l 2 i e ) 2 - (' IW 1082-2-Q - RL-2024 (Cop - T:E UNIVERSITY OF MICHIGAN OCf-.M — IIMRlING DPA III Of mCT*ICAL ENGINEERING Rodiaten l b..rly DOPPLER RADIATION STUDY Quarterly Report No. 2 1 October 1967 - 1 January 1968 By Chiao-Min Chu, Joseph E. Ferris and Andrew M Lugg 15 January 1968 Contract No. N62269-67-C-0545 Contract With: The U. S. Naval Air Development Center Johnsville, Warminster, PA. 18974 Administered through: OFFICE OF RESEARCH ADMINISTRATION * ANN ARBOR

0 TUE UNIVEB RSITY OF MICHIGAN 1c42-2-Q TABLE OF OONTZEWT ABTRACT i TlRODlUCTOKU 1 l EXPE-RI1MDIZI~fTAL ITUDY 3 ID TIEORETICAL DY 10 3. 1 SpsuLa1 1rinr1~l~u Of Speovrarly Rfi~cted 3.1 ~~Al Of R~iatI10 3.2 Thmpiori VarltLuis of Ref1aM RdIation 15 3.3 Bazl. rni Pwr Levul 18 3.4 Ampimat. RAdiatim Paserns 20 APPENDIX A: SPATIAL VARIATIONS IN FIELD STRENGTH FOR A GAU~SSAN BEAM 23 APPENDIX B: TEMPORAL VARIATIONS IN STRENGTH FOR A GAUSSIAN BEAM 33 I

THE UNIVERSITY OF MICHIGAN 1082 -2 -Q ABSTRACT In this, the Second Quarterly Report on "Doppler Radiation Study", some results of experimental and theoretical investigation are reported. The measured antenna pattern data for an AN/APN-153 system obtained during the last quarter has been put into digital, three-dimensional format. This digital data will be used in the theoretical calculation of the direct and and reflected radiation. In the theoretical study, a numerical scheme for calculating the spatial and temporal variations of the reflected radiation, based on a perfectly reflecting ground, is formulated. Schemes for presenting the numerical results in terms of 'nora lized coordinates' are also considered. For the minimum detectable 8ignal and gain of the possible receiving system, the approximat e signal ranges are calculated. ii

THE UNIVERSITY OF MICHIGAN 1082-2-Q I INTRODUCTION This is the Second Quarterly Report on Contract N62269-67-C-0545, 'Doppler Radiation Study" and covers the period 1 October 1967 through 1 January 1968. The primary objective of this project is to characterize the radiation from airborne doppler navigational radar systems, and the probability of detection of such radiation. During the present research period, the following research has been accomplished. 1. The measured radiation pattern of an AN/APN-153 antenna has been put in digital form. Sucha form is useful for the numerical calculations of the spatial and temporal distributions of the radiation transmitted from the antenna. 2. Techniques for the computation of the spatial and temporal distribution of the radiation reflected by a smooth perfectly reflecting ground is formulated. After various considerations and trials, a means of presenting the power level of the reflected radiation in terms of normalized coordinates are introduced. The possibility of approximating the actual rda^ipatte by an quivaleut Gaussin ditribtion is also carried out. 3. Techniques for the computation of the radiation reflected from a diffusely reflecting ground is also being started. This phase of the work, however, is not included in this report because of the somewhat moretedious algebraic expressions involved. During the next research period, the numerical scheme for the calculation of the reflected radiation from a diffusely reflecting ground will be continued. The possibility of approximati ng the actual radiation pattern 1 A

THE UNIVERSITY OF MICHIGAN 1082-2-Q by an equivalent Gaussian distribution to simplify the computational scheme will be carried out. The computational scheme for the reflected radiation from a perfectly reflecting ground will be extended to include some typical cases of trapped radiation and multiple reflections due to overcast. The antenna pattern measurements for the second system of doppler radar will be carried out when they are made available by NADC. 2

THE UNIVERSITY OF MICHIGAN 1082-2-Q II EXPERIMENTAL STUDY During the last quarter the three-dimensional pattern data, collected for the AN/APN-153 doppler antenna, has been digitalized and placed in a three-dimensional format. The pattern data was collected for each of the two input ports and is presented in Figs. 1 and 2. All of the pattern data has been normalized with respect to the pattern maximum so that the numbers are represented in db below the pattern maximum. In addition to the data collected by The University of Michigan, four sets of three-dimensional data are presented in Figs. 3 - 6 for the Ryan doppler antenna, APN-182. Provisions are now being made to equip the NIKE system, available at The University of Michigan, to make fly-by tests of a doppler system during the latter part of March or early April. This refurbishing requires a tbroutg system checkout of the tracking and missile tracking radar systems and of the communication equipment that will be required to maintain communications between the radar site and the airborne crew. 3

