1959
`
`Brennan: Linear Diversity Combining Techniques
`
`10975
`
`niobium films have produced units with low breakdown
`strength. Further work is being done with these other
`film-forming metals as well as with tantalum.
`
`CONCLUSIONS
`A new tantalum capacitor, essentially two-dimen-
`sional in structure, has been made and has properties
`superior to other types of tantalumn capacitors.
`Capacitances obtained are comparable to the capaci-
`tance-area relationships for tantalum electrolytic ca-
`pacitors formed to the same voltages. Using counter
`electrodes of 95 mils to 250 mils in diameter and the
`single-layered structure, capacitors have been prepared
`with capacitances ranging between about 2000 ,Af and
`0.25 ,uf.
`DC leakages measured at three-quarters the forma-
`tion voltages are of the order of 4X10i-
`a for 250-mil
`diameter electrodes. Expressed in terms of insulation re-
`sistance, this amounts to about 60,000 ohm farads.
`Dissipation factors are in the neighborhood of 0.008
`at 100 c and increase to between 0.1 and 0.8 at 100 kc.
`
`The relatively high losses at the higher frequencies are
`caused by the high series resistance of the tantalum
`filns. Thicker films of tantalum should reduce these
`losses.
`Room temperature breakdown voltages approximate
`the formation, voltages for these utnits, anid successful
`models have been formed to 5, 10, 20, 30, 40, 50, 100,
`150, anid 200 v. A suggested working voltage is one half
`the formation voltage for temperatures up to 650C. Volt-
`age derating characteristics for elevated temperatujre
`operation have not been determinled as yet. The units
`operate well at very low temperatures, however, even
`downi to -1960C.
`This type of capacitor should find many applications
`in the lower capacitance areas, anid seems ideally suited
`for printed circuit applications.
`ACKNOWLEDGMENT
`The authors are indebted to D. A. MIcLean and N.
`Schwartz for many helpful discussions, and to H. Bas-
`seches for aid with the sputtering technique.
`
`Linear Diversity Combining Techniques *
`
`D. G. BRENNANt, SENIOR MEMBER, IRE
`
`Summary-This paper provides analyses of three types of di-
`versity combining systems in practical use. These are: selection
`diversity, maximal-ratio diversity, and equal-gain diversity systems.
`Quantitative measures of the relative performance (under realistic
`conditions) of the three systems are provided. The effects of various
`departures from ideal conditions, such as non-Rayleigh fading and
`partially coherent signal or noise voltages, are considered. Some dis-
`cussion is also included of the relative merits of predetection and
`postdetection combining and of the problems in determining and
`using long-term distributions. The principal results are given in
`graphs and tables, useful in system design. It is seen that the sim-
`plest possible combiner, the equal-gain system, will generally yield
`performance essentially equivalent to the maximum obtainable from
`any quasi-linear system. The principal application of the results is to
`diversity communication systems and the discussion is set in that
`context, but many of the results are also applicable to certain radar
`and navigation systems.
`
`* Original manuscript received by the IRE, April 21, 1958; revised
`manuscript received, January 14, 1959. The research reported in this
`paper was partly supported by the Army, Navy and Air Force under
`contract with the Massachusetts Inistitute of Technology.
`t Lincoln Lab., Lexington, Mass., and Dept. of Mathematics,
`M.I.T., Cambridge, Mass.
`
`1. INTRODUCTION
`Wa HEN a steady-state,
`single-frequency radio
`wave is tranismitted over a long path, the en-
`velope amplitude of the received signal is ob-
`served to fluctuate in time. This phenomenon is known
`as fading, and its existence con'stitutes one of the bound-
`ary conditions of radio system design. It is observed
`that if two or more radio channels are sufficiently sepa-
`rated in space, frequency, or time, and sometimes inr
`polarization, then the fading on the various channels is
`more or less independent; i.e., it is then relatively rare
`for all the channels to fade together. The standard tech-
`niques for reducing the effect of fading--known as di-
`versity techniques-make use of this fact. The object
`of these techniques is to make use of the several received
`signals to improve the realized signal-to-noise ratio, or
`to improve some other performance criterioni.
