (2D DOA) angle estimation is proposed. In the method, firstly, the array antenna is divided into a series of subarray antennas based on the total number of receivers; secondly, the subarray antennas’ output covariance matrices are estimated; thirdly, an equivalent covariance matrix is synthesized based on the subarray output covariance matrices; then 2D DOA estimation is performed. Monte Carlo simulations showed that the estimation method is effective. Key words Spatial signal processing; Direction Of Arrival (DOA) estimation; Eigenspace decomposition; Time Division Multiplex (TDM); Subarray synthesis
I. Introduction In satellite communications, there are rigorous restrictions on the power consumption, geometry dimension and weight of satellite-borne electronic equipment. These limitations would result in some difficulties for system designers, and the following may be one of them: the total number of channel receivers is so restricted that it is less than that of array elements in the antenna equipped on the satellite, and, even less than impinging signals’ Directions Of Arrival angle (DOAs). As a result, due to the limitation of the receivers, generally, only some array elements in the antenna can be utilized to convey sensed signals to the receivers. Thus, in most existed DOA estimation methods that based on eigenspace decomposition, the maximum number of DOAs that can be estimated is limited by the total number of channel receivers, not that of array elements in the antenna. Those methods have two deficiencies. Firstly, because receivers are less than array elements, they cannot make full use of the antenna’s potential array element resource. Secondly, although there is a need for estimating more DOA with fewer channel receivers, just as in some satellite-borne DOA estimation applications, they cannot work well in this situation[1−5]. To solve the conundrum, we propose a new Two-Dimensional DOA (2D DOA) estimation method, a subarray-synthesis based method. 1
Manuscript received date: November 18, 2004; revised date: August 4, 2005. Supported by the National Natural Science Foundation of China (No.60462002 and No.60302006). Communication author: Xu Wenlong, born in 1962, male, Ph.D. candidate. Apartment #6-2-602, No.11, West Street of Wanshou Rd., Haidian District, Beijing 100036, China.
[email protected].
The proposed method is based on the subarray synthesis of an antenna. By dividing an array antenna into subarray antennas, each subarray antenna contains equal number of array elements; the total number of array elements in any subarray antenna is equal to that of receivers equipped on a satellite. Based on Time Division Multiplex (TDM) processing scheme, we make the receivers to receive snapshots from each selected subarray antenna, via a program-controlled switch, as shown in Fig.1, TDM receiver architecture for DOA estimation. In Fig.1, we assume that there are M array elements and M r channel receiving units, labeled from Rx 0 to Rx m, respectively, where M ≥ M r and m = M r − 1. At each snapshot sampling time, the complete set of receivers only need to sample all the array elements’ output in a selected subarray antenna; and at different snapshot sampling times, the same set of receivers may sample different subarray antennas. In this way, the receivers can acquire the snapshots of all the array elements in different subarray antennas. In theory, in a set of sampling times, they can acquire the output snapshots of arbitrary number of array elements in an antenna. Once completing the sampling and cumulating process, we could estimate output covariance submatrices of each selected subarray antenna, and synthesize an equivalent covariance matrix. Finally, we can perform DOA estimation based on the equivalent covariance matrix. The proposed DOA estimation method is different from that in Refs.[5−7]. Firstly, the problem discussed is different. While they focused on the decorrelation of the impinged correlative signals, we aimed at estimating more DOA with limited channel receivers. Secondly, though both papers adopted the concept of subarray antenna, the definitions are different in essence. Finally, refer to Fig.1, we assumed
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JOURNAL OF ELECTRONICS (CHINA), Vol.23 No.5, September 2006
fewer receivers than array elements existed, and adopted the TDM concept to acquire the snapshot on each subarray antenna.
