6' (417 / / / / UNI VERS ITY OF MI CH I GA N C(O~7~T ~ 4 %JMR L 040 (31 N OJI?. 2)476-1..F 18 April 1956 Final Report on 2476-1-F - RL-2053 I' \9~ 7, a~ A ANALYS IS OF PADAR AND ITS MflIOA1T ion T. 13. ("Ourtz, M. L. IBarasoh, H.s Dryoks To A. Kaplan, As. E. Kleinman, lv. C. Orthwoin, and H. Weil Purchase Order 130-35 uM~w Air loroo Coiftraot AF 33(616),.31M6 hOP;94j Ckk~I0 Sbjctimo a THE UNIVERSITY OF?4ICHtOAN Enigineering Rtesearch Inetift~ti Willow Run Laboratories Ypsilanti, Michiga I /I DWGA Al 12"~YEAR no Thinm 1d me-'0~ons793 and 79) INTERVALS; NV ~ J; y GLV *ion 01 It$ COMOer OBGLASSIFIED. Qa -UR 520i1 -, - 6-..j.L .1, ;__ . -7-117-7 — I.18 U'. IL C., ton or t* so '9 *n I j - a" minummmsm~~Wm~SS1mUM

UNIVERSITY OF MICHIGAN TABLE OF CONTENTS ITEM ] Nomenclature Text Introduction Ranging Capabilities of PADAR Tactical Considerations Appendix 1 - Bistatic Radar Cross-Sections of the F-86D at X-band Appendix 2 - Basic Considerations Relating to PADAR Appendix 3 - Further Applications of PADAR References - I, - I C_- I_- r PAGE i 1 3 5 6 21 42 65

UNIVERSITY OF MICHIGAN NOMENCLATURE I G M N 0 P P1 p P2 R T h h h2 ht r u = antenna gain missile position point of reflection on earth, Itbounce point" = center of earth PADAR receiver position = peak transmitted power = minimum detectable received power = radius of earth = transmitter position velocity of light =true height of PADAR receiver = height of PADAR receiver (flat earth) height of transmitter (flat earth) = height of PADAR receiver minus height of transmitter distance from PADAR reoeiver, P, to "bounce point", N distance from transmitter, T, to "bounce point", N distance from transmitter, Ts to PADAR receiver, P 1 r v 1- cos o = angle between 7 and the normal to TO measured positive countero loc kwise = angle between the horizontal (at the "bounce point") ant ray to the PADAR receiver ~ = angle between the horizontal (at the "bounce point") an( ray to the transmitter d the d the -- ---- -- r --- — I i I pmmw —L,~; I If~r_ &^UMM C.mmm I- - 5w- =v l

UNIVERS ITY OF MICHIGAN = angle from h (height of PADAR) to r (transmitter-receiver direction), measured positive clockwise = angle between Al (receiver-reflector direction) and -2l (transmitter-reflector direction) (f = ct t = path length difference between direct and reflected paths 6 == a higher order error term in Taylor series for range error 9 = angle from horizontal (at receiver) to transmitter direction measured positive counterclockwise 9 - angle between the horizontal at the "bounce point" and r (transmitter-receiver direction) measured positive counterclockwise X. = wavelength 0- = radar cross-section = angle between r (transmitter-receiver direction) and e (receiver-"bounce pointt direction) = angle measured from the earth's radius at the "bounce point" to TO, measured positive counterclockwise 6o s= angle from VO, normal to earth from PADAR receiver, to 1 (receiver-"bounce point" direction), measured positive counterclockwise [r, ] = angular coordinates of transmitter for cross-section calculationsdefined in Appendix 1. The remaining notation used in the Appendix is entirely contained therein and is defined when used. At = time difference required for energy to travel from transmitter to receiver over two different paths Lxi = error in xi where xi is a measured variable, e.g., G, hl,... &xi = standard deviation of error in xi _ _ _ _ _- -— i ii _____i__ _ __ i -

UNIVERSITY OF MICHIGAN ANALYSIS OF PADAR AND ITS MODIFICATIONS Introduction PADAR came into existence as the result of Fairchild's effort to fulfill the need in fighter-fighter and fighter-bomber duels for the U. S. aircraft to obtain range information about the enemy at greater distances, Their idea is based on the comparison of a direct and a reflected path. In Appendix 2 the following considerations for this range finding system will be analyzed: general slope and roughness of the terrain, the curvature of the earth, propagation effects and measurement errors. As discussed below, some modifications toward greater complexity of the system are necessary for successful ranging under operational conditions; limitations remain. It has been suggested that bombers might use a method based on the Fairchild idea for defense against missiles of the beam-rider type. The University of Michigan was asked to examine this possibility under the assumption that the enemy defensive system would be similar to Tales. The results of Appendix 3 show that in principle this modification of PADAR will usually give accurate passive range information at greater distances than active radar could obtain if both signals are detectable, It is strongly recommended that study of the application of PADAR to the bomber defense problem be continued., -. Fairohildts development of this idea is discussed in References 1 to 5. 1 ----- RX- [ [ [[

J [TV: l W I1 U NIVERSITY OF MICHIGAN We do not recommend immediate implementation, because of the following points (a) When the missile is on the transmitter-PADAR line of sight, no range determination can be made by PADAR methods. (b) We lack radar cross-sections of enemy missiles. (The bistatic cross-section needed is at least 1 to 10 square feet for typical interception trajectories. It is likely that all enemy missiles will have cross-sections of this order of magnitude; however, a few calculations should be done to verify this.) (c) An analysis of enemy radar patterns is required to determine whether sufficient side lobe energy will exist to permit PADAR ranging, (d) We do not know whether there is actually a tactical advantage in the use of PADAR over presently conceived active radar defense systems against beam-rider missiles. An experimental check of some of the concepts involved has been suggested by WADOC using two F-86D aircraft and the present Fairohild PADAR equipment. The very poor accuracy to be expected in such an experiment and the probability of detecting the reflecting F-86D are discussed in Appendix 3. We do not recommnend that this experiment be performed, The F-86D orois-@eotions used in this discussion are given in Appendix 1. I. I I 2 1 fmml[ C =

I C- A l 1 r UNI VERS ITY OF MICH I GAN Ranging Capabilities of PADAR 1. PADAR, as presently described in the Fairchild reports and using the equations given therein, is capable of successful ranging provided: a) The ranges are fairly short (of the order of 80 miles). b) The terrain at the specular reflection point does not have large scale irregularities, o) The terrain has an average slope which is perpendicular to the earth's radius. d) Sufficient data smoothing is used. 2, The limitation to short ranges can be removed by inserting the data in the equations of Appendix 2 (2.3.1) which take the curvature of the earth into account exactly. 3. Limitation l.o) and, in some instances, limitation l.b) can be removed if, in addition to the measurements proposed for PADAR, the angle between the horizontal plane at the PADAR and the ray from the specular reflection point to the PADAR is measured. The data are then to be used in the equations of Appendix 2 (2*5). 4. Over very rugged or heavily built up terrain, PADAR should not be expected to give usable range data. 3

E== l I1 UNIVERS ITY OF MICHIGAN N 5, Optimum conditions for application of PADAR to detecting and ranging on airborne reflectors ocbur when the angle between the reflectorPADAR direction and the transmitter-PADAR direction is large, and if the altitude difference between the PADAR plane and the transmitter is a maximum. To obtain range againstn n airborne reflecting target it is necessary to measure an additional angle over PADAR. go h C - r m-_ a r\N.l

