Sector Beamforming with Uniform Circular Array Antennas Using Phase Mode Transformation

Transcription

1 Sector Beamforming with Uniform Circular Array Antennas Using Phase Mode Transformation Mohsen Askari School of Electrical and Computer Engineering Shiraz University, Iran Mahmood Karimi School of Electrical and Computer Engineering Shiraz University, Iran Abstract Phase-mode transformation is a frequently used method in beamforming with uniform circular arrays By using this technique, the circular array is converted to a virtual uniform linear array Consequently, beamforming techniques which need vandermonde structure of the steering vector, can be applied to the uniform circular array indirectly It is necessary to use all sensors of the circular array in order to deploy the phase-mode transformation In this paper, it is shown that when sensors of the array are directional, and also an optimum beamformer is deployed, it is sufficient to use only a sector of the array instead of all the sensors Simulation results show that performance of the sector transformed beamformer is better than that of the ordinary transformed beamformer Index Terms phase mode transformation; sector beamforming; uniform circular array I INTRODUCTION Uniform circular arrays (UCA) are of importance in applications like beamforming and direction of arrival (DoA) estimation Compared with other arrays (especially uniform linear arrays), arrays of this type profit some advantages and suffer from some drawbacks By using circular arrays we can coverage entire azimuth angle in both beamforming and DoA estimation methods Also we can have two-dimentional DoA estimation with these arrays [1] In spite of Uniform linear arrays (ULA), steering vector of UCA does not have vandermonde structure So, beamforming and DoA estimation methods that are developed based on the vandermonde structure of the steering vector of the array (eg ESPRIT, root-music and spatial smoothing method) cannot be applied in circular arrays directly and some methods are applied with modifications [] Phase-mode transformation [3] is a method that is used to overcome this limitation With this transformation, steering vector of the UCA is converted to a virtual steering vector with vandermonde structure In other words, the UCA is converted to a virtual ULA So, all methods applicable for ULA, can be used in this virtual array Also an important virtue of using phase mode transformation is that we can have wideband beamforming and DoA estimation There are some limitations in using phase-mode transformation One of the limitations is that all sensors of the array must be used in order to extract the modes In some applications a sector of the array is used Based on the phase mode equations, in the scenarios in which a sector of the UCA is used, phase mode transformation cannot be utilized In this paper we show that when sensors of the array are directional and also an optimum beamformer (eg ) is deployed, we can make use of only a sector of sensors instead of all sensors of the array Using this technique, in addition to reduced the computational costs, performance of the beamformer is improved slightly In the entire part of the paper, vectors are denoted by boldface lowercase letters and matrices by boldface uppercase letters The superscript () H denotes the conjugate transpose, and () T denotes the transpose II SIGNAL MODEL A UCA with N sensors is considered Sensors of the array are assumed to be directional with known radiation pattern The array observation at time n, x(n) N 1 can be expressed as follows: x(n) N 1 = s(n) N 1 + i(n) N 1 + n(n) N 1 (1) where s(n) N 1 and i(n) N 1 are the desired signal and interferences components respectively, and n(n) N 1 is the array observation noise s(n) N 1 can be written as follows: s(n) N 1 = s(n)a N 1 () where s(n) is the desired signal and a N 1 is the desired signal steering vector It is assumed that the arrival signals and the array are coplanar, So, the steering vector a is a function of φ and shown by a(φ 0 ), where φ 0 is the azimuth angle of arrival of the desired signal Also, it is assumed that there are q interferences in the environment In the same manner, i(n) N 1 can be written in the following form: i(n) N 1 = A N q s i (n) q 1 (3) where A N q is a matrix that its columns contain steering vectors of the interferences s i (n) q 1 is a row vector that its components are interferences signals: A N q =[a(φ 1 ) N 1 a(φ q ) N 1 ] (4) s i (n) q 1 =[i 1 (n) i q (n)]

