Simple Gain Probability Functions For Large Reflector Antennas Of JPL/NASAu
Vahraz Jamnejad Jet Propulsion Laboratory Caliiomia Institute of Technology Pasadena, CA 91 109 818-354-2674 [email protected]
Abstract - Simple models for the patterns as well as their Station of the JPUNASA Deep Space Network [ 1,2]. In this cumulative gain probability and probability density functions scenario a large number of microwave transmitters will be of the JPUNASA Deep Space Network (DSN) antennas are installed in large urban centers in the future to provide high- developed. These a~ needed for the study and evaluation of density fixed services (HDFS). The frequency band proposed interference from unwanted sources such as the emerging for use by these transmitters overlaps the Ka-band (27-40 terrestrial system, High Density Fixed Service (HDFS), with GHz) allocated for NASA’s Deep Space Network @SN) the Ka-band receiving antenna systems in Goldstone Station receivers. Interference signals from these transmitters can of the JPLNASA Deep Space Network. propgate over the horizon and interfere with the DSN through various ”s, such as ducting, rain Several different models for the high gain ground antennas scattering, and diffraction. Both the present 34-meter of the JPUNASA Deep Space Network @SN) are antennas which are equipped with Ka-band feeds, as well the considered. The model definitions are then used to evaluate 70-meter antennas which might be outfitted to provide cumulative probability functions and probability density receive capabilities in the future will be affected by the functions for the gain values at a receiving antenna randomly HDFS interference. In support of the study of this located anywhere in the far field of the model antenna. Both interference, simple models for the patkms of the DSN two-dimensional models which simulate the pattern in a antennas as well as their cumulative gain probability plane cut through the boresight of the antenna and the range functions and probability density functions will be required. of the angle from boresight to *180 degrees and three- This study will also be helpful in many similar cases such as dimensional ones in which the pattem is given anywhere in those outlined in [3,4]. the half-space above the ground, with the range of angle from zero at boresight to 90 degrees are considered. In both Here, we first consider several different models for the high cases, regular as well as smooth “interpolated” models have gain ground antennas of the JPL/NASA Deep Space been developed. Network (DSN). The model deffitions are then used to evaluate cumulative probability functions and probability The gain value as seen by a receiver at a random location density functions for the gain values at a receiving antenna and exposed to the radiation from a transmitting antenna randomly located anywhere in the far field of the model with a given gain profile, can be thought of as a uniformly antenna. The models are all assumed to be circularly distributed random variable. Formulas are derived and symmetric around the boresight axis. We distinguish two various plots are presented for the gain models and their cases for the gain models. A two-dimensional case, in which associated cumulative probability functions and probability the model simulate the pattem in a plane cut through the density functions. boresight of the antenna and the range of the angle from boresight is assumed to be from zero to 180 degrees. A TABLE OF CONTENTS three-dimensional case, in which the pattem is given anywhere in the half-space above the and the range INTRODUCTION ground, 1. of the angle boresight is assumed to be fiom zero to CLOSED-FORMANTENNA from 90 2. PATTERN MODELS degrees. In both cases, regular as well smooth 3. GAIN FUNCTIONS as PROBABILITY “interpolated” models have developed. 4. SUMMARY AND CONCLUSIONS been The gain value seen by a receiver at a random location, 1. INTRODUCTION as exposed to the radiation fiom a transmitting antenna with a There is a growing concern that emerging terrestrial systems, given gain profile, can be thought of as a uniformly distributed random variable. Fonnulas are derived and such as High Density Fixed Service (HDFS), may interfere with the Ka-band receiving antenna systems in Goldstone various plots are presented for the gain models and their
