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The Radi Directin Finding Research Labratry Department f Electrical Engineering University f Illinis -TYPE DIRECTION FINDING J. L. L. Bulet J. M. Andersn T. R.OMeara Octber 1> 1948 Technical Reprt N. 8 Cntract N6-ri-71 Task Order XV Office f Naval Research Prject N. 076161 ** p *^^'^^srj^r^-~'x»".' " > " ^ j

r Cpy N.*,,, DOPPLER-TYPE. DIRECTION FINDING J.L. L. Bulet J. M..Andersn T. R. O'Meara Radi Directin Finding Research Labratry Department f Electrical Engineering University f Illinis TECHNICAL. REPORT N..8 Octber 1,, 1948 Office f Naval'Research Cntract N6~ri-7l Task Order.'N..XV Apprved by yf J/ (L/^.^,-^ 2, 0. fedan : - \

liiy-.^.i.a-jah^ ^ TABLE OF CONTENTS * FOR TECHNICAL REPORT KQ, 8 Page I FORWARD II ABSTRACT III INTRODUCTION 1 (a) General Discussin...» 1 (b) Nrt-Cperative Systems *..«,. 4 (c) Cperative (Beacn) Systems 4 IV CIRCULAR ANTENNA ARRAY FOR DOPPLER 6 (a) Advantages. '. 6 (b) Law f Cupling.,... 7 V DISCUSSION OF SITE ERRORS. 10 (a) Cnstant Fhase Frnt Surface.... 10 (b) Similarity f All Small Aperture Systems.... 12 VI ANALYSIS OF EFFECT OF AN INTERFERING WAVE,... 14 (a) General Mathematical Discussin... 14 (b) Figure f Merit Fr Dppler 18 VII CONCLUSION 24 VIII APPENDIX I * 25 APPENDIX II, 29 IX REFERENCES...... 34 X DISTRIBUTION LIST... 35 \

-,-, - «*--*--H--M-«-. SlL^i-A*J ^"t^ *: I FORETORD As part f its general research prgram, the Radi Directin Find- ing Research Labratry at the university f Illinis has undertaken t study new systems fr radi directin finding. One f the new systems prpsed, -was a "Dppler-Effect" system. The initial theretical wrk by this grup n this system was carried ut by Mr. J. Linel Bulet and (10 was written up in the Masters' Degree thesis referred t in this reprt. Unfrtunately, due t a prlnged illness, it became necessary fr Mr. Bulet t withdraw frm schl and further wrk n the prblem. The wrk was cntinued by Mr. Jhn Andersn and Mr, Thmas O'Meara, and this present reprt has been written by them. Sn after the prblem was undertaken, it was fund that sme experi- mental wrk n this idea had already been carried ut at Camp Cle Signal Labratry, the wrk being summarized in a series f patent applicatins by Paul Hansell, Cnsiderably later, there came t the attentin f this (3) grup the British reprt by C. W. Earp and R. M. Gdfrey n»radi Directin Finding by the Cyclical Differential Measurement f Phase." This excellent paper discussed in sme detail certain aspects f the thery and described a wrking system which had been built. The theretical apprach in this reprt is aimed chiefly at determin- ing the ability f the system t discriminate against interfering waves such as thse prduced by site reflectins. Several definite cnclusins in this respect have been drawn and are stated herein. A 'mdel» Dppler Effect system has been built and is in peratin. At present, it is being tried ut and cmpared with a simple Adcck systsa n the labra- try* s R-D-F System Analyzer. The results f these tests are t be given in a later reprt. E, C» Jrdan \

II ABSTRACT This reprt develps and discusses the basic thery f the Dppler Type f Directin Finder. The mst practical antenna system fr this purpse appears t be a circular array f fixed antennas with gradual r abrupt switching frm antenna t antenna arund the circle, A mathematical discussin f site errrs is included fr the purpse f shwing the effect f antenna aperture upn reductin f such errrs» It is shwn that the Dppler Directin Finder ffers n 4 advantage ver the Adcck System when the antenna aperture is a small fractin f a wavelength, but that an increase in aperture f the Dppler sn reduces site errrs t the rder f instrumentatin errrs. This cmparisn with Adcck Systems is made n a figure f merit basis. Sme discussin f-antenna cmmutatin difficulties and number f antennas necessary t place the Dppler in the class f wide aperture systems is included. \

T crrelate this with the previus discussin it must be recalled that the rat f change f phase with time is prprtinal t the instantaneus frequency. That is* A f. = _JL J± (8) lnst where j fr this case - u c t + cs ( 0 - u r t) TC d 2TC dt Then 2rt f inst = w c + "^ sin ( P - u r t ) (9) f inst = f c + -^- sin ( 9 - a) r t) (10) A The instantaneus frequency may be expressed as a functin f the linear velcity, v r = w r r, f the antenna and as a functin f the free space prpagatin velcity, c = Af c, f the incming wave, f inst " V A + -p- sin (B - u> r t)l (" > Ntice the similarity between (11) and (5), the nly difference being that the relative velcity is varying as a sinusidal functin. The expressin (7), if phase detected, will yield a term rcd cs ( 9 - w_t ) ( 12 ) \ which is sinusidal in nature at the frequency f rtatin f the antenna, and has a phase dependent upn Q, the angle f arrival f the incming wave. It is pssible t btain a sinusidal vltage frm the antenna mtin directly rd,^ s cs (u r t) (13) which when cmpared with (12) will give a means f determining «Thus the basic requirements fr a directin finder are fulfilled, - 3~ _^^f \

(b) Nn-Cperative Systems In rder fr the directin finder t be classified as a Dppler type it must pssess the equivalent f a mving antenna element f sme frm. Hwever, this des nt stipulate the path the antenna must fllw. In the thesis, previusly mentined, Mr. Bulet analyzed the case f an antenna mving at a unifrm speed n a linear path and retracing its curse peridically. This prduces a frequency change depending upn speed f mtin and upn the angle between the path f the antenna and the incming wave. If the speed is cnstant the frequency change is a measure f the angle. One difficulty arises when the signal arrives frm the same angle n the ther side f the axis. T remve this ambiguity a secnd antenna is made t traverse a path perpendicular t the first. What appears t be the mst practical case t cnsider is the revlutin f the antenna abut a vertical axis as was discussed in the intrductin. Greater linear circumferential speeds are btained in this case and the prblem f getting the antennae back t their starting pints in infinitesimal time des nt enter, as it des in the linear path. Certain transients present in the linear case are nt present in the circular case, making fr smther peratin«in the final analysis the rtating antenna will be replaced by a circular array f antennas, with electrnic switching frm antenna t antenna. (c) Cperative Systems These tw types f Dppler Directin Finders are classified as nn-cperative, since n knwledge is knwn f the nature f the transmitting surce. It shuld be mentined that this type f directin finder can be cnverted t the s-called beacn system by reversing the rle u mmffjulvff^.mn y t JW" u, ij,, *g Jm* "**,\»mkr. "t.t' JIB" 11^ W? w?l-isl^^±^*" '"*"'

f the rtating antenna frm receiving t transmitting. It becmes necessary t transmit a synchrnizing pulse, which -will be received alng with the phase mdulated wave«, Since the phase f the phase envelpe will vary accrding t the azimuthal angle frm the statin, a cmparisn with the synchrnizing pulse will give the radial line frm the statin. 5 -

