THE UNIVERSITY OF MICHIGAN RADC-TN- L, -' 5Q0 '<t I ASTIA Document No. EVALUATION OF THE PINCUSHION SYSTEM by C. Chu, A.W. Wren and J. LaRue February 1960 2872-1-T = RL-2089 Report No. 2872-1-T on Contract AF 30(602)-1982 Prepared for ROME AIR DEVELOPMENT CENTER AIR RESEARCH AND DEVELOPMENT COMMAND UNITED STATES AIR FORCE GRIFFISS AIR FORCE BASE, NEW YORK

THE UNIVERSITY 2872-1 -T OF MICHIGAN PATENT NOTICE: When Government drawings, specifications, or other data are used for any purpose other than in connection with a definitely related Government procurement operation, the United States Government thereby incurs no responsibility nor any obligation whatsoever and the fact that the Government may have formulated, furnished, or in any way supplied the said drawings, specifications or other data is not to be regarded by implication or otherwise as in any manner licensing the holder or any other person or corporation, or conveying any rights or permission to manufacture, use, or sell any patented invention that may in any way be related thereto. Qualified requestors may obtain copies of this report from the ASTIA Arlington Hall Station, Arlington 12, Virginia. ASTIA Services for the Department of Defense contractors are available through the "Field of Interest Register" on a "need-to-know" certified by the cognizant military agency of their project or contract. ii

THE UNIVERSITY OF MICHIGAN 2872-1 -T TABLE OF CONTENTS Page I Introduction 1 II Discussion a. Diversity Systems 1 b. Angle Diversity 2 III Performance of Some Diversity Systems 4 a. Angle Diversity System 4 b. Equivalent Space Diversity 10 c. Comparison of Angle and Space Diversity 12 d. Single-Frequency Angle Diversity 12 IV The Effect of Correlated Signals 16 V Summary and Conclusions 25 Appendix 28 References 35 iii

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THE UNIVERSITY OF MICHIGAN 2872-1 -T I. INTRODUCTION The stated purpose of this report is an evaluation of the Angle Diversity System proposed jointly by Harry Davis and Dr. Joseph H. Vogelman (Ref. 1). This evaluation is necessarily made in the light of present theoretical and experimental knowledge. It is definitely felt that a better theoretical model and additional experimental data are necessary for a precise evaluation. II. DISCUSSION a. Diversity Systems The general concept of an nth order diversity system requires the combination in some fashion of n signals S,(t) which are locally coherent from the standpoint of the R. F. carrier but whose amplitudes are slowly varying functions of time which are statistically independent (Ref. 2). Thus, predetection diversity systems require circuitry for the control of RF (or IF) phase. From a practical standpoint the problem of RF (or IF) phase control can be avoided by the use of post detection combining. The signal amplitudes encountered in practice may be correlated and have unequal means. Although the case of correlated signals has been studied by Staras (Ref. 3) and Packard (Ref. 4) and the case of unequal medians has been treated by the Bell Telephone Laboratories group (Ref. 5) and others (Ref. 6), the case of unequal medians and correlated signals has not received extensive analysis. Bolgiano, Bryant and Gordon (Ref. 7) give a treatment of correlated

THE UNIVERSITY OF MICHIGAN 2872-1 -T signals with unequal medians. Unfortunately this analysis results in an integral which must be integrated numerically. However, it is shown in this report that the method used by Packard (Ref. 4) to treat correlated signals with equal medians can be extended to the case of correlated signals with unequal medians. See Appendix. It is evident from these results that satisfactory diversity systems can be realized with correlated signals of unequal means provided the correlation coefficient is small enough and the ratio of the signal means is not too large. The operation of a diversity system then requires a suitable number of signal channels which provide signals which are at most partially correlated with as little difference in means as possible. b. Angle Diversity In an angle diversity system, offset feeds are used to obtain the necessary "independentt" channels for combination. Such a system is attractive both from the standpoint of economy and space (land) since one antenna can be used to provide a number of channels. It is apparent that successful use of such a system must rely on a scattering mechanism which is volume extensive. In particular, While this report was in preparation a report by the Hermes Electronic Co. (Ref. 12) came to our attention. Similar statistical methods to those given in the Appendix are used in the Hermes report. However, the results given in the Appendix are in a form which can be readily applied to the problem of combining signal channels when the mean power in each channel and the correlations between channels are known. In addition an explicit form is given for the case where the eigenvalues are not distinct. 2

