IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. EMC-23, NO. 1, FEBRUARY 1981

Remarks on Anti-RFI FencesJ. J. GERALD MCCUE, FELLOW, IEEE

Abstract-In anti-RFI applications such as shielding two radarsfrom one another, the protection obtained from a fence may besignificantly less than the diffraction loss introduced by the fence. Theeffect can be alleviated by using an inner and an outer fence. Thistandem design is also advantageous for anticlutter fences, because agiven diffraction loss can be obtained with smaller and less costlystructures. If the diffraction loss is to be augmented by an edgetreatment, the tandem design makes possible the equivalent of staggertuning.Key Words-Radar fences, interference reduction, diffraction,

radar clutter.

8

INC IDE NTWAVE

\\ \\\\\\I

Fig. 1. Diffraction at a fence; (r, 0) are polar coordinates with respectto the top of the fence, and e is the angle of diffraction.

where

I. INTRODUCTIONMETALLIC FENCES are in use at radar sites for shielding

against ground clutter, for reducing reflections from theground, and for the protection of personnel. Their use in eachof these functions has been discussed by Becker and Sureau[1]. Fences for reducing ground reflections were treated byPreikschat [2], and a detailed study of a large fence at theAMRAD radar in New Mexico was made by Ruze et al. [3].More recently, there has been interest in the use of fences toreduce interference between two radars, which may be a fewmiles apart and operate in the same, or nearly the same, fre-quency slot. The purpose of the present article is to addressthe problems controlling that application, to mention briefly anovel design that may be useful, and to call attention to theeconomies that are possible when a single fence is replaced bytwo smaller ones.

II. BASIC DESIGN

The simple fence is an upright conducting sheet with ahorizontal top edge and a lower end buried in the ground. Itseffect can be calculated by using the well established theory[4] of diffraction by a "knife edge." Though in practice thefence is likely to be an arc, its radius of curvature will be largeenough so that it can be treated as a plane.

The plane of the fence will be taken as normal to the linejoining the transmitter and the receiver, or the radar and aclutter patch. Take polar coordinates (r, 0) with origin on thepoint in the vertical plane between the two sites that also lieson the edge of the fence (Fig. 1). Then the ratio of intensityI at (r, 0) to intensity Io incident on the edge of the fence is

I 1=-I G(x7)+\W- G(yV7t_/22 (1 )Io 7T

Manuscript received February 5, 1980; revised October 1, 1980. Thiswork was supported by the Department of the Army.

The author is recently retired from Lincoln Laboratory, Massachu-setts Institute of Technology, Lexington, MA 02173. (617) 862-3754.

x =- r/Xcoss(6- a), y= -V 7 Xcoss (±+ a)

and

G(x\/2) = e- ifx2I/2{I - C(x)] + i[ -S(x)] }.2 2f

Here, C(x) and S(x) are the Fresnel integrals

f cos (rt2/2) dt and f sin (7rt2/2)dt.

In (1), the + sign pertains to polarization such that the electricvector is parallel to the edge of the fence, and the minus signapplies when the magnetic vector is parallel to the edge.

Useful insights into G(xV/7r/2) are revealed by the ingeniousgraph called Cornu's spiral [1], [4], which takes C(x) as ab-scissa and S(x) as ordinate. With the spiral, one can demonstrategraphically [4] that G is a decreasing function ofx when x >0, and hence, that I is a decreasing function of e in Fig. 1 whene>0.

For anti-RFI applications of a fence, a wave incident on itfrom clutter or a ground-based transmitter is propagatingnearly horizontally, so that a _ 7r/2; moreover, if obscurationof the receiver by the fence has to be limited to small elevationangles, e must be small (say < 0.1 rad), and we can replace-cos ( -a)/2 by e/2 and cos (6 + a)/2 by -1, which meansreplacing the exact values of x and y by x = eVl2-X and y =N/8r/X. Thus y > x, and G(yV7r) can be neglected1 in com-parison with G(xNv'/2). The relative unimportance of G(yV/i/)tells us that when a is near 1r/2 and e is small, the polarizationof the incident wave has little influence on the intensity of thediffracted wave. Dropping the second G in (1) leaves

I 2 {[ 2C(x)] 2 + [ 2-S(X)] 2}O~~~~~~~~~~-1 Specifically, if e < 0.1 rad, then the error made by dropping

G(yN/ir/2) is less than 10 percent as long as r > X/8.