`Several diversity combining anid switching techniques
`are known., and there have been numerous papers onz
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`1076
`
`PROPEDINIGS OF THE [RE
`
`June
`
`this subject in recenlt years. (A sample of these with
`comments, is indicated in a Bibliography at the enid;
`these papers will be referen-iced by inumiibers in square
`brackets, runninig footnotes by supericript.) However,
`very few of these have provided quantitative coimipara
`tive data on the relative perfor:-manice of the various
`techn-iques, especially the two significant techniques
`(maximal-ratio and equal-gain) iinvestigated sifice 1954.
`The major exceptioni to this is a paper by Altman-. anid
`Sichak [81, which is not widely kniown anid eveni less
`understood.
`Furthermore, there has been little attempt to explain
`the fundamental concepts and principles involved For
`such reasons, therefore, it appeared desirable to provide
`an expository treatment of a comparative anialysis,
`within a unified frarmework, of the three most promisin-g
`diversity techniiques presently knowni. Ani earlier metno-
`randuml aimed at these objectives indicated that such a
`treatment might be of fairly general initerest.
`Of course, in aii utidertaking of this kind, several
`previously published results are naturally iincluded as in-
`dividual cases, though the available information will
`also be rounded out in a number of ways. Specifically,
`this paper inicludes the following material that the au-
`thor has not seen- published elsewhere:
`1) A careful statemenit of the idealized circumstances
`required for canjoniical performiianice of coherent comr
`I),
`biu1ers (Section
`2) Simple expressionis for the ml-ean signal-to- noise
`power ratios of various combiners [(18), (28), anld (44);
`Fig. 8; Table I ],
`3) Probability distribution curves for equal-gain
`comubineers for 3, 4, 6, anid 8 channels (Figs.
`10--13,
`Table If),
`4) Estimates of the relative perforni-iance of three
`standard combiners for non-Rayleigh fading (Section
`VII1),
`5) A discussion of the relative performance of three
`standard combiners for correlated fading (Sectioni VI II),
`6) Estimates of the degradationi of the average per-
`fornmance of equal-gain and maximal-ratio combiniers
`caused by correlated noise voltages (Section X),
`7) A bound (due to Stein) on the degradation of
`coherent-type combiners with i'mperfectly coherenit sig-
`nals (Section XII),
`8) Certain aspects of the determination, meaning, and
`use of long-term distributions (Section XIII).
`In additioni, some previously published material has
`been- simplified or otherwise clarified.
`It should be mentioned that the criteria employed be
`low are expressed entirely in terms of SNR. This has
`sometimes been taken to mean that the results were
`principally applicable to continuous signals, although
`they are also applicable to certain binary or other dis-
`
`crete signals anid caii be tranislated into error rates ounce
`a suitable detectioi characteristic is either theoretcaally
`or exper-imreentally ktiowIi, But in, the case of bin.ary Sys*
`is possible to obtai-ni miiore specitic ati(ipdrecis
`tens, it
`results oni error rates for specific systeni:s. Such :results
`have been extensively sttudied by Pierce [fiq, [15s] and
`others anid are not considered below. Neither us there a
`discussioii of the considerable beniefits obtaiiiable bv
`codinig or other signial. preprocessiIug techniques designed
`to countiteract fadiiig, several of iNhich are currently
`unider investigationi by other wxorkers.
`On the other hand, it should be i-ioted that radar axid
`navigationi systenms in which a repetitive-addition] signal.
`enhancement technique is emnployed are closely similar,
`in some respects, to certaini diversity systemi:s. Althoulgh
`radar atnd ulavigatio i systen:ms are niot discussedl ini detail
`below, many of the results anid discuss:ions set fortlh
`there are directly applicable to such syste nis,
`II. BASic AssUxMPTIONS AND OTInER PIRELIMINARTES
`The principal background required of the reader isa
`basic acquaiiitaiice with certain eleniietary n-otions of
`probability and statistics, essentially equivalent to those
`developed iii the first six pages of a highlycreadable
`tutorial paper given by Benlnett2 No advantced tech
`niques are required here. However, we shall miiake fre-
`quent use of a few ideas aind techn-iques that were not
`particularly emphasized, by .Bennett, amid a brief exposi
`tionI of these is giveii in Appendix I1. All probabiltv
`distributions used in this paper will be int-LerpIreted as
`explainied there.