2 As it was mentioned, sensors of the array are directional So, steering vector of the desired signal and each of the interferences has the following form: a(φ i )=[g 1 (φ i )e jka cos φi π(n 1) jka cos (φi g N (φ i )e N ) ] T (5) where g n (φ i ) is the radiation pattern of the nth sensor and φ i is the azimuth angle of arrival of the ith signal III PHASE MODE TRANSFORMATION In this section, first it is shown how to extract phase modes when sensors of the array are omnidirectional Then the results are extended to a scenario in which sensors with directional pattern are utilized A Omnidirectional sensor Phase modes can be extracted by using a transformation matrix T (h+1) N as follows [4]: where and T (h+1) N = J (h+1) (h+1) F (h+1) N (6) 1 w h w h w (N 1)h 1 w 1 w w (N 1) F = w 1 w w (N 1) 1 w h w h w (N 1)h J = Nj h J h (ka) 1 Nj h J h (ka) Notice that J m () denotes the bessel function of the first kind and order m and h +1 is the size of the virtual array The matrix F in (7) is spatial fourier transform matrix There is a trade off in choosing the proper value for the parameter h The larger the h, the better the resolution of the array Also the smaller the h, the less the errors caused during phase-mode excitation An acceptable formula to choose the parameter h is as follows [4]: max{h h N 1, J h N (ka) ε} (9) J h (ka) Using phase-mode transformation, the steering vector of the virtual array is in the following form: (7) (8) a v (φ i )=[e jhφi e j(h 1)φi e jhφi ] T (10) Since F is a spatial fourier transform matrix, we can reduce the computational cost by passing the array signals through an N point spatial FFT Then the output of each beam is passed through an equalizer that its coefficients are frequency dependant [5] This procedure is shown in Fig1 Fig 1: Optimum Beamformer using phase-mode transformation B Directional sensors In [5] the phase-mode transformation technique is extended to a scenario in which sensors with directional patterns are deployed Patterns of sensors are assumed to be identical and equal to g(φ) Since g(φ) is a periodic function with period π, it can be expressed by its fourier series expansion as follows: g(φ) = l= c l e jlφ (11) where c l are the fourier series coefficients It was shown that when directional sensors are used, phase mode transformation matrix ˆT must be equal to [5]: ˆT N (h+1) = Ĵ(h+1) (h+1)f (h+1) N (1) where F is equal to that defined in (7) and Ĵ is as follows: A h (ka) 0 Ĵ = 0 0 (13) 0 A h (ka) and A m (ka) has the following form: A m (ka) = Ne j( N 1 )πm l= c l j l J m l (ka) (14)

3 Note that in extracting (14) only the dominant term of an infinite sequence is considered It can be seen from (1)-(14) that the transformation matrix ˆT consists of a spatial fourier transformation matrix F (like that in the previous subsection) and a diagonal matrix Ĵ IV OPTIMUM BEAMFORMING IN VIRTUAL ARRAY A Concepts In this section optimum beamformers are investigated when phase-mode transformation is applied to signals of the actual array Considering that directional sensors are used, the transformed signal is as follows: ˆx (h+1) 1 = ˆT (h+1) N x(n) N 1 (15) There are two basic methods for beamforming; conventional beamforming (delay and sum) and optimum beamforming In conventional methods, after applying the appropriate delays, the weight vector is obtained by using common windowing methods like chebyshev, Buttler etc In optimum beamformers, the weight vector is obtained by solving an optimization problem in order to optimize a specific criterion Minimum Variance Distortionless Response (MVDR) is the most famous optimum beamformer In this beamformer the optimum weight is found such that the output power of the beamformer is minimized while desired signal is passed without distortion In other words array gain is equal to unity in the direction of arrival of the desired signal The optimum weight is found by solving the following optimization problem: where ŵ MVDR = argmin ŵ H ˆRŵ (16) ŵ (h+1) 1 subject to ŵ H â v (φ 0 )=1 â v (φ 0 )= ˆTa v (φ 0 ) (17) and ˆR (h+1) (h+1) = E{ˆx(n)ˆx(n) H } is the virtual array correlation matrix and a v (φ 0 ) is steering vector of the desired signal in the virtual array Based on the definition, the virtual array correlation matrix, ˆR, can be written in the following form: ˆR = E{ˆx(n)ˆx(n) H } = E{ ˆTx(n)x(n) H ˆTH } (18) = ˆTE{x(n)x(n) H } ˆT H H = ˆTR ˆT where R N N = E{x(n)x(n) H } is the actual array correlation matrix Substituting (18) and (17) in (16), the optimization problem can be written in the following form: ŵ MVDR =argmin ŵ H ˆTR ˆTHŵ (19) ŵ subject to ŵ H ˆTa(φ0 )=1 Defining the weight vector w N 1 = ˆT H ŵ, it is directly applied to signals arrived from the actual array In practice there are not exact knowledge of â(φ 0 ) and ˆR It was shown that performance of the degrades severely if there are even small errors in the steering vector of the desired signal, or in the array covariance matrix [6]- [8] Robust optimum beamformers are developed to reduce the degradation of performance in these scenarios A powerful robust optimum beamformer that is developed based on worstcase performance optimization is considered in [9] In this method steering vector of the desired signal pertains to a ball set which covers all possible choices for the steering vector of the desired signal Optimization problem for the mentioned beamformer is as follows: ŵ wc =argmin ŵ subject to ŵ H ˆRŵ (0) ŵ H ĉ 1 where ĉ N 1 is any vector belongs to the set ˆTA and the set A is defined in the following form: A {c c = a + e, e ε} (1) where a is the presumed steering vector for the desired signal in the actual array After some simplification, the optimization problem (0) can be written in the following form: ŵ wc =argmin τ () τ,ŵ subject to Uŵ τ ε ŵ ŵ H â 1 Im{ŵ H â} =0 where U is the Cholesky factorization of ˆR and Im{ } denotes the imaginary part The problem () is a second order cone problem (SOCP) It can be solved with some open source solvers available like CVX [10], or by using interior-point method [11] It was shown that the mentioned robust beamformer works properly when there are errors in the steering vector of the desired signal or in the array correlation matrix Note that, like the weight vector w N 1 = ˆT H ŵ defined for the, by defining the weight vector w wc = ˆT H wˆ wc, the weight vector w wc is applicable to use directly in the actual array As equations in section III show, all sensors must be used in order to extract phase modes We know that practical sensors are directional Radiation patterns of these sensors are such that the maximum gain is achieved in the front side of the sensor and signals arrived from the back side of the sensor are attenuated severely Fig shows a circular array geometry Suppose that all signals and interferences arrive in the azimuth interval with green sensors specified by the lines (search zone) Based on the sensors radiation patterns, sensors located far from the search zone (signal free zone), only received noise components When we use conventional beamforming method (delay and sum beamformer), all sensors must be used in order to deploy the phase mode transformation But when an optimum beamformer is utilized, it is expected that components of the weight vector w that related to sensors in the signal free zone are approximately zero In the next section, results of