1082-2-Q.~ —. 1 — A *f A A A A P A ~R 1 9 A a *R * 4 A * A 4 43 * A 43 p sc k cl, R K1 P4 4 S < 4. 4 P P, a * A A 3 4 4- A A A -4- r; 4 4 2 4 C 3 t 4 4 3 P 4 4 4K 44 4 t 4442~ 3; tt42 443433K 43344; ~ 444 2 423~; 3 - A S A 44K 3 3 4 4; ~ ) 34 P 3 4 2 P? * 3 4 2a 4 4 4 * 44;; K 4 3 ~ 4444 44344tK 3! 444 4 4 P4 4 33 P S B 4 4 4 44!l K 444a P 333 - 4!!<; 3~ 44! P 443K: 4 P 445 444444a 3 4 4 2 4 434234K 3443!! a34434 44 A; 24 3t; 42 44 % <* 4 P3 44 4 f 243! P 4 4 4 A. 4 P 3 4 i 443 A 343 S 4 3 4!a s 4334 c 4.;; 3 4 4 A; A 4 434444S 3 4 4 4 P 2 3?; A 4; 33X; 4 3; 3433Kli 23434 X 44434; R 444433 R 44 232% 42a 44f P Pt Pt P Pt Pt 4t 4 *, 44 4 44~ 44 I1 4 A A A A; 4 A A 4 2 4 4 4 P 4 ft 2 4 4 4 9c 3 4 3 3 3 A 4 4 3 3 4 A 2 4 4 4 Hi 4 4 P A A P 4 4 8 4 a a 3 3 3 P P 3 ie 4 8 A 8 S! 8 SI A s i R I it I* II P. SI A A; 0 A ~ P. ft 14 AI ) A SA A A A )R 4 9 3 * It 4 a A A a x 4 X r. X a 3 4 P 4 )i 4 4 a 8 3 3 ~4 S 4 9 4 4 A 4 II 4 4 2 3 11, 4 2 3 2 3!R A A A St s A X it aR 19 M 84 A 4I 4I 4 A VI 4 31, A 4 4 3 I 4 3 a A A A 3 P 4 4 4 A 3 3 a 3 3 4 4 it 8 X A A a it X ~ ~ x I 49 A A 3 3t 3 4 4 St it 3I 3 3 it 3 3 3 * 3t 3t 3 3 A A A 3 4 4% 4 4 pi 3 3 3 A 3 A 41 Pt 4 4 4 Pt P P 4~J~ A A 4 A 42 R R S! A SI * A A 9 A 8 11 s s? 4 4 1 C it 9 A <. 43 A le JR M 34 A 4 A 44 43 A 43, 3! 4 4 2 4 2 A 1t 3 3 3 3 ft 4 IC K A 3 3 3 4 3 3 4 3 3 4 4 4 x x x $4 x 3 x 4 A 4 4 4 4 4 4 KS 4 A. 4 4 R1 1!C A II rl II X1 X1 x1 n II ~c It A xR JR s A 4 3 4 2 P3SS p 3 4 P2 X 4; 43 343 f 344f 444t 43K 44K 43 f 442 334 24X 344; XP t 4 P 23S 34 P 44 ~ X 4 4;S 433 433 * 434 444$ 348 3 A 4 4 4: " 4? X X 4 A API 34 II * 34 II 44 33 43 A I!K 4 P Xt 44 24 4it I I ot 1 a PI Q 44 I 2 3 3 3 A x R <2 3 3 3 4 $3 A1 3 4 4 x 4 4 II 4 4 8 3 it 4 3 2 4 ft X 4 Pt!R A IR it P. X P AI R it P! c 4; 24 tR 1, X 43A 44A 3 44A 34; 3 x A 4 A 4 3 4 3 3 4 A R 8 4 A It 4 14 A 2 4 A A1 -3 * 4 4 4 A P A 4 A II 1 P 4 4 4; 4 4 4: 4~ 4 4 2!4 4 2 a 4 AI 4 3. 4I A 3 3 4 P A It A 2 A ft A X 4 4 ft 3 3 x 4 4 X! 3 3 le 3 A f4 9 ft 5S 9 II 9 S X 3 it x x P 4 A A 2 3 4 A 'A 4 x 4 6 A A 2 14 14 A A A 4 4 4 3t A A A 4 x i f P It a. A A 4 3A I - I -i Ii'o 2c, fC 43 4! 0 A -t- 2 ft I I R 4 3 P. ft VIf A %4;2t >4 1n o ~ 04 'C' z kC fla A A P 4 4 3 0 4 i -T A 4 4 4 II W; 9 II AI A A II F~ A A 4% 2 29 4t 49 P P A A 4 4 JR x 3 4 P 4 i r~ x 4 - 3I 4 JR 3 3t it 3 3 4 A A A A 4 2 A 49 2 4 4 rK 4 A 4 A P1 A A 4 A 4 AP 4 4 4 A~ A x 4 -OL0 ova 4 015 A 0891 (nu113) (~iFAia) — 4 -TS — 4 — 0Co i C 06 4.