Fig.1
TDM receiver architecture for DOA estimation, m = M r
II. The Complete Input and Receiving Formulas for Array-antenna Systems Assumptions A Circular Antenna (CA), shown in Fig.2, M array elements distributed in and around the center of the coordinate in the x-y plane, and choose the array element in the center as benchmark. The distance between element m and benchmark is d m . P narrow band farfield planewave {s p (k )}, k = 1," , p," , P, impinged are complex value at the benchmark; f 0 denotes the carrier frequency. {θ , ϕ} denote 2D DOA, with elevating angle θ ∈ (−π / 2, π / 2) and azimuth angle ϕ ∈ (0, 2π ). nm (k ) ∈ { N (k )}, k = 1, 2," , N are complex value Additive White Gaussian Noise (AWGN) on the array element m at the k - th sampling time; E{nm nnH } = σ 2 δmn ; δmn denotes ⎧1, m = n ⎪ . The noises are Kronecker delta, δmn = ⎪ ⎨ ⎪ ⎪ ⎩0, m ≠ n uncorrelated to each other, as well as to impinged signals; and the impinging signals are uncorrelated to each other, too. The subscript p in s p ( k ) denotes the sequential number of the impinging signals, p = 1, 2," , P; and m and n are the sequential number of the array elements, m, n ∈ {0,1," , M −1}. The superscript H denotes conjugative transpose. Thus, the snapshot on the m array element at the k - th sampling time is[8]
E{nm } = 0,
P ⎛ 2π ym (k ) = ∑ s p (k ) exp ⎜⎜− j (d m sin θ p cos ϕ p ⎜⎝ λ p =1
⎞ +d m sin θ p sin ϕ p ) ⎟⎟⎟ + nm (k ) ⎟⎠ It can also be expressed in vector form: Y (k ) = AS (k ) + N (k )
(1)
(2)
In Eq.(2), A denotes direction steering vector matrix, A = A(θ , ϕ ), S (k ), denotes the complex envelope vectors of the impinging source signals; and N (k ) denotes an additive noise vector.
⎫ ⎪ ⎪ ⎪ ⎪ ⎪ Α (θ , ϕ ) = [a1 " a p " a P ] ⎪ ⎪ ⎪ ⎡ ⎪ ⎛ ⎞ 2 sin π d θ 1 p T ⎪ (cos ϕ p + sin ϕ p )⎟⎟⎪ a p = ⎢1 " exp ⎜⎜− j ⎪ ⎟ ⎜⎝ ⎢ ⎠⎪ λ ⎣ ⎪ ⎬ ⎛ 2π d M −1 sin θ p ⎞⎟⎤ ⎪ ⎪ " exp ⎜⎜− j (cos ϕ p + sin ϕ p )⎟⎟⎥ ⎪ ⎝⎜ ⎠⎥⎦ ⎪ λ ⎪ ⎪ ⎪ ⎪ S T (k ) = [ s1 (k ) " s p (k ) " sP (k ) ] ⎪ ⎪ ⎪ ⎪ T N (k ) = [ n0 (k ) " nm (k ) " nM −1 (k )] ⎪ ⎭⎪
Y T ( k ) = [ y0 ( k ) " y m ( k )
" yM −1 ( k ) ]
(3)
III. A DOA Estimation Method Based on Subarray-synthesis 1. The covariance matrix In some DOA estimation systems, the total number of signal processing receivers is equal to that of array element in the antenna, i.e., one receiver corresponds to one array element. In this case, at every snapshot sampling time, receivers can directly sample the output Y (k ) of all array elements in the antenna, acquiring needed snapshots. Thereby, the system can estimate the output covariance matrix R of the antenna directly. R = E {Y (k )Y H (k )} (4)
where E{} denotes statistic expectation, and k denotes sampling time. 2. The output covariance matrices of subarray antennas In the case that total number of channel receivers is less than array elements, we cannot directly acquire the needed snapshots of all the array elements, and of the output covariance matrix R . Refer to Eq.(4), to estimate all the covariance elements rmn in matrix R, we need to obtain the output snapshot of all array elements in the antenna, as shown below ⎡ r00 " r0 n " r0( M −1) ⎤ ⎢ ⎥ ⎢ # ⎥ % # # # ⎢ ⎥ rmn " " rm ( M −1) ⎥⎥ (5) R = ⎢⎢ rm 0 ⎢ # ⎥ # # % # ⎢ ⎥ ⎢r ⎥ r r " " ( M −1) n ( M −1)( M −1) ⎦⎥ ⎣⎢ ( M −1)0 In Eq.(5), rmn denotes the covariance between the output of elements m and n of an antenna. In traditional DOA estimation method, if the total number of channel receivers M r is less than that of array elements M, the receivers can only receive the given M r array elements in the antenna, and cannot receive the output of the remaining M − M r array elements, only M r array elements in
XU et al. A Subarray-synthesis Based 2D DOA Estimation Method
