IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 38. NO. 9, SEPTEMBER 1990 1496

Communications

Linear Antenna Array Pattern Synthesis with Prescribed Broad Nulls

MENG HWA ER, MEMBER, IEEE

Abstmct-A new technique is presented for synthesizing an antenna array pattern with prescribed broad nulls. The array pattern synthesis problem is formulated as a least-square null constrained optimization problem. Numerical techniques based on the matrix factorization method are developed for reducing the computational complexity of determining the optimal weight vector. Subsequently, a set of eigenvector constraints are used to approximate the effect of the quadratic constraint. Numerical results are presented to illustrate the performance achievable.

I. INTRODUCTION

Array antennas constitute one of the most versatile classes of ra- diators due to their capacity for beam shaping, beam steering, and high gain. One of the most basic problems in array design is de- termining the radiation pattern of an array. Synthesis techniques for determining weights which result in a desired pattern response for a given array have been available for more than 40 years [I]. The great majority of this work has focused on methods which result in patterns having prescribed main lobe width and reduced sidelobe levels.

The problem of synthesizing antenna patterns with controlled nulls has been extensively studied in the literature. Prescribed nulls in the radiation pattern can be used to suppress interferences from specific directions. Null steering systems have found application in radar, sonar, and communication systems for minimizing degradation in signal-to-noise ratio performance owing to unwanted interference.

In recent years there has been considerable interest in designing antenna arrays with broad null sectors [2]-[7]. The need for a broad null often arises when the direction of arrival of the unwanted inter- ference may vary slightly with time or may not known exactly, and where a comparatively sharp null would require continuous steering for obtaining a reasonable value for the signal-to-noise ratio.

In this communication, a technique for synthesizing a linear an- tenna array pattern with prescribed broad nulls is presented. The array pattern synthesis problem is formulated as a least-square null constrained optimization problem. Numerical techniques are also de- veloped for reducing the computational complexity of determining the optimal weight vector. Numerical results are presented to illus- trate the performance achievable.

11. PROBLEM RRMULATION A N D SOLUTION

Consider a linear array of N isotropic antenna elements with uni- form spacing. The antenna far-field pattern is described by

wo, 8 ) = w H s ( f O r 8 ) (1)

where W is the N-dimensional complex weight vector given by W = [w, , w2,. . . , W N ] ~ and S(fo, 8 ) is the N-dimensional space

Manuscript received January 20, 1989; revised January 30, 1990. The author is with the School of Electrical and Electronic Engineering,

Nanyang Technological Institute, Nanyang Avenue, Singapore 2263. IEEE Log Number 9037553.

vector given by S(f0. 8 ) = [exp(j2?rf071), . ' ' ,exp(j2?rfo~,)IT, where {7;, i = 1, 2, . . , N} are the propagation delays between the plane wavefront and the antenna elements with respect to some reference point. The superscripts H and Tdenote complex conjugate transpose and transpose, respectively.

The array pattern synthesis problem is formulated as the following least-square null constrained optimization problem:

minimize(W - W , ) H ( ~ - w,) ( 2 4 W

subject to WHQW 5 t where WO is a predetermined vector obtained via, say, Chebyshev synthesis, to achieve a specified sidelobe level and Q is the N x N dimensional Hermitian matrix given by

0" +A012

Q = 1 s ( fo , e ) s ~ ( f ~ , (3) .9"--.1.9/2

where A8 defines the spatial region in the interference direction Bo over which a broad null is to be formed. The constant (( 2 0) given in (2b) defines the mean-square null depth over the spatial region of interest.

The optimization problem defined in (2) can be interpreted as finding a weight vector which is to be as close to WO as possible subject to an integrated power constraint over the nulling sector.

Using the method of Lagrange multiplier [SI, it can be shown that the optimal weight vector W is given by

W =(I +aQ)-IWO (4)

where a 2 0 is the Lagrange multiplier for the quadratic constraint and is the solution of the following equation:

The computational complexity of determining a that satisfies (5) can be reduced significantly by using the matrix factorization method described next.

Since Q is a Hermitian matrix it can be factorized as

Q = rArH where r is the N x N dimensional unitary matrix given by

r = t E i , E z , . . . , E ~ l (7)

and {E,, i = 1, 2, . . . , N} are the eigenvectors. A is the N x N dimensional diagonal matrix:

A =diag[hl, h 2 , . . . , h N ] (8)

with {A,, i = 1, 2 , . . . , N } ordered such that hl 2 Xz 2 . ' 2 XN 2 0. Substituting (6) into (5) gives

w X r ( z + ~ - I A ( I + a A ) - l r H w o = t . (9)

Any root finding method can be used to solve for a that satisfies (9). This was further found to be made very efficient by using the half-interval method followed by the regula falsi algorithm [IO], as it has shown excellent convergence properties.