UNIVE R S ITY OF M I CHIGAN Tactical Considerations The purpose of this investigation has been consideration of problems relating to the feasibility of determining range by PADAR in certain given situations. Range information on enemy aircraft or missiles is clearly always potentially useful. At the same time, a number of important questions concerning the tactical use of PADAR range information naturally suggest themselves. Although no effort will be made here to answer these questions, their statement may prove helpful in evaluating the usefulness of PADAR. 1) Passive detection can work at ranges considerably beyond the detection range of ordinary airborne radar, With PADAR, passive range information may also be obtained with the accuracies indicated below. This presupposes, however, that the enemy has a pulsed transmitter operating; one must examine under what circumstances this will be true. 2) If the detection range of the enemy radar can be estimated with reasonable accuracy, passive range information can be used until the PADAR aircraft is known to be detected. After this, is there any advantage in using PADAR rather than a more accurate active radar? 3) Is the additional information worth the possible weight, space, and cost penalties? 4) In the beam-rider missile case, the missile will presumably be launched at ranges short enough to preclude evasive action by the bomber. Can successful countermeasures be taken at ranges greater than those attained by active radar? If the answer to this question is "lnol", do other possible advantages of remaining passive outweigh the greater accuracy attainable in active operation? I — -- X~PIl I

MI C iG U NIVERSITY OF MICHIGAN APPENDIX 1 BISTATIC RADAR CROSS-SECTIONS OF THE F-86D AT X-BAND 1.1 The Cross-Sections Required One of the problems specified in this contract was the determination of some of the bistatic cross-section values of the F-86D. This information is employed in the analysis of the proposed experiment described in Appendix 3 (3.3). In this Appendix we shall discuss the derivation of these cross-sections, and present the results obtained. A bistatic radar cross-section is defined as one for which the transmitter and receiver are not coincident. In this problem, the receiver is nose-on to the F-86D while the direction to the transmitter lies on one of a set of cones about the rear of the plane, These cones have halfangles from 200 to 60~, varying by 10~ steps. Further, horizontal polarization is specified for both transmitter and receiver, that is, the transmitted E vector is parallel to the plane of the wings, and the receiving antenna accepts only this polarization. Finally, an X-band frequency of 9400 Mc, or wavelength of 0.105 feet was used. For a comparison with experiment of theoretical results obtained by the method employed, see Reference 6. A full discussion of geometric and physical optics methods is also given in Reference 6. 1.2 Break-up of Plane As a first step in obtaining the cross-sections, the airplane is simulated by a set of components of simple geometric shapes, whose dimensions and orientations are recorded. A random phase argument enables us to 6 6 (m 7 = lr - _ --

U N I V UNIVERS ITY OF MICHIGAN treat these components as independent scatterers. Their separate crosssections are therefore obtained and added, taking shadowing into account. We now present the break-up which was used. The wings were considered as truncated elliptic cones having dimensions as shown in Figure 1.2-1 b L1^ -— =Figure 1.2-1 L1= 255, L2 = 470", a = 4611, = a/b = 6 and a sweepback angle d = 35. Angle of attack and dihedral were taken as zero. The horizontal and vertical tail surfaces, like the wings, were modeled by truncated elliptic cones. The dimensions are: ____~7 __

I UIUNIVERSITY OF MICHIGAN _ I (a) Horizontal tail surfaces La a 80% L2 160c, a " 19n, 6 and a sweepback angle a0 280 (b) Vertical tail surface LI, 55- t L2 - 1 a * 31% Y" 7 and a setback angle from the vertical of d " 35~ The nose of the fuselage was treated as a prolate spheroid and an elliptic oylinder combined as indicated in Figure 1.2-2, Figure 1,2-2 Here, d " 49H9, d2 a 48, d3 20", d * 89", d * 24t, d6 * 62". I 8 - - i~- E: I- C I

I UNIVERSITY OF MICHIGAN The midsection of the fuselage, measured from the front end of the canopy to the point at which the trailing edge of the wing joins the fuselage, was considered the exposed surface of two intersecting elliptic cylinders, as in Figure 1.2-3. t- 1- L1 - -bl Figure 1.2-3 For this component a1 = 48", bl = 29", a2 = 27.5,s b2 = 5.5 L1 = 180, L2 68.". Only the front half of the wing tanks contributes. These were modeled by prolate spheroids whose dimensions are (see Fig. 1.2-4): M I me. L, — on I 1L L mm 51", L2 = 12" Figure 1.2-4 9 - C= ~lr Ill I Ill

-.U N I VERS ITY OF M I CH I GAN All dimensions were obtained from the configuration drawing contained in "Standard Aircraft Characteristics", Wright Air Development Center, 3 August 1951. This drawing was neither large nor detailed, so that we were forced to estimate certain dimensions not explicitly given. 1.3 Shadowing The cross-section formulas to be employed are all either of physical or geometric optics type. In the former case, it is the dimensions of the illuminated area of the component which appear in the expressions, while in the latter, a contribution to the cross-section is obtained only if the so-called "stationary-phase point" is illuminated. Thereforeit was necessary to determine how components shadow others as a function of transmitter aspect. 1.4 Cross-Section Formulas The prolate spheroid cross-section was obtained from the expression 2 2 2 (cosQR+cos;T) +(sineRsin0R+sinQTsin0T) (sinGRcosR+sinGTcosT) (cos +cosQT)2+(sin2sin2+sin@Tsin0T) 2+ (sin9cos0&+sinQTcos0T) 2 Here a is the semi-major axis, b the semi-minor axis, and 7= a/b. 9 and 0 are the usual spherical coordinates of the vectors to the transmitter (T) and receiver (R), as illustrated in Figure 1.5-1. The Z-axis is taken vertical; the X-axis points forward through the nose of the plane. We are 10 -- lO..c...r I LiI

UNIVERSITY OF MICHIGAN therefore interested only in the case 9R = 77/2, 2 R = 0. The spheroid is assumed horizontal, with its nose pointing forward. This is a geometric optics formula. That is, it is equal to f R1R2, where R1 and R2 are the principal radii of curvature at the specular reflection point corresponding to a choice of transmitter and receiver aspect. Such an expression is valid for short wavelengths such as we have here. It is independent of transmitter and receiver polarizations. The contribution it yields should be used only when the specular point is not shadowed. For the elliptic cylinders, physical optics cross-sections, which are polarization-dependent, were employed. The expressions are: For the lower middle fuselage and the lower front of the fuselage 4? cos4 gT cos2 2T a2 b2 sin2 L CT(H,H) = /2 2 T a2 sin2 T sin2 aT + b2Cos2 where D = l+sineTcos0r, a = semi-major axis, b = semi-minor axis, and L = length of the cylinder, \ = wavelength and k = 21T/. T and OT are the usual aspect angles. R = tr/2 and lR = 0 have been inserted explicitly. For the upper middle fuselage, which is rotated through t/2 about its axis from the orientation of the others, a and b are interchanged in the expression, so that 11

UNIVERSITY OF MICHIGAN os4 T c02 0T a2 b2 in2 kD o'(H,H) h-? —o-_43 tr [2cos2 T + b2sin2:/ sin2 T D2 i? is very large, being of the order of 200Tr to 400rr. Therefore, 2 any slight error in specifying the aspect, which corresponds to a small uncertainty in D, may change the argument of sin2 kL significantly, 2 Our procedure is then to average the sin2 kDL to 1/2, yielding a 2 oross-seotion value, which, assuming the validity of the magnitude predicted by physical optics, would be approached by the average value of a large number of observations "at" each aspect considered. For the wings and tail surfaces, which we represent by truncated elliptic cones, we use again a physioal optics formula in which an avtraging has likewise been performed. For the starboard wing and starboard horizontal tail, 2, 2(L +L2)t an(.[. * d] - [( o no J a J] k(A2+ 2B2)3/2 (tan R 2 + 232 + C ) wherein k 2 L1 and L2 are the two truncation lengths, measured from the vertex of the cone, tan /3 L and tan e( =, where b and a are the semi-minor and L2 L2 semi-major axes of the ellipse at L2, Y = tanoC/tan(3 12 I -rPI I