`We shall be concerned throughout vith raimdom va -i
`ables given as functionis of timlae (waveform s) in various
`intervals In this settinig, tim-le anid distributmoti averages
`are one an-d the same thiiig sot or (f) or a or (x) for such
`averages will be writteu-m interchanigeably, buit it iS mii--
`portant to note at the outset that our averages will refem
`to imitervals of differenit duratioris Specifically, iiitervals
`of three different durations will be considered: 1) Short
`intervals, whose duration will be denioted by 7 t'The re-
`quirement for T is that it must be short in comparison to
`the time required for the fadiiig amplitude to chanige ap
`preciably, but long in comparison to the period of the
`lowest frequency of initerest in the sigmial. Specific repre
`sentative values of I' wxould mange froml'l a few micro
`seconds to a few mi-ilhisecoclds. 2) Intermmediate iimtei-vals,
`I he requiremients
`whose duration will be deiioted by;
`1,
`T1 must satisfy are rather complicatedcland will be ex-
`plained at various poimnts below. Specific suitable valcies
`of T) would range from a few milinutes to a few hours. 3)
`Long intervals, whose dIui-ationi -x7ill be deniot-ed by TF.
`Values of IT would ran-ge froml- oime nmloiith to omtme
`,eacr or
`moire
`These intervals will be employed as follows. The short
`intervals of lenigth T will be usecd to form "local" statis-
`
`1 D. G. Breniniani, "Liinear techiniquLes in diversity conininica-
`tion," Marchi, 19,56 (Unpublished meniorandLiMr.)
`
`2 W. R. Bennett, "Methods of solvinig nioise problem1-s, " Plzoc. IIRE,
`vol. 44, pp. 609 638; Man, 1956.
`
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`1959
`
`Brennan: Linear Diversity Combining Techniques
`
`1077
`
`tics. For example, suppose ei(t)
`is the instantaneous
`signal voltage and e2(t) is the instantaneous noise voltage
`on some circuit. Then
`~~ 1 r t
`x(t)= [ ef(T) dj
`T
`
`lle~~~1
`
`= V(el2)
`
`(1)
`
`-T
`
`and
`
`y() =
`
`T [e2(r)]2dr
`
`= V(e22)
`
`(2)
`
`would be the local rms signal and local rms noise, re-
`spectively, and x2 and y2 would be the local mean-square
`signal and noise voltages. Letting R denote the circuit
`resistance, x2IR would be the local average signal power
`at time t, obtained by averaging e12 over the last T
`seconds to find x2(t). This averaging could be performed,
`for example, by feeding e12 into a suitable linear filter.
`Alternatively, one could determine the distribution of e1
`in the interval ft -T, tj and obtain x2(t) as the second
`moment of the distribution, though distributions in
`initervals of length T will not actually be of concern here.
`Local statistics such as (1) and (2) will generally
`fluctuate in time because of fading and other effects.
`For example, the local rms SNR x(t)/y(t) and the local
`signal-to-noise power ratio
`
`P(t)
`
`x2(t)
`y (t)
`
`(3)
`
`will usually vary over wide limits, though they will be
`much better behaved than the (meaningless) instan-
`taneous ratio el(t)le2(t). The behavior of variables such
`as the local statistic (3) in intervals of length T1, where
`T1>>T, will be studied. In particular, various distribu-
`tions and averages relative to intervals of length T, will
`be considered. Such Ti-distributions and Ti-averages
`will also change with time, in ways discussed in Sections
`VII and XIII. Performance relative to T1-intervals uii-
`der standard conditions is summarized in Section VI.
`Finally, the variability of certain Ti-averages will be
`considered in intervals of length T2, where T2>>T1. This
`is done in Section XIII. It is usually assumed in system
`design that, for suitable values of T2, all distributions for
`the system in question will be essentially the same in
`every corresponding interval of length T2. (A suitable
`value might be one year, for example.) This is in marked
`contrast to the situation for Ti-distributions. However,
`it is found experimentally that this assumption is a
`reasonable first approximation; moreover, if this as-
`sumption were not satisfied, there would be no method
`available for predicting the performance of the system,
`at least at the present time.
