R646 Philips Res. Repts 22, 568-576, 1967 OPTIMUM RADAR-ANTENNA-GAIN PATTERNS FOR ATTENUATION OF FIXED BACKGROUND CLUTTER Abstract 1. Introduetion by S. B. WEINSTEIN The antenna-gain patterns which maximize the clutter attenuation, subject to a constraint on beamwidth, are numerically determined for a detection radar using a delay-line canceler to attenuate fixed background clutter. Plots of these optimum patterns are presented for radar systems using single and double delay-line cancelers. For the former case, it is,shown that Gaussian patterns are almost optimum. It is frequently necessary to detect a moving target against a background of fixed reflectors, as, for example, a vehicle or low-flying aircraft against the ground. Detection of such targets is possible, even when the target echo is much weaker than the return from the background clutter, by detecting the Doppler shift in the target echo and attempting to cancel out the non-doppler-shifted clutter signal. In this study, we determine the optimum antenna-gain patterns, under a reasonable beamwidth constraint, for a pulsed detection radar using a delayline canceler to attenuate the clutter. It is shown that the performance of these patterns can often be closely approximated by simple Gaussian patterns. 2. Delay-line cancel ers The delay-line apparatus described by Skolnik 1) subtracts the echo of the current pulse from the delayed echo of the previous pulse. If the antenna did not rotate between pulses, the fixed clutter would be completely canceled. However, if the rotation angle ij</> between pulses is very small, substantial cancelation will occur, and may be numerically defined by the following expression for the attenuation Al ofthe received energy contributed by statistically uniform clutter in the observation plane.: where G(B) is the antenna-voltage-gain pattern. -It A 1 =------- (1)
OPTIMUM RADAR-ANTENNA-GAIN PATTERNS FOR CLUTTER ATTENUATION 569 If an additional delay-line section is introduced to take advantage of the clutter coherence over 3 pulses, the clutter attenuation is defined as - A 2 =------- (2) For either the single (eq. (1» or double (eq. (2» delay-line canceler, clutter attenuation is unbounded for G(O) = constant. It is clear, however, that a radar system with an antenna gain which is uniform in all directions has no resolution. In the following sections, beamwidth constraints are formulated and the antenna-gain patterns G(O) which maximize (1) and (2) are determined subject to these constraints. 3. Maximizing clutter attenuation with a beamwidth constraint A beamwidth constraint in integral form, resembling (1) or (2), is desirable in order to make use ofthe Lagrange-multiplier formulation. As a first thought, we propose a beamwidth (J2 defined by a2 = _-_---- (3) which is equal to the variance of the square of the gain pattern. The G(O) which is peaked near 0 = 0 and small near 0 = ± n will result in a small beamwidth. In sec. 5 we will shortly propose a slightly modified definition of beamwidth which is better suited to physical radars, but the formulation is very similar to that using (3)., If G(O) is narrow compared with the interval (-n, zr), the problem of maxi- mizing (1) with the constraint (3) is already solved. The Weyl inequality 2) 00 00 00 (4) -00-00 - 00 can then be approximately rewritten: 00 00 J G 2 (6)d6 (LI cf»2 Al = _-_00 ~ 4 _-_00 = 4 (J2, 00 00 J [G'(B)]2d6-00 -00
570 S. B. WEINSTEIN and equality is obtained when G(B) is Gaussian, i.e. G(B) = A exp (-rxb2), rx ~ o. (6) The Gaussian solution is no longer optimum when the (-:lt,:lt) limits of expressions (1) and (3) cannot be replaced by (-00, (0) with negligible error, but it still, as will be seen, comes very close to the optimum in performance. 