, UNIVERS ITY OF MICHIGAN -.A = cos GR + cos 9T, B = sin 0R cos (0 +&) + sin 9T cos (~0 +J), C = sin 9R sin (Oa + d) + sin &j sin (Or~ + d), & is the sweepback angle from broadside, no = X C0s 0 + Y sin 9R cos(0R +g) + Z sin 9R sin(0R +g), A A A ^ ^ o A a = - sin 9T X + Y cos 9r cos) + sin(Wr +J), A A A A d = - sin 9R x + Y cos a oCos(0R +~) + Z oos 9R sin(~ +0), N = X A + B - tan A2 + 2B For the port wing and port horizontal tail, 6 is replaced by I7 - everywhere in this expression. For the vertical tail, we use the same form with the following replacements: A = sin 9T sin 0T + sin R sin 0R, B (cos ( O + cos PT) sinc+ (sin 9 s 0, + sin '9 ) 00 s 0 - (cos @R + co 9, c0os & - (sin 9& cos $Z + sin c oos 0 ) sin is the sweepbaok from the vertical, ' ho - X sin 9 sin A + Y [ o-, 9, sin d + snu C os ~~ 00os $ + Z [oos 9, ooa - Bin Q cos PR sin dj, ( c= osQ@Rinr,-t [ tiuJ^B &-ao8UOS }os 2,JZ[sin{rcosi+cos9^ossin4 A C C oosQsi in$ -Z VOinftuinfosUo scooon J os4 o n4 '-1 sntationary phags evaluation whioh yields these elliptic cone oroB;,sreotio.:, pressions wai found to be valid for A 200 although it is not I I1~ l u i'rther baok. __ ---- _ — 13 —.k ~ - fc f= - r - - - I, A.s;,, = - IIITI i,1-, - j ___ j ^ ^ j j" "" I jt:-9; I r 1: =. i

UNIVE RS ITY OF MI CHIGAN Finally, the shadowing procedure used for elliptic cone surfaces will be explained. The physical optics formulation used prescribes an integration over the illuminated surface only. The contribution to the crosssection depends explicitly upon the limit of integration corresponding to the edge of, the shadow. However, the surface currents at the shadow boundary really go continuously to zero, and do not, therefore, contribute to the cross-section like the currents at the tip of a body. For this reason we have treated the factor L1 + L2 in the following way, which should be a satisfactory approximation: 1. If both ends of the body are illuminated, we use the full value L1 + L2. 2. If one end is shadowed, only the L corresponding to the other end is used. 3. If both ends are shadowed, the contribution to the cross-section is negligible. 1.5 Results In this section curves for G (H,H) of the F-8Dxlre given in Figure 1.5-2 through 1.5-6. The cross-section in square feet is plotted as a function of 5 (the rotation angle around the cone) for fixed values of.t, the cone half-angle. The definition of these angles may be clarified by the following sketch. 14 r_- r- Lr —_ - i r

L-, r--, C — r- a a r-—.i U N I V E R S I T Y UNVRIYOF MICHI-GAN - 7 200i 6! 6&0 / / 7 / rf 4. gurt-,3,, 1*5,ol 15 C.. I

I' E=7r - U N I VERSITY OF M ICH IGAN - b i. i I~~~ ~ I l l t.20' I I I I I, I, -. II l l l a %....... N....... ' ii X I I / 102 10:1 ^ Bistatic position 4U ou OU 1Uo lie 4U I.OU jOU I (Degrees) Figure 1.5-2 cross-section of the F86-D, as a function of on the "transmitter cone" for receiver nose-on. - - 16 I I l l r

UNIVE Y - — I I. I — U NIVERSITY OF MICHIGAN '10 N \1% L4 102..L... t.. S.. _,............- - --,.... 1=300 - - - - - I - - -i i i i - -- jl i -- -,. -- 4~~~~ \............. —,. ---.... __.. _... I, \ / 0 \,.. I II I II I'. WI I-..W.^I A,. I! In On }~~~~~.n r Q n e 1.....e n 10 I In v c-. LU4v uo uu JLUV jLZU S (Degrees) JL4U JQOU OLU Figure 1.5-3 Bistatic cross-section of the PF6-D, as a function of position on the "transmitter cone" for receiver nose-on. 17. I r - -. r m c m"-'1'1-...

[LTLLF, =-11'! lr\\ Or I > i I I. UNIVERSITY OF 10. =o 10 - b -, U =l 1-., MICHIGAN I 5 (Degrees) Figure 1.5-4 Bistatic cross-section of the F86-D, as a function of position on the "transmitter cone" for receiver nose-on. ---- 18 - c lFEi r- I

ILLF7 L- r I \ 77 U N I V E R S I T Y -- U V OF MICHIGAN 10-i 102 K 6 'Ssol 1I0 1. 0 20 40 60 00 100 120 (Degrees) JJ14 160 I0o Figure 1.5-5 Bistatic cross-section of the F86-D, as a function of position on the "transmitter cone" for receiver nose-on, 19 I

c,-r =1 77 U NI VE R SI TY ~,j.3 4 I,..I OF MI CH IGAN I-u io 2 N 1.0!u600 - m m m m m - - - -Ftmm m m - A i el t ik Id W( L4U ou CIO log~J 12( *3 (Dogream) Figure l#5-6 a 7UI4U itio 4 Bistatic Conao seotion of the F!86mDO as a ~\motion of position on the,lU~r~nsmittar cone"' for receiver' none-son. 20 l j '=11=-inIr\L.H

UNIVERSITY OF MICHIGAN APPENDIX 2 BASIC CONSIDERATIONS RELATING TO PADAR 2.1 Prelimina Results and PADAR Equations Fairchild's present experimental set-up differs from PADAR as described in References 1 through 4 in that the target elevation angle is not measured and the target plane flies at a known altitude. Figure 2.1-1, which is reproduced from Reference 5, shows a typical A-scope presentation photographed during one of the first runs. The sharp, nearly vertical, leading edge of the pulse makes possible accurate reading of the time delay. The available data show that the errors in range presently being obtained can be tolerated. It will be observed that the experiments do not involve the angular measurement which would actually be required in practice. However, the analysis of ranging error shows that as range increases so does the importance of error in the angle measurement, and it will eventually be the dominant source of error for ranges that should be of interest. Figure 2.1-1 21 PI~j~mn I =.Mo C=r

r~ UNIVERSITY OF MICHIGAN The analysis of ranging error also shows that the range accuracy attainable at long ranges can be greatly improved by adequate consideration of the curvature of the earth. To determine the PADAR range equation consider Figure 2.1-2. P represents the PADAR receiver and T the transmitter, whose existence and range is to be determined. The dotted line is parallel to the ground and the reflection is assumed to be specular (ol =OC2)' Once T is detected, the information measured by the PADAR is;hhl, and At= ( + - r)/c, where c is the velocity of light. To obtain r from this data note that * Figure 2,1-2 22 - = v C=

LL= 1K mI ro > UNIVE R S I TY OF MICHIGAN I 41 s in cp(1 + h1 + r sin 91 sin~C1 (2.1.1) (2.1.2) od c At = + r + hi + r sin -r A2- r = +............. - r, sino < sinoC1 r cos h + h r sin 1 r cos 9 - =+1 +- n 1 +. onCi'. +., o,f (2.1.3). V CLl - 1 MU A11 1 From (2.1.2) and (2.1.3) we get Gcos C<1 r os - =) (cos 9s - cos <1) (2.1.4) which may be converted to the form used by Fairchild, namely 2 hl(hl + r sin 1) 1_ CC 2 (2.l.5) 2,2 Accuracy of Range Determinations In the measurement of hi, ^, and 01 there are unavoidable errors. Their effect may be expressed by the first order terms of a Taylor expansion of r, namely r=r ao + a h Ahh+r hl + cf (2.2.1) 23 - -- - -- -- --- - -- ----- --- rI1-L \ -