THE UNIVERSITY OF MICHIGAN 2872-1 -T such a system will be effective when the beamwidth of the antenna used in the system is small compared with the angle subtended by the effective scattering volume. In other words, angle diversity becomes effective when the antennato-medium coupling loss becomes sizable. This point is clearly recognized by the Rome group in their proposal of the pincushion system (Ref. 1). An angle diversity system then uses a number of narrow antenna beams to illuminate and receive from a volume of space larger than the individual beamwidth. The gain of such a system depends on the variation in scattered power as a function of angle from the transmitter and/or receiver. Both the Booker-Gordon theory (Ref. 8) and the BTL layer theory (Ref. 9) may be used to predict the average scattering law. In either case, however, assumptions are made as to scale of turbulence and anisotropy or layer size and height dependence. Since there is neither strong experimental evidence nor adequate theories of turbulence to make other than an arbitrary choice, these atmospheric parameters are adjusted to produce average signal parameters which fit observed scatter propagation data. For this reason, it is felt that a choice of theory for analysis of an angle diversity system, at this time, is mostly a matter of preference. However, the recent publication by BTL of an extensive study of beyondthe-horizon propagation data (Ref. 5) and the fact that the BTL layer theory produces general expressions in closed form for the variation in received power as a function of antenna beamwidth and scattering angle makes this theory a more 3

THE UNIVERSITY OF MICHIGAN 2872 -1 -T attractive choice in our opinion. For this reason, the calculations contained in this report are based on the BTL layer theory (Ref. 9). III. PERFORMANCE OF SOME DIVERSITY SYSTEMS a. Angle Diversity System To determine the performance of an angle diversity system, it is necessary to determine the reduction in the mean signal level for the off-axis feeds and the correlation between adjacent beams. If we consider the case of different frequencies for each beam, there is insufficient data to determine the correlations to be expected. For one beamwidth separation, it seems reasonable to expect the correlation to be less than 0. 5. With identical frequencies, observations at Cornell (Ref. 7) have shown average correlations of 0. 5 for a limited data sample taken with one beamwidth separation. The use of different frequencies in the individual beams should provide additional decoupling resulting in a decrease in correlation between adjacent beams. In order to determine the decrease in the mean received signal level for off-axis feeds, we consider the case of symmetrical beams with a 3 db beamwidth a separated by one beamwidth. It is assumed that the on-axis feed is aligned along the optimum great circle path. See Figure 1. Let 90 be the angle between the chord, connecting the transmitter and receiver, and the tangent to the optimum great circle route. Further, we define a quantity K to be equal to the ratio of the beamwidth a to the minimum half scattering angle 00, i.e. 4

THE U NIVERSITY 2872-1 -T OF MICHIGAN Figure 1 Scatter Path Geometry 8 Figure 2 Geometry For 0i ij 5

THE UNIVERSITY OF MICHIGAN 2872-1 -T K - 00 We may now determine 0..,the angle between the chord and the bottom of the i, jth beam, where i is the elevation beam number and j is the azimuth beam number. From Figure 2 for small angles 0 we find 012 = 02 + S2 (1) For separation of one beamwidth and a = K 00 0' = 0o+(i -1)K0o+ K00 2 O' = ao[l+(i_ 1)K] and = (j - 1) K0 Since K 0" = 0.. + K0 iJ 2 (1) becomes Ki + 2 = 2 (i - 2+ (j-1)2 or 0ij = 0o (l+(- K) j 1)K2 (2) o2 The ratio of the power in the i, jth beam P.. to the power in the optimum great circle beam as predicted by the BTL theory is (Ref. 9) 6

THE UNIVERSITY OF MICHIGAN 2872-1 -T 5 Z+ a f( a( Pij So 2+ a f( a ) - 00 ij where 4 f(x) = 1+ 2 + x (1 +x)4 8 1 +x Using (2) and a- = K we obtain 0o 2+ K _11 ( DDiJ K f(K) p (Di K) (4) Pij 2+K f K Dij K where DijK ([(1 + (i- ) K)2+ (j 1)2 K 1/2 K We have in equation (4) an expression for the ratio of the mean received power in any beam with respect to the mean received power in the optimum great circle beam which is a function of the beam number (i, j) and K only. Figures 3, 4, and 5 show the ratios of mean received power in the various beams to the mean received power in the optimum beam for K = 1, 1 and 1 respectively. 4 From a comparison of the power ratios for the indicated values of K, the importance of the ratio of the antenna beamwidth to 00 can be seen. The number of available channels increases rapidly as K decreases. It would appear that higher order angle diversity systems become feasible for values of K equal to or less than 1. 2 7