IV CIRCULAR ANTENNA ARRAY FOR DOPPLER (a) Advantages Cnsider the afrementined rtating antenna and pick a set f representative values fr the variables invlved. Assuming a rtating arm, ne meter in length, an angular velcity f 60n radians/sec. (1800 R.P.M.) the maximum instantaneus frequency difference will be given by (9) A f niax =.628 x 10* 6 f c cycles/sec. (13) The immediate imprtant bservatin is the small value f Af unless the frequency invlved is very large, r the practical length f the arm is exceeded. The pssibility f prducing an effect similar t thstt described fr the single rtating antenna, by using fixed antennae cnnected successively t a receiver wa3 investigated. This system shuld be strictly referred t as indirect Dppler since the mtin is simulated by rapid switching. One f the advantages f this indirect apprach is the speed at which electrnic switching can be accmplished, thus eliminating the prblem f small Af«In additin, the use f fixed antennas makes it pssible t build arrays f large aperture a very necessary requirement fr accurate bearings in the presence f site reflectins. Bth British and American systems t date utilize a circular array f antennae with electrnic switching. T appreciate the shape f phase variatin invlved, plate (3) shws an array with a signal arriving at an angle f 45 azimuthal, and als the phase f the r«f. vltage picked up as cmmutatin prgresses arund the circle. If the number f antennae used is sufficient^ the step functin f phase, when detected,, will yield a fundamental sinusidal variatin similar t the rtating antenna. This variatin will change phase with the angle f incming wave in a linear manner, again giving the basic requirement fr a Dppler System* \

-f^-lt- '' f». «1pl L _ (b) Law f Cupling The abrupt switching frm ne antenna t anther presents prblems in detectin inasmuch as discrete jumps in phase cause accmpanying infinite frequency change, Eq. (8). T alleviate this a law f cupling t the antennae was studied. Cnsider tw elements in a circular array f fixed antennae. Call ne element the "nth»' element and the next, the "(n + l$h)" element* It is desired t find a law f cupling as switching is cmpleted between the tw antennae such that the vltage utput will have a sinusidal variatin f phase. Let the vltage received by the "nth" antenna be. 2nr r n 2K (n - 1),,, /,. \ e n = cs { w t + - cs [9 r J / and vmj by the»( n+ 1) th "antenna be e n+l = cs t wt + ~~\ cs ( 9 - )] (15) where the ntatin is the same as plate (3). The additin f these tw vltages with the apprpriate law f cupling wuld be required t take the frm: That is; A cs [ ait + - cs ( 9-2n;at)] (l&) A. G(t) cs{,t * ^~ 'cst 9-1^1} } (17) A k r,/ N r 2rcr, n 2ixn._ r 2rtr, n n... + H(t) cs[wt + - cs (9 ) = A cs[wt + cs (9-2rcat)j A k A Cnsider plate (3)«The difference in phase between e n and e n + i is cp = a{ cs(9 ) - cs[9 ~ -] } (18) K K and the phase angle (6) f the resultant R is given by H(t) easing (19) ' * -7-

rrrr tan(3= H(t) e n+1 sin <p (20) G(t) e n + H(t) e^ cscp If the arbitrary phase f the e vltage is called y the phase n-i n f R at any instant t, where ; < t < r xs grv en by 0(t) - Y + arc tan[. ' rrrr 1 (21) G(t)2n_ + H(t) cstp v e n+i _SQL = cnstant = c e n+i This variatin must be equal r clse t the variatin a cs (6-2Kat).The integral t be minimized is thent / ka [a cs(9-2nat) - 6(t)] dt (22) ka The necessary cnditin fr a minimum must satisfy Euler equa- tins, that is if J" G(x,v,y') dx is t be minimized a d 36 9G d dx Qy 1 9y dx (G- y,,)=4i (23) ay' x must be satisfied. In ur case ' = X * G(t) (24) H(t) l?(t) = y and. G = a cs(9-2rcat) - Y - arc tan [ ~T* ] **'* cp(t) + cscp d 9G 9G ',, N Then +- - : = -TTTTT" ^2ö ' dt 3F'(t) 3F(t) Since F! (t) is nt written explicitely in G 9G 9 B"(t) (27) Then 8G 9P(t) - 0 (28) and c sincp [ci?(t) + cstp] 2 + sin 2 cp - 8 - (29) \

T-, Because c i O.sirsp = 0 which leaves F (t) undetermined. Hence, there appears t be n general law f cupling which will give the desired result. Hwever, Bulet has shwn (4.) that linear variatins f G (t) and H (t) will give a phase curve which clsely apprximates the ideal curve. -9 - -_ ; *.- v-jptt* *"*' \

V DISCUSSION OF SITS ERRORS (a) Cnstant Phase Frnt Surface The errrs encuntered in Directin Finders can be classified in general as fllws: (1) Prpagatin Errrs The greatest surce f this type f errr is the nnunifnn nature f the insphere which gives rise t a lateral deviatin f the main r desired ray» Fr the mst part these errrs are randm and may vary rapidly (within the minute) r slwly (within the hur)«the apprach t the eliminatin f errrs due t these variatins seems t be the applicatin f mathematics f prbability» Spt readings are taken ver a perid f time and the reading f greatest prbability calculated«the insphere can als be respnsible fr the arrival f several rays f slightly different azimuth» Minimizing f these errrs als fllws statistical lines. (2) Instrumentatin Errrs These are errrs that ccur in the equipment and as a result f the type f equipment used» Plarizatin errrs, that is errrs due t randm plarizatin f the incming signal, are in general classified under this heading since their effect can be virtually eliminated by prper antenna arrays. Errrs due t bservatin by the peratr can als fall under this heading. - 10 - \