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McCUE: ANTI-RFI FENCES

118 3 15500]

21

40 T t ol/ I'

30 l i 0 4

20 A

o0 10

0.1 10 loo

i/2AFig. 2. Intensity of diffraction at a knife edge, as a function of angle,

distance, and wavelength.

which is the same as the result obtained on the basis of theFresnel-Kirchoff (i.e., scalar) formulation [4] of the problem.Fig. 2 gives (Io/l) as a function of x. Using I/Io as a parameter,we derive the diffraction-loss plot in Fig. 3, relating e and r/X.The linear part of Fig. 2 is associated with the fact [4] thatwhen x > 0, G(xV ) i/x ,the error in this approxima-tion being on the order of x-3. This means that, for e\/27Xlarger than about unity

I X - 2^ ~~~~~~~~~~~~(3)Io 41T2re2

The behavior of the curve in Fig. 2 shows that this approxima-tion is excellent when the diffraction loss is 15 dB or more.2

As a possible design to fence a radar operating at X = 0.10 m,suppose that we want I/NO = -20 dB and e = 0.087 rad (5.0deg). The antenna aperture might be about 3 m in diameter,with its center 3.5 m off the ground. Fig. 3 shows that, forI/o = -20 dB and c = 0.087, one needs r/X = 320, so thefence needs to be 32 m from the antenna. For the fence top tobe at an elevation of 0.087 rad when seen from the center ofthe aperture, the fence top needs to be 6.3 m above theground.

Taking the center of the aperture as a reference level isusually justified; if a circular aperture is divided into hori-zontal slabs, the middle slabs are the big ones, and in any casethe tapered illumination of the aperture limits the importanceof the top and bottom.

When the antenna beam must be movable in azimuth, as istrue of a radar, the fence usually occupies something close tothe arc of a circle. Relation (3) shows that I/Io varies only

2 Because in our geometry y > x, the approximation can be appliedto G(y\7/2) whenever it is valid for G(xlir/ 2); thus one can see that,for e < 0.1 under these restrictions, dropping G(y\/7jl) from (1) pro-duces an error of less than 1 percent.

r/X

Fig. 3. The diffraction angle at a knife edge, as a function of distancein wavelengths, with diffraction loss I/10 as a parameter.

inversely as the radius of the arc; if e is fixed, decreasing I/IOby 5 dB means tripling the radius of the arc, and also triplingthe rise above the antenna, thereby greatly enlarging the areaof the structure. Better ways out will be suggested below, aftera look at some of the implications of the simple design.

III. SOME CAVEATSBecause of the effect called "height gain" [5], Io at the top

of the fence may be different from the field intensity at thecenter of the antenna before the fence is erected. The basictransmission loss, i.e., the transmission loss between isotropicantennas when there is no loss caused by polarization [6] , canbe calculated [7] using the theoretical development by Fock[8]. Suppose that we have two 10-cm radars 25 km apart on asmooth earth, with antennas 3.5 m off the ground. Take ac-count of refraction by assuming a "4/3 earth" [9]. At awavelength of 0.10 m, the basic transmission loss is 168 dB[7]. The diffracting edge of the "20-dB" fence just discussedis 6.3 m off the ground, and the basic transmission loss be-tween two of them, placed so as to protect the antennas, isonly 157 dB [7]; at each, there is a height gain of 5.5 dBcompared with the center of the antenna. Therefore, thoughthe diffraction loss at each fence is 20 dB, the pair of themincreases the RFI protection by only 29 dB.

