IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 31, NO. I , FEBRUARY 1989 8 5

Short Papers

Antenna Coupling Model for Radar Electromagnetic Compatibility Analysis

TERRY L. FOREMAN

Abstract-This paper presents a statistically based antenna coupling model. The model predicts the Probability of interference based on the antenna coupling statistics. In addition, a modification to the model is presented that takes into account sidelobe blanking on the victim antenna. This model has advantages over an existing model because in addition to considering sidelobe blanking it makes no assumption about the antenna pattern statistics and readily incorporates the main beam region.

I. INTRODUCTION

This paper presents an antenna coupling model to be used to predict the amount of power coupled between two radars. The model is a statistical model that predicts coupling levels and their probabili- ties. The probabilities can be used to predict the probability of interference between radars when used in conjunction with other models. This model can be adapted to use either measured antenna pattern data or theoretical antenna patterns. This model is similar to "A medium gain model for rotating radar antennas" [l]. It has two significant advantages over this existing model. First, it makes no assumption as to the probability distribution of the random variables. Second, it automatically includes the main beam region of the antenna pattern. This is not possible without causing significant errors with the other technique [ 11.

This model derives a random variable e, which represents the coupling loss in decibels (dB) between the source and victim antennas referenced to main beam to main beam coupling. If the interference to noise ratio (INR) in decibels at the victim with respect to main beam to main beam coupling has been calculated, the total INR may be calculated as follows:

I R R ~ = INR - e (1)

where

IRRT is the total interference to noise ratio; INR is the interference to noise ratio excluding antenna coupling

loss.

The next step is to determine the threshold below which interfer- ence can be ignored. This can be taken as the 50-percent probability detection threshold.

The probability of interference may then be defined as the probability that the total INR is greater than the detection threshold [2]. This is stated mathematically as

P = P ( IRRTZT)=P(esINR- T) (2)

Manuscript received April 4 , 1988; revised September 13, 1988. The author is with EG&G Washington Analytical Services Center, Inc.,

P.O. Box 552, Dahlgreen, VA 22448-0552. Tele. (703) 663-9314. IEEE Log Number 8825188.

is the probability of interference; is the probability of event A ; is the 50-percent probability of detection threshold (in decibels).

n. DERIVATION OF MODEL

In this model, it is assumed that the antennas for the victim and the source radar have pointing angles that are statistically independent. This assumption is justified when the antennas are scanning antennas (search radar) and scan at different rates. In the case of scanning antennas with the same scan rate, this assumption can be invoked if it is assumed that the scan rates will be slightly different (which would be expected in practice) and performance predictions can be averaged over times much greater than the scan period.

The analysis begins by defining two random variables as and 8,. (See Fig. 1 .) These random variables represent the scan angle of the source and victim radar antennas, respectively. These angles are referenced to the direction of the other radar such that when qua's zero, then the source antenna is pointing at the victim and when 8, equals zero, the victim antenna is pointing at the source. From the previous paragraph, it is assumed then that 8, and as are statistically independent.

The gain of an antenna can be defined as a function of the scan angle. This function can be measured or it may be calculated. In either case, two new random variables are defined. These are the gain of the source antenna (GS) in the direction of the victim antenna and the gain of the victim antenna (4) in the direction of the source antenna. Mathematically they are as follows:

(3)

(4)

where G,(@ and G,(8) are the gain functions of the antennas either measured or calculated.

We define the gain functions G,(B) and G,(8) as normalized to the main beam gain and inverted. By this we mean that the functions Gs(8) and G,(8) have a minimum value of 0 dB for the main beam angles and positive values for values of 8 that correspond to the antenna sidelobes. Note that 0 5 G,(Q and 0 5 G,(8). Also, note that

G,(O) = G,(O) = 0

which is the case when both antennas are looking at each other. This is the maximum coupling case. The gain functions are defined in this way to account for the antenna main beam gains separately, and so that the antenna coupling loss e can be defined in terms of 8, and 8,. Therefore, we have

e = €s + 8".

Since 8, and 8, are statistically independent, 8, and 8, are also statistically independent.