«w«gotiawj«^ re'^fejjii^ij^j^^^^c!v~^ Ti^" (3) Site Errrs These are errrs due t reradiatin and reflectin frm bjects in the lcalef the directin finder» Scattering and reradiatin by inic cluds are classified here since their effect is the same as that fr near-by bjects. This sectin is mainly interested in the subject f site errrs since their effect can be materially reduced by the chice f the prper type f directin finder«. When examining the field due t a single incming wave, a cn- siderable distance frm its surce, it will be fund that the surface defining a cnstant phase is a plane perpendicular t the directin f travel. Any directin finder that perates n the phase frnt principle, (such as the Dppler) will indicate an errrless bearing in the directin frm which the wave is arriving. Hwever, if there is present anther signal f appreciable amplitude and f arbitrary phase and directin f arrival, the phase frnt indicated abve will n lnger be plane. The effect f the interfering signal is t crrugate the surface in a manner depending upn the relative characteristics f the tw signals«. When the resultant vltage due t the space additin f the tw signals is examined, it is fund t describe a standing wave pattern with pints f minimum and maximum amplitude spaced at unifrm intervals* The phase f the resultant vltage als fllws definite space patterns and, t get a line f cnstant phase, it is necessary t set the expressin fr phase at any pint equal t a cnstant and plt the resultant curve. (6) This prcedure has been utlined in a previus technical reprt. issued by this labratry and nly the resultant expressin is given heret A 1 + k 2nx. n X / n \ : arc tan [ - tan ( csa )] - \30J 2rt sin«1 - k X sitkx

.VIM- ;^«w^ «;^«iui^'m '«^ - yjrwmitff- «juffs^jbtsaäff^ ^jgy^ ^^J^SSR... affair't^ '^''P, "^""T..' Zl~-~i~JJ^iS*i^z^^~i -^E^^CZ^-~L^^^JL~T '~ 1,^U " ^l, 1~ ~, where n= (0, 1, 2, 3-1, -2, -3 «. ) k s rati f amplitude f signal #2 t signal #L and 2«. - azimuthal separatin f the tw rays«a plt f cnstant phase surfaces is shwn n plate (5)> this plate being a reprductin f plate j/1 in the abve mentined Summary Technical Reprt. (b) Similarity f all Small Aperture Systems If a directin finder with small aperture, that is small dimensins f antenna array, were perating at different pints alng the cnstant phase frnt surface, it is clear that the directin f arrival indicated wuld be a perpendicular t the tangent f the phase frnt curve at the specified pint. Since this phase frnt curve is nt a straight line, different pints wuld indicate different directins f arrival«it will be seen in a later sectin that maximum errr in bearing will ccur at pints f maximum and minimum amplitude f standing wave pattern and that at ne pint between a maximum and minimum there will be indicated' the true bearing. Frm phase frnt cnsideratins, it can be cncluded that any directin finder perating n phase frnt principle and with aperture small in wave length will be subject t the same site errrs. It might seem that a directin finder using an antenna system with a very narrw beam might be an exceptin t this statement if the antenna culd be made t have small aperture. Hwever, it has been shwn* that such antenna* systems are nt practical, because f extremely lw sensitivity. Thus t get the desired sensitivity, it must be made large in wavelengths and falls in. the grup f wide- aperture systems. *»Small High Gain Arrays fr Directin Finding», N. laru, Technical Eeprt N.6, Univ. f 111. Directin Finding Research Labratry, September 1, 1948-12- :l_z--i

Nw if the aperture f the system is made larger such that the antenna perates ver a greater distance n the phase frnt curve, the maximum errr variatins wuld be less». This is smewhat analgus t the length f wheelbase n an autmbile. On a rad f given rughness, the lnger the wheelbase, the smther the ride. The Adcck System des nt lend itself t wide aperture peratin due t ambiguities which arise. Hwever, the Dppler System is theretically nt restricted n this accunt and can be increased in aperture t the limit f physical struc- ture and electrnic equipment techniques. This tends t put it in the class f wide aperture systems f which there are several ther examples» The site errrs f the Dppler System under different apertures will be given in a later sectin. When increasing the aperture f the Dppler, frm a practical viewpint, the tw prblems encuntered when fixed antennas are used are: where t sample the field, and hw t detect the resultant phase variatin. Cnsider the phase frnt diagram (Plate 5) and ntice that with a Dppler System f nly fur antennas, prperly spaced, it is pssible t get n phase variatin whatsever as the receiver is switched frm antenna t antenna. That is, lcate the antennas at pints f equal phase* This is an extreme example, but it is evident that t get a phase variatin that is at all clse t the desired, the number f antennas must be suffi- cient t prevent ambiguities and give a phase pattern that is gemetri- cally cnsistent. Because f limitatins n the phase discriminatr emplyed, the steps between antennas must nt be greater than apprximately 90, This requires that the number f antenna be prprtinal t the aperture when the aperture is very large» When using small apertures, the number f antennas must be enugh t cause the fundamental f the step phase variatin t fllw the angle f arrival f incming signal. The least number f antennas that will give a shifting phase variatin is three but aperture requirements set the lwer limit at arund 6 r 8 antennas. -13- \

.I..»* VI ANALYSIS OF EFFECT OF AN INTERFERING WAVE (a) General Mathematical Discussin When ne r mre interfering signals arrive with randm directin and time phase -with respect t the desired signal, the effect upn the Dppler Directin Finder indicatin is t change the effective psitin f the desired phase variatin and thus bring abut a bearing errr. Fr wide aperture Dppler Systems, the larger in amplitude f the tw signals will make by far the largest cntributin t the resultant phase variatin. The strnger signal will «take ver" the bearing in a manner analgus t the "capture effect" shwn by the strnger signal in a wide-band frequency mdulatin system. The case which cnsiders a single interfering signal (f the same frequency) is treated mathematically in this reprt. A general frmula has been develped t give the bearing errr ( Ap ), Fig. (la) Plate 6 gives the space diagram illustrating the arrival f the primary and interfering waves. The rati f the interfering signal magnitude t that f the primary signal is designated by a, is the arbitrary angle f arrival f the interfering signal with respect t the ther, and Y is the time phase difference between the interfering and primary signals«fr the primary wave and fr the interfering wave ea = cs( ct + ß cs<4t) (31) e 2 = a cs[wr, + 6 cs(w t t - 9) + y] (32) - 14 - ^5' ' " "

rife where w s frequency f the carrier 3 = «. electrical aperture f the system A Wi s rtatin r switching rate a - radius f circle. If e_ and e 2 are represented as vectrs they can be cmbined as shwn in Eig» (16) f Plate 6. The magnitude f the vectr, R, is f little imprtance as the Dppler System perates purely n a phase cmparisn basis» In rder t determine the change in phase, the angle Ae must be determined? Ae = arc tan ( ) \Ji) 1 + a cscp where q> = [<,)t + R cs^t - 3) + y] - [ut + ß csu>it] = ß[ cs^t - 9) - cs ^t] + Y (34) The cmplete expressin fr the vectr resultant f e. and e is : R cs[wt + S cswit + arc tan ( )] (35) 1 + a cscp Let \y be the phase angle f this vectr: n. a sincp.,, N V = wt + (3 csujt + arc tan ( ) (36) 1 + a cscp Withut any interfering signal, the phase wuld be: W = Ut + 3 COS Wit (37) Hence, the extra arc tan term must be an errr term:,. a sin cp.. Ae = errr term = arc tan (- - ) (38) 1 + a cscp - 15