1082-2-Q P.8 A 8 8 8 -, R 1 Z2 1 8 1 8 8 8 8 8 8 8 8 8 8 M it 8 8 8 8WI -- 8 R q 8 it~ 8 8888 it it it It It 8 81 A 8 A it 8 8~ 8 8 8 It8it A 8 i 8 888 88 A 88 888 8t I 88 8 pi 888888A )8 A 88 888 8 8 8 8 8 8 8 8t 88 8 8 * - IC88 88.it::: T.i 1, 8 A it A 888 8 R A 8 A Yj A it 'ICG 8, A 8 8 88, A8 8 K Aiiop A8 8 8 x8A8 A8C88a a 9 88 8 8 A88 888 1 89 8888 88:)R 8888 888 I8t 8 R8 8A 8 8~ 88It 8 8 8 88 8 8 8 8 8 8 8 8 It 9 JR.E T R 9 S A A " 7 CZ It8 8It88 88 '4A A it 88 I P. 88 9t 8 8 8, A A 8888 88 88 8 88 88 it 8884 8A - 8 88 8 8R 8 8 Lq W 84 8C 5

9 t %II U IV Ia II i I; i "I I I -,/ A I. t -.4 -I I 1, i. I 1 I j -;l-,:s, ' — 0 I (7 I I i t, I,:,,: 1 1 ll -t - I - I I I 1, 'I:- "kt I. PS: I - -- I, I I I 1. 1 1 8, / 4I 4/ I- K (Elevation) 4y1AtIA-,Afk 1; I I I 11 11 A.. I I I 4-, 11 +6, i T — - i i i i..0, -y "I s It Iw I.1 1 -- "" I I qlll --- "I" I i I —

1082-.2-Q.11 it z 1% 1117. I - 1% I, - Il- 4 i IZ t i.;i —\\ 11 OAI & S I.C I Ao% ii,4 'II ~ — -4-. I-4 II~ ".)t, I II I /" 11 I I 0 I 'N ( {2 4 0 i I I -j

I v 19 ji 4t I,,Mr iT I" 9 i II I I I. -4 i w,%NW.11' -4 - -, -j,, ) \ lw I,"I,.1-~ ~-",-, N '. 0 KII. (. I, f, -14 " I 11, 7 ---, -* - %, )l -, - --- i' "I' ( --- - ('q,,C), ' 11, o i — f"l -/,O II) k I ' -- '4 --— "4 -- 2+- - 14 — -- - - - -- - 4- - - — 4 -i co I ) iii < ' r Y KJA (< -'D\ k 9 N, N, cz79J'N r K K N p14) 4, I; , / 5 9 . L (; ii ) 40 /-) i, I / >4,9 I I I I (Azimuth) -4 --- 4 -.bl 4~ —. - w' — -* — i I"" p~' AF t A0 t~/ A ## - 24 FIG. 5: APN-182 Doppler Anteuu Beam Number Three (Courtesy Ryan ConipaRY) Three Dlmmseaal Patbrn (in db)

1082-2-Q i4*lp 4 II i i tf -. t tR ti i I t, i R T II - R! f I (.r i IL w M it I I" L, " 1, a - "'-I z A 9 "I I -1 >1'I I -/ _ ' i~~ N (uorwASiI) N I -Th lob II r4 I --- - -1..; 116 ' I-Ar- — 't II I 2 / '-, / N, c- I jI N; I/; i - 'I 1. i I 1. j 1, k I -i -3 r I I I 4- Ir 4 a 0 I I 0 9

THE UNIVERSITY OF MICHIGAN 1082-2-Q I THEORETICAL STUDY The numerical scheme of calculating the spatial and temporal distribution of the ground reflected radiation from a doppler radar, and the method of clearly presenting the numerical results, are investigated. Based on geometrical optics, and the digital data of the radiation pattern, the scheme calculation of The choice of the geometric optics approach is based on the anticipation that this scheme is amenable to extension to trapped and multiply reflecting oases. The scheme for the same calculations in the case of a diffusely reflecting ground is also being investigated but is not completed. This phase of the work is therefore not reported here. Anticipating that future development in the antennas of doppler systems may be osing morenarrow beams, approimat calculations for the ground reflected radiation assuming a Gaussian distributed beam is also being carried out and the results are given in Appendices A and B. 3. 1 Spatial Distribution of Specularly Reflected Radiation From the radiation pattern of the transmitting antenna, the spatial distribution of the transmitted radiation reflected specularly from the ground may be calculated by the method of Images. Figure 7 shows that the radiatfnm from a transmitter at a position (xa, ya Za) in the direction (0, 0) would, after reflection from the ground, reach a height h at the point (xh, yh, h), A simple geometric argument yields xh = -(z+h)tan.cos4+x Yh= (Za+h)tan0sin-+ya The distance travelled by the reflected radiation is then R(h, 0, ~)= /(za+h)2+{xh-xa)2+(yh-ya)2 (3.2) 10.