UNIVERSITY OF MICHIGAN provided the remainder is negligible. Here Ar is the error in r due to errors in 91@ f, and h1. If A\, Ao1, and6hl are considered to be random errors, their standard deviations 0 ^ Ca, and hlare the quantities of most interest. If we assume that the errors in 91, d, and hI are mutually independent we have (2 2 h r 2 6 r ( l) ((2.2.2) ar01 +hl If the remainder after the first order terms in the Taylor expansion is not negligible, Ar cannot be linearly related to the errors in the input variables and the situation is obviously much more complex. In general it is necessary to examine the remainder and determine for what ranges of the variables it is negligible. In most instances in the subsequent error analyses it will prove adequate to consider only ranges of the variables where the linear relationship is valid. The remainder in the expansion (2.2.1) can be written as F 2r 2r 2r_ 7 ^$)( 1)+ * ((-)P (Ahl) +(* )(,. +2 L2, A, + 2 2Ah 2 - AAh 1 a 2 lhlh s 9l a hi ~1^1> 0 24 %Nma C =l l

UNIVERSITY OF MICHIGAN GA N In this expression the derivatives are to be evaluated at some point 91, h1, satisfying 91- 015 91 + A91 h1 h1 _ h1 + ^ hi but not otherwise determinable. It is well to note that the sensitivity of the range to a given parameter as exemplified by the partial derivatives is a function not only of the geometrical configuration as shown in Figure 2.1-2 but of the choice of variables to be measured for the determination of r. A simple example which very clearly illustrates the importance of this point in an error analysis, and which also illustrates the determination of a region of validity for a linear error relation, is the following. Suppose we are given the right triangle shown in Figure 2.2-1 and are asked to determine the length a by measuring b and 9 or b and c. Let us assume our only errors occur in measuring b. In the first case we must use the equation a = b tan 9 and the error Figure 2.2-1 ------------- 25 ---------- i,....

UNIVERSITY OF MICHIGAN analysis yields ha = Ab tan Q, The remainder is zero. In the second case a =c2 -b2 and a - Ab cot Q l + b, bcbb +Ab. "42 2 We shall assume that any measurement in which b exceeds c is automatically rejected. Then the condition that the magnitude of the remainder be negligible leads to sin2 J 9 b Subject to this inequality we have a = - Ab cot Q. The formulas show formally that the effect of errors in the measurement of b and either 9 or c are serious for small 0 in the second case but not in the first case; conversely for large 9. That this should be expected is evident from the fact that for small 9, tan 9 is a slowly varying function of 9 but c and b are nearly equal so that in the second case one is taking the difference of two nearly equal numbers. On the other hand for 9 near 90~, tan 9 is very rapidly varying but b and o are quite unequal. Hence, from Equation (2.1.5) 1 26 __I [C"= l- - I 1001 r"". C 1

L Le oll 7N FE 7K I- U IUNIVE RS ITY OF MI CHIGA N Ar^ r2cosG hi 1+ A - hi -1 0 -cosQ1- 2 tanl-~ -ha r 2 h 2 hi -^ hv (2H+l) (2.2.3) 2rh 1 ---h2 hi h [1 Ahl H + -- I r r h" 2hi2 -aJ6 - 41.2 -2 h* an 2 1 0 - 2hi + r sin =1 hl + r sin 1 Y 2hi + r sin w1 In these equations h 1=.1.., H = h...... h = 1 sin 2 hi 9 ) hi + r sin 91. ^ H =, and r = r(Ql,hl,6 ). In order to achieve h this relatively approximately "' 2 -2 simple form we have assumed that r cos 2rh hi h1 constant in the region from 91, hly to 1 +A r2 and - were 2h12 i1l hI +hl, and 6+ A. This requires that in these intervals h1 is bounded away from Oand Q1 from 90~. In addition it was assumed that 2 << 4h 2. Conditions under which the expansion is simply linear in AG1, h1, and AS are (a) 2Ah) j1 1 << ----1 -— % 2 -i hi 2 (b) rA<1 - r + <1 hi- hi-0 (2.2.4) (0, — I h, (2f+1) << 1 ( A hi (d)t H2 - < 1 2 r I 1 (e) ~0 1 ~ cos 01 — tan O, 1[ (e) 'A r O tan 1 h1 2 l. - 27 ' j i - -~ AG, |y(91;?/hl) | c 1. I ~ - -[ r=E —n- =

UNIVERSITY OF MICHIGAN With any reasonable height measurements condition (a) is always satisfied. From the basic geometry we can certainly expect that I^r < r so that b may be replaced by (b') AS i 2r which requires that Abe more accurately measured as range increases for the first order approximation to hold. For 91 = 0, however, it is not a very restrictive bound, becoming Ac 1 Next consider (e). The function Y(91) y(1; r/hl), Y= r cos 9 - tan 01 (see Fig. 2.2-3) hl 2 h1 T ^^* ----J1 -f,/2 I m Figure 2.2-3 28 F- E n -r 3~~ X C a Cl F

U NIVERS ITY OF MICHIGAN is monotone decreasing as shown. For Gi~O it crosses the Y 0 O axis at ~ arcsin (1 ) * 4r ~ Also we have Y(-.9) = h 008 o = h(- ). If 1 |< 0o = arasin (1 - ) 4r and 2r h( condition (e) is satisfied. Now r < 1 so that 9o>480 always generally it is much larger. Furthermore, let us say r->30~ then the second hl inequality yields A Q4<4.04 rad, another restriction which should always be satisfied in any practical situation, but which may be lightened for smaller r, h 2.3 Effect of the Curvature of the Earth Before making use of the expression obtained for the error in the preceding section we discuss the effect of neglecting the curvature of the earth, ieferring to Y'igure 2*3l we see that one effeot of neglecting the angle Y (i.e., assumingy is zero) is the same as making an error in determining ",* 29 L[:ml

r- c JF~~E r UNIVE RS ITY OF MICHIGAN T I r Frigure 2 *)'4 30, ___m r=3 C= - r"""- CM.OM -

- - UNIVE RS ITY OF MI CHIGAN If \ is not taken into account the resulting computed range is always too short by an amount 2 2 r drr -- Lr 2=_ (2.3.1) h h 2R 2Rh for small 1. R = radius of the earth. 1 Equations which take the curvature of the earth into account exactly are derived with reference to Figure 2.3-1 as follows: The quantities which are measurable in the PADAR aircraft are really 9 and h rather than 01 and h1. The quantities 91, hl, and~C1 are related to 9, h, and~Cby o9- Y+ o, (2.3.2) h = (R + h) cos W-R, (2.3.3) and =0 (- r- (2.3.4) In addition to these equations we have (R + hl) tan \ttan(<-Y) = h1. (2.3,5) Equations (2.1.2), (2.1.3), (2,3.2), (2.3.3), (2.3.4) and (2.3.5) are six independent equations involving the six unknowns, r, 9'l, XC, h1,<C and /; the parameters to be measured,6, 9, and h; and the known quantity R. Hence, in principle, they can be solved simultaneously and an equation obtained for r in terms of &, 9, h and R. However, 01 and ~C1 may be eliminated immediately leaving only four equations in the unknowns, r, hl, oC and y/. 31 it BLr

UNI VE RS ITY OF MICHIGAN 2.4 Magnitude of Errors 2.4.ol Bias Errors Due to Earth's Curvature An approximate determination of the effect of neglecting the earth's curvature is given in the following table, computed using Equation (2.3.1) and r and h combinations satisfying Equation (2.1.5) with At = 2.5/sec. h r r/h Ar miles miles r 1.9 1504 8.1.02 3.8 61.6 16.2.1 5.7 139 24.4.4 7.6 222 29.2.8 Table 2.4-1 2.402 Random Errors To get orders of magnitude to be expected for -r let us assume r Q0,o We shall assume that the altimeter reading for h will lead to -.01 Ol *o Let us assume that At is being read near the smallest h readable value so that we might expect 2-, ol. The accuracy of reading S will depend greatly on the steepness of the leading edge of the transmitted pulse and on the terrain roughness. There will be a minimum error This error will be larger over hilly terrain or cities with tall buildings. 32