’ 0-7803-7651-X/03/$17.000 2003 IEEE 2 EEEAC paper #1339, Updated Jan. 09,2003 associated cumulative probability functions and probability angle fiom boresight is assumed to be density functions. The probability density functions for the 0 < 6 < 180". non-interpolated models have impulse responses both at the bewing and the end for both the 2-D well 3-D points as as ii) A three-dimensional case, in which the pattern cases. In the interpolated cases, however, there are no given is anywhere in the half-space above the impulse functions, but the function goes to infimity at both ground, and the range of the angle from boresight the beginning and end points for the 2-D cases, and only at is assumed to 0 < B < 90". the beginning points for 3-0 cases. be Now we proceed with the definition of the models. 2. CLOSED-FORM ANTENNA PATTERN MODELS 1) The first model is a simple standard model commonly and is represented as: Actual and realistic patterns involve many factors, too used, complicated and diverse, to be exactly accounted for in a simple theoretical computation. For example, position of G = Go O<6<8, the (3) nulls and peaks in the sidelobe regions vary as a function of G = c,- c2iog(e) e, < e < e, antenna gravitational loading, winds, etc., and are best represented by an envelope. Over the years many pattem G=C3 e, < e < e, models have been suggested for large reflector antennas, see e, = 90", for 3-D case e.g., [5-71. e, = 180°, for 2-D case A simple but effective method of characterizing an actual with, antenna is to use a model which is based on pattern many Go = a given peak boresight gain theoretical and experimental results and provide an upper- andor lower-bound or envelope for the antenna which can 32.0, for D>lOOh (gain>48&) be easily applied to many situations. Ideally, following the definition of directivity of an antenna, the gain model G = 29.3, for D4Wh(gain<48dB) , given in dB, ignoring the difference between gain and C2 = 25, the slope parameter directivity, should obey the formula -10.0, far D>lOOh (gain>48dB) 2ZZ -G(e,+) c3= -12.7, for D 2) The second model is a "smoothed" version of the first model and given 0 as: However, in the models that are usually an upper G = Go-a,(8/8,)2 o Go = a given peak boresight gain C2 = 25, the slope parameter -10.0, for D>100h (gain>48dB) c, = -12.7, for D4OOA (gain<48dB) 3) The third model is more realistic in term of the main lobe and near sidelobes and is given by 1 a, = alog(e)C2 G=Go 0<6<6, = c, - c2log(6) 6, < 6<62 (5) c, = co+ c2log(-) 6, , = c3 62 < 6 < 6, & 6 - 10(c1-c3)/cZ with 20 - 2' 62 6, (8, 6,,)e (2n+17 Go = a given peak boresight gain = - - C2 = 25, the slope parameter 7 and, -10.0, for D>1003L (gain>48dB) a2 = C,- C, - C2log(6,). c3= -12.7, for D 3. GAIN PROBABnITY FUNCTIONS The gain value as seen by a receiver at a mdom location, exposed to the radiation hma transmitting antenna with a given gain profile, can be thought of as a uniformly 4) This fourth model is a "smoothed" version of the third distributed random variable (same is true if the receiver is model and is given fured but the transmitting antenna makes a random rotation. as The gain pattern is, in general, given by Assuming circular symmetry around the boresight axis, only the gain in one cut boresight axis is 0, 90",n pattern through the = = 1, for 3-D case needed and is given by 0, = 180", n = 2, for 2-D case G = G(B), 0" c e < 180°, for anycp,Oo < 4 < 360" Here we consider two distinct cases. P,(G)= cos('(G)) (10) I) In a two-dimensional case, the receiver is assumed to be in Following the procedure in the two-dimensional case, the plane through the boresight of the antenna. For example, if a probability density function is the transmitting antenna is horizontally directed, then the gain obtained as receiver is also in a horizontal plane around the antenna. In this case, the probability that the gain at the receiver is below a given gain level (the cumulative probability) is given by: 0, F de The factor (d180) is introduced because the angle arguments in cosine and sine functions assumed to be in degrees. 