4 Normalized absolute value of weight sensor number Fig : circular array geometry Fig 3: Plot of w and ŵ versus number of sensors the numerical examples confirm this approximation So, there is no need to use sensors that are located in the signal free zone B Sector transformed optimum beamforming In the sector beamformers which are investigated in the last subsection, first the optimum weights are calculated using all sensors of the array and then some weights are neglected Another approach switches the sensors in the signal free zone off and then find the optimum weights For this end, the matrix ˆT defined in (1) is rewritten in the following form: ˆT =[t 1 t N ] (3) where t n is the nth column of the matrix ˆT and N is the size of the UCA Because of the fact that the size of the sector array must not be smaller than the size of the virtual array, a new matrix named ˆT r is defined in the following form: ˆT r =[t 1 t k t N k+1 t N ] (4) where k is size of the sector array and ˆT r is the transformation matrix for the sector of the array Using the matrix ˆT r, the transformed signal vector is as follows: ˆx r = ˆT r x (5) Consequently the virtual array correlation matrix is calculated in the folllowing form: R = E{ˆx r ˆx H r } (6) In this case the optimum weights of MVDR and worst-case sector beamformers are obtained as follows: w r,mvdr = R 1 â r â H R (7) r 1 â r ŵ r,wc =argmin τ,ŵ subject to τ (8) Ũŵ τ ε ŵ ŵ H ã r 1 Im{ŵ H ã r } =0 where Ũ is a high-triangular matrix which is the cholesky factorization of the matrix R and ã r is the desired signal steering vector in the virtual array which is achieved by using only a sector of the array The parameter ε is chosen such that all possible choice of the steering vector of the desired signal in the new virtual array is covered V NUMERICAL EXAMPLE A 35 elements UCA is considered In order to construct the virtual array, h =9is choosen In other words the size of virtual array is 19 The distance between two adjacent elements is half of the wavelength Each element is directional with radiation pattern equal to: g(φ) = 1 (1 + cos(φ)) (9) The desired signal and the two interferences are coming from 0, 30 and 0 respectively Signal-to-Noise-Ratio (SNR) and Interference to-noise-ratio (INR) in each sensor are 0dB and 30dB respictively The desired signal is always present in the data cell Also there are enough samples of array observation So, we do not have considerable error in estimating the array correlation matrix The desired signal s steering vector is assumed to be known exactly In Fig 3 absolute value of the weights w and w wc are plotted versus the number of the sensors It can be seen that only the components of the weights that related to the beginning and the end of the array are significant So, the other weight components can be neglected in order to reduce computational cost For simulation studies, when the sector beamformer is deployed, a sector of 0 elements is used instead of 35 elements UCA In Fig4 the beampatterns of are plotted when all sensors are used and when the sector beamformer is deployed This figure demonstrates that both beamformers have nulls in the direction of interferences It is also clear from this figure that the beamformer using only a sector of the array have an acceptable beampattern compared to that using all sensors In Fig5 a close-up image of the null of