THE UNIVERSITY OF MICHIGAN 1082-2-Q A z T(^aya, Za) Direct Signal \ 0(,0) I I I I I I I Image 9 FIG. 7: GEOMETRY FOR REFLECTED RAY 11

THE UNIVERSITY OF MICHIGAN 1084-2-Q Thus, the power radiated from the transmitter T would, after reflection, reach the point B after a time delay of,(h, ), =) R- J) (3.3) C where c is the velocity of light. Moreover, the power received from the ground reflection is different at different points of observation. To investigate the spatial variation of the reflected signal, it is therefore more convenient to investigate the variation of the reflected power observed at points in a horizontal plane of fixed h (i. e. for fixed height of receiver). For the convenience of numerical calculation and display of results, the following normalized coordinates are defined. a) The relative height Z -h a (3.4) z +h a b) The relative horizontal coordinates Xh-X 2 X X(00, OB J =- 2- tano oosj (3.5) a Y=Y(0,, )- = + tan sin. (3.6) a Using these relations, one finds that R=z lx2 + )2'. (3. 7) a For the variation of radiated power across a horizontal plane we use Eq. (3.18) of the First Quarterly Report+, which states that the amplitude of electric field is +Chu, C-M, J.. Ferris and A. M. Lugg (1967), "Doppler Radiation Study, " The University of Michigan Report 1082-1-Q. 12

THE UNIVERSITY OF MICHIGAN 1082-2-Q E(xhy f(e, 0) (3.8) Yh) -(a+h)tanO a where p is related to the gain of the antenna, Gt, and the total power radiated, P. From the radiation pattern in the digitized form, we find that there is a direction (08M, M ) at which f('eM, M) = 1, correspoading to 0 db in the chart. It is evidently more convenient to refer to the received power level (in db down) to this direction. Thus, we may denote the power level corresponding to a ray in the direction of OM, OM, i. e., in the direction of the main lobe, at 0 db. At a height h, this ray intersects the point: 2 X = - tan 0 cos (3.9) m 1+ I m m Y +- tan 0 simf (3.10) m 1+ m m The ratio of the power level corresponding to a ray in any direction (0, frt) to that corresponding to a ray in the direction of maximum power is, f tan. (3.11) 2 m tan e Expressed in decibels, it is therefore: 10)+2g f2e log1 tan m)-20 log (tan 0). The term 10 log f2 (0, 0) describes the radiation pattern and its value is obtained from geometrical considerations (cf Appendix A) or from the digitized output of the experimental study. We- dmie this term Wa follows: dbo(e, ) = 10 log f2 (0, O ). (3.12) 13

THE UNIVERSITY OF MICHIGAN 1082-2-Q To express the power level of reflected rays n from the transm er in any direction db (0, 0)=db(8, 0)+20 logtane -20logtan. (3. 13) Equation (3. 13) together with (3. 5) and (3. 6) can be used in computing and displaying the relative power level of the specularly reflected raiation from the ground. The variation of the reflected power level can be cast in a slightly different form to obtain the power level of the direct field strength. This latter power level is usually appreciable when h < Z. For a direct radiated sinal in the direction (0, 0) to reach a height h, the normalized coordinates are easily shown. to be (d) (za-h) 2 X (,,) = - tanOcose = - 2~gtan0cos Y (d)( ) = — tan sin n Thus Xd(o,, ) = X(,, ) and yd(o, ) = e Y (0,, ). Furthermore the ratio of field strength of the direct ray to that of the reflected ray is given by field strength of direct ray a 1 field strength of reflected ray z -h a or, in decibels, (db)d= (db)r- 20 log e. (3.14) (3.14 14

THE UNIVERSITY OF MICHIGAN 1082-2-Q Thus, for a fixed e > 0, the variation in the relative power levels with x and y can be used for both the direct radiation and ground radiation by a change in the X and Y scales. This justifies the use of the parameter g in the evaluation and graphical representation of the relative power level. 3.2 Temporal Variations of Reflecte Radiaton The analysis given in the previous section for the power level does not take into account possible time variations. In other words, if a short pulse is transmitted from the antenna at a certain time, the power level s are observed at a point in phase, after a time delay given by Eq. (3. 1). This time delay, together with the actual motion of the vehicle carrying the radar, determines the temporal variation of the radiation. To bring out some essential features of the temporal variation, let us assume that the vehicle is moving with a horizontal velocity. The trajectory of the moving transmitter is given by Xa(t) and y (t). For a receiver at height h and X, Y att =0, at any time t> 0 xa (to)-xh xa(tx a() and yA(to)-ya(0) Y(t)=Y + a o a a Thus, for a horizontally moving vehicle, the relative power level of the reflected signal at any fixed point may be easily obtained from the power level in the x-y display constructed in Section 3.1. To illustrate this, let us refer to Fig. 8. For a point defined by (X0 Y0) at t0=0, the trajectory of the moving transmitter may be represented by the locus as indicated. From this curve, we may obtain the variation of the reflected power level as a function of to. Let us denote this by dbr (t. 15

THE UNIVERSITY OF 1082-2-Q MICHIGAN. oTraiecYe o ryfFtlig in ormalzed;ooronates / / / 7 FIG. 8: DIAGRAM SHOWING FLIGHT TRAJECTORY FOR FIXED e. X 16