0 are Now we are in a position to apply these equations to the gain or with 0, = 180" models developed in the previous section. For the four cases '(GI that we developed in that section we obtain the following P,(G)= 1 -- results. Notice that in each case, reference is made to the 180 parameters C,, C2, C,, a1, a2, e,, and 6 from the respective The probability density function of the gain at the receiver is model definition: then given by 1) The simple original gain model: 2-0 case: 6 10 c, ae P,(G)= 1--=1-- ,C, < G < Go (12) For the P, function given above we obtain, { 180 180 -1 P(G)= ',G'(') P(G)= (1 - -)S(G '2 - G3) 180 II) In a three dimensional case, the receiver is assumed to be anywhe~in the half-space above the ground and for + '(G) +-S(G-Go) '1 simplicity, the transmit antenna boresight is considered in the 180 - log(e) - C, 180 zenith direction normal to the ground plane. However, The results thus obtained are approximately correct even for cases in which the transmit antenna is pointing in an arbitrary direction as long as the antenna is a high gain pencil beam antenna and is pointing at least a few beamwidths above the horbntal ground plane. In this scenario, the -I- probability that the gain at the receiver is below a given gain level (the cumulative probability) is given by: 180 - log(e) - C, -Cl -GO 360 Id/ sin(8)d' +- lo S(G-Go), G3 00 In which 8, = 90" (9) and results in 3-D I C3 I C,+% 90-(90-8,)(-)4G-G, I G2 G, 2-D case: G3+a2 3) The simple new model: The forms of the equations 180 for this case are identical to the ones for the original Go-q 180 Go-a, The research described in this paper was carried out at the Jet L Propulsion Laborato~~,California Institute of Technology, G C3 < < C3 +a, under a contract with the National Aeronautics and Space e Administration. The author would like to acknowIedge the kind encouragement and helpll suggestions of Miles Sue of cos (0)= cos(l0 c, ), (22) JPL. The efforts of Charles Ruggier, Daniel Bathker, Peter C3+a, REZICRENCES Go-a, [4] C. Ho, D. Bathker, M. Sue, and T. Pen& “ Adjacent Band Interference hmSan Diego Area Transmitters to Goldstone Deep Space Network Receivers Near 2300 Megahertz,” JPL TMO Progress Report 42- 142, October-December200 1. [5] “Mathematical Model of Average and Related Radiation Pattems for Line-of-Sight Point-to-Point Radio-Relay Figures 3 through 10 show the gain models and their System Antennas for Use in Certain Coordination Studies associated cumulative probability functions and probability and Intederence Assessment in the Frequency Range from 1 GHZ density functions. The probability density functions for the non-interpolated models have impulse responses both at the to about 70 GI-Iz,” Recommendation ITU-R F.1245-1, ITU- beginning and the end points for both the 2-D as well as 3-D R, 2000. cases. In the interpolated cases, however, there are no impulse functions, but the function goes to infiity at both [6] “Generalized Space Research Earth Station And Radio Astronomy Antenna Radiation Pattem For Use In Interference Calculations, Including Coordination Procedures,’’ Recommendation ITU-R SA.509-2,1978-1998. [7] “Methods For Predicting Radiation Patterns Of Large Antennas Used For Space Research And Radio Astronomy,” Recommendation ITU-R SA. 1345,1998. Vahraz Jamnejad is a principal engineer at the Jet Propulsion Laboratoy, California Institure of Technology. He received his M.S. ad Ph.D. in electrical engineeringkom the University of Illinois at Urbana- Champaign, specialumg in electromagnetics and antennas. At PL, he has been engaged in research and somare and hardware development in various areas of spacecrajy antenna technology and satellite communication systems. Among other things, he has been involved in the stu&, design, and development of ground and spacecrajy antennas for fiture generations of Land Mobile Satellite Systems at L bad, Personal Access Satellite Systems ai WKa bad, as well as feed arrays and reflectorsfor fiture planetay mirsions. Htr latest work on communication satellite systems involved the development of ground mobile antennasfor HKa band mobile terminal, for use with ACTS satellite system. In the past fav years, he has been active in research in parallel computational electromagnetics as well as in developing antennas for MARS sample return orbiter. More recently he has studied the applicabil?~ of large arrays of small qerture reflector antennas for the NASA Deep Space Network, and is presently active in the preliminay design of a prototvpe array for this application. Over the years, he has received many US patents and NASA certificates of recognition. I llllllll I Ill/ll11 111111111 I11111111 11111111 -20 ' """" '""1"' ' "llll'l I lllllll I "lru 10- 10'' 10'' I$ 10' 10' -1"- -1"- Figure 1. A comparison of 3-D gain models for a 85 dB gain Figure 3(b). The cumulative probability function for the antenna. original 3-D gain model of an 85-dB gain antenna. I l1llll1l I llllllll I llllilll I llllllll I11111111 I lllllll I11111111 I11111111 I11111111 I llllllll I llllllll I11lIO1 -20 1 I'llU I"""" ' ""1'1' I11111111 I illlllll I 11111 10" 10.' 