4<Li^.T..«MWMJjH>i.M ; i*. W^*? This Ae term will cntain sme cnstant terms independent f u-t (which- will cause n phase shift f the (3 cs Wjt term), and sme cs w t terms as well as harmnics f these terms. The cntent f A e can be reslved by expanding it in an infinite series as fllws (see Appendix I fr mathematical derivatin),. a sincp... Ae = arc tan ( ) (39) 1 + a cscp <* 2. «3.,. = a sincp - sm 2cp + sin 3<p... +... (40) The expressin fr 0 can be simplified t sme extent.. Refering t Fig. (lc) <p - ß[cs (Wtt - 9) - csu^t] + Y (34) Cmbining ß cs(wit - 9) and -ß csi^t as vectrs gives a vectr X X = 2-2 cs9 cs («i t - - ) (41) Let 3 v 2-2 cs 9 = g anci fat - - ) = r = 2-2 cs 9 sin fat - - ) (42) 2 then <p = g sin r + Y (43) A, r a sin fe sin r + Y^ > 11) > Ae = arc tan L : ; : r J Khh) 1 + a cs(g sm r + Y).2 = a sin(g sin r + Y) r sin(2g sin r + 2Y) +... (45) By using the trignmetrric expansin s.in(g sin r + Y) = sin(g sin r) COSY + cs(g sin r) sin Y (4; A e becmes a sin(g sin r) COSY + a cs(g sin r) sin Y.2.2 a 2 sm (2g» sin r) / cs 2Y r-^ cs(2g sin r) sin 2Y + 2» (47) ~ 16 -

It is knwn that cs(g sin r) = J 0 (g) + 2 2 J* n (a) cs (2nr) (48) n-x sin(g sin r) = cc 2 2 J 2n+1 C g)sin[(2n + i)r] n=.0 6 There is need t cnsider nly the terms invlving ( w«t ) 1 2 as these are the nly terms which can cntribute t a phase shift f the a riginal 0 cs «^ t term. The [2(u 1 t-- )] and higher frequency terms are generally remved after detectin by a band-pass filter, t facilitate phase cmparisn» Let the prtin f Ae 9 (u>it ) terms be called Ae' invlving nly The expansins f cs (g sin r) and sin (g sin r ) give fat - - ) terms nly fr n - 0 Ae' = <x[2ji(g) sin r] csy + 0 a a r~ [2Ji (fr) sin r] cs2y + 0 + -y t 1 + 0 ( 49 ) i- i Q Ae* = 2a csy J a (ß V 2-2 cs9 ) sin^t ) - a 2 cs2y J a (2ß 2-2 cs9 sin^t - ) (50) Ae' n [ 2 (-l) n+1 cs ny J 4 (nß /2~T2~cs9lTsin(w 1 t--) t 51 ' n=i n 2-17-

1_«A_... _ ji»i^_: - ;.. v_ Let the infinite series which is the cefficient f g sincwit ) be called D. Then Q Ae = D sin(u>it - - ) (52) D cnverges sufficiently rapid.. that nly fur terms ( a, a 2, a 3, a*, ) usually need be calculated t give three r fur place accuracy fr D. As bearing is given by a shift in the phase f the riginal ß cswit term, then bearing errr ( Ap ) will result frm any ube desired shift in the phase caused by adding the D sin (t,, t ) term» The magnitude f this additinal shift can be determined by refering t Fig. (Id), let y be the resultant vltage f cmbining ß cs «it and D sincwjt - ) y = ß cs Wit + D sincujt ) (53) = E csl^t - Ap) Q (54) Bearing errr = Ap r arc tan [ isüllaj -j (55} ß - D sin(-#-) It shuld be emphasized that this bearing errr assumes a linear phase detectr and n instrumentatin"errr. In practice a phase detectr is linear fr small phase deviatins, but appraches a sinusidal respnse fr larger deviatins» (b) Figure f Merit fr Dppler Systems T give sme idea f the amunt f errr encuntered in the Dppler System under different cnditins f the interfering wave, the expressin (55) was pltted. Three different apertures were chsen with the express purpse f shwing the effect f increasing the aperture and cnsequent reductin f site errr». In rder t cmpare the relative errr f the Dppler and Adcck Systems, the expressin (23) ' f Appendix II was pltted under similar circumstances. (23) is the bearing ' - 18 -

*mmimm-ma&z&bäl**ti < ' ^^ mnuhimmam/mmm^m***?^* (5) errr fr an Adcck System whse- aperture is very small in wavelengths, ' " The curve's are divided int the fllwing grups* 1, Plates 7, 8, 9, -Adcck Small aperture 2. Plates 10, 11, 12 «Dppler ~ Ä/8.Aperture 3«Plates 13, 14* 15 Dppler X Aperture 4. Plates 16, 17, 18 Dppler 5X Aperture The wrd Aperture abve, as thrughut the reprt, refers t the electrical radius f the system. These grups are further divided int different ratis f signal magnitudes, namely a ä,2, *.5* and «8. Different values f Q, space separatin f the waves are pltted against y» the time phase difference. facts: A general bservatin f the curves will reveal the fllwing 1» Maximum errr ccurs at Y = 0, 180 2* At tw pints the bearing errr is zer as was pinted ut earlier in the reprt. 3«As the aperture f the Dppler increases the maximum errr decreases» Cmparing the curves fr Adcck and Dppler (ß s 45 ) it will be nted that there is a marked similarity. Since an aperture f 45 is reasnably small, it is just what is expected when the phase frnt curve is cnsidered. It is interesting t let the Dppler Aperture apprach zer as a limit and then cmpare the resultant mathematical expressin with that fr the errr f the Adcck. Frm Appendix II we extract (31) which is the bearing errr f the Dppler as ß - 0... r a sin 9( csy + a)( 1 + a csy + a cs9 + a cs9 csy ), Ap= 72 arc tan l : *- ; * J i + 2a csy (1 + cso) + 2a cs0 (2 + cs' y cs9) + 2a 3 csy (2 cs 2 0 + cs9 - i) + a* (2 cs 2 9-1) ^^ -'19 -