4. Single delay-line canceler Instead of attempting to maximize (1) directly, subject to constraint (3), we shall perform the equivalent, but much easier, task of minimizing subject to the two constraints 7< J [G'(B)J2dB, -7< and -To: 7< 7< (7) (8) The normalization to unity in the first constraint implies no loss of generality, since expressions (1), (2) and (3) are not affected by a scalar multiplication of G(B). By the use of Lagrange multipliers p, and Ao, the problem reduces to the minimization of -7< 1t (9) The variational calculus may be used to derive the Euler- Lagrange equation 3) for the extremal G(B), with a polynomial in B as a solution, but the coefficients of the polynomial are functions of p,and Ao. The evaluation of p, and Ao from the constraint equations i s extremely difficult. A more practical approach is to write a truncated Fourier series for G(B) in the interval (-:lt,:lt) and solve for the coefficients. It will be seen that -p, becomes an eigenvalue of a linear transformation, and that the corresponding eigenvector is the desired solution. Let 6. ao 1 N G(B) = -- + - l: an cos (nb), (2:lt)1/2 JI:lt n= 1 -:lt ~ B ~:lt, (10) where the symmetry in () is a consequence of the symmetry of expression (9), in which no bias with respect to contributions from G(B) as compared with
OPTIMUM RADAR-ANTENNA-GAIN PATIERNS FOR CLUTfER ATIENUATION 571 G(-e) is evident. Odd symmetry is ruled out because of its physical unreasonableness, particularly near e = 0, but mathematically a sine series would also yield, an optimum G(e). Substituting (10) into (9), N N N 11 n=o n=o m=o -11 N N (11) where -n n=o m=o.6. 11 Rnm = J 0 2 cos (no) cos (mo)do = n=m =0 2:n(_1)n+1II :n3 ----+-, n=m=l=o (n + m)2 3 (_I)n+1II (_1)n-mJ (12) 2:n +, n =1= m, [ (n + m)2 (n- m)2 and.6.,10,1=-, :n n=o n =F 0. The set {al}~=o of Fourier coefficients yielding a minimum of (11) is found as a solution of the simultaneous equations i = 0, 1,..., N. (13) These equations can be written as a vector equation o = {B(A) + f11}a, (14) where the elements of the (N + 1) X (N + 1) matrix B(A) are (15)
572 S. B. WEINSTEIN It is evident that the solution vector a is an eigenvector of B()'). A computer programme using the Jacobi numerical solution method can be used to find the N + 1 eigenveetors associated with each choice of).. For each eigenvector, one may readily compute the beamwidth (3) and dutter attenuation (1) generated by the function G(8) whose Fourier coefficients constitute the eigenvector. One of these eigenveetors is the best, in the sense ofyielding the highest clutter attenuation that can be achieved with the resulting beamwidth. Because values of ). are chosen without precise knowledge of the functional relation between ). and beamwidth, the resulting values of beamwidth are -not well rounded. Figure 1 pictures the optimum antenna-gain patterns, all renormalized to the value 1 at 8 = 0, constructed from a Foutier series truncated to 15 terms (N = 14). For small values of the beamwidth 0-2, the curves are very close to the (almost) optimum Gaussian, and the ratio of clutter attenuation x(li cfo)2 to 0-2 is about 4, as required by inequality (5). For larger values of 0-2 the curves are still close to Gaussian, even though the inequality(5)is no longer applicable. G (e) 1 1.a 0 5 0 4 0 2 00 05 --+e 3 5 Fig. 1. Antenna-gain patterns which maximize the clutter attenuation Al ofthe single delayline canceler (eq. (1» subject to the constraint of eq. (3) on the beamwidth u 2 On the interval -:Tt < IJ < 0, the gain patterns are the mirror images of the above curves. The patterns produce the following beamwidth - clutter-attenuation pairs:. curve a2.a1x(llcf»2' 1 1-048 4'564 2 0 642 2 583 3 0 282 1 128 4 0 163 0 6515 5 0'089 0'3568 6 0'0515 0'2060 7 0'0258 0 1031
OPTIMUM RADAR-ANTENNA-GAIN PATTERNS FOR CLUTI'ER ATTENUATION 573 ( S O A,x(tlcpl 1 2 0 1 0 O S 0 2 / '/ 1/ / 1/ 1 / Ó-Ol 0-02 O-OS 0 1 0 2 O S 1 0 2-0 _C)2. Fig. 2. Clutter attenuation (eq. (1)) vs beamwidth (eq. (3)) for solutions shown in fig. 1 (upper curve in this figure) and for Gaussian antenna patterns. Figure 2 shows the plots of clutter attenuation vs beamwidth corresponding to the solutions of fig. land to the Gaussian patterns. 5. A revised beamwidth constraint The initial beamwidth constraint (3) tends to discourage large values of G(f) for f) far from f) = 0, but does not very much suppress large values of G(f) for f) closer to f) = 0. Since large values of G(B) occurring close to, but not within, the nominal width of the central peak are as undesirable as large values which are farther away, it is better to discourage G(f) more or less equally for all 1f)1 ;;;:: f)o, where 2f)o is roughly the desired width of the central peak. We therefore propose the following beamwidth constraint, where (12 again denotes the beamwidth: J f (O)G 2 ( a2 = _-_----- O)dO (16) where f(f) is the piecewise-linear function sketched in fig. 3. -T( Fig. 3. Function.t.:(O) used in beamwidth constraint (eq. (16)).