_- UNIVERSITY OF MICHIGAN 10 of approximately - = 0014 rad in a stable table determination of the 4 horizontal, The accuracy of determining G, once a reference horizontal is fixed, is at best about 0 =.001 rad so that G( =.005 is an estimate of the accuracy in the @ measurement. We obtain the net result of these errors from the equations of (2.2) /1/2 ] r -r2 6 12 r =(h 25 x 10 + 4 x 104 +.01) This yields the results given in Table 2.4-2. r/h r 2.1 10.1 25 i.2 50 o.3 Table 2.4-2 The portion of these errors which is not due to error in the stable table determination of the horizontal can be further reduced by smoothing. However, at the longer ranges this unreducible angular error is the dominant source of the error. These errors give an idea of the accuracy which might be expected over fairly level horizontal terrain or moderately calm seas. 33 --— 3 ---

- UNIVERSITY OF MICHIGAN It should be pointed out that the derivatives such as c-E can be ah obtained from the equations for a curved earth by differentiating each of the final set of equations arrived at in Section 2.3 with respect to the independent variable in question, h, then solving the resulting set of four linear equations in _ il, j 1, a r and. In this way ah ah ah ah ah a more accurate but far more complicated expression than Equation (2,2.3) for the effects of errors can be obtained. 2.5 Terrain Considerations Over rolling, or flat but sloping, terrain the above numerical error analysis is not applicable. It is based on the ground level lying perpendicular to the radius of the earth, Any deviation from this plane is essentially a non-measurable error in 6 in addition to the ones onsi dered previously. It is therefore evident that quite small slopes of only 10 or 2~ can easily shoot the relative error in range above 100 per cent. If the ground is simply rough or rolling up and down on a short enough scale the error will very likely be essentially random but often with a non-zero mean which puts a bias on the results even when the fluctuations are smoothed out. Long paths along a smooth slope in one direction will similarly give a bias error, If PADAR is to be used with good results over sloping terrain (which would be quite common over land) it is neoessary to measure the 3l. IB[:l: r — =;

r 0 | r 1711 II 7K U N I V E R S I TY OF MICHIGA N angle oC. The new geometric situation is shown in Figure 2.5-1 * A straightforward generalization of the previous set of equations leads to the replacement of Equations (2.1.2) and (2.1.3) by S h + sinlC h + r sin r sino2 2 (2.5.1) T Figure 2.5-1 *In what follows the normal to the sloping surface has been chosen to be in the plane defined by P, N, and T. If the slope has a component perpendicular to this direction, the equations involving oC are no longer valid. However, the effect of such a component on range error is much less than that of a comparable component in the plane (for small slopes) and thus the equations derived will still be approximately correct. 35 XEC -M

UNIVERS ITY OF MICHIGAN hl h + r sin Q1 r cos Q,, - + -1..... (2.052) -tanoC1 tan oC2 Equations (2,3.2) through (2o3.5) still hold. These equations now involve the unknowns ra, Q1, 0 C~2' h, and A in tenms of R and the parameters &, 0, h, and ~C to be measured. In general an accurate measurement of rcC will be difficult and it is not easy to give an estimate of the effect of this error. The following relatively simple set of equations may be used instead of the above to determine range when oC is measured. In this case, the significant parameters are 0 6 + c and GJ ff - oc From Figure 2.5-2 we have T r P Figure 2,5-2 36 -ull

UNIVERS ITY OF MI CH IGA N (R + h) + 2 - 221 (R + h) cos J - R= 0, (2..3) + r2 2rl cos 2 = (2.5.4) Combining these with + 2 - r -&= O (2.5.5) yields three independent equations for the unknowns fl, ^~2' and r in terms of the measurable quantities O, J, h, and & and the parameter R. By implicit differentiation of these equations we find o;r r^1 sin 6 2 - (2.5.6) e2 r' +4 cos ~ (r (R + h) QlBinw(2 +^ - r cos )( aJ L[el-(R+h) oo8W]( 2-r+ 00 ) Lr - [-(h) +1ooa (+ l -, r eo oo) 5 2 h) -4 + CrO 000, r [t -R ( h) oos60(0 ) - r +a oo, 3 [ -(R+h)eoa on (2 - r + oos P) I 37 FX- Ir

U N I VE R S ITY O F -MI CH I GAN We shall compare the orders of magnitude of these derivatives with those obtained for a flat horizontal earth when 090 (see Eq. 2.2.3). We let $ be small, and Ur: - 0. Then Pf 2h/r. Also we shall assume,2 f2 41,r/2 (i.e.,o =oC2). Then we have r. 2 2R r2 h 5 ) rg 4 URh ar 4Rr r r r r2 r 16h3 2;4 R 3 hah r2-lRh h 2C 6 2h2 BR 3r(r2-LRh) A _ r ar - r r In the limit as R-.-oo, for 8 = 0 and 'l =oC=, these coefficients are of the same order of magnitude as, but not equal to, those given in Equation (2.2.3) for the same oase. In Equation (2,2.3) we had (for first order terms and 9~%0) r r ^+ Ah.. In the present case we obtain r = - 2h (^ +A) ' h 1 r 2h ~ - 38 XPE::Cl

U N UNIVE R S I T Y OF The reason for grouping the angular terms in +a +Aco= (0 +A) =D(1r + 9), is precisely 2 determined by the more complicated technique relative error equations gives LHir MICHIGAN ___, this fashion is that A Q We denote the range as as r. Then combining these. L A r 2h 2 r It will probably not be possible to measure 0 (which is the direction to a diffuse reflector) with nearly as good accuracy as can be obtained in measuring G (direction to a transmitter). We regard 9 and 0 errors as uncorrelated and obtain _ r r 2 r This shows that if the assumption concerning the accuracy with which 0 can be determined is correct, using r instead of r will usually lead to increased range errors for the flat earth, equal altitude case. It is worth mentioning that conventional antenna pointing methods such as conical scan should not be expected to give very good results in determining 0 since there will generally be no very clearly defined bright spot on the ground standing out from the surroundings and subtending an angle much smaller than is enclosed by the antenna half power points. The effect of index of refraction variations in the atmosphere in deteriorating the accuracy of oC measurements is discussed in the next 39,...~.t.-X... WSS 1X B O IM -

UNIVERSITY OF MICHIGAN section. Here we merely point out another source of error due to refraction; namely, refraction of rays through the radome. This, however, is to a considerable degree under the control of the PADAR equipment designer. 2.6 Effect of Index of Refraction Variations The index of refraction of the atmosphere is not a constant from point to point. The rays representing the direction of energy propagation are therefore not straight lines but bents Because the index of refraction is affected by variations in the temperature and moisture content of the air, it fluctuates considerably near the earth's surface. In the most extreme case, the phenomenon known as ducting occurs. This is a situation in which rays below about 1~ to 2~ from the horizontal are to a considerable extent trapped and guided along the earth's surface. There are large regions of the world where this situation is almost always present, others where it occurs at night only, and others where it occurs irregularly, Thus, it appears that PADAR will be unreliable for situations where the angle Oc must be measured, if the indirect ray is at angles ~ less than about one degree. Whether or not o4 is measured, poor results are to be expected from this source for extreme ranges, or for receiver and transmitter at low altitudes (i.e., whenever the direct ray comes close to grazing the earth), The characteristics of these random effects are undergoing investigation in various quarters. ------ 1o --

UNIVE RS ITY OF MI CHIGAN For greater C01, the ground layer effects will not predominate. At higher altitudes, fluctuations in the index of refraction tend to be less violent, though ducting is still present at times (and when it occurs may preclude ranging). On the average, the index of refraction falls off fairly continuously with altitude, though large fluctuations are possible. The data appear to be very scarce above a mile or so. The error introduced into the angle measurements by the bending of the rays becomes comparable with the non-refraction angle errors at ranges of the order of eighty miles. Since refraction tends to lengthen all the ray paths, direct and indirect, barring dusting the effect on swill usually be smalyl. This discussion is of necessity preliminary. Further analysis and correlation of data and muoh new experimentation would be required to chlear up the propagation picture. -- -- -- ------ ------ --- 41 f m mmmmlm PI iiri tI muST~l~t,,3'f