Anticlutter fences are basically like anti-RFI fences, butthey commonly need to protect against a virtually continuousdistribution of sources, perhaps a range of hills. The roleplayed by the height of an anticlutter fence at the AMRADradar has been discussed by Ruze et al. [3] for a situationwhere the wavelength (0.23 m) was long enough, and the near-by terrain smooth enough, so that reflection from the groundwas not negligible. In such a case, the field strength may notincrease monotonically with height.

In high-frequency propagation over long paths, troposphericscatter [10] may contribute more to the field strength at afence top than diffraction over the earth's bulge does. Exceptwhen reflection from the foreground produces interference

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IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. EMC-23, NO. 1, FEBRUARY 1981

with the directly propagated wave, tropospheric-scatter propa-gation does not exhibit height gain; the protection offered bya fence is equal to its diffraction loss. The signal arrives with aspread of elevation angles. Indeed, one expects that, in prin-ciple, some of the scattering paths must bring radiation thatarrives at the radar with elevation angle higher than that ofthe fence top, and against this the fence offers little or noprotection. However, the loss over the troposcatter path risesvery rapidly with angle above the horizon. For example,Chisholm [11] found that on a 3.67-GHz transmission over a300-km path, the signal arriving at a 2-degree elevation anglewas more than 20 dB weaker than the signal arriving along thehorizontal, i.e., with az = 7r/2. The diminution of diffractionloss with elevation angle, for any plausible anti-RFI fence, ismuch less rapid than that. Against transmissions propagatedby troposcatter, therefore, the protection offered by the fenceis essentially equal to the diffraction loss as given by (1) witha = ir/2, as above. In determining whether the more importantmode of propagation is the ground wave or troposcatter, [61is a useful guide.

Another consideration not to be overlooked is the gain3 ofthe antenna for points on the fencetop. For a simple example,suppose that an antenna is pointed with an elevation of 10 degand an azimuth the same as that of an interfering station justover the horizon. Suppose that the sidelobe toward the hori-zon in that azimuth has a gain 5 dB above isotropic. Nowinterpose a fence such that N/1o = -25 dB and e = 6 deg. Thefence top is only 4 deg off boresight, and the gain there maybe 10 dB above isotropic; the rise in gain will partly offset theloss introduced by the diffraction.

IV. FANCY FENCES

To improve the protection offered by a fence withoutjust making it bigger, a number of authors have suggested moreelaborate designs than a plain edge. Ruze et al. [3] studied afence with a crenellated top, having two sets of horizontaledges whose diffracted waves could be made to interferedestructively. To avoid some drawbacks arising from thevertical edges of the crenellations, these authors suggested, andBecker and Millett [12] analyzed and tested, a fence withcontinuous horizontal slots near the top. The results lookedencouraging. Another suggestion [1] has been to use a meshfence that gets gradually coarser and leakier at the top. Theauthor has studied, by means of computer simulation, a fencewith a dielectric strip along the top. The choice of height,thickness, and dielectric constant of the strip controls theamplitude and phase of an auxilliary diffracted wave whichinterferes with that from the metal edge. The analysis wasdone by superposing the solution for diffraction by a semi-infinite conducting plane, as above, and the solution fordiffraction by the strip [4], taking into account the phaseshift caused by the dielectric. Fig. 4 shows the result of

3 The concept "gain" is applicable at points not in the far field ifgain is defined as the ratio of the power density, at a point in the fieldof the antenna, to the power density that would exist there if the actualantenna were replaced by an isotropic one radiating the same powerand located at the phase center of the actual antenna for the field point.In the near field, gain is a function of all three space coordinates.