THE UNIVERSITY OF MICHIGAN 2872-1 -T i 4 3 2 1 2 3 4 1 -26.0 db -16.5db - 6.4db 0 db - 6.4db -16.5db -26.0db 2 -30.0db -24.1 db -18.8db -16.5 db -18.8 db -24.1 db -30.0db 3 -35.3db -31.2db -28.2db -26.8db -28.2db -31.2db -35.3db..... II Figure 3 Power Ratios for K = 1 i 4 3 2 3 4 1 -13.7db - 7.7db - 2.3db 0 db - 2.3db - 7.7db -13.7db 2 -18.8db -14.7db -11.6db -10.4db -11.6db -14.7db -18.8db 3 -23.6 db -20.8db -18.8 db -18.1 db -18.8db -20.8db -23.6 db Figure 4 Power Ratios for K = 4 3 2 1 2 3 4 1 - 5.6db - 2.6db -.7db 0 db -.7db - 2.6db - 5.6db 2 -10.0db - 8.0db - 6.6db - 6.1db - 6.6db - 8.0db -10.0db 3 -14.1db -12.5db -11.5db -11.2db -11.5db -12.5db -14.1db Figure 5 Power Ratios for K = 1 4 8

THE UNIVERSITY OF MICHIGAN 2872-1 -T If the correlations between the various beams were known, the method outlined in the Appendix could now be used to compute the reliability for different order diversity systems for specific system thresholds. Since the assumption of system threshold and correlation would be arbitrary, we simplify the calculation by assuming zero correlation and a system threshold 10 db below the mean power received in the optimum beam in order to compute reliabilities. With these simplifying assumptions, the calculated reliabilities are as indicated in Figure 6. K 1.5.25 n 3 99.43 % 99.9 % 99.94 ~/ 4 99.47 % 99.96 % 99. 993 / 99.5 % 1 99.98 % 99. 999 % 6 99.52 % 99. 99/o 99. 999+ / Figure 6 Angle Diversity Per cent Reliability for Pmean 10 db above System Threshold The reliabilities for n = 4, K = 1/4 and n = 5, K = 1/2 are not optimum since a more efficient system could be achieved by straddling the optimum great circle path for these values of n and K. 9

THE UNIVERSITY OF MICHIGAN 2872 -1 -T b. Equivalent Space Diversity In order to have some basis for comparison of angle diversity with space diversity, we make the assumption that the cost of an antenna is proportional to its effective area. With this assumption, the beamwidth of an equivalent nth order space diversity system is found to be \IfK0o. The power ratios become P = ( i)3 2+K f(K) Pn 2+K f (-K) (5) where K and f(x) are as previously defined. The ratio of P1 for various Pn values of n and K are shown in Figure 7. K 1.0 0.5 0.25 n 1 0 db 0 db 0 db 2 -4.2 db -3.4 db -2.7 db 3 -6.8db -5.6 db -4.4 db 4 -8.7 db -7.2 db -5.7 db 5 -10.2 db -8.5 db -6.7 db 6 -11.5 db -9.5 db -7.6 db Figure 7 Power Ratios Equivalent Space Diversity 10

THE UNIVERSITY OF MICHIGAN 2872-1 -T The calculated reliabilities of equivalent space diversity for various values of n and K are given in Figure 8. nK 1.5.25 n 2 96.8 0 97.6 0%/ 98.3 0 3 97.3 % 98.6 0/ 99.3 %,, 4 96.6% 98.7 0/0 99.67 % 5 95. 0 98.7 ~/ 99.76 0/ 6 91.0 % 98.44 0/o 99.78 /o Figure 8 Per cent Reliabilities for Pmean 10db above System Threshold Equivalent Space Diversity Diversity Diversity K Order Type 1.0.5.25 2 Equiv. Space 3 db 5 db 6 db Space 11 db 11 db 11 db Angle 11 db 19 db 21 db 3 Equiv. Space 4 db 7 db 10 db Space 23 db 23 db 23 db Angle 12 db 23 db 28 db+ 4 Equiv. Space 3 db 8 db 12 db Space 28 db+ 28 db+ 28 db+ Angle 12 db 26 db 28 db+ 5 Equiv. Space 2 db 8 db 14 db Space 28 db+ 28 db+ 28 db+ Angle 12 db 28 db+ 28 db+ 6 Equiv. Space -1 db 6 db 15 db Space 28 db+ 28 db+ 28 db+ Figure 9 Diversity Gain above a Single Channel Note: System Reliabilities above 99.99 o/o are Indicated as 28db+ 11

