Pattern synthesis method for arbitrary arrays based

Pattern synthesis method for arbitrary arrays based on LCMV criterion

If patterns with arbitrary sidelobe envelopes are desired, only a little modification is needed. The sidelobe envelope is denoted by D(y), and then the reference level Prk is modified as:

Q. Guo, G. Liao, Y. Wu and J. Li

Pr ak ðyÞ ¼ Prk DðyÞ:

A pattern synthesis method for arbitrary arrays based on the linearly constrained minimum variance (LCMV) criterion is presented. Given mainlobe regions and an arbitrary sidelobe envelope, this algorithm searches the pattern with the lowest sidelobe levels. Its iteration coefficient is robust to synthesis conditions, and patterns with a flat top mainlobe can be obtained using phase-independent derivative constraints.

Introduction: Recently, pattern synthesis for arbitrary arrays has been a focus of research and many approaches have been proposed. One method [1, 2] is to apply adaptive array theory. Iteration coefficients are important, which determine stability and convergence speed, but the iteration coefficients cannot be chosen easily. Especially in [1], the proper value of the coefficient is dependent on the synthesis conditions, and needs modifying by an iteration process. The value is obtained by trial and error. Although the method in [2] is for a mainlobe control mechanism, the appropriate reference pattern Pr(y) (y is angle of arrival) in [2] cannot be chosen easily because it is complex-valued. Its magnitude can be easily determined according to the mainlobe shape, but its appropriate phase cannot be chosen easily. For example, we should not make Pr(y) (y in mainlobe region) equal to a real constant in order to obtain a pattern with a flat top mainlobe. Its phase is constrained to zero in all the mainlobe region if we do so, which is unnecessary because we do not mind its phase in the problem of pattern synthesis. This may cause undesirably high sidelobes and the requirement of more array elements to meet the same synthesis specifications. In this Letter, a new algorithm is presented. Its coefficient is robust to synthesis conditions and flat mainlobe patterns can be obtained easily using derivative constraints. Given mainlobe regions and sidelobe envelopes, the algorithm searches the pattern with the lowest sidelobe levels, which is different from the algorithms in [1] and [2]. These fix the difference of the sidelobe level and mainlobe peak beforehand without regard to whether it can be achieved for the given array.

Substituting Pr ak(y) for Prk in (1), we can obtain the corresponding iteration formula. The LCMV criterion and its optimal solution are familiar, so it is omitted here. Derivative constraints can be used in the LCMV beamformer to achieve a flat mainlobe pattern. Tseng [3] proposed a set of phase-independent derivative constraints that only constrain the magnitude response of the beamformer. Similarly to the derivation in [3], the minimum variance optimisation problem with the second-order derivative constraint can be expressed as:
w~ opt ¼ arg min w~ T R~ x w~ ð3Þ st C~ T w~ ¼ g~ ; V T w~ ¼ h where the definitions of w˜ , R˜ x, a˜(yi), and aˆ(yi) are the same as those in [3], C˜ ¼ [a˜(y0) aˆ (y0) a˜ 0(y0)], y0 is the constrained angle, a˜ 0(y0) is the derivative value of a˜(y) at y0, g˜ ¼ [1 0 0]T, V ¼ [aˆ 0(y0) aˆ 00(y0)], aˆ 0(y0) and aˆ 00(y0) are, respectively, the first- and second-derivative value of aˆ (y) at y0, h ¼ [h
h2], and h is an unknown variable to be determined. Putting Q ¼ [C˜ V], and f ¼ [g˜ T h]T, the solution of this problem is:
1 T ~
1 w~ ðhÞ ¼ R~
1 x QðQ Rx QÞ f

ð4Þ

Therefore the minimisation problem reduces to that of finding h, which minimises w˜ (h)TR˜ xw˜ (h), namely, f T(QTR˜ xQ)
1f. It can be solved quite efficiently using numerical nonlinear optimisation techniques such as found in [4]. Once h is determined, the optimal weight vector can be computed using (4). Although only one constrained angle is discussed here, it is straightforward to extend it to several constrained angles. The steps of the proposed algorithm are as follows: (1) Specify mainlobe regions and sidelobe envelope D(yi). Set the initial value of jammer powers f0(yi) ¼ 1 if yi is in the sidelobe region, and f0(yi) ¼ 0 in the mainlobe region, i ¼ 1, 2, . . . , N, where N is the number of uniformly distributed jammers with one degree spacing. (2) Calculate the data covariance matrix Rx using the following formula: Rx ¼ A diag½ fk ðy1 Þ fk ðy2 Þ    fk ðyN ÞAH þ sI

Proposed pattern synthesis algorithm: How can we search the pattern with the minimum uniform sidelobe level? When the uniform sidelobe level has not been reached, the sidelobe peaks are rugged. Our aim is to make all sidelobe peak levels equal to the lowest level that the given array can reach. We should increase the powers of jammers in high sidelobe regions and decrease those of jammers in low sidelobe regions in order to depress the high-level sidelobes. In this algorithm, the lowest sidelobe peak level of the synthesised pattern (the location of the peak usually varies in the iteration process, but can be found easily) is found at each iteration and used as the reference level denoted by Prk, where subscript k denotes the kth iteration. The iteration formula controlling the powers of jammers at different angles is 8 y <
 y in mainlobe region Kfk ðyÞ½Pk ðyÞ
Prk  fkþ1 ðyÞ ¼ ;0 y in sidelobe region :max fk ðyÞ þ Prk