the antenna are being utilized. This means only up to M r − 1 DOA can be estimated[1,2], not the potential M −1 ones. Analyzing Eq.(5), it is obvious that the crux of estimating the covariance matrix R is to acquire all the covariance elements rmn in the matrix R, m, n = 0," , m," , L," , M −1, respectively. We cannot acquire all the needed snapshots simultaneously. Nevertheless, refer to Fig.1, we can acquire them in a TDM manner under the control of a program-controlled switch. At each sampling time, we choose L = M r from the M array elements in the antenna, constituting a subarray antenna, and sample the output of the L array elements synchronously with the receivers, thereby, acquiring a series of output snapshots of the subarray antenna. In the similar way, we can obtain the output snapshots of another set of M r array elements, by switching the same M r receivers to another L array elements set, which constitutes another subarray antenna, and so on. In this way, we can acquire the needed snapshots of all M array elements in the antenna, and can estimate covariance submatrix of every subarray antenna. Having acquired all covariance submatrices of a selected subarray antenna set, we can synthesize a covariance matrix equivalent to R of the antenna, based on the submatrices. 3. The subarray-synthesis based equivalent covariance matrix We take an antenna that composed of small number of array elements for an example. Refer to Fig.2, a plane CA contains M = 7 array elements, element spacing is d = λ 2, λ is the carrier wavelength of impinging signal; assume each subarray antenna compose of M r = 5 array elements; and P = 6 arrival signals impinged on the antenna, S (n) is the complex envelope vector of arrival wave in the benchmark, DOAs are (θ p , ϕ p ), p = 1, " , 6, respectively. They are uncorrelated to each other, N (n) is AWGN vector, with equal mean zero and covariance σ 2 . Obviously, the DOA cannot be estimated based only on the output of an individual subarray antenna.
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Based on Eq.(4), we can define the output covariance matrix RYY of the antenna: RYY = A(θ , ϕ ) RS AH (θ , ϕ ) + σ 2 I
(6)
where matrix I in Eq.(6) denotes an M × M identity matrix. Once having estimated all the covariance submatrices of the selected subarray antennas, we can construct a matrix Req that equivalent to the covariance matrix RYY based on the subarray-synthesis method. Take Rl as general expression form of a covariance submatrix of the subarray antenna. The matrix Req will approach RYY under some given conditions; therefore Req can substitute RYY to estimate DOA of impinging signals. Refer to Fig.2. Take the array element in the coordinate benchmark on the x-y plane as benchmark, and number it zero. We can number the other elements from one to six sequentially, with positive angles in counterclockwise. The total number of subarray antenna is C75 = 21. Nevertheless, there is no need to calculate all the output covariance submatrices of all the twenty-one subarray antennas. In fact, selecting six out of the twenty-one subarray antennas will be enough to comprise a complete subarray antenna set for constructing the equivalent covariance matrix Req . The six subarray antennas selected can be numbered from A1 to A6 , which also represented the direction steering vectors of the subarray antennas, comprised by the array elements of [0 1 2 3 4], [0 1 2 3 5], [0 1 2 3 6], [0 1 2 4 5], [0 1 2 4 6], [0 1 2 5 6], respectively. In addition, the corresponding covariance matrices can be denoted by R1 , ", R6 , respectively. Define variables vi , i = 0,1, " , 6 as the following: v0 (θ , ϕ ) = 1 v1 (θ , ϕ ) = exp( jπ sin(θ )sin(ϕ )) ⎛ 1 ⎞ v2 (θ , ϕ )=exp ⎜⎜− j π 3 sin(θ ) cos(ϕ )− sin(θ )sin(ϕ )⎟⎟⎟ ⎜⎝ 2 ⎠ ⎛ 1 v3 (θ , ϕ )=exp ⎜⎜− j π 3 sin(θ ) cos(ϕ )+ sin(θ )sin(ϕ ) ⎜⎝ 2
(
)
(
v4 (θ , ϕ ) = exp(− jπ sin(θ )sin(ϕ ))
Fig.2 Plane CA of seven elements, DOA is {θ , ϕ}