THE UNIVERSITY OF MICHIGAN 2872-1 -T c. Comparison of Angle and Space Diversity The reliabilities calculated in the preceeding sections are based on independent signals and selection diversity. In practice, either equal gain or maximal ratio diversity are probably preferable. To some extent the two assumptions will offset one another. In any case, the increase or decrease in reliabilities will be essentially the same for the different types of systems used in this comparison. In Figure 9, the diversity gains above a single channel are listed for the two systems previously examined. In addition, a third system is added which is an nth order space diversity system utilizing n antennas with beamwidths, and therefore areas, equal to the single antenna used in the comparable nth order angle diversity system. The data in Figure 9 indicates an angle diversity system will out-perform an "equivalent" space diversity system by from 7 to 15 db depending on the value of K and the order of diversity. In addition, it should be noted that a 3rd order angle diversity system will give equivalent performance to that of a space diversity system utilizing two antennas of the same size for K = 1. d. Single-Frequency Angle Diversity In the angle diversity calculations which have been made up to this point it has been assumed that the individual beams are fed from transmitters with 12

THE UNIVERSITY OF MICHIGAN 2872-1 -T different frequencies. In the course of performing such calculations one is struck with the possibility of obtaining a more efficient use of the scattering volume by utilizing the same frequency for each transmitter. The principal advantage gained is in the vertical plane where the use of a single frequency results in a substantial increase in the scattering volume common to a given receiver beam. The use of a single frequency results in common volumes which are unsymmetrical. The beam power ratios can be calculated however using the method indicated in the BTL article of September 1959 (Ref. 5). The ratio of the power in the first beam to the power in the jth beam is a ( 0o 4 aT -aR o ) f(TR) P _00 Go O+a R Oo+aR R Pij 4 aT +00-0 ea+0 R-0 OR-o / 00 \4 a T+O-a R - Pij ) )f( TR ) + f( ) -f( ) - R)f( - R (6) where aT = nk 0 aR = k0 OR = 00 [l+(j-l)k] f(x) -=1+ 1+ -- () (1 +x)4 8 v+x In addition it is found that the ratio of the power in the on-axis beam for a single frequency operation to the power in the on-axis beam for multiple frequency operation is 13

THE UNIVERSITY OF MICHIGAN 2872-1-T eo 4 a TR f( P11 (single frequency) _ 1 + f% ) 0+ )f(T +aR ) P11 (multiple frequency) 2 a f( -R) 00 (7) Using equation (7) the following power ratios are calculated for 2,4 and 6 vertically stacked beams for the indicated values of K. K No of 1.5.25 Beams 2+.8db + 1.db + 2.0db 4 + 1.2db + 2.2db + 3.3db + 1.4db + 2.4db + 3.8db Figure 10 Mean Power in On-Axis Beam (Single Frequency) Mean Power in On-Axis Beam (Multiple Frequency) In addition, from equation (6) the decrease in mean power for the upper beams is found to be smaller. See Figure 11. Using the data given in Figures 3,4 and 5 for multiple-frequency operation together with the data in Figures 10 and 11 for single-frequency operation the increase in mean power for single-frequency operation can be determined. This data is shown in Figure 12. It should be noted at this point that the power ratios given in Figure 11 are a function of the number of vertical beams used. The calculations show, however, that the change in the power ratios between the vertical beams is small 14

THE UNIVERSITY OF MICHIGAN 2872-1 -T K Beam 1.5.25 Number 1 0 0 0 2 - 7.2 db - 4.6 db -2.7 db 3 -11.3 db - 7.8 db -4.9 db 4 -14.2 db -10.2 db -6.6 db 5 -16.3 db -12.2 db -8.3 db 6 -18.4 db -13.8 db -9.4 db Figure 11 Power Ratios for Vertical Beam (Single Frequency) Separation K eam 1.5.25 Number 1 + 1.2 db + 2.2 db + 3.3 db 2 +10.5 db + 8.0 db + 6.7 db 3 +16.7 db +12.5 db + 9.6 db 4 +21.0 db +16.0 db +12.0 db Figure 12 Increase in Mean Power for Single-Frequency Operation 15