THE UNIVERSITY OF MICHIGAN 1082-2-Q This function, however, is not the relative power level of the received signal because; a) The transmitted power may be varying. This variation may be represented in db by dbt(to), b) There is a time delay due to finite velocity of propagation of electromagnetic waves. The position of the signal from the transmitter at to, is received at the point of observation at a time X(to)2+Y(to)2+( )2 t=t + (3. 15) O o From the above observations, it is seen that the power level of the received signal as a function of to is given by relative power level of receiver system = db (t )4dbtft ) This variation, of course, can be transformed into a function of t by using Eq. (3. 15). For an arbitrary direction of flight, i. e., in the ascending or desoal g phase of flight, x a Y a z may all change. Hence for each t, X(t ), Y(to), f(to), should be evaluated for each observation point. The relative power level of the ground reflected radiation as a function of time can then be computed. At present, a computer program for calculating the relative power level of the reflected radiation is being formulated. However, before attempting any actual calculation, we are investigating possible approximations, and the ranges of parameters for the actual antenna to be used in the doppler system. 3. 3 Rage and Power Level It is well known that for a transmitting antenna of output Pt, gain Gt and radiation field pattern f(6, j), the Poynting vector at a distance R from the 17

THE UNIVERSITY OF MICHIGAN 1082-2-Q transmitter is P Poynting vector= - 2 Gt R 2(9, ) (3. 16) ' 4R2 For specularly reflected radiation from-the ground, this formula also holds provided R is the actual distance traversed by the radiation, such as given by (3.2). For a receiver at distance R, if the pin of the receiving antenna is Gr, and the frequency of radiation is X, then the power received is PtGt (e, ) X2Gr p= -. (3.17) 4r R 4ir The detectability of the radiation depends on the noise level, the scheme of detection, etc. At present these quantities are not easily specified. However, in general, we may assert that there is a minimum power level above which the signal may be detected. If this minimum detectable signal is denoted by PN then the significant range at which the radiation may be detected is given by R= 4-P G t For the antenna AN/APN-153, which is currently being studied; P = 5 watts G = 19.5 db f = 13. 320 GHz. The values of G and PN for a fixed range are plotted in Fig. 9. For the typical r N values such as Gr = 20 db above isotropic, and Pr = - 100 dbm, it is seen that the reflected signal from the maximum direction, corresponding to the man lobe is detectable at ranges included in the shaded area. 18

THE UNIVERSITY OF MICHIGAN 1082-2-Q isotropic) Transmitter Power = 5 watts Transmitter Gain = 19.5 db Operating Frequency = 13. 320 GHz >'. 2/ 0 Pr(dbm) FIG. 9: RECEIVER GAIN VS MINIMUM DETECTABLE SIGNAL POWER FOR VALUES OF SLANT RANGE. (Shaded area indicates region in which aircraft is detectable.) 19

THE UNIVERSITY OF MICHIGAN 1082-2-Q Now the range indicated in Fig. 9 is the slant range. For any other direction, of course, the range will be smaller. To find the horizontal range the lant range is multiplied by loos el. From Eq. (3. 17) in a direction (9, f), the horizontal range is: Gr - A- - to, ) I, sinO' This corresponds to the region, XI < X Gjr sineOlosl ft(9,) IlY < PtGt sinO sinf(, ) 4n -— 1 3.4 Approximate Radiation Patterns For a narrow beam antenna, it is common practice to approximate the actual radiation pattern in terms of a Gaussian distribution f2(,e = exp a a2 b 821 (3. 18) where a, B are the deviation from direction of maximum radiation along two mutually perpendicular directions normal to the maximum direction. In Appendices A and B, the calculations of spatial and temporal distribution of radiation from an antenna with an approximate radiation pattern is carried out. To derive the approximations from the actual measured radiation pattern, relations between a, i3 and 0 and j are necessary. In Appendix A, it is shown that if the direction of the maximum radiation is given by (0WM, OM)' then, tan(O,+ A9)cos(M,+ A=tan(04-a)cos0 -sec OMtanB sinoM (3.19) M In'M *m M 20

THE UNIVERSITY OF MICHIGAN 1082-2-Q tan(8M+ A)sin(M+ )=tan(i+a)sin0M+sec 0Mtan osM (3.20) where a and,B are angular deviations in the direction of increasing 0 and # respectively. In the neighborhood of the maximum direction, we assume that a, 1, AO and AO are small. Then, approximately, by using appropriate small angle approximations of trigonometric functions, it can be shown that AO0 a (3.21) sin ef s - 2 O (3.22) cos oM Now, at 0 = OM, 0 = M, we know that and af2( f, 0 =0 Therefore, for most radiation patterns we may approximate f2 (o., ) l-S A2 -Sa z 2 where S1a2 f2(. =a2[.f2(. 2 a20 2 a2 2 a20 2a 32 evaluated at OM0, M. Comparing this approximation with the Gaussian distribution for small a and 3, which is given by f2(0, ) 1-a a2 -b 32 21

THE UNIVERSITY OF MICHIGAN 1082-2-Q we see that a and b are related to the differential quotients S1 and S2 by S A02 a — = S (3.23) a and SA sin2 M 2~ M b = =S M (3.24) 2 =2 4 cos 0 For narrow beam antennas, the approximate distribution given by (3.16), together with the values of a and b determined from the actual radiation pattern is reasonable provided sidelobes do not exist. For patterns with sidelobes we may approximate the pattern by the sum of several terms: f(, ()(e.xp aa2b 02] +Al exp [a2-b^ +. where A1 is the value of f2(0, f) at the peak of a minor lobe, and al. b are determined from (3.23) and (3. 24) by using corresponding values of 0, 0, S at the minor lobe peak. 22