System (23), and frm the same appendix the expressin fr the Adcck a ' v - r 2a sin6 (csy + g cs8).. 9 ne = % arc tan [ - - -pr. 2- a Q. ].(57; 1 + 2a csö csy + ö (cs 9 - sin 6) When Y = 0 r 360 which is the case fr the rigin f the phase frnt curve f Plate (5) the tw expressins (56) and (57) be- cme equal, namely,,., a sin9 + a sin9 cs9 ftö\ 9^ ne = Ap = Vi arc tan ( -? - ), ^0/ 1+2«cs6 + a cs 29 ' 0 If Y is placed equal t 180 there results anther set f equiv- alent equal frmulas- Hwever, if y is set"equal t any ther angle, the tw bearing errr frmulas are nt e^ual. In the derivatin f the Adcck frmula attentin was.paid t the magnitude f the interference pattern as well as the phase. At the 0, 180, and 360 the magnitude is symetrically dispsed and causes n blur in Adcck reading, while at ther pints the blur causes the minimum reading t be shifted. The difference in actual bearing between Adcck and Dppler is very slight and fr all practical cases it wuld seem that the result culd be extended generally t in- clude all small aperture systems, the cnclusin being that any, small aperture system suffers frm, the same site errrs.«when cmparing Directin Finding Systems it becmes necessary t frmulate sme standard by which a quick cmparisn f site errr reductin can be made, A figure f merit (r demerit) used by H. G. Hpkins f the Natinal Physical Labratry (England) is the fallwing: The maximum errr is fund fr each angle f arrival f the interfering signal as the time phase varies frm 0 t 360. The r.m.s. value f these maximum errrs is used as the figure f demerit f the system. 20 -

-*- «. r» Fllwing this methd, (with the aid f Plates 10 thrugh 18)j the r.m«-s.-errr angle fr the Dppler under different aperture has been calculated and appears in the fllwing cncise tablet. ^V APERTURE RATIO 0F^\ SIGNALS \. ßlT 45.ß - 360 ßr 1800 a ~ 2. 7.68.59.0321 a -.5 23.8 1,66.0892 1 <* Z.8 39-7 3.1 1.1712 I TABLE I The imprtant bservatin f the abve table is the large reductin f errr which is btainable in an aperture f nly ne wavelength«as was pinted ut earlier, the errrs fr Dppler ( ß z 45 ) and Adcck were almst identical, s the abve table indicates the superirity f wide aperture Dppler ver the Adck in this respect.» Theretically, when the aperture is increased t ß - 3.800 there is a further reductin f errr. Hwever, there is an upper limit n the aperture (due t the cnstant instrumentatin errr) abve which it becmes unprfitable t further increase ß,. Frm the table it wuld appear that &his limit lies smewhere between ß ~ 180 (a r V2) and ß 55-360 (a =X). Mst f the site reradiatin signals have amplitudes which are much less than 1 («<<!) and this further helps the -21.««-»SS^SSSW»:^-»

r~ situatin» It is interesting t cmpare Dppler and Adcck systems under cnditins -where the interfering wave is arriving frm almst the same directin as the desired signal and has cmparable magnitude (this is the usual case f sky-wave multipath prpagatin). Apply the fllwing cnditins t expressin (57)> which is the errr fr the Adcck with small aperture. Let y = 180 and 6:Ae, y ~ 180 is a pint f maximum errr as can be seen frm plates (7 thrugh 18). The result is: 9 re % - % arc tan (-^~ ) (59),ß 1 -a Apply the cnditins t expressin (55), the errr fr the Dppler. Make the assumptin that the aperture f the Dppler is very small and the result is: Ap ~ - arc tan ( ) (60) 1-a Cmparisn f (59) and (60) shws that fr small apertures the errrs f the tw systems have the same rder f magnitude. In particular fr nearly equal t unity the errrs apprach 90 in bth cases» It is cn- cluded that when the apertures f the tw systems are small, the bearing errr fr the case f multiple path transmissin is the same. Hwever, when the aperture f the Dppler is increased the expressin fr errr must be apprached alng different lines. Gnsider expressin (55-)* and let 9 r A Q f Ü cs( ), (KK\ Ap** arc tan [ *"- * ] ^'i 8-D sin(-s-) 2 Ap % arc tan ( -r- ) (61) - 22 -

CXJ 2tt n where D = 2 (~l) n+1 cs n y J. (n<3 2-2 csag ) n=l n when y = 180 c a n 0 = _ v j ( n ß V2-2 csaö ); n=*l n If t<l the summatin will cnverge t a finite value that is nt directly dependent n ß. Thus Ap frm (6l) can be made t decrease by increasingß «The cnclusin is that discriminatin against the weakest f tw signals arriving frm almst the same angle can be made arbitrarly great by increasing the aperture. -23- K^- j* V f- - "sr

- - VII : CONCLUSION It is cncluded that a Directin Finder perating n the Dppler Principle can ffer a very practical and accurate slutin t the prblem f determining the directin f arrival f ivave energy» when the dimensins f the antenna system are made cmparable with that f the Adcck, it seems t ffer n imprvement ver the later in the matter f site errr reductin. Hwever, the Dppler System lends itself admirably t wide-aperture peratin under which cnditins it can be made t give marked reductin f bearing errr caused by site reradiatin. Of curse, as a wide aperture system, its use wuld be limited t grund installatins and"shipbard fr the medium and high frequencies, with the pssibility f aircraft applicatin nly at very and ultra-high frequencies. ~ 24 ~ - (.j. \

J^l VIIT-. '.APPENDIX I COMBINATION OF PHASE MODULATED WAVES A mre rigrus analysis f the cmbinatin f the vectrs is presented. e t ~ cs (tt + ß cs Wj. t) e 2 = a cs [wt + 3 cs (tjt - 0) + Y3 e x * cstt cs (ß cs w t t) - sin tt sin (ß cs t x t) e 2 = a cswt cs[ß cs(tit - 0) + y3 - asintt[sinß es(wit-g) + Y! e t * e 2 * cswt { cs Iß cs«! t)-+t'a cs[ ß cs(u 4 t - 0) + Y 3 } - sin ut{ sin(ß cstit) + a sin[ß cs(t x t - 6) + y' } In cmbining tw expressins E = A cs ut + B cs wt = i/a 1 * Bg cs (tt + y) where f \ / = arc tan \~~r~) B^ < v). We are mainly interested in the phase f the resultant-.expressin, sin(ß cstit) + a sin[ß cs(tit - 0) + Y3,, x\f " tt + arc tan I rr r 7 hr) - cs (p cswjt) - a cs[ß cs(wit - 0) + yj which can be written sin (6 cswit) + a sin(ß cstit) cs(ß cstit) cs[3 cs^t-b) + Y3 - a sin(ß cstit) cs(ß cswit) cs[ß cs(tit-0) + y] + a sin 2 (6 cst x t) sin [3 cs(w a t - B) + y ] + a cs' (3 cstnt) sin[ß cs (tit - 0) + y], y - tt + arc tan { r : ~ r j- -- -} cs(15 cstit) + a sm(p cs t x t) cs(ß cstit) sm(.p cs(tit-0) + yj - a sin(3 cstit) cs(3 cstit) sin[8 cs(tit-0) + y] + a cs 2 (S cstit) cs [ß cs(tit - 0) + y3 + a sin 2 (ß cswit) cs[ß cs(wit - 0) + y] - 25 -