HEIGHT BELOW RIM (ft)10 20 30

118-3 155021

- O --<LAIN FENCE0

\ 9:l.Ocm000 0 0 0 -O--O.1.0

1BlO Ocm

ARRAYTOP BOTTOM

6

HEIGHT BELOW RIM (m)

Fig. 4. The calculated effect of a dielectric rim on diffraction from afence described in the text. The points indicate the degradation ofperformance when the wavelength changes by 10 percent; the im-provement at the center of the antenna is even then about 8 dB.

adding a dielectric strip to the plain fence mentioned earlier,32 m from the antenna center, designed for X = 0.10 m, e =5 deg, and I/Io = -20 dB. To keep reflection from beingimportant, the dielectric constant was taken as 1.5, obtainablewith foam plastic and a thin weatherproof skin. The strip was19.8 cm high and 16.6 cm thick; these dimensions were chosenin a few minutes of trials at a computer console, the criterionbeing to place a maximally deep null at the center of theantenna aperture, 3.5 m off the ground. For an actual optimi-zation, one would need to know the aperture illuminationfunction of the antenna, and also the frequency interval to becovered.

The calculations leading to Fig. 4 must at present be takenas first-order approximations, for two reasons. They are basedon scalar theory, and on superposition of the individualeffects of the metal half-plane and the dielectric slab. Super-position is not valid rigorously; the field in the dielectric isperturbed by the fence, and if the perturbation depends on thepolarization of the wave, scalar theory may fail. It would beinteresting to know, either from measurements on a model orby a rigorous solution of the boundary-value problem, whethera rimmed fence is less sensitive to the polarization of theradiation than a crenellated [3] or slotted [12] fence.

V. TANDEM FENCESThe designs described in Section IV respond to the fact that

decreasing I/Io to less than -20 dB by means of a single knifeedge requires either a very large fence or a renunciation of lowangles of elevation. An alternative to a more elaborate fence istwo simple ones in succession-"in series," one might say. Itwill be shown that this alternative offers impressive advantages.

Diffraction by two parallel half-planes has been treatedrigorously by Heins [13] , and more accessibly by Millingtonet al. [14], and by Deygout [15]. The work of Millingtonet al. includes an evaluation of the error that is made whendiffraction by two knife edges is estimated by iterated applica-tion of the result for a single one. We will use that approachand then apply their correction. If we are interested in diffrac-

a n2

30

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l0

-j301

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McCUE: ANTI-RFI FENCES

TABLE ISCREENING BY SINGLE AND TANDEM FENCES

GEOMETRIES OF STATIONS AND FENCES ARE GIVEN INTHE TEXT

One Two One TwoFencing None Single Single Tandem Tandem

Basic Transmission Loss BeforeFence Diffraction (dB) 158 146* 134 150 142

Basic Transmission Loss Between 158 171 184 185 212

*This is the basic transmission loss between one antenna and the top of

the fence around the other radar; 142 dB is the loss between the tops

of the outer members of two tandem fences, one at each radar.

tion losses on the order of 20 dB per fence, the approximation(3) is valid; iterating it gives

I k~~2-= ~~~~~~~~~~~~~~(4)Io 16ir4r1r2e12e22

where r, is the distance to the first fence and r2 is the distancebetween fences, while e1 and e2 are the angles of diffraction atthe first and second fences, respectively.

The distance from the antenna to the outer fence is r, + r2 =R, and if we continue to regard the antenna as concentratedat some height h from the ground, the antenna is blind belowsome elevation angle e = e1 + e2. For a sample design, we willconstrain R and e to some assigned values4 and use the meth-od of Lagrange multipliers [16] to maximize log (Io/l); theresult is that I/Io is a minimum (given the stated constraints)when e1 = e2 and r, = r2. Millington et al. find that whenr, = r2, the diffraction loss is 3 dB more than the value givenby iteration, which is (4) if the individual diffractions havesufficient loss to make (3) valid for each of them.

Consider the dimensions of a single fence with e = 5.0 degand I/Io = -25 dB when X = 0.10 m. From (3), one finds thatr = 104m;then the fence top must be 104X .087 = 9.0mabove the antenna, say 12.5 m above the ground. If this fenceis to cover a considerable sector in azimuth, it is a formidablestructure, though smaller than some that have been built [1],[3]. For a tandem fence with the same obscuration and thesame (outer) radius, with r, = r2 and e1 = e2, the inner andouter portions would have heights 5.8 and 8.0 m, respectively.The approximation (4) gives I/Io = -32 dB; applying the 3-dBcorrection for interaction of the two fences raises the diffrac-tion loss to 35 dB, a 10-dB improvement over the single fence.