ð2Þ

ð5Þ

where A ¼ [a(y1) a(y2)    a(yN)], s is a small quantity, and I is the identity matrix. sI is added to prevent the covariance matrix from being ill-conditioned. Obtain the weight vector w using the LCMV criterion and then compute the synthesised pattern. (3) Calculate the jammer powers using the iteration formula and the data covariance matrix Rx. (4) Recalculate the weight vector w, and then the synthesised pattern. If it is satisfactory, stop; otherwise, go to step 3. The algorithm can be extended in a straightforward way to the synthesis of 2D array patterns.

ð1Þ where fk(y) is the jammer power at angle y, K is an iteration coefficient, and Pk(y) ¼ jwH k a(y)j, where w is weight vector, a(y) is steer vector and H denotes conjugate transpose) is the synthesised pattern at the kth iteration. The iteration formula is more efficient than those in [1] and [2], because more comprehensive factors that affect the variation of jammer powers are considered. The first is the difference between the synthesised pattern Pk(y) and the reference level Prk, which is denoted by Dk(y). The second is the correlation of jammer powers of the current and next iteration. The jammer power at one angle is greater than that of other angles at the current iteration, so is its variation at the next iteration. Thirdly, the absolute levels of the pattern may vary greatly in the iteration process; it is the ratio of Dk(y) and the absolute level of the current iteration that plays a key role in controlling the variation of jammer powers at the next iteration, so Prk appears in the denominator of (1).

Fig. 1 Synthesised pattern of example 1

Results: Through many experiments, we have found that 0.1 is an appropriate value for k and it is independent of the synthesis conditions. Only three examples are shown here because of limited space. In the first example, we synthesised a pattern for a 33-element linear

ELECTRONICS LETTERS 13th November 2003 Vol. 39 No. 23

array of dipoles with nonuniform spacing and orientation, which is also an example in [1]. The element pattern of the dipoles and their parameters is listed in [1]. Fig. 1 shows the final result. The mainlobe region is [
6, 6] (deg), and the iteration number is 20. The iteration coefficient does not need modifying using an iteration process.

[35, 55], and the iteration number is 40. The dashed line is the sidelobe envelope. The third example is the synthesis of a 2D pattern for a nonuniform planar array of 100 elements, shown in Fig. 3. The 2D pattern is shown as a function of x ¼ sin(y)cos(f) and y ¼ sin(y)sin(f), where y is elevation angle and f is azimuth angle. Fig. 4 shows the final synthesised pattern with a flat top mainlobe. The iteration number is 40.

Fig. 2 Synthesised pattern of example 2

Fig. 4 Synthesised pattern of example 3

Conclusions: An efficient pattern synthesis method for arbitrary arrays is proposed. Given mainlobe region and arbitrary sidelobe envelope, this algorithm searches the pattern with the lowest sidelobe levels, which is different from the former algorithms based on adaptive array theory. The iteration coefficient is robust to pattern synthesis conditions, and a pattern with a flat top mainlobe can be easily obtained using phase-independent derivative constraints. # IEE 2003 Electronics Letters Online No: 20031068 DOI: 10.1049/el:20031068

21 July 2003

Q. Guo, G. Liao, Y. Wu and J. Li (Key Lab. for Radar Signal Processing, School of Electronic Engineering, Xidian University, TaiBai South Road #2, Xi’an, 710071, People’s Republic of China) Fig. 3 Element positions (in multiples of wavelength)

References 1

Example two is the case of multiple flat top mainlobe pattern synthesis. Fig. 2 shows the synthesised pattern for a 20-element linear array with element positions (unit in wavelengths) {0, 0.2940, 0.8049, 1.2738, 1.5891, 2.3555, 2.7873, 3.0794, 3.7287, 4.0124, 4.6466, 4.9622, 5.5998, 6.1606, 6.8551, 7.1672, 7.5713, 8.1920, 8.5184, 9.0958}. The two mainlobe regions are [
55,
35] and

2 3 4

OLEN, C.A., and COMPTON, R.T., JNR.: ‘A numerical pattern synthesis algorithm for arrays’, IEEE Trans., 1990, AP-38, pp. 1666–1676 ZHOU, P.Y., and INGRAM, M.A.: ‘Pattern synthesis for arbitrary arrays using an adaptive array method’, IEEE Trans., 1999, AP-47, pp. 862–869 TSENG, C.Y.: ‘Minimum variance beamforming with phase-independent derivative constraints’, IEEE Trans., 1992, AP-40, pp. 285–294 DENNIS, J.E., JNR., and WOOD, D.J.: ‘New computing environments: microcomputers in large-scale computing’, SIAM, 1987, pp. 116–122

ELECTRONICS LETTERS 13th November 2003 Vol. 39 No. 23