Divide the numeratr and denminatr by: cs (3 csu 1 t){l + a cs(ß csuit) cs[s cs(u!t -9) + Y3 + a sin (6 csuit) sin[ß cs(u 1 t - 9) + y]} Then a sin[ß cs^t - 9) + y - ß csu^t] tan(ß cswjt) + 1 + a cs[ß cs^t - 9) + Y - ß cswxt] \y = wt + arc tan { } tan(ß cswit) ctsin[ß cs^t - 9) + y - ß csuat] i- 1 + a csfß cs((0it - 9) + Y - ß cst^t] But / x + Y \ arc tan ( ) = arc tan x + arc tan y 1-xy, a sin[ß cs(wj.t - 9) + Y "" ß cswtt] Let y = 1 + a cs[ ß cs(u)!t - 9) + Y _ ß csuit], x = tan(ß cswit), and cp = ßcsCujt - 9) + y - ß cswit Then y = wt + arc tan (tan ß cswit) + arc tan y. ce sin <p. y = wt + ß csujt + arc tan( ) 1 + a cscp -2 6- ^

Frm identities it is knwn that Applying this: 1.. 1 + ix arc tan x = - In( -====) 2i 1 - ix i a sincp 1 _. 1 + 1 + ft cscp. y = wt + ß cswjt + -~*r- lnf : ) 2x i _ ia sincp 1 + a cscp 1,. 1 + a cscp + v. ftsmqp, \y = wt + ß cswit + - - In ( -= : : ) 2i 1 + a cscp - ia sincp v = wt + ß csw a t + -i [ ln(l + ae 1^) - in (i + cxe" icp )] 2i N value f 0 can make e icp greater than + 1. Als a was chsen ~U a & Under these cnditins-.«e 1(p cannt be greater than lr less than -1. V = ut + ß csw t t + ~ [( ae i( P- J^V** + -J- av^ -+,-...) -(ae-^ - % av 2i <P + 4- «V öi< P -+...)] 2 3 4 «. a. n a., vy = wt + p cs w x t + a sincp sin2cp + - sm?cp sm4cp +.... 2. *± It will be nticed that if a = 0 (n interfering wave), the first tw turns will give the bearing» Hwever, any interfering signal will cause the infinite series t have a value and thus prduce errr«this errr is the deviatin f phase frm the desired as the single element prgresses arund the circle* Analyzing any ne f the infinite series terms a sin[(ß cs(wit - 9) + y - ß cs w a t] ß w * _ 2^ - Y

l^j >W»Wt^W»P*lPWMI»^^TII ^ Wpinr.uiiU'WffWu.'l-" ***** M «MMhP'^M«' '.<.»*>»(*' it is fund that the errr is due t (a, ß, w, 6, and y ). The factrs which carry the greatest weight are a, ß, and y The final bearing errr will be fund by extracting all terms f the expansin cntaining w 3t and it will be fund (Appendix II) that the reductin f errr is mainly dependent n ß (the aperture f the system). - 28 -,*CS~iiP 3p!w&'

APPENDIX II BEARING ERROR (DUE TO SITE RERADIATION) OF THE DOPPLER SYSTEM AS THE APERTURE APPROACHES ZERO (ß - 0), AND CO^IPARISON T/KTH ADCOCK ERROR* AS previusly shwn, the general frmula fr the bearing errr f a Dppler system is given by: e. r Ap = arc tan [ D cs C" -* ) r i j (1} ^x' ß - D sin {$_) where D.J! «- Tf«1 * 1 JlW (2) n=l n and Ji (ns) = Ji (n ß /2-2 ccs6 ) (3) Fr any value f n which can be chsen ß can still be "made small enugh t make valid the fllwing apprximatin This causes D t nw becme :> = 2 _«"cusny(-l) n+1 _ ( _n _ } (5) n=i n 2 Px = 1 «"csryf. (-D n+1 n=l ^ Di = i (a <-CSY - f 2 ccs?. Y + a" cs 3y + ) ^) The series can be synthesised as fllws: It is true that:* 1 -f j = 2 (a 2 COS2Y + a 4 COS4Y + a 6 COS6Y +...) 1-2a : cs?y + «. /a\ where a" < 1 \ ) * (Dwights Table f Integrals). ~ 29 -

->*a 1 - ~ B 5 = - 2( a 2 cs2y _ s* cs4y - a 6 cs6y ) 1-2a cs2y + a 2 (9) where a < 1 w/ 1 - a csy, 2 n 3 ; 5 1 = s csy + s cs2y + s COS3Y +.... 1-2«COSY + s, 2, (10) where a < 1 v * ' Adding the last tw identities r- (1 7 fl4)., * * I; acosy ]»(ID 1 + 2a - 4a cs y + a 1 + a 2-2a COSY ' Thus: = ( a COSY - a 2 COS2Y + a 3 cs3y ) Dl. ( -*Lz± g., + I^w } (12) 6 1 + 2a + a - 4a cs' y 1 + a - 2a csy The imprtance f this equatin is that it reduces the previusly given bearing errr ( A p ) expressin t an analytic functin» T simplify D-i rewrite its Ap = arctan t D i _c«i_j 1 (13) ß - Di sm/ 9 \ \ / 2, a* - 1 ( 1 - a csy) ( 1 * a 2 + 2a csy ) (14) 1 " g (1 + a a ) 5 -'4a^77~" + ~~Tl"+"?^"lä~csYJ ( 1 + a a + 2Ö"cösY ) Cmbining a* + a 2 + a csy - 9 a 2 cs'y ~ s" csy (15) D = g(. j 5 5 1 j (1 + a) - 4a cs y 1 = agl ( 1 + a 2-2a csy )( a + csy ) (16) ( 1 + a- 2a COSYH" 1 + a K + 2a csy ) ' ^4_ c ±fy } (17) 1 + a + 2a csy aß v>2-2cse ( - «V f Y ) (18) 1 + a + 2a csy -30-, -^!_~w«^r^-.«*-~«-t^-«'m^ rtwewvffls&e» *9Tqr "Wl»< jf