The increased diffraction loss is by no means the onlyadvantage of the tandem design. Because it is lower, its Io maybe less than for the standard design. Table I gives the trans-mission loss [7] between two stations 18.5 km (10.0 mi) aparton a smooth 4/3 earth, when the heights of the stations arethose of the radar (3.5 m) or those of the fence tops (12.5 mfor a single fence, 8.0 m for the higher member ofthe tandem).

4 The optimization criterion used here is not, of course, the only de-fensible one. One alternative would be to constrain the obscuration andthe cost. The example used sufflciently illustrates the principles thatgive tandem fences advantages over single ones.

Except in the case of the two single fences, which extendabove each other's radio horizon, use of [6] in place of [7]yields the same transmission losses. The table shows that,when height gain is taken into account, the transmission lossbetween radars, when each is equipped with the tandem fence,is 28 dB greater than when each has the single fence. Indeed,in this particular example, a tandem fence at one radar is asgood as a single fences at both. Yet the tandem should be lesscostly than the single: it is not as high a structure, it can have alarger leak-through at each structure, and it has a smallerarea.5 The reduction in height reduces the wind loadings.These advantages give the tandem fence a claim to attention.

VI. TANDEM FENCE WITH EDGE TREATMENT

A further advantage of the tandem fence is that it has twoedges, which can be modified separately to improve the per-formance. By using two dielectric rims, for example, the rimon the outer fence could be treated so that the sharp dip inFig. 4 cut down the illumination at the top of the inner fence,for some chosen frequency. The inner fence would be rimmed(or slotted or crenellated) in such a way as to place a minimumat the center of the antenna, not necessarily for the same fre-quency. Thus one has the capability of achieving selectivesuppression of RFI from sources at two different frequencies,or of producing a broadbanding effect by the equivalent ofstagger tuning.

The designer of an anticlutter fence is concerned onlywith the frequency band occupied by the radar. However, ifthe clutter arises from a mountain range, he can have theproblem of waves arriving with assorted angles of elevation,and, with a plain fence, the diffraction loss is less for thehigher elevations. With a tandem fence, he can treat the edgeof the outer member so that waves arriving at the highestangle of elevation produce an interference null at the top ofthe inner fence, thus tending to level out the illuminationthere as a function of clutter elevation angle.

The degree to which diffraction at a second treated edge isdependent on edge treatment at the first fence is at present anopen question.

S When (3) is valid, a tandem fence has less area than a single onewith the same ground extent and same obscuration, unless the antennaheight h is so large that h > 3eR/4.

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IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. EMC-23, NO. 1, FEBRUARY 1981

REFERENCES

[1] J. E. Becker and J.-C. Sureau, "Control of radar site environmentby use of fences," IEEE Trans. Antennas Propagat., vol. AP-14,pp. 768-773, Nov. 1966.

[2] F. K. Preikschat, "Screening fences for ground reflection re-duction,'" Microwave J., pp. 46-50, Aug. 1964.

[3] J. Ruze, F. I. Sheftman, and D. A. Cahlander, "Radar ground-clutter shields," Proc. IEEE, vol. 54, pp. 1171-1183, Sept. 1966.

[4] M. Born and E. Wolf, Principles of Optics. New York:Pergamon, 1975.

[5] H. Bremmer, Terrestrial Radio Waves. New York: Elsevier,1949.

[6] P. L. Rice, A. G. Longley, K. A. Norton, and A. P. Barsis,"Transmission loss predictions for tropospheric circuits," NationalBureau of Standards Tech. Note No. 101. Washington, DC: U.S.Government Printing Office, 1967. AD 687 820 and AD 687 821.

[7] I. H. Gilbert and A. J. Curtis, "Compatibility between linear-FMradars: Two programs," 1976 Microwave Engineers Handbook,Microwave J., Dedham, MA, 1976.