5 10 Beampattern of the Beampattern [db] azimuth angle worst case Beamformer worst case with sector array Fig 4: Beampattern of s in two cases; when only a sector of array is used and when all sensors are used Fig 6: Output SINR versus number of snapshots when INRs=10 db 5 Beampattern of the Beampattern [db] worst case Beamformer worst case with sector array azimuth angle Fig 5: Close-up image of the null of the s in the direction of one of the interferences the beampatterns in the direction of the interference arrived from 30 is plotted It is shown that as opposed to the, the sector cannot place a deep null in this direction and this is the shortage of the sector Note that the same result is achieved for the interference arrived from 30 Aslothe worst case beamformer has the same results in the case of sector beamformign and in the case that all sensors of the array are deployed So, the beampatterns for the worst case beamformer are not shown in this paper In Fig6 and Fig7, output SINR of the mentioned beamformers are plotted versus the number of snapshots when INRs=10 db and INRs=30 db respectively For each beamformer, the code runs 00 times to obtain the output SINRs It can be seen from these figures that when the interferences are weak, the performance of the sector beamformers is better than that of the beamformer in which all sensors of the array are used But, when the power of the interferences are high, the feeble suppression of the interferences caused degradation Fig 7: Output SINR versus number of snapshots when INRs=30 db in the performance of the sector beamformers Also it is clear from these figures that when small number of snapshots are available, the performance of the worst-case beamformers is better than the s In Fig8 the null of in the direction of one of the interferences are plotted when all sensors of the array are used and when the sector transformed beamforming method is applied in the case that INRs=30 db It can be seen that the performance of the sector beamformer is improved in comparision with Fig7 In Fig9 output SINR versus the number of snapshots are plotted when INRs=30 db It is clear that the performance of the sector transformed beamformers is better than that of beamformers in which all sensors are deployed The same scenario is tested in the case that INRs=10 db The results are the same as Fig9 CONCLUSION In this paper it is shown that when sensors of the UCA are directional, optimum beamforming weights of some sensors in the array are near zero and consequently they can be neglected

6 Beampattern [db] azimuth angle Fig 8: Close-up image of the null of the s in the direction of one of the interferences when INRs=30 db [4] M Wax, and J Sheinvald, Direction finding of coherent signals via spatial smoothing for uniform circular arrays, IEEE Trans Antannas Propagat, vol 4, pp , May 1994 [5] H Steyskal, Digital beamforing aspects of wideband circular arrays, IEEE Aerospace conference, Mar 008 [6] L Chang and C C Yeh, Performance of DMI and eigenspace-based beamformers, IEEE Trans Antennas Propagat, vol 40, pp , Nov 199 [7] D D Feldman and L J Griffiths, A projection approach to robust adaptive beamforming, IEEE Trans Signal Processing, vol 4, pp , Apr 1994 [8] K L Bell, Y Ephraim, and H L Van Trees, A Bayesian approach to robust adaptive beamforming, IEEE Trans Signal Process, vol48,pp , Feb 000 [9] S A Vorobyov, A Gershman, and Z Q Luo, Robust adaptive beamforming using worst-case performance optimization: A solution to the signal mismatch problem, IEEE Trans Signal Process, vol 51, no, pp , Feb 003 [10] M Grant and S Boyd, CVX: Matlab Software for Disciplined Convex Programming (Web Page and Software) Apr 011 [Online] Available: boyd/cvx [11] S Boyd and L Vandenberghe, Convex Optimization Cambridge, UK: Cambridge Univ Press, output SINR [db] worst case Beamformer worst case with sector array number of snapshots Fig 9: Output SINR of the sector transformed beamformers versus number of snapshots when INRs=30 db The performance of two important optimum beamformers are investigated by using this technique It is shown that when power of the interferences is high, the performance of the sector beamformers are reduced because of weak interference suppression In order to have better performance and the lower computational cost, another approach is introduced and the problem of finding optimum weights for the tested beamformers are reformolated in this technique The performance of the mentioned beamformers is investigated and it is shown that the performance of the beamformers are improved by using this technique REFERENCES [1] U Baysal and R L Moses, On the geometry of isotropic wide-band arrays, in Proc IEEE Int Conf Acoustics, Speech, Signal Processing (ICASSP), vol 3, Orlando, FL, pp , May 00 [] M Askari, and M Karimi, Quadratically Constrained Beamforming Applied to UCA, ICEE 01, pp , Tehran, Iran, May 01 [3] D E N Davies, A transformation between the phasing techniques required for linear and circular aerial arrays, Proc Inst Elect Eng, vol 11, pp , 1965