0018-9375/89/0200-0085$01 .00 0 1989 IEEE

86 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 3 1, NO. I , FEBRUARY 1989

VKTIM ANTENNA

D I R E C " OF SOURCE ANTENNA r-- / REFERENCE LINE MRMlON OF VKTlM ANTENNA

2 /

Lf SOURCE ANTENNA

Fig. 1 . Source-victim geometry

At this point it is necessary to determine the probability density functions for Gs and e,. Two approaches have been used by the author. First, they can be compiled from the antenna pattern, if available. Second, if the antenna pattern is not available, a theoretical pattern can be calculated with the knowledge of the antenna beamwidth and aperture illumination and reasonable assumptions about the sidelobes.

To aid in computation it is convenient to represent the random variables as lattice random variables [3]. This causes their density functions to be expressed as follows:

n

f ~ ~ ( G ) = z Ps ( i )6 (G- i -0 .5 ) ( 5 ) i = O

where

Ps(i)

n

is the probability that the source antenna gain G is i I G < i + 1 dB and 6(x) is the Dirac delta function. is the maximum antenna coupling loss for the source antenna (dB).

Similarly, we would write the density function for G, as

where

P,,(i)

m

is the probability that the victim gain G is i I G < i + 1 a; is the maximum antenna coupling loss for the victim antenna (in decibels).

From the expression of the density function it appears that Gs and 6, can take only discrete values. However, in reality, GS and 6, can take on any value between 0 and + W. The maximum error caused by this is 0.5 dB for the above formulation. For the intended use of this model it was felt that this level of error was acceptable. The accuracy can be increased by determining the probabilities Ps(i) and P , ( j ) to any desirable interval provided that sufficient data points are available.

It can be shown [3] that the density function for e( f c ( C ) ) can be determined by convolving the density functions for

This gives and e,.

Combining (5)-(7) we have

f c ( C ) = [Ip (i Ps(i)6(G-i-0.5) i = O

--m

P v ( j ) 6 ( C - G - j - 0 . 5 )

By straightforward factoring and the interchanging of summations and integral we have

I-m 6(G-i-0.5)6(C-G-j-O.S) dG. (9) - m

From the theory of distributions [4] it can be shown that the convolution of two delta functions is

6(7-t1) . 6 ( t - 7 - t f 2 ) d7=6(t - t1- t2) . (10) -cm

Therefore, we have

n m

fC(c)=e P S ( i ) P u ( j ) 6 ( C - i - j - 1). (1 1) i = o j = o

Since e is the sum of two lattice random variables, it too is a lattice random variable and its density function can be expressed as follows:

n + m

f c ( C ) = Pc(k)6(C-k- 1). (12) k = O

Therefore. we have

n

Pc( k ) = PS( i)P" ( k - i). (13) i = O

The cumulative probability distribution function for e is as follows:

IlC- 111 F,.(C)= Pc(k ) , C r l

k = O

0, otherwise

where

11 11 =greatest integer function.

Recall the probability that e I C is Fc(C). From (2) then, the probability of interference is

P = Fc (INR - T) (15)

where

P is the probability of interference.

III. SIDELOBE BLANKER MODEL

Next we would like to extend this model to include the case where the victim has a sidelobe blanker (SLB). The SLB minimizes detections caused by energy entering the sidelobes of the victim

IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 31, NO. 1, FEBRUARY 1989 8 7

ANTENNA PAITERN

-10

-20 SIDE LO&€ BLANKER

-30

-40

-Y)

do

-70

Fig. 2 . Antenna pattern with sidelobe blanker.

antenna. This is accomplished by having a second antenna with a nearly omnidirectional pattern as shown in Fig. 2. The signals between the normal and SLB channel are compared and, if the signal in the SLB channel is greater, then the normal detection channel is blanked.

As a first approximation we will assume that the SLB works perfectly and no detections are made in the sidelobes. We will begin by changing the definition of C to

e= Gs. (16)

This means that C represents the coupling loss between the source and victim given that the victim antenna is pointing at the source. From this, it follows:

f C ( C ) =fcs(C) (17)

IlCll F,(C)= P J K ) , CrO

K=O

=o, c<o (18)

(19) PsLB = F, (INR - T ) where

P ~ L B is the probability of interference given that the victim antenna is pointing at the source.