`By concentrating on system behavior relative to such
`prescribed lengths of intervals, it is possible to keep the
`relation between theory and experiment clearly visible,
`including, in particular, the practicable experiments re-
`quired to verify theoretical predictions. This procedure
`is therefore vital to a complete and realistic analysis of
`
`communication systems in getneral and diversity systems
`in particular.
`In general, the term "diversity system" refers to a
`system in which one has available two or more closely
`similar copies of some desired signial. For example, cer-
`tain radar systems operate by storing the signal received
`durling one scan and adding this to the signal received
`during the next scan. If fi(t) is written for the output
`of the storage device and f2(t) for the signal currently
`being received, then the composite signal is simply
`fl(t) +f2(t) =f(t). Now, fi(t) may consist of a desired
`message component sl(t) and ani undesired additive
`noise component ni(t), so that fi(t) =si(t) +nl(t), and
`s2(t) +n2(t).FHence, the composite signal
`similarly f2(t)
`may be written
`(4)
`f(t) = [Sl (1) + S2(t) ] + [nl(t) + n2(t) ] I
`i.e., in the form of a resultant message component
`(SI+S2) plus a resultant noise component (ni+n2). If the
`message components si and S2 are closely similar, their
`sum s1+s2 will simply approximate an enlarged copy of
`either si or S2* On the other hand, the lnoise components
`ni and n2 may be quite dissimilar; one may be negative
`part of the time the other is positive, and vice versa,
`so they may partially cancel for part of the time. The
`sum (4) may then be a better signal than either f' or f2
`alone; in particular, f(t) may have a higher local SNR
`p(t), defined as in (1)-(3), with ei=si+s2, e2=nli+n2
`than either fi or f2 alone. Thus, one way of usinlg two
`similar or suitably related copies, fi and f2, miay be
`simply to add them together. Certain navigation sys-
`tems in which a periodic signal is transmitted also use
`this storage-and-addition principle.
`More generally, one may have N such copies fi(t),
`f2(t)? * f*,fN(t), each of the form fj(t) =sj(t) +nj(t), and
`one may form the sum
`f(t) = fl(t) + f2(t) + *
`
`.
`
`f(t) = aifi(t) + *
`
`N
`
`(6)
`
`N
`
`(5)
`
`* + fNv(t) = E fjQ(),
`J3L
`which may outperform, in some sense, the individual fj.
`However, in view of the fact that the fj will have fluctu-
`ating local statistics, it will be convenient to consider
`weighted sums of the fj; that is, the general linear com-
`bination will be considered:
`* + aNfN(t) = E ajfj(t),
`j-l
`in which each f; is weighted by a combining coefficient
`a1, which is proportional to the channel gain and miay
`be allowed to vary in accordance with the fluctuating
`local statistics of the fj(t). However, the cases to be
`considered will be those in which the aj are locally con-
`stant, or at least approximately so. The adjective
`"linear" in the title of this paper stems from (6). Since
`the aj may be allowed to vary, depending on the fj,
`one should perhaps speak of (6) as locally linear or
`"4quasi-linear." Evidently (4) is simply the case of (6) in
`which N-2, a, :=a2=-1
`
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`1078
`
`7PROCEEDINGS OF THE IRE
`
`In diversity communication systems, there are several
`known methods of obtaining two or more signals fA,
`and several kiiown methods of comrbining these to ob-
`tain an improved signal. However, all of the latter meth-
`ods in current practical use are special cases of (6). Let
`us first consider briefly niethods of obtaining several
`suitableAj. The simplest of these is that in which a single
`transmitting antenna furnishes a sign-ial to several well
`separated remote receiving antennas; this method is
`called space diversity. A variant of this, suitable for use
`in systems operating at UHF and above, uses two sepa-
`rated transmitting antennas, one of which transmits
`vertically polarized radiation and the other of which
`tranismits horizontally polarized radiation, anid a single
`receiving reflector with two feed horns or dipoles to
`separate the vertical and horizoontal received signals. By
`combining these two methods, Altman and Sichak [81
`obtained a fourth-order, bidirectional, full duplex space
`diversity system that requires orly two reflectors at
`each end, as inLdicated iin
`Fig. 1. (However, it should
`be added that recenit experimental evidence irldicates
`that the fading on. the crossed pair of paths is m-ore
`highly correlated than oni the other pairs of paths.) In
`one form oi- aniother, space diversity has been the most
`commonIly used form of diversity comuinuiicationi.