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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 38, NO. 9, SEPTEMBER 1990 1497

111. APPROXIMATION OF QUADRATIC CONSTRAINT BY A SET OF LINEAR CONSTRAINTS

In the previous section, a numerical method based on a matrix factorization is used to solve for the optimal Lagrange multiplier, to reduce the computational load. This section presents another ap- proach to this by approximating the quadratic constraint defined in (2b) by a set of linear constraints, also known as eigenvector con- straints.

Ideally the array should be constrained so as to have zero response over the spatial region of interest in the interference direction, thus

WHQW = 0. (10)

Substituting (6) into (10) gives

wH rArH w = 0. (11)

If Q has a rank equal to nl , then it is clear that the conditions for ( 1 1) to be satisfied are

WHE, =0, i = 1, 2 , . . . , n 1 . (12)

Therefore, a set of linear constraints of the form given by (12) can be used to ensure that the optimal array has zero response over a spatial region of interest in the interference direction.

So far it has been assumed that the matrix Q does not have full rank. When Q has full rank or because, in practice, the eigenvalue evaluation will not yield exactly zero eigenvalues, one can impose no linear constraints of the form

WHEj =0, i = 1 , 2 ; . . ,n0 (13)

where no is the smallest integer such that the percentage trace of the Q matrix defined by

/ n n N \

is greater than or equal to some threshold value. Note that (13) can be thought of as a set of constraints that yield

a small value for WHQW. The value of no determines the number of degrees of freedom lost.

It follows from (6), (7), and (13) that

N

WHQW = hiIEHW12 i = n o + l

Using the Cauchy-Schwartz inequality [9] it can be shown that

/ N \

(16)

Since {hi, i = no + 1 , . . . , N } are the smallest eigenvalues, it can be seen from (16) that the set of linear constraints defined by (13) ensure a low gain with respect to 11 W 11; and provided 1) W 11; is not too large, then WHQW will be small.

Replacing the quadratic constraint defined in (2b) by the set of lin- ear constraints defined in (13) and using the method of the Lagrange multiplier, it can be shown that after some simplification the optimal weight vector is given by

W = [ I - DDH]Wo (17)

where D is the N x no dimensional matrix:

(18)

0 .0

I

-30.0

w

n_ - 4 0 0

2 -50 0

-60.0

-45 -70.0

-90

I I

0 45 90

BEARING ANGLE lOEGl

Fig. 1. Polar response of a 20-element linear array with a prescribed broad null at 00 = 30°, A0 = 5 O and mean-square null depth E = I O p 5 .

BEARING ANGLE IOEG)

Fig. 2 . Polar response of a 20-element linear array with a prescribed broad null at 00 = 30°, A0 = 5' and mean-square null depth = IO-'.

It is worth mentioning that the approximation of the quadratic con- straint by a set of eigenvector constraints can be viewed as an efficient representation of array signals of specified angular occupancy. This approach, in a way, is similar to that taken in [3], although both approaches are completely different.

The effects of choosing different values of %tr on the null depth will be illustrated by numerical examples in Section IV.

IV. NUMERICAL RESULTS

To demonstrate the performance characteristics of the new design approach, computer studies involving a linear array having 20 equally spaced elements have been carried out. The interelement spacing is set at 0.5h.

Fig. 1 shows the polar response using the quadratic constraint approach. WO is designed using Chebyshev synthesis to achieve a sidelobe level of - 30 dB. A controlled broad null at Bo = 30' with A0 = 5 O (centered at 30') and mean-square null depth C; = IOW5 is designed. It is clearly shown that deep nulls are achieved over the spatial region of interest at the prescribed null position.

In Fig. 2, the design parameters are same as that in Fig. 1 except that the mean-square null depth C; is changed to lo-'. It can be seen that a null depth of 70 dB is achieved over the spatial region of interest.

1498 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 38, NO. 9, SEPTEMBER 1990

-20 l3 C O I

-30 0

a a -40 0

-50.0

-60 0

BEARING ANGLE I O E G I

Fig. 3. Effects of using different values of %tr on the null depth of the polar response.

Fig. 3 illustrates the effects of using different values of %tr on the null depths of the polar responses. The controlled broad null is designed at 30’ with A0 = 5’ centered at 30’. Three different values of %tr are used, i.e., 99.999% (solid line), 99.00% (broken line) and 90.00% (dash-dot line) with the corresponding no of 8, 4, and 3, respectively. From the diagrams and (16), it can be seen that using different values of %tr is equivalent to specifying different values of 5 for the quadratic constraint. Hence the mean-square null depth will change accordingly.