⎛ 1 ⎞ v5 (θ , ϕ )=exp ⎜⎜ j π 3 sin(θ ) cos(ϕ )− sin(θ )sin(ϕ ) ⎟⎟⎟ ⎜⎝ 2 ⎠ ⎛ 1 ⎞ v6 (θ , ϕ )=exp ⎜⎜ j π 3 sin(θ )cos(ϕ ) + sin(θ )sin(ϕ ) ⎟⎟⎟ ⎜⎝ 2 ⎠ Thereby, we have the direction steering vector of subarray antenna A1
(
)
(
)
⎞
)⎠⎟⎟⎟
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JOURNAL OF ELECTRONICS (CHINA), Vol.23 No.5, September 2006
A1 (θ , ϕ ) = [a1 , a2 , a3 , a4 , a5 , a6 ] where
(7)
a Tp = a Tp (θ p , ϕ p ) = [1 v1 (θ p , ϕ p ) v2 (θ p , ϕ p )
v3 (θ p , ϕ p ) v4 (θ p , ϕ p )]. We can also have the remaining direction steering vectors A2 (θ , ϕ ) to A6 (θ , ϕ ), respectively, and estimate the submatrices Rl , l = 1, " , 6. Rl = E{[ Al (θ , ϕ ) Sl (n) + N l (n)]
⋅ [ Al (θ , ϕ ) Sl (n) + N l (n)]H } = A(θ , ϕ ) RSl AH (θ , ϕ ) + σ 2 I (8) Matrix RSl and RS are in the same form, both are P × P matrix and represent the covariance matrices of impinging source signals. The impinging signals are uncorrelated to each other, RSl and RS are diagonal matrices. In essence, the vectors of source signals received at different snapshot sampling time are not completely the same, and the additive noise vectors N l (n) also are not completely the same, so, RSl ≈ RS , suffix l denotes the sequential number of the subarray antenna. Therefore, in order to make use of the subarray synthesis method, we need to assume the arrival wave signals are wide sense stationary and ergodic processes in the period of sampling time T×L, in both spatial and time domain. Where T denotes the sampling interval of one snapshot sampling time. L denotes the total number of the selected subarray antennas, in the example L= 6. Fortunately, in communications, most impinged signals meet the restrictions. In this case, RSi = RS and σl2 = σ 2 , σl2 denotes the additive noise variance of subarray antenna l. We propose a sampling scheme for the output snapshots of subarray antennas: the output signal of all the array elements in subarray antenna A1 is sampled at the first snapshot sampling time; then the output signal of all the array elements in the subarray antenna A2 , and so forth. In this way, in T × L periods of sampling times, we can acquire for once the output snapshots of all subarray antennas in a complete set. When finished sampling all subarray antennas in the complete set, we can begin a new round of sampling process again, starting from subarray antenna A1 . The sampling process can take place for N times, acquiring N snapshots of correlative output of each subarray antennas. N can be determined by simulation according to prescribed estimation precision of error rate. We define one processing cycle of a complete subarray antenna set as follows: the time needed for sampling all the L selected subarray antennas for one cycle. Therefore, a processing cycle of a subar-
ray antenna complete set is equal to T × L, L times the sampling time T. Refer to Eq.(8) and the definition of subarray antenna A1 , we can computer its output covariance matrix R1 , as shown in Eq.(9). Thereby we obtained part elements of the equivalent covariance matrix Req , shown in Eq.(10). ⎡ r00 r01 r02 r03 r04 ⎤ ⎢ ⎥ ⎢ r10 r11 r12 r13 r14 ⎥ ⎢ ⎥ R1 = ⎢⎢ r20 r21 r22 r23 r24 ⎥⎥ (9) ⎢r ⎥ ⎢ 30 r31 r32 r33 r34 ⎥ ⎢r ⎥ ⎢⎣ 40 r41 r42 r43 r44 ⎥⎦ ⎡ × ×⎤ ⎢ ⎥ ⎢ × ×⎥ ⎢ ⎥ ⎢ ⎥ × × R 1 ⎢ ⎥ Req = ⎢⎢ (10) × ×⎥⎥ ⎢ ⎥ × ×⎥ ⎢ ⎢ ⎥ ⎢× × × × × × ×⎥ ⎢ ⎥ ⎢⎣× × × × × × ×⎥⎦ Symbol “× ” in Eq.(10) denotes unknown covariance to be estimated. Similarly, we can estimate the remaining covariance submatrices of selected subarray-antennas, R2 , ", R6 respectively. A subsequent submatrix in a complete set contains at least two new elements. In essence, those elements that were not included in the previous covariance submatrices are just what we were to estimate from succeeding matrices. Based on the covariance submatrices and given processing criterion, we can synthesize an equivalent covariance matrix, and furthermore, perform DOA estimation.