`
`Fig. 1-Foour-chann-iel bidirectional space diversity system suitable
`for UHF and SHF systems. Signal paths are iiidicated for one
`only. The circles marked D deiiote diplexing filters.
`directioii
`The transniitters are on) different freqiieicies.
`
`practice of sending each word twice, as used by many
`commercial CW stations, is actually a primil-live but
`useful form of time diversity At the other extremee
`a
`very sophisticated communication systemn, currently
`un-der development,3 which is designed to eliminate
`effects due to multiple transm-nLission paths between fixed
`antennas, actually sorts out the various multipath con
`tributions and recombines them with suitable delays
`and may be regarded as a form. of time diversity in which
`the diversity is provided by the tranusmission mediumn
`itself.
`A miethod that will sometim-ies yield two approxi-
`mately inidependent fadiiig signials is called polar-izatioii
`diversity. In I-iormal ionospheric transmission at Ire-
`quenicies of a few mnegacycles, it is founid that the re
`ceived signial includes both vertically arod horizointally
`polarized coimponeints, and the fading of these compo
`is approximately ii]dependent" Hlowever, in
`lnents
`tropospheric transmission at UHF aiid above, the polar-
`ization of the transmitted signal is quite well preservedJ'
`and very little effect of this type takes place. Further
`more, even if both horizontal and vertical conlipon.ents
`are transmitted and separately received, the fadinig of
`th-e twlo com-lponients is far from-i indepeniden-t if only a
`sin-igle transmissioii patli
`is involved.6
`Another method that has been used (in-ifrequently) in
`the high-frequei-icy regioii inivolves the coimbination of
`sign-ials arriving with different anigles of ai-rival (the
`Mlusa system).' A sonmewhat similar approach at J3fll
`and above is currently uinder investigatioii by seveeral
`workers,8 10 but the efficacy of this technlique is niot yet
`firmly established.
`Whichever of these methods is used, the signials ob-
`tained will initially be at radio frequentcy. 'Fhe diversity
`combining techniquies employed subsequen-it to this stage
`conmbinriiig
`may be classed in twlo groups: predetectioi
`
`Another method, called frequeiicy diversity, iniivolves
`transmitting the sanme informnation on two or more
`carrier frequencies
`If these are sufficiently separated,
`the fading on the various chainnels is approximately
`in-idependent, as in the case of space diversity. This
`tnethod. is economical in termLs of antennas and real
`estate, but expensive in terms of transmitters anid re-
`quired bandwidth. It has been discussed more ofteii
`than used. (However, there are circumstances in which
`it is uiseful and has actually beetn used.) This is also true
`of the method called time diversity, so far as coemiiniuni-
`cation systems are conlcerned; however, it is not true of
`radar and navigation systems, as the method discussed
`in the opening paragraph of this sectionr is essentially
`time diversity, although this ternminology has not been
`much used in the radar field. In radio communication
`systems, time diversity involves transmittinig the same
`information two or mnore distinct times. When this is ir-
`strumented foi automatic operation, its chief disad-
`vanitage is equipmient complexity; however, the simple
`
`I R. Price and P. E. Greein Jr., "A communication technlique for
`iultipath channels," PnOC. IRE, vol. 46,
`Pp. 555-570; March,
`1958.
`4 J. L. Glaser and L. P Faber, 'Evaluation of polarization diver
`sity performance," PRoc. IRE, vol. 41, PP.1p7741778; Deceniber,
`1953.
`G. L Grisdale, J G. Morris, and D. S.