THE UNIVERSITY OF MICHIGAN 2872-1-T as the number of beams is increased from 2 to 6, the maximum change being less than 0.2 db, which indicates as expected that a more efficient system could be realized if the transmitter power used in the vertical beams above the tangential beam were used instead in the tangential beam. From the increase in mean power for the various beams which results from single-frequency operation it would appear that a single-frequency angle diversity system may more closely approach an optimum scatter communication system. Dr. J. H. Vogelman has pointed out in a private communication that the problem associated with single-frequency feeding may be avoided while obtaining the full use of the scattering volume in a multiple-frequency system if the receivers associated with each feed have bandwidths which are wide enough to cover the spread in transmitter frequencies. With this type of system suitable filters would be used in the individual beam receivers to separate the different transmitter frequencies for combination in diversity combiners. Thus a system with n feeds would be a diversity system of order n2. IV. THE EFFECT OF CORRELATED SIGNALS For the case of two correlated Rayleigh signals with unequal means it is shown in the Appendix that maximal-ratio combination will give a signal whose statistics are described by - R _ R 2X1 x X2 e 2 2 P( R) - (8) 16

THE UNIVERSITY OF MICHIGAN 2872-1-T where 2 2 2 2(1 2 CT + o-2 + l (oj + c2 ) - 4oT 2 (1 - P12 ) 2 2 2 2/,22~ 4r 2~ i 2 T+ 02 - (o + 2 ) - 4CT 12 (1 - PT2 ) X= - --------- 2 Using equation (8) and the relation between c and p (see equation (7) in the Appendix), the probability of the combined signal exceeding a given level may be calculated for various ratios of o-T to 0-2 and correlation coefficients c. By choosing the median of the higher channel as a reference level and assuming the local noise powers are constant and equal to 1, we obtain the graphs shown in Figures 13 through 19. The odd values of c result from assuming convenient values for p. A comparison of the curves for the various values of c indicates useful diversity gain for signal correlations as high as c =.78. It can also be noted that the effect of correlation on the combined signal is essentially the same for the unequal means as it is for the equal means. For example, in the region of interest (i. e. high reliability) the local signal-to-noise ratio for a correlation coefficient of c =.59 is approximately 2 db below that for uncorrelated signals for each of the three ratios of signal means (0 db, 6 db, 12 db). To show the effect of correlation on the combination of more than two channels we consider the case of three channels with equal means and-equal 17

THE UNIVERSITY OF MICHIGAN2872 -i -T WCI 1 I ff" I. C0 LO) I i+.r-j co;-4 - Q).r.4 Q 0.r-4 v - x x - x -P4 x x " —f x x N %, Cd x x - x P.4 - x Cd 'IL 'IL. ht- a () (a) o4 '. 0. II Q CQ C) N It i % % N L AL ' il Ar 4 4 - i It IMIt AC 0_ 0 C.)w a1) a) H '+-4 C)a) a1) CO ' —4 a1) Z44 0) 0) co 0) 0) 10) C) I I --- I I 0) 'I I I I I I I I I I I 0) LO CIQ 0 C) ITguutBqD ia-qt-H JO U'81PgOk 9AoqV q cijl OJl as-ioN-o4-1Pu~iS ieoorj 18

6T Local Signal-to-Noise Ratio db Above Median of Higher Channel C) IN C' I CI I. C) CC) co Co co 00 C." C) I I I I I I I I I I I I I I I -i I I I I I I I I I I I I I I. ' * ' ' I i I i i -I1 I I 4 Co0 Co0 Co CD C ________________________________ ____________ C C I. C,, CD C 0 -II ______ _ __________ ___ + - rN - 7 -4 CD p CD 0 CD R i ~0 I-h I +f 1- 1 )-J CD pi C) CD 0 CD 0 -CD CD CD I x I --- I/ CD CD CD CD CC) NNI C.' CD cj -t 1CJD - 0 )0 Co C C." C NVOH3IN ao AIIS~aaAlNf1 m NV la I HD In a H I

THE UNIVERSITY 2 872 -1 -T OF MICHIGAN 0 LO. Ir Ii I IL '2 Wi i + ",O*,\ VIN I I - A - -1 x x x IL x x I x x % x IIL I x x x N'th, - I I Nt,:! w m F —4 k 0 0 w bl). r-4 p M (1) -0 vj.0 C) v P-4 Cd (1) 004 r —q;-4 'a) Cd k 0 0 D 5 u.r-4 x 4 Cd k:t 0:5 44 : co.0 - lk 3% i lqk 3%r i i i ""*N I lk 3% -. i 3L I%- I I I lk lk I N A 1 '9 I i _ _ _ _ _ _ _ _I+L _ _ _ _ _ C.0i it + r —l 0 0 a, 0) 0) 0)3 0) la) C.) 0) E p 0) 0) 0) (M OD 0) I I I I I LO) 'I T-T 0 N\ IF 0) 0) co 0) I 0) 0) LO) PTIUU'tI.IltI~1H JO T1r8T7p0W 9Aoq\T qpf OJTI-H 0STON-04-BTUMS ile0orj 20