fc = 2.3 sin (-!-)(, g : c SY ) M) 2 1 + tf + 2a csy Frm the Thesis by R. W. Annis, mentined in the text, the bearing errr fr the Adcck System is 9 = V + %~ (20) ne 2n If placed in ur ntatin, it wuld appear a i/ r a sin ( 9 Z Y) -, x w".. rasin(9 + y), (21) 9 = y 2 arc tan [ r] + % arc tan[,.1 *- Äj ne -' 1 + a cs (9 - Y) 1 + a cs(0 + Y) which is a sin (0 - Y) a sin (8 + Y) T, 1 + a cs(9-- Y) 1 + a cs(9 + Y), (\ 2 9_ e = arc tan{ g...,,.. r } ^Z2 ne >, a sin (8 - Y) sm(9 + y) [1 + a cs(0 - Y)1 [1 + a cs(9 + y)l Expanding and reducing: 2a sin9( csy + a cs9). (23) 9 = Vi arc tan 1 : >*-, 57; nrz? J ne 1 + 2a cs9 csy + a (cs 9 - sin 9) Putting the Dppler errr expressin in this frm, the result is: 8-2ß Di sin ( 9 ) - Di cs9 Nw it is necessary t place the expressin fr D (19) in (24) and cmpare (24) with (23). Expanding the numeratr f (24) 2 W-4-) - «0*2 cst-?-, s.i (i-) frfifa,. a + csy,. n - a3 (-; * sin9 1 + a + 2a csy (25) - 31 - r$xz*»'* mt *rsy~~

n 2»/ a A,2ß2 2 / e + 2<X - \ aa r * cs Y + cs2 Y, Dl ^2 s*n9 - - 4 a 3 «in (y-)» sm9[ - ;? ; -. ], 2 Q 2 /,/ W1 \/ n\ r 0 ' 2+ 20! COS Y * 'v! = - 4«ß ( X ) (1 - cs6) (sm9) [ /T,5-0 rr-1 (1 + a + 2a csy) Adding 2 ",.,., «2 + 2a csy + csv - - a ß ( sin9 - cs9 sxns) [-7: 5 rh (26) (1 + a + 2a COSY) N ' 6 Diß cs ( ) and - D^ l A sin<9 Numeratr Z [-7; 2 x «H I (<* + COSY) (1 * a + 2a COSY) sin9 < (1 + a + 2a COSY) 2 - a(sin9 - cs9 sin9)(a + 2a COSY + cs Y)1 a 2 ß1*1 2 = [ r 5 r 2 ] [sin9(csy + a) (1 + a COSY)+ a cs9 sin9(a + COSY) 2 ] (1 + a + 2a COSY) ' ; raß sin9( COSY + a)..., 2 v, fpn\ = [ 7; =5 T5 I [( 1 + a COSY + a cs9 + a cs9 COSY)1 \ ü 'i (1 + a' + 2a COSY) Expanding the denminatr f (24) - 2ß D X sin (4-) - - 2ß 2 2a sin 2 (- -) ( ± + g*l ) 2 2 1 + a + 2a COSY = -2aß 2 (l-cs9) ( - M^Y 1 + a + 2a COSY ) <28) - D cs - - 4a e «ui ( T ) cs, ( 1 + tf6 + 2g COSY ) - 2aß 2 ( 1 - cs9) cs9 (- a ^ c f ) 2 (29) 1 + a + 2a COSY Cmbining these tw terms Q g 2 ß 2-2ßD sin(-r-) - D 2 csb = [- 5-- rj ] [1 + 2a COSY(1 + cs9) 2 (1 + a + 2a COSY) 2 2 + 2a cs9(2 + cs Y cs9) + 2a s csy(2 cs 2 6 + cs9-1) + a 4 (2cs 2 9-l)] (30) -32-

This gives as the resultant expressin fr A p. r c sin9( csy + a) ( 1 + «csy + a 2 cs9 + t cs9 csy), /^) Ap - / 2 arc tan[ j_ + 2fl csy( x + cs0) + ^ csq ( 2 + C s z Y cs9) ^ + 2a 3 csy ( 2 cs 2 9 + cs9-1) + a 4 ( 2 cs 2 9-1) Ap "was placed in the frm (31) in rder t cmpare it with equatin (23) which cmes frm the üdcck system. It is pssible t express A p in a smewhat simpler frm by placing the expressin fr DjU9) directly in (1).., 9 v,9,, «+ csy Ap = arc tan[ ] (32) «. a, 9 w a + csy ß - 2a(3 sin 2 ( -) - rs -- - 2 1 + & + 2a csy. a sui9( a + csy). = arc tan[ : =5 rs ~ 7-5 rr~j 1 + a" + 2a csy - 2 sin (JL) ifl + a csg) v 2 '. a siii9(«. siiji-h«+ csy; csy) Ap = arc tan I : I 1 + a COSY + a (a + csy) cs9 (33) ~ 33- \

IX REFERENCES H(I)»Thery f Sund», Lrd Raleigh, (bk). 7(2) Patent Applicatin by Paul G. Hansel, Serial N,627,272 (Directin Finder f the Dppler typs utilizing a revlving vertical antenna and direct reading phasemeter. The pssibility f using a circular array f fixed antennas t replace the ne rtating antenna is suggested.) X(3) "Radi Directin Finding by the Cyclical Differential Measurement f Phase", C. til. Earp and R. M. Gdfrey, Jur. I. &. E., v.94, Part IIIA, «=15, Mar-Apr, 47«J\k) "Investigatin f Dppler Effect in Determining Dir- ectin f Arrival f Radi Waves", Masters' Thesis by J. L. L. Bulet, Univ. f 111., Sept. 20, 47. (5) "An Analysis f Antenna Arrays Having Harmnic Patterns", Masters' Thesis by R. vv". Annis, Univ. f 111., June, 48«(6) Summary Technical Reprt #U, Univ. Of 111, Directin Finding Research Labratry,- April 1$, 1948. -34-