`P'almiier, "Fadling of
`long-distance radio signals and a comparison of space an-d polariza-
`tion-diversity reception in the 6 18 nic ranige,' Proc. IEE, pt. 13,
`vol. 104, pp. 39--51; january, 19570
`1J. H. Chisholm, P. A. Portma in, J. T. deBettencourt ancd J. F,
`Roche, "Investigations of the angular scattering aid nmultipath
`properties of tropospheric propagationi of short radio waves beyonid
`the horizon," PROC. IRE, vol. 43, pp. 1317-1335; October 1955c See
`especially Fig. 20, p. 1331.
`'7 F. E. Teirman, "Radio Eingineers IIandbook,'7 McGraw-Hill
`Bool Co., Iec., New Yorlc, N.Y., pp. 660-661, 1943. See also papers
`by Polkiiigton- and Friis and Feldman cited thereii.
`I R. Bolgialno, Jr., N. H Bryant, and W. E. GordonL, "Diversity
`Receptionn
`in. Scatter Corrmmunication with Emphasis on Angle
`Diversity," Cornell 4Jniv., Ithaca, N. Y., Elec. EF grg 1es Rep. 359
`Janiuary, 1958.
`A. T. Waterman, Jr, "A rapid bearaimswinginig experim.ent in
`transhorizon propagation," IRE I RANS. ON AN'TENNAS AND PROPA-
`GATION, vol. AP-6, pp. 338-340^ O)ctober, 1958.
`10 J. H. Chisholm, L. P. Rainville, J. F. Roche, and H. G. Root,
`"Angular diversity reception at 2290 mcps over a 188-mile path,"pre-
`sei,ted at Symp. on LExtendLed Ran-ge and Space Comi-ntiiiication
`s
`George Washington Univ., Washiiigton, D. C.; October 68, 1958.
`(Published in the Symp. Rec.)
`
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`1959
`
`Brennan: Linear Diversity Combining Techniques
`
`1079
`
`methods aiid postdetection combining methods. In
`those methods in which, at any given time, only one of
`the aj in (6) is different from zero, i.e., a switch of some
`kind, the distinction is basically unimportant. However,
`important differences arise wheni the conmbining method
`is one in which two or more of the aj may be different
`from zero at the same time. For example, it is clear that
`the simple addition scheme (4) can fail grievously if the
`message components si(t) and s2(t) are not in the same
`phase, and RF or IF diversity signals will niot usually
`be in the same phase unless special measures are taken
`to insure this. Consequently, such combining methods
`require special phase-control provisions when used in
`predetection applications, while this is not always the
`case in postdetection applications. An even more im-
`portant difference arises in the case of FM or other
`bandwidth-exchange systems, where predetection com-
`bining can lead to substalntial improvement over post-
`detection methods, as will be seen.
`Once the method of providing a multiplicity of sig-
`nals is decided, the basic problem confronting the de-
`signer of a diversity system becomes one of choosing
`the most appropriate method of combining these sig-
`nals on the basis of reasonably accurate quantitative
`estimates of the performance of the various techniques.
`The balance of this paper is principally devoted to this
`problem. Instrumentation problems as such are not
`considered here; however, papers which describe certain
`instrumentation techniques are indicated.
`We shall find it most econoniical to consider first a
`particular class of circumstances, and then indicate the
`way in which the results are modified by other circum-
`stances or, in some cases, indicate where such modifica-
`tions are treated elsewhere in the literature. The circum-
`stances initially considered are those often applicable to
`postdetection combining in an AM system, or a single-
`sideband system in which provision is made for main-
`tainiing coherence of the postdetection signals.ii These
`conditions are as follows: assume that N simultaneous
`, fN(t) represent the signals
`functions, fi(t), f2(t), *
`received in N different diversity channels as corrupted
`by noise and fading; each f, j = 1, 2,
`N represents
`the corrupted signal in the jth channel containing the
`originally transmitted signal m(t). For convenience, sup-
`pose that m(t) is a steady test tonie at a representative
`midband frequency. or some other steady test signal
`with a constant local mean square m2 =1. That the fol-
`lowing conditions are approximately satisfied is also
`assumed:
`(A) The noise in each channel is independent of the
`signal, and additive: fj(t) =sj(t) +nj(t) where si and nj
`are the signial and noise components, respectively, in the
`jth channel.