THE UNIVERSITY OF 2872-1 -T MICHIGAN 1 _ 1 I I I O 0~,'EL I.1 S ' \ --- —--- IL p x -x - 4:, \ \ - N. K w- - ll 0 1 U2 1 P —4 1 Cd! a Ul 0 i.,A cd I co (L) 0 1-1 4 (3) ) -t P-4 1 Cd I- '.4;j 4 (2) V P (1) I k 0 0 ) 4 0 4 4 A 4;q i I (:) -P 4.4 P4 i C. I - 0 - I 0 oM;>7 <I + 1-4 + __ 0 0 CdC a) 0 H 0 C)) C) (M a) CD *-4 )-I 9b C) 00 Co 0. I I I I I I I I I I I I O LO 0 r-4 I LO '-4 0 C9 j I[- I I LO I I r I CO 139UU't3l J1;9I[.2H JO U16JPOWq gAoqV cip OTWIVI 9STON-0 —BCU2TS J-eoo'J 21

THE UNIVERSITY OF 2872 -1-T M IC HI GA N I I I i IC lqqkrl i a a i i XX %4 I 'M 'Rim, 'NS Ll x x x N, x x x N, x x 1% '%I. x x Ix k \ x x N%, I 11 i P. U) U2) - "- bD -rC Ir4 ) P4 a) 11 - af) bC1 0" 0 g00 a) a2) a1) a) a) m0 C)) C) C) C) C)) I- - I I I I I I I I I 0) 'I I I I I I I I I Cl LO CO I I I I Iou~e.o TH JO UTXT.PJA OAoqVcff)1 OTIWU C)STjqO4I'-jV12S I-60r 22

THE UNIVERSITY 2872-1 —T OF MICHIGAN C0 lb I r I I i >43 a) 0 r43 -s ~LZZX C,' a) r43 a1) 0 Cd a) a) 14 00 XN4 P=l IL NO 4- C4-4 0 0 c0c0 C) Q a2) a1) (a) C4-4 0 43 C) Q a) co v —4 (1) 9b.r.q P4 C) C) a) C) co C) C) C) C) I I I I I I I I I I I I I I I I I I I I I '-4 LO '-4q C)Z LO c~l I I 0) TIDuu'etE[ J9TIBJH JO uejTPoW GAoqVr qp OTV OSTON-01- rIUU2S reoorj 23

-5 a) 4 --10 +12 db -1- - -- Maximal Ratio Divgher Channel 0 MeanFor Correlated Signa - - 25 ~ Per cent of Time Level Exceeded 99.998 99.99 99.9 99 90 Per cent of Time Level Exceeded -I-' * 10r i^ ity 50

THE UNIVERSITY OF MICHIGAN 2872-1 -T correlation coefficients. The correlation matrix is 2 p <X2 p a.2 p 2 o-2 p o-2 L 2 2 21 p (r2 p 2 o2 The roots of the characteristic equation can easily be shown to be 1 = o-2(1 +2p) x = x3 =- 2 (1-). Using the results from the Appendix for the case of non-distinct roots we find 2 P (r < R) = 1- (1 +2p 9 p2 R - Re 20-2 (1+2p) + e 2'2(1-p) (1+5p)(1 -p) 9p2 This result is plotted in Figure 20 for c = 0,.78 and 1. An examination of the curves indicates substantial diversity gain at the higher reliability levels for correlation coefficients as high as c =.78. V. SUMMARY AND CONCLUSIONS An analysis of the performance of the pincushion system using the BTL layer theory indicates reasonable diversity gains over an "equivalent" space diversity system. Although the Booker-Gordon theory would produce different numbers the conclusions can be expected to be the same. Both theories predict average performance and from this standpoint suffer from the same limitation. The analysis shows that the pincushion system as originally proposed does not efficiently utilize the effective scattering volume. It can be concluded 25

Local Signal-to-Noise Ratio db Above Median of One Channel 01 I I II CAD N N O 01" 01C. 01 CD 01 CC) Co co CO0 CO0 CO co CD 0 (D 0 CDt 0 CD CD CD co Co co CDA 01 Lii I-7ZL8? N VOI H3II1N 3AINfl alll

THE UNIVERSITY OF MICHIGAN 2872-1 -T that a more efficient system can be obtained either by use of a single frequency or by utilizing the method proposed by Dr. Vogelman. In the analysis of the effects of correlated signals a general method is derived for computing the combined reliability of n correlated signals with unequal means. Curves have been plotted for the special cases of two correlated channels with unequal means and three correlated channels with equal means. These curves indicate useful diversity gains at the higher reliability levels for signals with correlation as high as c =. 78. It is felt that the pincushion system when used with a single frequency or with wide-band receivers as proposed by Dr. Vogelman will more closely approach an optimum scatter communication system. 27