i* X DISTRIBUTION LIST OF TECHNICAL REPORT #8 Chief f Naval Research (Cde 413) Navy Department Washingtn 25, D. C, (5 cpies) Chief, f Naval Research (Cde 427) Navy Department Washingtn 25, D,.C. (5 cpies Directr, Naval Research Labratry Anacstia, D, C. (5 cpies) Directr, Special Devices Center Office f Naval Research Sands Pint, Prt Washingtn lng Island, N. Y, (1 cpy) Cmmanding Officer Office f Naval Research Branch Office America Pre Building 616 Nrth Rush Street Chicag, Illinis (2 cpies) Chief f Naval Operatins (Op 20-2) Navy Department Washingtn 25, D, C. (1 cpy) Chief f Naval Operatins (Op 2Q-EZ) Navy Department Washingtn 25, D.,C. (1 cpy) Chief f Naval Operatins (0p~413-B) Navy Department Washingtn 25, C, C. (1 cpy) Chief f the Bureau f Ships (910) Navy Department Washingtn 25, D. C. (1 cpy) Chief f the Bureau f Ships (913) Navy Department Washingtn 25, D. C. (1 cpy) Chief f Naval Operatins (Op 413-Z) Navy Department Washingtn 25, D. C. (1 cpy) Chief f Naval Operatins (Op 50-Z) Navy Department Washingtn 25, D.# C. (1 cpy) Chief f the Bureau f Aernautics (R-7) Navy Department Washingtn 25, D, C. (1 cpy) Chief f the Bureau f Aernautics (EL-3) Washingtn 25, D. C. (1 cpy) Chief f the Bureau f Aernautics (El-31) Washingtn 25, D. C«(1 cpy) Chief f the Bureau f Aernautics (El-33) Washingtn 25, D. C. (1 cpy) Chief f the Bureau f Aernautics (El-4) Washingtn 25, D. C (1 cpy) Directr, U. S. Nasal Electrnics Labratry San Dieg«Califrnia (2 cpies; Chief f the Bureau f Ships (903) Navy Department Washingtn 25, D. C. (1 cpy) Chief f the Bureau f Ships (911) Navy Department Washingtn 25, D. C. (1 cpy) - 35 ~» -S^*^

I. Chief f the Bureau f Ships (925D) Chief f the Bureau f Ships (920) Navy Department Navy Department Washingtn 25, B. C Washingtn 25, D. C. (1 cpy) (1 cpy) Naval Liaisin Officer Chief f the Bureau f Ships (930) Signal Crps Engineering Labratry Navy Department (CSL) Washingtn 25, D. C. Frt Mnmuth, N. J» (1 cpy) (2 cpies) Signal Crps Engineering Labratry Chief Signal Officer Bradley Beach, 'New Jersey U. S, Army, Cde Sig TA2S (2 cpies) Pentagn Building Washingtn 25, D. C. (2 cpies) U. S. Naval Academy Central Radi Prpagatin Labratry FG Schl, Electrical Engineering U. S. Bureau f Standards Dept., Annaplis, Maryland Washingtn, D. 0. (1 cpy) (2 cpies) Cmmanding Officer Cmmanding Officer U. S. Navy Office f Naval Research U, S. Navy Office f Naval Research Branch Office Branch Office 495 Summer Street Van Nuys Building, Suite 50? Bstn 10, Massachusetts Ls Angeles, Califrnia (1 cpy) (1 cpy) Cmmanding Officer Cmmanding Officer U. S. Navy Office f Naval Research U. S. Navy Office f Naval Research Branch Office Branch Office Building 3, U. S. Naval Shipyard 801 Dnahue Street Brklyn, New Yrk San Francisc 5, Califrnia (1 cpy) (1 cpy) Assistant Naval Attache fr Research Cmmanding Officer Naval Attache Office f Naval Research American Embassy, Navy 100 Branch Office c/ Fleet Pst Office 1030 East Green Street New Yrk, New Yrk Pasadena 1, Califrnia (1 cpy).(1 cpy) Cmmander Bureau f Aernautics General Rep», Operatinal Develpment Frce U.S.M., Central District Fleet Pst Office Wright Field, Daytn, Ohi N9vs Yrk, N. Y. "Fr Transmittal t Air Materiel (1 cpy) Cmmand" (3 cpies) Attentin: Dr. Straitn Electrical Engineering Res, Labratry university f Texas Austin, Texas (1 cpy) - 36 -

DIRECTION OF PROPAGATION WAVE OF C.A.= Reference Axis B.= Psitin f Antenna at time t=t ( THE CASE OF THE SINGLE ROTATING ANTENNA PLATE TECH. RPT. 8 I

-*s : \- < z Ü ÜJ <r Q cc LÜ a. Q -j a: < Lü a.? UL < a: < XL O _l m

r DIRECTION OF ARRIVAL ANTENNA ELEMENTS UJ a. en UJ 2 3 \ 2 i r~ Q. U. O LU 4 -«TiMP 1 INIL 8 Z < 2 Q Ui <r 2 5 7! * PHASE RELATIONS OF COMMUTATED ANTENNAS PLATE 3 TECH. RPT. 8

FIG. CIRCULAR ANTENNA ARRAY OF N ELEMENTS /«(t.).n/ p, G 2 LAW OF COUPLING-VECTOR DIAGRAM PLATE 4 TECH. RPT. 8 ' \

/e(t,)e n / p JQ LAW OF COUPLING-VECTOR DIAGRAM PLATE 4 TECH. RPT. 8 t_ tlp-~- -«r~ ' \

-I.A. --».Lfi»Tinill-iiirlt -.^SJ MAXIMUM OF STANDING WAVE NOTE! a) K STWEWCTH OF WAY 2 nv ic a/ r> ST i NCTH 0F RAY, b) ELECTRIC VECTOR ASSUMED PERPENDICULAR TO PAGE ;EQUIPHASE SURFACES MAXIMUM OF STANDING WAVE EQUIPHASE SURFACES FIG. I -Y V I FIG. 2 -Y STANDING WAVE PATTERN IN THE FIELD OF TWO INTERSECTING RAYS PLATE 5 TECH. RP?. 8

A DIRECTION OF ARRIVAL OF INTERFERING WAVE 6^= acs wt+ßcs(üj ; t-0*y) DIRECTION OF ARRIVAL OF PRIMARY WAVE e,= es/ü>t+ßcsc,t] FIG. I DIRECTION OF ARRIVAL OF PRIMARY AND INTERFERING SIGNAL a cs/wt + ßcs(u),t- 9) + yj A»=nrr.inn Qsin ö / l+acs^> f38(wt+/8csa» / t)j / ' ^[acsdj < (t)= wtt-/3cs(tdt-ö+y J~ ä>t+ßcscu,f] -/Ö/cs(cj,t-ö)-cscü ) tj+/ FIG. 2 VECTOR DIAGRAM OF SIGNAL PHASE SHIFT PLATE 6a

^ -C0S(6J,t) COS(tO,t) COMBINATJON OF TWO VECTORS yß COS(qt) DSIN(W,t- t ) FIG. 4 Ecs( u,i-a P ) VECTOR DIAGRAM OF BEARING SHIFT PLATE 6 b TECH. RPT. 8

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