THE UNIVERSITY OF MICHIGAN 1082-2-Q APPENDIX A SPATIAL VARIATIONS IN FIELD STRENGTH FOR A GAUSSIAN BEAM In general, a normalized electric field strength may be defined. For CW transmitted signal, a suitable normalization is as follows |(| E (X(Q ) Y(,0)) h V ' M MOm), M' ^WO From thl discussion in Section 3. 1, we find sec 0M Fhsec= f(opA (A. 1) As suggested in Section 4t, it is convenient to define another coordinate system. Figure 10 shows the system that we have in mind. The transmitter is situated at T, Bo is the intersection point of the beam center and a horizontal plane, and the point B is fixed and arbitrary. This observation point, B, is described in terms of the angles 6 and, the depression angle and the azimuthal angle respectively. Thus, the ooordinates of B are {"X a-(z+h)tancos {YA-(zA)tanain0| ' -h (A.2) Alternatively B can be described by utilizing the angles a, 3, where a is measured in the cOAB plane about AB and is measured about AB in a plane perpendicular t OAB. This coordinate system is introduced to facilitate the computation. It will be seen later that the radiation pattern is expressed more simply when a and, are used than when 0 and f are used. In this system, the coordinates of B are 23

THE UNIVERSITY OF MICHIGAN 1082-2-Q z T 0 Y X FIG. 10: GEOMETRY FOR DETERMINING POSITION OF OBSERVATION POINT IN TERMS OF ANGLES a, 3. 24

THE UNIVERSITY OF MICHIGAN 10o82-.2 -Q XA (zA +b [tan(oM-+cosoA-se00M tang1 sin~ -(z h [tan(OM4a)sln A+sece Otsan osoJIs -hI a and 0 can be expressed in termB of 8 and 0 by equating distances (A. 2) and(A. 3)and recallin gttate04Om O +and e O+ -, are given by Eqs. (3. 17) and (3. 18). Alternatively we may write: tan(OM -a) 1 os Bin 1 OB sec 6Mt(Om+SAtan La' -08 snAJ (A. 3) in exressions The results (A. 4) usin this transformation (A. 1) may be written as isecO f(a, 0 lFh(XVY)j M {se2(OM-fa)+.sec2O tn~ 1/2 where for a Gaussian beam f a )Aexp I(.-b land for a beam with sidelobesI fAce ) A64J2(aa) J2(b2I3) (aa) (b23)2 where a,, b etc may be calculated from Eqs. (3. 2 1) and (3. 22) or fromth 1 following expressions: 1.388 1.388 1. Z7r 1. 27r a1 b-; a2=9;b 25

THE UNIVERSITY OF MICHIGAN 1082-2-Q and ea, 0a2, etc are the respective beamwidths. Figure 11 shows the pattern functions for beamwidths of i0 and Fig. 12 shows a plot of the power distribution of these functions. Consider a Gaussian distributed beam; then 1 (a22 2+2) -2^) sec e Ih(x Y)l = c2( )ec2 tn2 1/2 (A. 5) h' hsec (8 — a)-f ec ~tan where the position X, Y is given indirectly by (A3). For small a, 3 the variation in the exponential term dominates the expreaak.. As a, 3 increase the magnitude of Fh(X, Y) decreases rapidly. Because we are particularly interested in the region in which the signal power is discernible by some monitoring device, we can write as a first approximation (Fh(XY)| exp[-i (aa22+b$2)] (A. 6) If the reflected radiation at some a, 3 is Adb down on the power at a = 3 = 0, the following relation results 2 Alge (A. 7) a 10 Equation (A. 7) defines contours of electric field strength in the X, Y plane for a fixed height, h. By way of example, wechoo a = b = 0. 277/0, i. e. a 5~ beamwidth in both the a and B directions. The contours are given by a2+B2= 3 A. Figure 13 shows such electric field strength variations in the X, Y plane. Furthermore a similar expression for the direct radiation can be found as described in the body of this report (see Eq. (3. 14) ). The direet radiation pattern is shown in Fig. 14 for e = 1/2. 26

THE UNIVERSITY OF MICHIGAN 1082-2-Q -1 O- Fi e+(b - - F 1(8 0)=e 1. --- F(0, 0)= 64 — -- J2(a0) J2(bS)1 0=0, a=O. 8/0 0=0, a=0.277/~ /.2 / / 0 (degrees FIG. 11: PATTERN FUNCTION FOR BEAMWIDTH = 5~. 27

CA 0) 0 O. r I ar Ci I.. 3eam with sidelobes 5~Beamwidth Gaussian Beam 50 Beamwidth C) I I0 to H W1 d I.-4 0 -4;> z 0 Q3 FIG. 12: POWER PATTERNS FOR BEAMWIDTH = 50~. (0=0, a=O. 277/~ for Gaussian Beam, a=O. 8/0 for beam with sidelobes.)