THE UNIVERSITY OF MICHIGAN 2872-1 -T Appendix The effect of correlation on maximal-ratio combiner diversity for signals of unequal mean may be computed in the following manner. Consider n Rayleigh distributed signals CZi P- (zi) (1) (Zi) e -i (1) 0T. with correlation cij between zi and z. Cij = i j - > (2) <(i - <Zi )2><(Zj - <Z > )2> Each of variables zi may be resolved into two uncorrelated variables 2 xi and yi which are normally distributed with zero means and variance i, z. - x2 + y. (3) The correlation between the x's and y's is such that <Xixj> = Pij Ti cj i 7 j (4) <yi yj Piji i i j (5) <xiyj> =0. (6) The correlations between the z's and the x's and y's are related by the equation (Ref. 7) 28

THE UNIVERSITY OF MICHIGAN 2872 -1 -T cii 4= ii F (znI).915 pij+.0572 pij 4 i 4 2n (n l )2 (2n-l) (7) Consider the linear change of variables u] = Ax] (8) v] = Ay] (9) where the symbol ] indicates a column matrix (Ref. 10). In addition we require the following conditions <u > = 0 i: j (10) <vivj> = 0 i: j (11) = Zx Zv II (12) <U2 = X (13) <v.2> - X. (14) Since the ui and vi are related to xi and Yi by the same transformation matrix A, it is only necessary to discuss one of the transformations. From condition (12) we obtain,, u] -= L AT Ax] =, x] (15) thus A1 A = U and A is an orthogonal matrix. Forming the product u] u, and taking the average we obtain 29

THE UNIVERSITY OF MICHIGAN 2872-1 -T U1 <Ul> u3>.... <ulun> r 1 P12^ 1T 2. -Pln 1Tn <UU> <U2 <U 3>. 2 222n 2 n.... <Ui n> < n> < n> n Pln> n 2' n (16) Using equations (10), (11), (13) and (14) this reduces to c Xai00....0 i Pequto (R *f 1n 1 )n 0X20.....0 A 1 1 2 (r2 nP 2 GnA (17) 2 000. n.. P1..... j A l n le n Since the u.'s and v.'s are linear combinations of normally distributed variables, they are also normally distributed and each ui and v is completely determined by its variance X'. Furthermore, by virtue of the condition imposed by equations (10) and (11), the new variables are uncorrelated. It can now be seen that it is not necessary to determine the matrix A explicitly for the X's may be determined directly from the solution of the characteristic equation (Ref. 11), 2 P12 / 1' ' Pn T ln 2 P12o2 02-....2n 02n 0= f(;) (a)n (18) Pln 'l "'n P2n2rn '* * n - 30

THE UNIVERSITY OF MICHIGAN 2872-1 -T where f(A) is a polynomial of degree n in X expressible as f(X) - An+a n1.... an (19) It is now possible to determine the distribution which results from the maximal-ratio combination of the n Rayleigh variables zi. The variable of the combined signals is n n z2 2 2 = = (xi + i (20) i=l i=l From equation (12) we have n n z2 = x2+ v2) = ( + ) (21) 1 1 i=l i=l Let r = u2+ v2 and j r. = r. Because ui and vi are normally distributed 1 1 1 variables with zero means and variances, the probability density function of ri is ri 15 r (r = e 0 ri < 03. (22) Pri (ri) 2i The characteristic function of (22) is 0 yi(r) eji p (ri) d X ( )-j (23) 0 ri ri 2 1i j Since the ri's are uncorrelated the probability density function of r (or z2) is =o -j1r1 n 1n 2 e 'yi(w) dw L. -. dw p(r) = 1 e e ___ 2 00 - 2 - 2 1 (24oo (24) 31

THE UNIVERSITY OF MICHIGAN 2872-1 -T The integral in equation (24) can be evaluated easily since the integrand is analytic everywhere except at the poles w = - 2j Xi If the X's are all distinct p(r) becomes n r n-2 p(r) =1 n-2 e i f 1. ~ (25) i=1 d = f(k kdi Equation (25) can be expressed in terms of the characteristic equation (18) as n -2 2ki p(r) ---- 1 d= Xi The probability that r < R is by definition R.R n X n-1 n-1' 2Xi P (r 4 R) = \ p(r) dr = A-l - n e -. (26) 0 i=l df(A) i= df(A) dX dX )x X=X The first term in equation (26) can be evaluated as follows. From equation (18) f(A) is a polynomial of degree n f(X) X + a1 )1 +.. + an (27) choose () = ai +.. + a (28) n-n The rational function 0(Xk can be expanded in partial fractions as f(A) 32