IV. Simulation Results Simulations are performed based on MUSIC algorithm. Define the orthogonal projection arithmetic operator on the noise subspace as ˆ ⊥ = Uˆ nUˆ nH ∏ (11) ˆ where U n denotes the estimation of the eigenvectors in the noise subspace. The MUSIC method’s spatial spectrum is defined as PMU (θ , ϕ ) =
a H (θ , ϕ )a (θ , ϕ ) ˆ ⊥ a (θ , ϕ ) a H (θ , ϕ ) ∏
(12)
Simulation conditions A plane uniform- distributed array antenna that contains 19 array elements, each subarray antenna is consisted of five elements. 10 incident source signals, with DOA (θ p , ϕ p ),
XU et al. A Subarray-synthesis Based 2D DOA Estimation Method
p = 1,",10. Ten pairs of 2D DOA angles are (θ p , ϕ p ) ={(3, 5), (13, 40), (23, 75), (33, 110), (43, 145), (53, 180), (43, −170), (73, −110), (53, −75), (33, −40)} degrees, respectively. Fig.3~Fig.6 show the simulation results. In Fig.3, the number of snapshots N =100, signal to interference and noise ratio SNR=20.0dB. In Fig.4, N=1000, SNR=0.0dB. Fig.5. and Fig.6 are the corresponding contour of Fig.3 and Fig.4, respectively.
Fig.3 N=100, SNR=20.0 dB
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the equivalent covariance matrix could be used to perform 2D DOA estimation well. The method proposed in this paper can be applied in satellite-borne DOA applications, so does in other signal processing systems as well, such as, satellite-borne beamforming, wireless communications, solving such problems as the scarce resources of signal processing receivers versus plenty sources of signal sensors, etc. Compared with direct methods, the subarray-synthesis based DOA estimation method will result in some processing delay. Nevertheless, if sampling frequency were high enough, the influence of processing delay on the performance of DOA estimation system can be ignored.
Fig.4 N=1000, SNR=0.0 dB
Acknowledgment The first author would like to thank Dr. Liang Xin; the author has been enlightened in discussions with him during the period of years. He also wishes to express appreciation to Dr. Guo Junqi for the help of finalizing the paper.
References Fig.5 N=100, SNR=20.0dB
Fig.6 N=1000, SNR=0.0dB
Fig.3 to Fig.6 showed that the proposed subarray-synthesis based DOA estimation method keeps effective in the above mentioned simulation conditions. In fact, if arrival signals satisfy stationarity and ergodicity restrictions, the DOA estimation method is equivalent to direct estimation methods in the case that the covariance matrix can be estimated directly from array sampling. Though either of the two methods can be applied to perform DOA estimation, the subarray-synthesis based DOA estimation method may be the best choice in some cases, such as in satellite-bearing DOA estimation applications, where the potential number of receivers is less than the array elements of the satellite-equipped antenna.
V. Conclusions Based on the analysis of characteristics of satellite communications, we proposed a 2D DOA estimation method for arrival planewave. The subarraysynthesis based DOA estimation method stems from the strict limitations to satellite-bearing equipment that the number of arrival signal processing receivers has to be less than that of array elements in an array-antenna. Simulations showed that, instead of main covariance matrix in traditional direct methods,
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