V. CONCLUSION

This communication has presented a new technique for synthesis- ing an antenna array pattern with prescribed broad nulls. The optimal weight vector is obtained by matching to a predetermined vector in a least-square sense subject to an integrated power constraint over a nulling sector. Numerical techniques based on the matrix factor- ization method have been proposed for reducing the computational complexity of determining the optimal weight vector. Subsequently, a set of eigenvector constraints are used to approximate the effect of the quadratic constraint. Numerical results show that the proposed techniques are very effective in the design of an antenna pattern, with a specified controlled null width and null depth.

REFERENCES C. L. Dolph, “A current distribution for broadside arrays which opti- mizes the relationship between beam width and side-lobe level, ” Proc. IRE, vol. 34, pp. 335-348, June 1946. P. J . D. Gething, “Linear antenna arrays with broadened nulls,” Proc. Inst. Elec. Eng., vol. 121, pp. 165-168, Mar. 1974. S . Prasad, “Linear antenna arrays with broad nulls with applications to adaptive arrays,” IEEE Trans. Antennas Propagat., vol. AP-27, pp. 185-190, Mar. 1979. H. Steyskal, “Synthesis of antenna pattern with prescribed nulls,” IEEE Trans. Antennas Propagat., vol. AP-30, pp. 273-279, 1982. - , “Wide-band nulling performance versus number of pattern con- straints for an array antenna,” IEEE Trans. Antennas Propagat., vol. AP-31, pp. 159-163, Jan. 1983. - , “Methods for null control and their effects on the radiation pat- tern,” IEEE Trans. Antennas Propagat., vol. AP-34, pp. 404-409, Mar. 1986. M. H. Er, “Technique for antenna array pattern synthesis with con- trolled broad nulls,” PIX. Inst. Elec. Eng., vol. 135, pt. H, no. 6, pp. 375-380, Dec. 1988. A. E. Bryson, Jr. and Y. C. Ho, Applied Optimal Control. Waltham, MA: Blaisdell, 1969.

191 R. Bellman, Introduclion to Matrix Analysis. New York: McGraw- Hill, 1960. B. Carnahan, H. A. Luther. and J . 0. Wikas, Applied Numerical Methods. New York: Wiley. 1969.

[lo]

RCS of a Partially Open Rectangular Box in the Resonant Region

Abstract-The radar cross section (RCS) of a partially open rec- tangular box in the resonant region is investigated. Two-dimensional numerical results are generated using the method of moments solution to the electric field integral equation. Resonance spikes are found in the RCS spectrum at frequencies which correspond to the resonant frequen- cies of the interior rectangular cavity. The dependence of the resonant behavior on the box dimension, aperture size, and incident polarization can be interpreted in terms of the field distribution inside the cavity. Experimental data for a three-dimensional box are also presented. They are consistent with the two-dimensional simulation.

I. INTRODUCTION

The scattering characteristics of partially open cylindrical and spherical cavities in the resonant region have been studied exten- sively recently [ 11-[6]. One particularly interesting phenomenon is the strong resonances observed in the induced current on the cavity walls. The resonances occur at discrete frequencies which corre- spond to the resonant modes of the internal cavity. Furthermore, the scattered field and scattering cross section also show sharp resonant behaviors. This phenomenon is not only of theoretical interest, but may play an important role in the area of target identification. In this paper, we present both theoretical and experimental data on the cavity resonance phenomenon for a partially open conducting rec- tangular box. First, a theoretical study of a two-dimensional par- tially open rectangular cavity is carried out using the electric-field integral equation and the method of moments. Numerical results, part of which have been presented in [6], are summarized. With the insights gained in the 2-D numerical simulation, RCS measurement for a three-dimensional partially open rectangular box is next carried out. Resonances in the RCS spectrum are detected, identified, and interpreted in terms of the internal modes of the box. In addition, all the phenomena observed in the 2-D simulation, including the effect of polarization, are confirmed experimentally.

11. NUMERICAL SIMULATION OF 2-D RECTANGULAR CAVITIES

Consider a perfectly conducting, infinitesimally thin rectangular cylindrical cavity, with an infinitely long slot aperture parallel to the axis of the cavity. Two types of polarization for the incident plane wave are considered separately. For the transverse magnetic polarization (E-polarization), the incident E-field with unit amplitude is parallel to the axis. For the transverse electric polarization ( H -

Manuscript received April 5, 1989; revised November 16, 1989. This work

The authors are with the Department of Electrical and Computer Engineer-

IEEE Log Number 9036715.

was supported by National Science Foundation under Grant ECS-8657524.

ing, The University of Texas, Austin, TX 78712.

0018-926X/90/0900-1498$01.00 O 1990 IEEE