U NIVERSITY OF MICHIGAN APPENDIX 3 FURTHER APPtICATIONS OF PADAR 3.1 General Considerations The situations in which the PADAR technique may be used fall naturally into two broad classes. The first involves determination of the range from receiver to transmitter; the second, determination of range from receiver to reflector. The most promising application in the first class appears to be the original PADAR scheme which has been discussed at length in References 1 through 5 and in Appendix 2. The subject of this appendix is applications in which the range of a reflecting object is to be dtermined. Two such problems will be treated in some detail. 3.2 A2plioations of PADAR Against BsamRider Missile Systems 3.2.1 Statement of the Beam-Rider Range Problem The situation to be considered is illustrated in Figure 3.2-1. Consideration of the geometry involved shows that missile range information may be bought only at the price of an increase in the complexity of the system. Determining the range of the attacking missile involves the measurement of the two angles, and %, analogous to those of Appendix 2 (2.5), rather than the gt required in the simple "bounce off the ground" scheme discussed first in Appendix 2. Furthermore, although these are essentially the same measurements made in the modified PADAR technique dsoussed in Appendix 2 (2.5), the situation is further oomplicated due to the addition of a now dimension. In the 'bounce off the greund" - ---- 42 --- ~ lr

UNIVR ITY OF MICHIG 1 -U NI VE R S I T Y OF M I C H I G A N p M r R 9 Figure 3.2-1 Notes The plane defined by MPT is the one in which 0 is measured and it is generally not the vertical plane. 43 ~ ~~ r!~~'r _ _ _ 1..

UNIVERS ITY OF MICHIGAN problem, a two-dimensional picture sufficed to approximate the situation since the PADAR airplane, the transmitting airplane and the "bounce point" were assumed to lie in a vertical or nearly vertical plane. Now, the same measurements as before can be made to yield range information. However, an additional measurement is required if the direction of the missile is desired. It should be kept in mind that accurate angular measurement was shown to be the most important requirement for improved PADAR range determination. Because beam-rider missile systems involve both a tracking radar and a guidance transmitter the question of which of these is providing the PADAR energy must be considered in this application. After the incoming bomber is in the direct beam of the tracking radar, the missile is launched in the beam of a guidance transmitter and the guidance beam is brought into conjunction with the tracking beam. These radars will be assumed to be operating at different frequencies. Thus, tuning of the PADAR receiver eliminates confusion of the two. Since the beam-width of the tracking radar will generally be very narrow in comparison to the beam-width of the guidance radar, the utilization of the guidance radar in the PADAR technique, despite its lower power, seems to be necessary. Whichever radar is used it is quite possible that because of the absence of sufficiently powerful side lobes, no range determination can be made in time to take any effective action. Information as to the pattern of an enemy transmitter is essential in deciding whether PADAR should be developed for this application. In the absence of this information it will be assumed that 44' I = - I- I \ = - I 44. L. [

UNIVERSITY OF MICHIGAN the direct path signal from the transmitter is detectable at the PADAR receiver throughout the interception. We estimate the minimum missile cross-section such that the incoming bomber can detect the missile at any point along its trajectory. The radar range equation in the form (4 1r) 3(' )2. 2 (3.2.1) X2G1 G2 P1 where subscript 1 refers to the transmitter and subscript 2 to the receiver, indicates that for fixed values of the minimum detectable received power P2, the wavelength, the gains, and the peak transmitted power, an upper bound for the minimum (s will be obtained for the maximum value of (~1 Rt )23 We will now specialize to a particular beam-riding missile system, Talos. Choosing the parameters for the. missile guidance transmitter on the basis of information given in References 8 and 9 we obtain for the Talos system ' - (,)3(.gl g2)2 P2 2 (4r3-. 2 P2 ft2 (3.2.2) (.0625) 103 G2 15 A =.25 ft GL *103 P2 = kw., = ft., T2 = ft. Typical maximum values of the product of the ranges, 2}2 should be of _____IIIIIII —U_ — _ III II --- I....'...

UNIVERSITY OF MICHIGANthe order of 25 * 109 square feet. The parameters over which the PADAR designer has at least some degree of control are P2 and G2. It seems reasonable that a receiver with a gain of about 1000 and a minimum -17 detectable power of 10 Kw. could be produced. Very roughly then, the order of magnitude of the minimum cross-section to permit ranging the missile throughout its attack is of the order of 1 to 10 square feet. It should be pointed out that this result is far more sensitive to variation of the value of the range product than to that of the other parameters and that if the beam-rider missile has greater range capability than Talos this minimum cross-section increases greatly. In ordinary PADAR the reflected signal moves away from the direct signal on an A-scope presentation as the range decreases. However, in such applications as this one, the path difference which is being measured tends to zero; henoe on an A-soope the reflected signal moves toward the direct signal, This probably increases the difficulties arising from the presence of more than one target, A preliminary oaloulation indicates that it may be possible to use the Doppler effect to discriminate between different targets* 3,2.2 Error Analysis for the Talos System With the above general oonsiderations in mind we proceed to determine ranging aaouraoy for those oases where the angle ' between the transmitter and receiver direotiona, measured at the missile, is equal to, or greater than 1200~ The PADAR carrying aircraft is referred to herein as a bomber. 4I _I '- 46...... -: I C= I [EN C r I

U N I VE RS I TY OF MICHIGAN On the assumption that it is more important to have accurate range information at ranges under 100 miles, only such missile-bomber separations will be considered. Referring to Figures 2.5-2 and 3.2-1 we observe that the algebraic manipulation involved in obtaining the unknown distance is the same in both oases. Substituting / for j and interchanging r and 4 in Equation (2.5.3) gives us, with the help of Equations (2.5.4) and (2.5.5) which still apply, a system of equations from which 1 may be determined. ^4+ e2 r (3.2.3),22 j2 2 - 2 r^ ~o~ (3.2.4) R2 (R + h) + r2 2r(R + h) co/n (3.2.5) from whioh we obtain '1 t r 7oo ) ' (3.2.6) i2(6 + r r oo. ') Writing D+ (r+Ad)(2r + 2?rm+ ) (3 2.7) 2 +1 a +(r +Ar)( + A )] where v - 1 - 00 O, we obtain 47 XB[:1XBI

UNIVERSITY OF MICHIGAN c2(2-v) Ar + [J2 2vr(r +d -rrS(2r+S)Av 2(C+ vr)2 (1 +E) (3.2.7) + 2(6+ vr) ^rA +&( +vr) A 2 (2r +&) vb^ r 2(6+ vr)2(l + ) where = r v + vA r + A + v Ar 0+ rv Neglecting second order terms in the errors gives the first order terms of the Taylor series -2, 2(2-V) r +[j2+ 2vr(r +J)] r..r(2r +)Av (3.. 8) 2( 0~+ vr)2 It is necessary to examine the conditions under which. this is a good approximation. They are a L<< l, 1<< 1, r<< 1 and Av<< v + - r r r The last condition corresponds to bounding 0 away from zero. This is a very natural restriction, since when the missile is along the PADARtransmitter line of sight no range determination can be made. The bound we will use on 0 will be: 20 | Aj|<0 (which leads to v-lO0v). 60 contains only random errors and should be measurable with good equipment o to an accuracy of at least +.1. This leads to v>.001 corresponding to the fourth condition. In virtually all situations of interest, the 48.... _____l I|_I ---- ----