To determine the overall probability of interference then, the probability that the victim antenna is pointed at the source must be determined. This is given as

where Ai3 is approximately twice the 3 dB beamwidth and Pp is the probability that the victim antenna is pointed at the source. The quantity Ai3 represents the angular extent that the main beam gain exceeds the SLB antenna gain.

The value of Ai3 may be increased to allow for the angular extent of sidelobes that stick through the SLB antenna pattern. This approxima- tion will cause the probability of interference to be slightly overestimated. The overestimation occurs because the adjustment includes the implicit assumption that the gain in the stick-through region is the same as in the main beam. Since the random variable 8, is uniformly distributed, (20) may be written as

-03dB A p--+-

U 200U

where

A = 3 dB beamwidth (radians) = percentage on antenna coverage with stick-through.

The total probability of interference, therefore, is

P=PsLB * Pp. (22)

It should be noted that a victim with an SLB will have a maximum probability of interference of Pp. This corresponds to interference detected in the main beam region and every uncovered sidelobe.

IV. SUMMARY AND CONCLUSIONS As was stated in Section 111, the probabilities Ps and P,, may be

determined from the antenna patterns. In using this method, no assumption is made as to the statistics of the antenna pattern. Ps and P,, are calculated by sampling the antenna patterns at uniform angles (since the scan angles are assumed to be uniformly distributed). The ultimate limit in accuracy of this technique is determined by the accuracy of the measured antenna patterns, the number of points sampled, and the quantization of the antenna patterns (1 dB assumed here). Thus, in principle any level of accuracy can be obtained with this model.

This model may be applied to antennas that also scan in elevation by using the three-dimensional antenna pattern and sampling eleva- tion and azimuth. If the antenna is not scanned uniformly (as is often the case with electronically scanned antennas) then the antenna pattern should be sampled according to the distribution of the scan angles. This will give the correct distribution for the gains. Thus, this model can be applied to a variety of antenna types.

REFERENCES

S. Guccione and H. Ricker III, A Median Model for Rotating Radar Antennas, ECAC-TN-74-07, DoD ECAC, Annapolis, MD, Feb. 1974. P. Newhouse, Radar EMC Analysis Handbook, ECAC-HDBK-M- 066, DoD ECAC, Annapolis, MD, Sept. 1984. A. Papoulis, Probability, Random Variables, and Stochastic Proc- esses. New York; McGraw-Hill, 1965. -, The Fourier Integral and Its Application. New York: McGraw-Hill. 1962.

Radio Wave Propagation Loss in the VHF to Microwave Region Due to Vehicles in Tunnels

YOSHIO YAMAGUCHI, MEMBER, IEEE, TAKE0 ABE, MEMBER, IEEE, AND TOSHIO SEKIGUCHI, FELLOW, IEEE

Abstract-Electromagnetic field intensities in tunnels are measured in order to determine the effect that vehicles may have on radio wave propagation in the frequency range from VHF to microwaves. A

Manuscript received May 24, 1988; revised August 5, 1988. This work was supported in part by the Grant-in-Aids of the CAS10 Science Promotion Foundation and the Inamori Foundation.

Y. Yamaguchi is with the Communications Laboratory, Department of Electrical Engineering and Computer Science, The University of Illinois at Chicago, Chicago, IL 60680, on leave from the Department of Information Engineering, Faculty of Engineering, Niigata University, Ikarashi 2-8050, Niigata, 950-21 Japan.

T. Abe is with the Department of Information Engineering, Faculty of Engineering, Niigata University, Ikarashi 2-8050, Niigata, 950-2 1 Japan.

T. Selaguchi is with Tokyo National Technical College, 1220-2, Hachioji, Tokyo, 193, Japan.

IEEE Log Number 8825185.

OO18-9375/89/0200-0087$01 .OO 0 1989 IEEE