`(B) The signals sj(t) are locally coherent; i.e., sj(t)
`
`-
`
`V. E. Morrow, Jr., C. L. Mack, Jr., B. E. Nichols, and J. Leon-
`11
`hard, "Single-sideband techniques in UHF long-range communica-
`tions," PROC. IRE, vol. 44, pp. 1854-1873; December, 1956.
`
`= xjm(t), where the xi are positive real numbers that
`change with time because of fading, but at a rate that
`is very slow in comparison to the instantaneous varia-
`tions of m(t). More precisely, assunie that the xj do not
`change appreciably within any period of duration T,
`where T is the duration of the interval emiployed for the
`local averages. Then, siiice m2 = 1,
`I '
`2 S--f ,j[52(T) ]2dT =
`T
`-T
`
`Xj2 [M(T) ]2dT
`
`(7)
`
`= Xj2.iT [m(T)]2dT = xi2,
`t -T
`so that xj = xj(t) is simply the local rms value of s, takeii
`over the last T seconds before the present time, t. It is
`clear that T muist be short in comparison to the time
`required for the fading amplitude to change appre-
`ciably, but long in comparison to the period of m(t).
`(C) The noise components nj(t) are locally incoherent
`(i.e., uncorrelated) and have zero means: nini=nj ni
`if ifj, where the duration of the averages is the same as
`in (7). We shall also assume that the local mean square
`noises nj2 are slowly varying, or, sometimes, constant.
`(D) The local rms values of the signals, xj(t) -\(Sj2)
`are statistically independent. Note that this assumption
`automatically implies that at least two intervals are
`considered: first, the period T [of (7) ] involved In the
`definition of the xi; and second, an interval of duration
`TF in which we observe the xj(t) as new random vari-
`ables. Evidently T»>>T; in practical cases, T might be a
`few milliseconds and T1 approximately 30 minutes. A
`discussioni of the requirements on T1 is provided in
`Section XI II. It is important that T1 cannot be too long.
`Assumption (D) is that, when observed in intervals with
`a duration on the order of Ti, the variables xj(t) are
`statistically independent.
`The circumstances characterized by assumptions
`(A)-(D) are illustrated in exploded fashion in Fig. 2 for
`N= 2. By "exploded," we mean that the actual signals
`given would be fj(t)= s(t) +nj(t), j = 1, 2, while the sj
`and nj are shown separately. The meaning of the
`locally coherent assumption (B) is that, over periods of
`length T, the signals si and S2 are essentially idenitical
`except in amplitude, which is approximately constant
`over such periods. The local rms values xj(t) are indi-
`cated by the dashed curves. Note that the assumption
`sj(t) -xjm(t) implies that the sj(t) have the same zero
`crossings, and are in phase. If the sj are RF or IF signals,
`the period F might be several microseconds or more, in
`which case no variation of the xj would be perceptible
`within the scale of Fig. 2. If the sj represent base-band
`signals, T might be a few milliseconds.
`In contrast to the sj, it will often be required that
`the noises nj(t) be essentially different; this is the nmean-
`ing of (C), as suggested by the waveforms n1(t) and
`n2(t) of Fig. 2. In particular, it will often be assumed
`that (njnj)=O (if i
`j) over every interval of length T.
`In addition, however, it will sometimes also be as-
`
`IPR2020-00038
`MM EX1013, Page 5
`
`

`

`1080
`
`1PROCEEDINGS OF TILE IRE
`
`June
`
`sumed that the noises nj have constant local average
`powei, i.e., that
`
`n2 = 1-J
`
`1[nj(Kr) jIdr
`T ;d-T
`is a constant, independent of t and j. This woulc
`lbe at
`d M,2 0f
`teast approximatekv true of the waveforms vi an-U
`Fig. 2.
`
`(8)
`
`i(t)
`x1t
`
`(t)
`
`S?(t
`
`AIM11
`.Ifd ilh A
`ng(t)l|lis *P
`AA
`-.MJj
`
`ah
`
`4 plli
`
`Fig. 2--Signals ai-nd noises in two diversity cha-in ies
`
`Assumption (D) is not particularly illustrated in Fig.