, — U NIVE RS ITY OF MICHI I GAN __ Taylor series adequately represents the error in range. Now Av = sin 0A0 - 4v(2-4v)' 0. Replacing d/r by u, A41 = 1 hl(uv) Ar 2 where - r h2(u,v) A0 + h (u,v)&,d, u (2 - v) (u + v)2 (3.2.9) h,(u,v) = h (u,v) = u(u + 2) 4 v(2-v) (u + v)2 hI(uv) 1 (U ) (u + v) Inigures 3.2-2, 3.2-3 and 3.2-4, h1, h2 and h^ are plotted as fnotions of u and v. Theoe figures may be used to obtain the value of the error in f or any assumed onfiguration and fixed errors in p,, and r, the measured quantities. Furthermore if we regard h1, h2, and h3 as the weights assigned to the errors in measurementthe three figures give an immediate qualitative pioture of the importance of oaoh of the error quantities. As before, the quantities in Zquation (3.2.9) are unoorrelatedj thus, the standard deviation of the distance 44 is 49. _ I

I,L- r--n C_-. U N I V E R S I T Y UNIERSTYOF MICHIGAN h1 (u, v) 2 I2. u I. Figure 3. 2-2 I I I i C= I

F r.r L-L+~~n\L-. ii h2 (Ulv) OF MICHIGAN A.-0 I — A-.4 K m*,o,-I0 v F~Lg-ure 3.-2 —3 51. L-EE (Fh E = rL~L- Wr\<L II

F I r iF h3(ulv) I / OF MICHIGAN FNote Reverseal of Axes Piprxe 3o,2,4 C= I- M r"\

!,j rr UNIVERSITY OF M I C H I G A N 1/2 2 2 h 2 -= 1 hli(u, v) + rh2 (u, v) l + [h3(uv) J9 (3.210) To give an explicit example of the sort of errors to be expected a few cases were computed. Here r was chosen to be 40 miles and 71 as 15 miles. For the errors, Gr = 1 mile, ( =.01 miles, and C =.1~. This is not a particularly favorable case. The results and the 0 and combinations used are shown in Table 3.2-1. 0 0 44 5~.091 1.1 10~.36.35 150.80.22 Table 3*2-1 In order to compare directly the errors in this case with those ooo l when the reflection is from the ground we rewrite A^ in the partial derivative notation. ^A ^1 r^r (3.2.11) ar +. h, (3.2.12)...... - I-r IL~ LCI

110 -C — r — 2F - UNIVERSITY OF MICHIGAN I and r =r(R + _h) _sinl"~.= 33 (R + h) cos,6-r ar _(a -+ h) z'r c os/ c)h (a + h) cos~, - r cs J 1 +~.j cosQ ( h (3.2,13) (3. 2.14) Or, analogous to the equation given in Appendix 2 (2.5) f'or the error h. O 40+ 04 iA + C)/ a)h Ah +- ai S I (3.o2.s15 ) where a 1 )o Qll = W C) r,P sin$0 C+ r - r Cos0' + COS 0 - JD~ (J~. r - r os 0 (3.o2 * 16 ) r(R + h) sinA [r -(a + h) c os~~ - (CF~- ki+4iCos 0) ( r - r c os 0) Er cos/~-(R +h)J V.0(R + h) c os/3] * (3.2.17) (3, 2.*18) (3.2.19) (3.w2.02 0) & + r - r cos 0' 24,2 2(1 + c Os 0) (44 2r)2 5)4 -

UNIVE RS ITY OF MI CHIGAN G A NIt can be seen that the coefficients in Equation (3.2.15) are generally smaller than the corresponding coefficients in Appendix 2 (2.5) In addition, the errors in this case are generally of the same order as those in Appendix 2 (2.5) and in particular the most important error, the one in /, is much smaller because in this case we measure a direction to the transmitter, in the other to a diffuse spot on the ground. Thus it is seen that Equation (3.2.12) yields much smaller range errors than does the corresponding equation of Appendix 2 (2.5) for all cases of interest where Equation (3.2.12) is valid. If we neglect 0/2 with respect to A~1, then Af1 ~12 sin sin (3.2.21) r~ 1 c22 (3.2.22) (32.23) r2 Q P2 rl,r........(3.2.24) h r2 ah S 1 Al2(l + cos (3)2.25) o)r 2r2 We omit ~1 R which is always small in this case. DR 551 -I-l ---- --.I.Ej C. Cj- ^ |

UNIVERSITY OF MICHIGAN These approximations are very useful in the case in which the beamrider missile follows a trajectory near the transmitter-PADAR line of sight. In the case of an up and over trajectory i/2 is not always negligible with respect to 1. 3.3 Applioation of PADAR to a Reflecting Target Experiment 3.3.1 Description of the Experiment It has been proposed by the United States Air Force that some experimental evidence clarifying the feasibility of obtaining range information on a reflecting target could be obtained with the use of readily available equipment. Referring to Figure 3.3-1, in the proposed experiment one F-86D aircraft irradiates another, and the present Fairchild PADAR equipment (modified so as to be able to obtain the appropriate angular measrements) makes range measurementse P-86 0 "82 Figu 33I -1 Figue 3.31 - -- 56 Zr%[ a CII

UNIVERSITY OF MICHIGAN In principle, range information can be obtained in this situation; the problem differs from the one discussed in Appendix 3 (3.2) in that one less piece of information is available -- the height of the transmitter (or equivalent information) must be supplied to the PADAR carrying airplane. The governing equations are very similar to those determined in Appendix 3 (3.2,2); they are 1 _ = (2r ) (3.3.1) 2 [J+ r (1 - 0oos )]l r- h. (3.3.2) 008/5 3.3.2 Detectability of Reflection The reflected power received can be determined from the following form of the radar range equation p =P 7 PI 1 G2 (.3 p = P0 2G (3.3.3) 2 (4)3(} J )2 where P = transmitted power (200 kw) 01 = transmitter gain (103, assumed) 02 = receiver gain (900) = wavelength (3 am) C= ross-section of the reflector:V 57 PI r ILI

UNIVERS ITY OF MICHIGAN Values of these parameters for the E-4 fire control system of the F-86D and the APS-4 receiver utilized in the Fairchild PADAR aircraft were obtained from Reference 10. The range r is assumed to be fifty miles or less. Reference to Appendix 1 shows that for the angle M restricted to a range greater than 120~, the minimum radar cross-section of the F-86D is about 3ft2. The largest value of )~1J2 is approximately 625 mi2. From Equation (3.3.3) we get P S. 10'1kw. 2 Comparing with the minimum detectable signal of 10 16kw for the APS-4 receiver, we see that for detection of the reflected signal at least a factor of 10 must be picked up. That is, the quantity 2 / 2 must be at least 10 [rmin/( 442)2max] Hence at ranges 4 ~ f2 2 25 mi, (Tmust be at least 30 ft2, Referring to the cross-section data of Appendix 1 one sees that for the cone, = 20~, the smallest cross-section is 98ft2; on the cone 3 = 30~ the cross-section is above 30ft2 for between 0~ and 20~ and between about 55~ and 180~; for g = 40~, 50~, and 60~,(OY 30ft2 only for a very limited range of the angle ~. Hence on the basis of these numbers, we would expect the reflected signal to be detected when ~1 = '2 = 25 mi, over all aspects of the cone f = 20~, over about 3/4 of the conef = 30~, and essentially not detectable on the larger cones. We point out that usually radar sets do 58 BEI. 'ir I! II!l M

UNIVERSITY OF MICHIGAN not operate at 100% efficiency, the degradation tending to reduce the expected ratio of received signal to noise. Also the cross-section values are only calculated to an accuracy of + 0ldb (a factor of 10). Taking into account these uncertainties, we make the following rough predictions concerning the detection of the reflected echo. Let us define the instrument degradation factor, X/7C ' where Xt5= P1 G1 G2 where P PN N noise power in the receiver as given in the instrument specifications, Pt' Gt, and G are the values also given in the specifications; t' = P1'G1 G2' - f --, the primes indicating actual values obtained in practice. We N will not consider high, narrow peaks in the computed cross-section as being significant, since comparisons (Ref. 6) of previous theoretical computations with experiment show that these peaks are generally too high. On the basis of the assumption that.2 I 2, we will predict sure signal detection if the computed received power, P2, divided by 10 t' is greater than PN; fair probability of detection if <PN <2 NJO ' N J 10?"l no prediction when - <P< -(2 2; low probability of detection if — 2 op. 7e N!R N = =-2 25 miles;;/N ' 2 Since it/X is assumed. 2,a6 60ft2 will be detected. u= 20~ If all the computed cross-sections were a factor of ten too large, there would be no detection. If G' 1 G C(GIc is the computed cross-section) the echo would be detected at about half the aspects on this 200 cone. Conclusion: Fair probability of detection at most aspects.