574 S. B. WEINSTEIN ------------------------ The formulation of the optimization problem is the same as in the last section, with eq. (14) again valid. However, the matrix R now has terms, s; 1t = J fee) cos (ne) cos (me)de. -1t As a numerical example, let BI and Bo of fig. 3 be arbitrarily chosen as 0 3 and 0 5 radians, respectively. Following the procedure of the last section, we find the antenna patterns of fig. 4. The plots of clutter attenuation (eq. (1» vs beamwidth (eq. (16» are given in fig. 5 for the results of fig. 4 and for Gaussian G(B). Again, the Gaussian gain pattern is close to optimum. (17) G (9) ro. 0 6 0 4 0 2 00 {)OS 3 1( 3 5 ----9 Fig. 4. Antenna-gain patterns which maximize the clutter attenuation Al of the single delayline canceler (eq. (1)) subject to the constraint of eq. (16) on the beamwidth az. The patterns produce the following beamwidth - clutter-attenuation pairs: 6. Double delay-line canceler curve a Z Alx(Llr/»Z 1 0 690 6 417 2 0 573 2 542 3 0'336 0 740 4 0 251 0 500 5 0 167 0'330 6 0'0897 0 212 7 0'0100 0'091 With the double delay-line canceler, one seeks to minimize 1t (18) -1t, which is a modification of eq. (9). Substituting the series expansion (10) into (18), we have /
OPTIMUM RADAR-ANTENNA-GAIN PATTERNS FOR CLUTTER ATTENUATION 575 S'O A,X(tJCP)2 1 2 0 1 0 O'S 0 2 V V ~I / V If' II ~ /; 1 Ö-Ol 0-02 0-05 0 1. 0 2 0 5 1 0 2-0 _û 2 Fig. 5. Clutter attenuation (eq. (I» vs beamwidth (eq. (16) for solutions shown in fig. 4 and for Gaussian antenna patterns. where 6. s.; = J f(o) cos (no) cos (mo)do (20) - and Il and H; are as previously defined. Then the vector equation, formally identical to (14), is o = {B(Il) +,ui}a, (21) where now (19) (22) The solutions of (21), for N = 14 and f (B) the piecewise-continuous function of fig. 3 with BI and Bo again chosen as 0 3 and 0 5 radians, respectively, are shown in fig. 6. One should not compare the clutter attenuations of fig. 6 with those of fig. 4 (single delay-line canceler) without noting that a factor of 1j(iJcp)4 is required in fig. 6 and a factor of 1j(iJ cp)2 in fig. 4. Since ij cp is ordinarily a small fraction of a radian, the clutter attenuation achieved by the double delay-line canceler is much greater than that of the single delay-line canceler. 7. Conclusions The exact shape of the antenna diagram does not appear to be of critical importance in achieving good clutter attenuation. Any smooth and almost unimodal function can be expected to perform well, and experimental antennas which suppress and smooth side lobes by utilizing an auxiliary non-directional
576 S. B. WEINSTEIN --...e Fig. 6. Antenna-gain patterns which maximize the clutter attenuation A 2 ofthe double delayline canceler (eq. (2)) subject to the constraint of eq. (16) on the beamwidth 0 2 The patterns of this figure produce the following beamwidth-clutter-attenuation pairs: curve 0 2 A 2 x(.dcf»4 1 0 660 3 257 2 0 543 1 042 3 0 403 0 279 4 0 271 0 091 5 0 104 0 0186 6 0 0366 0 0068 7 0 0166 0 0039 antenna 4) have been very successful as clutter attenuators, although they have no claim to optimality. Since the ideal antenna patterns can only be approximated in practice, it appears worth while to maximize clutter attenuation within the constraints of a physical antenna. This work is now in progress. Acknowledgement The author wishes to thank Professors C. J. Bouwkamp and N. G. de Bruijn, Ir A. Meijer, Ir J. Oosterkamp, Dr F. L. H. M. Stumpers and Ir F. Valster for their valuable suggestions. Eindhoven, May 1967 REFERENCES 1) M. I. Skolnik, Introduetion to radar systems, McGraw-Hill Book Co., Inc., New York, 1962, pp. 149-150. 2) H. Weyl, Theory of groups and quantum mechanics, Dover Publication 5269, Appendix 1, p.393.. 3) O. Bolza, Lectures on the calculus of variations, Dover Publication 5218, chapter 1. ~) J. Croriey and P. R. Wallis, A side-lobe suppression system for primary radar, lee Symp. on signal processing in radar and sonar direction systems, Birmingham, England, 1964.