THE UNIVERSITY OF MICHIGAN 2872-1 -T n f(A) i1 df(X) -(29) i=1 df(i) dX The desired result is then obtained from equation (26) as n n-1 -R p (xi e zXi P(z2 < R) = P(r < R) = 1 - e 2 (30) i=l df() dX X The extension of the above formulation to the case of multiple roots is straight-forward. The result can be generalized to the following: If f(A) has roots Xi of order Pi, then n n-1 e —R 2 aPvi 1 e J P (z2 < R) = P (r < R) = 1 -, p i - 1 ei i= l i aiPi-1 f(Pi)(Xi) ( (31) where f(i) (Xi) denotes the p.ith derivative of f(A) evaluated at X = \. Specifically, for a double pole, the term corresponding to ki in the summation is R n -i n-2 n-3 __R n-1 (3) ) 1 ie-' L 1 f(2)(xi) ( f (2i) ]2 J 2 [ 2_ ~)(i )] ',,.,. 33

THE UNIVERSITY OF MICHIGAN 2872-1 -T Similarly, expressions can be written for the contributions due to higher order poles. For the special case n = 2, equation (30) becomes - R R XI e 2X1 X2 e 2X P(z22 R) = P(r R) = 1- 1 - (33) 2Xi-(o +or-) 2X2-(2-T+or) where.( -+o)+;(o+ ) -4o |( -P1) - ( + - - 4 1 - P ) X2 2 2 2 2 2 In the special case c = -r2 l 1 [ + P12 2 = '1 [ -P12] (30) becomes R R 22 (l+p) P(r =R) 1 + e 2T(1 -p) (l+p)e (34) 2p 2P which is in agreement with Packard (Ref. 4), when the Gaussian variance used above is replaced by the variance of a Rayleigh distribution; i.e., 2cr2 = 2 =0R By a series expansion it may be shown that equation (34), in the special case p = 0, reduces to R P(z2 < R) = P(r < R) = 1 - e 2 1+ R ] 34

THE UNIVERSITY OF MICHIGAN 2872-1 -T REFERENCES 1. J. H. Vogelman, J. L. Ryerson and M. H. Bickelhaupt, "Tropospheric Scatter System using Angle Diversity", Proc. IRE, Vol.47, pp. 688-696, May 1959. 2. D. G. Brennan, "Linear Diversity Combining Techniques", Proc. IRE, Vol.47, pp. 1075-1102, June 1959. 3. H. Staras, "Diversity Reception with Correlated Signals", J. Appl. Phys., Vol. 27, pp. 93-94, January 1956. 4. K.S. Packard, "Effect of Correlation on Combiner Diversity", Proc. IRE, Vol.46, pp. 362-363, January 1958. 5. A.B. Crawford, D.C. Hogg and W.H. Kummer, "Studies in Tropospheric Propagation beyond the Horizon", The Bell Sys. Tech. Jour., Vol. XXXVII - No. 5, September 1959. 6. Z. Jelonek, E. Fitch and J. H.H. Chalk, "Diversity Reception, Statistical Evaluation of Possible Gain", Wireless Engineer, Vol.24, pp. 54-62, February 1947. 7. R. Bolgiano, Jr., N. H. Bryant and W. E. Gordon, "Diversity Reception in Scatter Communications with Emphasis on Angle Diversity", Contract AF 30(602)-1717, Final Report, Pt. I, School of Electrical Engineering, Cornell University, January 1958. 8. H.G. Booker and W. E. Gordon, "A Theory of Radio Scattering in the Troposphere", Proc. IRE, Vol.38, pp. 401-412, April 1950. 9. H.T. Friis, A.B. Crawford, D.C. Hogg, "A Reflection Theory for Propagation beyond the Horizon", The Bell Sys. Tech. Jour., Vol. XXXVI, pp. 627-644, May 1957. 10. E.A. Guillemin, The Mathematics of Circuit Analysis, Chapter II, John Wiley and Sons, Inc., New York. 11. E.A. Guillemin, The Mathematics of Circuit Analysis, Chapter III, John Wiley and Sons, Inc., New York. 12. S. Stein, D. E. Johansen, and A.W. Starr, Final Report on Antenna Performance in Scatter-Type Reception, Contract No. AF 19(604)-5214, Hermes Electronics Co., September 30, 1959. 35

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