rczr 7F U N I V E R S I T Y OF MICHIGAN r = 30~, 00, 4 50, 60~ No prediction. -', ^-I -J! - miles; -/t/ -2 2 Detection if Co a80ft2 &= 20~ Detection at all aspects. = 30o Detection at I:J10~, 60~ to 100~, 1700 to 180~. ~ Fair probability of detection at most aspects. f= 600 No prediction. j~.. 10 miles; 2 Detection if Cc l5ft2 f= 20~ Detection at all aspects. Y= 30~ Detection at all aspects. t= 40o Detection at ~: 00 to 25~; 70~ to 85; 1400 to 180~ Detection at S: 10~ to 200; 160~ to 180~ Fair probability of detection at most aspects, a 6o -- -- I C.- CC=lj -= 1=-[-" \ & 1

UNIVERSITY OF MICHIGAN,t= 60~ Fair probability of detection at most aspects. 1=2P -5 miles Detection if 0co31 square foot. Detection at all aspects even if t/ ' is as great as 6. 3.3.3 Errors in Range for the Experiment The equations governing range determination in the proposed experiment differ from those derived for the problem of determining the range of a beam-rider missile in only one particular, In the experiment it would be necessary to provide the PADAR with information to make up for the fact that its altitude relative to the transmitter can no longer be computed since it is proposed that an airborne transmitter be used. One of the simplest ways of making the problem determinate would be to have the transmitting aircraft supply its height to the PADAR. The following discussion is based on the assumption that this would be done if the experiment were performed. Neither the analysis nor the results would change muoh if another technique was used. Assuming that the transmitter is operating at a known altitude the distance r is now obtained by means of the relation r = h' instead of by an application of the law of cosines. That is, the situation in the experiment is that which would obtain in Appendix 3 (3.2) if it were assumed that the earth is flat. If we'take the limit as R —oe in eaoh 61 r[:[PI I

U.-NIVERSITY OF MICHIGAN equation of that section the resulting equations are precisely those which apply here, However it will be seen below that if the experiment were performed with the present PADAR equipmentvvery large errors would generally be found. This is so because the APS-4 antenna of the present PADAR equipment cannot provide angle measurements accurate to better than about three degrees, One should not rely too heavily on the validity of the Taylor expansion in this case because of the size of the angular errors. Nevertheless, for angles satisfying the restrictions imposed in Appendix 2 (2.5) it will still be valid. The exact error, Equation (3,2.7), may alway be examined in any specific case to decide what approximations may be made, We examine an interesting speoial oaset Oonsider the quantity Ar. We measure r by r 0 h —' cooA Hence iAr hi, ~/ ~ ooe/oos4+ in/eina f h 0 3oo( solJAosAA - in+S sin a/) oo,4 or, oc0/A- ooA ooe dA inLns ini n Ah' r oe (p+L@) hl I T61

UNIVERSITY OF MICHIGAN For the APS-4 receiver, a3 3~<:.1 rad. Clearly for /3 near 90~, -r can be infinite. (A-r ---oo as +I/3-..90~). However, this does r r He, tI not necessarily preclude the possibility of obtaining range information. If r is actually much greater than (and 0 $ 0), the range determination is rather insensitive to r so that one could obtain an accurate value of l1 in spite of a very large error in r. To determine the expected errors in Q l in this situation we see that -v) Ar2(2vr+ )A< r- (2r+d)vlr I P+ 93 +o0 2(vr+ c) (vA r+AvAr) or ^ J2(2 v)+ (2- r )A&-S (2r+ ) [ 2-v)v 0+ ( )2] 2(vri-)[ v + (2-v)v' 0 + 2 (0)2] cf (2-v) [iF+ +2(L+ (^ )2].,, A1 v + (-jv 0v + + (he)2 2 NOTE: 0 is not necessarily small when = 90~, since the latter is true when the plane of transmitter, reflector, and receiver is horizontal. Computations based on the above error equations show that in almost all cases the errors obtained in the experiment would be intolerably large. Because of this and such factors as the non-comparability of the crosssections and equipment we do not believe that the proposed experiment' could give results which would be capable of interpretation in terms of the miss ile problem.. 63:[, --— I LE I

UNIVE RS ITY OF MI CHIGAN 3.4 Bistatic Radar Systems The growing complexity of the defense and early-warning problems has led to the adoption of the bistatic method of operation in parts of the defense system. In the light of probable future developments in the decoy field and the advantages of bistatic detection which are discussed in Reference 11, it seems probable that more use will be made of bistatic radar. It was observed in an investigation made in 1952 that the principle used in PADAR could be made to give range information in a pulsed bistatic system, (Ref. 12). Its use for this purpose was suggested by an article on the ranging of atmospheric irregularities in which the basic idea appeared, (Ref, 13). One can say without computation that in most cases the accuracy of the range information obtained should be better than that obtained in any of the proposed applications of PADAR. This is so because only one angular measurement is required and that one is made from the ground. f - 64 ^ ij i <r

J [! ( I r < 1 r UNIVERSITY OF MICHIGAN REFERENCES 1. PADAR Investigation, Fairchild Guided Missiles Division, Progress Report 1, 64M-0001, 1955, CONFIDENTIAL. 2. PADAR Investigation, Fairchild Guided Missiles Division, Progress Report 64M-0002, 1955, CONFIDENTIAL. 3. PADAR Investigation, Fairchild Guided Missiles Division, Progress Report 64M-0003, 1955, CONFIDENTIAL. 4. PADAR Investigation, Fairchild Guided Missiles Division, Progress Report 64M-0004, 1956, CONFIDENTIAL. 5o PADAR Investigation, Fairchild Guided Missiles Division, Special Progress Report, Report 64R-5, 1956, CONFIDENTIAL. 6. "Studies in Radar Cross-Sections XV - Radar Cross-Seotions of B-47 and B-52 Aircraft", C. E. Schensted, J. W. Crispin, and K. M. Siegel, The University of Michigan, Engineering Research Institute, (2260-1-T, August 1954) AF-33(616)-2531, CONFIDENTIAL. 7o "Propagation of Short Radio Waves', Volume 13, Kerr, D. E., MIT Radiation Laboratory Series, McGraw-Hill Book Company, 1951. 8, "A Study of Electronic Countermeasures for Application Against Air Defense Guided Missile System", The University of Michigan, Engineering Research Institute, UMM-111, April 1953, SECRET. 9. 'Talos in the Air Defense of the United States", The Johns Hopkins University, Applied Physics Laboratory, TG 190-2, April 1953, SECRET. 10. "Battlefield Surveillance Radar Systems", The University of Michigan, internal report, January 1956, SECRET. I -6 I C= i r=0

UNIVERSITY OF MICHIGAN 11* Stuadies in Radar Cross-Sections XVIII - Airborne Passive Measures and Countermeasures", K. M. Siegel, M. L. Barasch, J. W. Crispin, R. F. Goodrich, A. H. Halpin, A. L. Maffett, W. C. Orthwein, C. E. Schensted, and C. J. Titus, The University of Michigan, Engineering Research Institute, (2260-29-F, January 1956) AF-33(616)-2531, SECRET. 12, "An Examination of Bistatio Early Warning Radars", K. M. Siegel, The University of Michigan, Engineering Research Institute, (UMM-98, August 1952) W-33(038)-ao-15222, SECRET. 13* "A Forward Transmission Echo Ranging System", D. B. Harris, Prooeedings of the Institute of Radio Sngineers, 37, 767-770 (1949), UNCLASSIFIED. _ I [ a