THE UNIVERSITY OF MICHIGAN 1082-2-Q Y-YA Y -ZA /.8.4 21db / / J Refleced Sidelobe ' Reflected Sidelobe ol g o f - I% 0=0 \ A -9 / / I I I I Sidelobe 12 I..36 db / Reflected / Radiation % d 'db " -V A'. Direct Radiation 30 db 0 x-x X-XA ZA FIG. 13: SPATIAL VARIATION OF ELECTRIC FIELD STRENGTH. Radiation Pattern with main lobe at 0A = 1200, OA = 300 and sidelobe down 15 db on main lobe and atA = 130~, A = 45~, = 1/2. 29

THE UNIVERSITY OF MICHIGAN 1082-2-Q -~ Za Direct Signal 0- ^ (at.'o h Bo FIG. 14: GEOMETRY FOR DIRECT ARRAY 30

Ti Hl *: K I l N E R S 1 "I v1 0 i F MI, H I T A N 1082 2 —.. It is of interest to know when an observer at (X, Y, Z) is measuring predominantly direct signal power rather than indirect or reflected signal power. Let ac, (' define the direction of the direct ray and a, a that of the reflected ray. From the geometry, it is found that tan(OMta) tan (OM- ) and tan:= tan 3 (r) (d) The locus of points at which E (X, Y) equals Eh (X, Y) is given by ^ (2Vea 2 2 ~+an 2 (OMa)+sec 2 tan 2 - ( expM [a2 (a?' )-a b2 2-2)] (A. 8) 2 +tan (6 M+.a)+sec M tan 3 It is noted that this equation may be solved by hand. To complete this description of the spatial variation of the electric field strength, the sidelobes must be included. As noted above this may be effected by using sedond-order Bessel functions in the beam pattern function. In this case, it is found that E(h (X, Y) sec O Eh )((a1=0), Y(a==)) fsec2 (9M+a)+ese c 2Mtan J2(a a') J2(b ') 64 *64 — 2 2 (aa') (b 0') E (XY) sec 0M E(r) (X=3=0), Y(a=:=0)) sec (OM2+,)+sec20 ta /2 h \ / ( M M ) J2(aa) J2(b ) * 64 2 2 — (aa)2 (b )2 31

T H E U N I V E R S I ' Y0 F M, C H 1 (I A N 1()082-2 -Q and E(r) (X(a=3=O), Y(a=B=O)) d (x^a=pO). Y(;=0=O)) Sidelobes are certainly introduced in the above equations. However it is not possible to specify arbitrarily the position of the center of the sidelobe pattern nor the number of db down of the sidelobe on the main lobe. In order to circumvent this problem we oompound the pattern from a collection of Gaussian beams, i. e. we use the principle of superimposition+ Figure 13 shows sidelobes determined in this fashion. There is a further advantage in specifying the sidelobes in terms of Gaussian Beams. If we wish to find the locus of points for which the reflected radiation equals the direct radiation we have to solve the following equation for 0 and 0: L la' J2(aa') J2(bfB') Clearly, Eq. (A. 7) is more accessible to hand computation. 32

T H E UNIV E R S I TY F M I C H I G A N 1. 82.2-Q APPENDIX B TEMPORAL VARIATIONS IN FIELD STRENGTH FOR A GAUSSIAN BEAM In Section 3.2, expressions describing the temporal variations of electric field strength were developed. Here, these equations are particularized to the case of a Gaussian beam and CW transmitted signal. Figure 15 shows an aircraft travelling with X-directed velocity. The fixed observation point is described either by reference to the initial position of the aircraft or to the position of the aircraft at time, t. If ca(t) and P(t) specify the ray direction from the airplane to the image of the observation point at time t, the following relationships are obtained: tan(OM+at(t)) =tan(0M+ (0))+S(t)cos A tan3(t) =tanP(0)+S(t)sinAcosO [Itlt (=(zA+h)i2 {S)+s(t) Ltan(0M to(0))cos4A+sec OMsinAtan3(Oi +sec2(o+a(o))+sec20Mtan2P(0) and LR(02 =(zA+h) sec (0Ma(0) )+sec 2Mtan2 3(0) where SA Vt -(zA+h) The ratio of the power level corresponding to a ray at any direction (a, 3) at time t to that corresponding to a ray in the direction of maximum power, t=O, at a fixed distance is R2(O) 2 R (t) 33

THE UNIVERSITY OF 1082-2-Q MICHIGAN A(O) 0m iau(o) B (x A' ' A) YA A) I C(t) C(O) '(o) A V (t) x FIG. 15: GEOMETRY FOR AIRPLANE WITH XDIRECTED VELOCITY, V. FIG. 15: GEOMETRY FOR AIRPLANE WITH X-DIRECTED VELOCITY, V. A z 34