10" 1d ld 10' -1"- -1"- Figure 2. A comparison- of 2-D gain models for a 46 dB gain antenna Mnmu rgb lomv I iiiiii~l I iiiiiill i IIIIIIII i ~IIIIIII I i~liilr I11111111 I I1111111 I11111111 I ll/lllll I IlllIl1 -20 ' I1'1"'' ' "11111' I I I111111 I I1111111 I I IIIILL to4 10= klohlIUm-CI1.10' *Id01$ 1d IO' Figure qa). The interpolated original 3-D gain model for an 85-dB gain antenna. 11I I 1-- r--- I I _--I ,- - -. 0.7- - I I - I I 1-- L - -- - - I I lo,- I I i-- r--- so&-- I I I I !0.4------I 7--- I1I I 1-- 0.3- - - ,I 7-- 0.2 - - - :"f I -__I 01 -- I IIIIIIIII 11,IIIIIl o.h (m) Figure 5(b). The cumulative probability function for the new 3-D +model of 811 85-dB gain antenna 111 0.7- - io.e-- -- :--:--I --:-I, - -;- - -1 I1 11 -r--i- I\I I I I I I I I I ?!o ?!o .:o -3- ;o :o A :o a m A m 1 Figure qc). The probability density function for the interpolated original 3-D gain model of an 85-dB gain antenna. Wnwwqbf"bmd$a I I iiiiiil i I IIIIII~ I i iiiii~i i i iiiiii 17I1 I I1 111111 I I lllllll I I1111111 1 I llllll -i -r rnnm- -i-riinin - i-iiti+iw - t i inn I11111111 I1lll11l1 I llllllll I I I1111 I I I111111 I I I llllll I I1111111 I 1111111 -II 111111 I I I IIIIII I I I Ill1 I111111 -T rim- I I11111111 I LIt\ LIU - -1-LIL UIU - 1I11111111 -1J Llllll I I I1111 I111111 y) -L l.LI.LIUI- -L -2 LIIIYI- I I I 1111l1 I llllllll 111111111 l1t111111 I 1111l1 I111111 I I11llll/ I 11111111 11111111 I llllllll I 111111 --l111lll 3 4O - r T rt-rInI- -r r rl ml- -1 17 nm - i-17 TIT^ -7 I I I Inl- I I1111111 I 11111111n I I llllll I I l11l111 I $50 --ClCIIIUI- -c C CIUHI- -1-I-I 111 - 4 -1-4 ClClU -1 I I I ,I1111 I I I111111 I I lllllll I I IIIII I I ,111 i I I11Il 111111 I I1111 1lLlU1 J-IJL I I1111 ~~-LlLIIIU~~~~~LLILlUI__l~Cl~U~U_I I I111111 I I111l111 I I I1lIl11 I I I1 I I1111 I I I111111 I I 1111111 I I I111111 I I I1 I I1111111 I I Illllll I I Illll11 I I1111111 I I 111111 I I I111111 I I , ,,lLll L , I I, ,111 I , I ,ill,, I I I I ,,,L 16 16' 10. 10, Id 2:* 1o* 10' Id I0, Id iWbfmmbol.iO* hb(W w-b-m "a Figure 5(a). The new 3-D gain model for an 8543 gain Figure 6(a). The interpolated new 3-D gain model for an 85- antenna. dB gain antenna. -(ds) Figure qb). The cumulative probability function for the Figure 7(b). The cumulative probability function for the interpolated new 3-D gain model of an 85-dB gain antenna. original 2-D gain model of a 46-dB gain antenna. I I, I L ?A-;o b tb ;o :o m m m I A (Yn (ds) A Figure 6(c). The probability density function for the Figure 7(c).The probability density function for the original interpolated new 3-D gain model of an 85-dB gain antenna. 2-D gain model of a 46- gain antenna. I llllliil I I111111 I1111111 - LI LlllU1111111 I1111111 I I111111 -e tI4IU+ I lIllllli I 111lIl1 I1111111 - t titint I1111111 I1111111 I 1 I1ll11 - r riiinr I1111111 - r r~m Figure 7(a). The original 2-D gain model for a 46-dB gain Figure 8(a). The interpolated original 2-D gain model for a antenna. 46- gain antenna. 0.71 - - - I I I - -4 ip- I I I I 3.5- - -1 --I- ---1- I I I Io..-- I -1 I I 0.3 - - -4 I I /I I I I , I I Figure 8(b). The cumulative probability hction for the Figure 9(b). The cumulative probability function for the new interpolated original 2-D gain model of a 46-dB gain 2-D gain model of a 46dB gain antenna. antenna. -- o.7 t - - I I I I I I I I 0.31 - - 0.2 - - 0.1 1- - Figure 8(c). The probability density function for the interpolated original 2-D gain model of a 464 gain antenna. (H YU WbtOmhW I ~imI IIIIIIII i iiiiiiii i ~IIIIIII i iiiiiiil i iiiiiii hrm;:1I11I iIIX 1 11111111 I ;I113 40 -1-11 UlUl- 1 UUlU - - LlllU I I1111111 1111111111 LI LI I111111 I I1111111 I 11111111 I I I llllll I llllllll l1111111l I I I111111 90 -I-I4UIyI- 4 A u UIU - c -e LlLlU I 11111111 I llllllll I I I111111 I llllll/I I11111111 I I I1111111 I I I 1111111 I kll111 I 11111111 I I1111111 I 11111111 I I1111111 , I 1111111 I I1111111 I I lI1l111 i7 nnm - - I I1111111 Ir 11111111rmnir I 111111111 I I 11l1111 I I lllllll I I11111111 *.obh"mkb(dq) -ab Imnbol.ll1.-(* Figure 9(a). The new 2-D gain model for a 46-dB gain Figure 1qa). The interpolated new 2-D gain model for a 46- antenna. dB gain antenna.