[8] V. A. Fock, Electromagnetic Diffraction and PropagationProblems. New York: Pergamon, 1965.

[9] B. R. Bean and E. J. Dutton, Radio Meteorology. Washington,DC: U.S. Government Printing Office, 1966.

[10] F. Du Castel, Tropospheric Radiowave Propagation Beyond theHorizon. New York: Pergamon, 1966.

[11] J. H. Chisholm, "Recherches experimentales sur les possibilitesd'utilisation de la propagation troposphdrique pour les commun-ications bien au dela de l'horizon," Onde Elec., vol. 37, pp.427-434, May 1957.

[12] J. E. Becker and R. E. Millett, "A double-slot radar fence forincreased clutter suppression," IEEE Trans. Antennas Propagat.,vol. AP-16, 103-108, Jan. 1968.

[13] A. E. Heins, "The radiation and transmission properties of a pair ofsemi-infinite parallel plates," Quart. Appl. Math., vol. 6, pp.157-166 and 215-220, July and October 1948.

[14] G. Millington, R. Hewitt, and F. S. Immirzi, "Double knife-edgediffraction in field-strength predictions," Monograph No. 507E,Proc. IEE, vol. 109C, pp. 419-429, Sept. 1962.

[15] J. Deygout, "Multiple knife-edge diffraction of microwaves,"IEEE Trans. Antennas Propagat., vol. AP-14, pp. 480-489, July1966.

[16] H. S. Wilf, Mathematics for the Physical Sciences. New York:Dover, 1978.

Computational Experience with a Dual BacktrackAlgorithm for Identifying Frequencies Likely to

Create Intermodulation ProblemsSUSUMU MORITO, HARVEY M. SALKIN, AND KAMLESH MATHUR

Abstract-This paper describes the results of a computational studyusing a particular enumeration procedure, called a backtrackalgorithm, to find the lowest order of radio-frequency inter-modulation. The average lowest order and its standard deviation, theaverage computer time and its standard deviation, along with otherrelevent statistics are obtained for a series of randomly generatedproblems with sets of five to 75 threat or source frequencies. Otherparameters, such as the guard band, the maximum number ofconcurrent threats, and the size of the frequency band on the lowestorder of intermodulation are varied during the computations.Statistics for these computer runs, along with those relating toterminating the algorithm when the lowest "acceptable" order isreached, are presented in some detail. Brief conclusions follow alisting of the results.Key Words-Intermodulation, problem frequencies, identification,

computational experience.

INTRODUCTIONTWO OR MORE electromagnetic signals, transmitted from

and/or received on a small platform (e.g., a ship, satellite,or airplane), tend to produce a type of cosite interference

Manuscript received October 2, 1980. This work is supported inpart by the Office of Naval Research under Contract N00014-78C-0028.

The authors are with the Department of Operations Research, CaseWestern Reserve University, Cleveland, OH 44106. (216) 368-2750.

known as intermodulation interference. It is dependent on,among other factors, the lowest order of intermodulation. Inmany applications (see, e.g., Chase, Li, Rockway, and Salisbury[1], Chase, Rockway, and Salisbury [3] , Lustgarten [5], [6],Morito and Salkin [7], and Morito, Salkin, and Spence [10])avoiding lower-order, and especially odd-order, intermodu-lation such as third, fifth, seventh order is crucial as thesetypes of intermodulation tend to cause serious threats tocommunication. Therefore, a capability for finding the lowestorder of intermodulation is useful to evaluate a given fre-quency assignment. This is particularly true in naval fleet radiocommunication environments where frequency assignmentsmust produce acceptable levels of intermodulation interference,even when most communication nets are operating.

Mathematically, the lowest order Q* of a given intermodu-lation signal FI, produced by NF distinct signals, F1, F2, ,FNF, can be obtained by solving the optimization problem(RFI) below, which is an integer program with a single con-straint and unrestricted variables.

NF(RFI) Minimize Q = xi I

i=l(1 a)

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