`2, and could not be successfully illustrated there because
`the period TF required for approximate indepeiidence of
`the xj is much greater than the total scale length of Fig.
`If the xj were plotted throughout an interval of
`2.
`length Ti and the graph were then compressed to the
`lenigth of Fig. 2, the resultinig cuLrves would resemible the
`waveforms illustrated for vi and a2, except that the xj
`would be non-negative and would niot usually be symn-
`metric about their mean values. In particular, the xj are
`not locally coherent in the sense of (B), where this
`"locally" refers to intervals of length 7 l. Note that dis
`tributions or averages of the xj or quantities dlerived
`therefrom, e.g ., (X2), refer to intervals of lerigth
`7P,
`Such averages could be distiniguished by suitable nota-
`)1, but it will simnplify the appearanice of
`(
`tionr
`e.g.
`various expressioins if the context is relied tiupo
`to miiake
`elear whether a short-terfim or intermediate-term aver
`age or distribution is min-eanit.
`Most of the work below is concerned with signaL.
`noise-ratios, and from here on the word "ratio," is to
`meani "SNR." This will be qualified as ai
`amplitude
`ratio or a power ratio as the context requires. pjxj2/ni 2
`will be written for the local power ratio in the jth chan-
`nel, and xjVa\ ;), is, sininilarly, the local amplitude
`ratio. We shall often- take nin2
`,
`1, j 1, 2,
`, in which
`*
`
`-
`
`case the local amplituede ratio is simriply xt nuniercally,
`and p - x42
`It will frequently be assuined. tha-t the varimbles xv
`follow a Rayleigh distributton wath den'sity aml
`dis
`tribut01n functions
`2xve-
`p(xj)
`(9)
`P(X) =
`2-$ri
`respectively. A plot of the Rayleigh deinsity furwctioi is
`given in Fig. 15
`All distributions considered. in this
`paper are zero for negative values, and expressiois such
`as (9) are to be understood as referring to positive valuLes
`only. Writiing the Rayleigh distribution in the to ru
`(9)
`i -mplies a particular cioice of scale; in pairticular, it iuri
`plies that (axj)
`1 The Rayleigh distr bt;X.lion osoften
`Wriitten with an iarbitrary scale factor, sav
`
`P(y)
`
`1
`
`e -y / R
`
`t(y) _2;PS
`.RI
`
`(10)
`
`in which case ty2)
`2R. However, the data below are
`given in a formi that is completely independent of such
`scale factors, un-til Section XIII. This saves conside,rable
`of the lammdscapc below. Simnilarly, a.2
`clutterinig
`will often be taken instead of nj12- no, for exanipIe.
`1, thee yj.is exactly the local amplitude
`For, when a)
`ratio, wvhich has the distribution (9), and ,x,=X2 has the
`simple distribution
`)(I.1)
`G(pj)
`1#- e-P'
`g(p
`r
`a'e
`(Distributioni funictions wvill always be written with up
`per case letters, density functions with lower case
`letters.)
`There are four pritncipal types of diversity combiiimug
`systemis in practical use. Many of the combiners an
`actual use are not puie examples of one of these types,
`e., they involve approximations to, or
`nodificatiois
`of oine of these types. However, the effect of such nmodi
`fications can often, be estimated, at etast oughly. (T1e
`termiin.ology uised heme is niot entirely standard.i-iideed
`there is n-o generally accepted standard terminoloprgy
`but is the result of careful consideration anid discussion
`xirith several colleagnes ) 'Ihe ourin
`pre" tecihiique1s are
`as follows:
`1) Scanniniig
`Diversity.
`this
`the
`techitque
`of
`s
`switc hed type, ie, at. any tinie, o01
`o -re of tihe aj in
`(6) is differemit fromi zero, an-d that onie is eqlual t'to
`. A
`selector device scans the channels n a fxed seqnuecec
`un-til finding a sigmial ahoye
`_t preset threshold, uses that
`A
`signal on-ily until it drops below threshold, atid then
`seains the other than iels iri
`the sanme fixed sequence
`until it again fiuids a signal abuove tthreshold
`t is often
`applied to the case of two antennas supplyinmg a simugle
`receiv