THE UNIVERSITY OF MICHIGAN 10 82-2-Q for CW transmitted signal. Denoting this ratio by F2(t) and recalling the expression for the transmitted Guassian pattern function, we have c2(@M+a(^O)ec2OMtan2(O) aexp a2(t)-b2 t) p2(t)= CL____ w___________.-. -- -- sW2( t)4 2(t(oM ())ooa sAolhi@ At4(O)l +s ec2( M^o~r(o))+sec Mtan2(0O) (B. 1) Assuming that S(t) is small for some small values of t, the exponential term dominates the expression for F(t) and (B. 1) reduces to F(t) _~ exp 'aa2(t)-b2(t). (B. 2) A1BO a(t) - S(t)cos20MCOSA 3(t) S S(t)cos OM OA From Eq. (B. 2) we find that the power level at a fixed point'(X', Y', Z') at time t is 8.7 [a2coes2M oospA+ b2sin2Aoos2 M8 2(t) (B. 3) db below the power level at t = 0. If (X%, Y', Z') is chosen such that at t= 0, a(0)=3(0)=0 we find from (B. 1) that the power level at time t is 8. 72cos2 M+b2sin2A] cos2OMS2(t)+ 10 log R2(0)-0 log R2(t) (B. 4) db below the power level at t = 0. Figure 16 shows plots of the expressions (B. 3) and (B. 4) against S(t). It is noted that the term 10 log R2(0)-10 log R2(t) is very small because the plot of the expression for the power in (B. 3) is almost identical to that for (B. 4). That is to 35

CA3 )C C N CoA a) \ O Normalized Power (db down on power at t=0) j I I-iI I I I 410 I I I I H z Pd rrl Cd) ~0 I m I,. —OO" 0 ITn. C) z \ Curve computed for expression (B. 4) Curve computed for expression (B. 3) FIG. 16: TEMPORAL 0M 1500, VARIATION OF ELECTRIC FIELD STRENGTH OM - 300 C 0c el OF n — I

THE UNIVERSITY OF MICHIGAN 1082-2-Q say, for the oase chosen, the approximation made above holds. From Fig. 15 the time an aircraft is 'visible' to an observer, can be approximately computed. For example, if the signal can be detected at 30 db below the maximum received signal and M = 150~ and OM = 30~ it is found that the normalized time 'visible' is So = 0. 44 and the actual time is given by (Z +h)S t= V (B. 5) V Thus, for an aircraft travelling at 200 knots, an observer at 10, 000 ft and an aircraft 70, 000 ft, the time 'visible' will be 10. 9 sees. Equation (B. 5) predicts the expected conclusions that the time 'visible' decreases with lower airplane altitudes and with -higher airplane velocities. )

UNCLASSIFIED Security Classification DOCUMENT CONTROL DATA - R & D (Security classification of title, body of abstract and ilhdldxilng annotation ntult be entered when the overall report Is clas fllied) 1. ORIGINATING ACTIVITY (Corporate author) 2a. REPORT SECURITY CLASSIFICATION The University of Michigan Radiation Laboratory, Dept. of UNCLASSIFIED Electrical Engineering, 201 Catherine Street, 2b. GROUP Ann Arbor, Michigan 48108 l 3. REPORT TITLE DOPPLER RADIATION STUDY 4. DESCRIPTIVE NOTES (Type of report and inclusive dates) Quarterly Report No. 2 1 October 1967 - 1 January 1968 5. AU THOR(S) (First name, middle initial, last name) Chiao-Min Chu, Joseph E. Ferris and Andrew M. Lugg 6. REPORT DATE 7mi. TOTAL NO. OF PAGES 7b. NO. OF REFS 15 January 1968 37 8a. CONTRACT OR GRANT NO. 91. ORIGINATOR'S REPORT NUMBER(S) N62269-67-C-0545 1082-2-Q b. PROJECT NO. c. )h. OTHER REPORT NO(S) (Any other numbers thot many he assirncd this report) d. 10. DISTRIBUTION STATEMENT Requests for this document should be directed to NADC, Johnsville, Warminster, PA 18974 11. SUPPLEMENTARY NOTES 12. SPONSORING MILITARY ACTIVITY U. S. Naval Air Development Center hnsville, arminster, PA 18974 13. ABSTRACT In this, the Second Quarterly Report on "Doppler Radiation Study", some results of experimental and theoretical investigation are reported. The measured antenna pattern data for an AN/APN-153 system obtained during the last quarter has been put into digital,three-dimensional format. This digital data will be used in the theoretical calculation of the direct and reflected radiation. In the theoretical study, a numerical scheme for calculating the spatial and temporal variations of the reflected radiation, based on a perfectly reflecting ground, is formulated. Schemes for presenting the numerical results in terms of 'normalized coordinates' are also considered. For the minimum detectable signal and gain of the possible receiving system, the approximate signal ranges are calculated. 1, o- A A I' rs ~ r UU 1 NOV T14 / UNCLASSIFIED Stcuriv Cl.s. i f t ion

UNCLASSIFIED Security Classification, 4. 14. LINK A LINK B L INK C OKEY LE W R OLE I WT I - E A IT ROLL |, t Doppler Navigational Systems Airborne Radar Radiation Patterns Geometrical Optics IR.1 I I I I I I lb - - -t- -.- -- 1..- -I --- I I L__ lt -m UNCLASSIFIED - c ---urt. — - ~l t'-i o S I fi ~ ~ 1) 1