IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY. VOL. VT-36. NO. 4. NOVE\IBER 1987
`
`149
`
`Antenna Diversity in Mobile Communications
`
`RODNEY G. VAUGHAN, MEMBER, IEEE, A?\"D J. BACH ANDERSEN, SENIOR ME\IBER, IEEE
`
`Abstract-The conditions for antenna diversity action are investigated.
`In terms of the fields, a condition is shown to be that the incident field
`and the far field of the diversity antenna should obey (or nearly obey) an
`orthogonality relationship. The role of mutual coupling is central, and it
`is different from that in a conventional array antenna. In terms of
`antenna parameters, a sufficient condition for diversity action for a
`certain class of high gain antennas at the mobile, which approximates
`most practical mobile antennas, is shown to be zero (or low) mutual
`resistance between elements. This is not the case at the base station, where
`the condition is necessary only. The mutual resistance condition offers a
`powerful design tool, and examples of new mobile diversity antennas are
`discussed along with some existing designs.
`
`I. INTRODUCT!Oc'I
`
`THE DEMAND
`
`in
`for better spectrum efficiency
`narrow-band cellular frequency reuse systems can be
`eased by the application of antenna diversity. The possible
`improvements from diversity are well known for reduction of
`fading, but there are other advantages potentially available in
`the case of mobile communications. These are the suppression
`of both the random FM, which limits BER improvement in
`angle modulated systems, and cochannel interference, which
`limits frequency reuse base station density.
`The signal conveyed through a narrow-band mobile channel
`becomes impaired by long-term (shadow) fading, short-term
`(Rayleigh-like) fading, random FM (including click noise),
`and especially in cellular systems, cochannel interference.
`Perhaps the most serious of these is the Rayleigh-like fading
`caused by the multipath environment. The random FM is
`caused by the Doppler shifts of the multipath signals, and the
`click noise component is associated with the deeper fade~. The
`shadow fading is caused by a lack of power density, and this
`problem cannot be solved by diversity action at the mobile
`alone. The macrodiversity action required, if necessary, to
`overcome shadow fading is accomplished by strategically sited
`base stations. Macrodiversity will not be addressed here.
`The simplest technique to maintain acceptable channel
`capacity (relative to the nonfading channel) is to increase the
`transmitted power. However, in doing so, the overall spec(cid:173)
`trum efficiency is reduced because the distance between
`frequency reuse transmitters must be greater to maintain
`acceptable cochannel interference levels. Moreover, the ran(cid:173)
`dom FM cannot be suppressed by simply increasing the
`
`Manuscript received May. 9. I 986: revised May JO. I 987.
`R. G. Vaughan is with the Department of Scientific and Industrial
`Research, Physics and Engineering Laboratory, Gracefield Road, Gracefield,
`Private Bag, Lower Hutt, New Zealand.
`J. Bach Andersen is with the Institute of Electronic Systems. Aalborg
`University, Fr. Bajers Vej 7, 9220 Aalborg 0, Denmark.
`IEEE Log Number 8718834.
`
`transmitted power. Alternative techniques to maintain channel
`capacity employ some kind of diversity scheme. Both antenna
`and signaling based diversity systems are well known (e.g.,
`.Jakes [13]).
`With antenna diversity, the problems of the mobile channel
`are attacked directly. Higher orders of diversity are readily
`available in principle. An existing mobile antenna can be
`replaced by a diversity antenna with combiner so that existing
`systems can be improved without the need for implementing a
`signaling diversity scheme. The random FM is suppressed
`to
`the order of diversity and
`the combining
`according
`technique.
`There are well-known schemes other than antenna diversity
`for improving the mobile channel capacity. Proponents of
`antenna diversity view the inherent advantages as follows.
`While covering "system" and "overall" spectrum efficien(cid:173)
`cies requires much discussion, it is sufficient here to note that
`
`1) antenna diversity improves the channel capacity at the
`expense of adding extra equipment (antenna, combiner)
`to the receive end of the link (no extra spectrum is
`consumed); and
`2) all other schemes consume extra spectrum to improve
`the channel capacity.
`
`Regarding the first point, it is worth adding that adaptive
`retransmission with feedback allows the diversity antenna to
`be at the transmitting end of the link. The price paid is the
`required coding and housekeeping functions at both ends of the
`link with a corresponding slightly degraded channel message
`capacity compared to the receive antenna diversity case. A
`possible exception to the second point is delay diversity, in
`which uncorrelated signals arriving at different delay times are
`aligned (in time) for combination (cf. Rake and Drake
`schemes). There is no guarantee, however, that the natural
`delay distribution is suitable in the general case and so the
`scheme is not deemed appropriate.
`The traditional disadvantage of antenna diversity is the cost
`and inconvenience of the extra equipment. There is much
`concern regarding efficient use of the spectrum, so it seems a
`matter of time until this concern forces greater use of antenna
`diversity. Much recent effort has been toward data coding to
`improve the information bit error rate (BER). Considerable
`progress has been made using a priori knowlerge of the
`channel. Specifically, the Rayleigh-like fading gi es rise to
`bursts of errors during the deeper fades. The channel is often
`treated as having "good'· and "bad" states of transmission in
`a scheme known as the Gilbert-Elliot model (e.g., Ahlin, [l]).
`Most coding schemes rely on the channel signal-to-noise ratio
`(SNR) being exactly Rayleigh distributed, so the calculated
`
`0018-9545/87/1100-0149$01.00 © 1988 IEEE
`
`Authorized licensed use limited to: Fish & Richardson PC. Downloaded on May 16,2022 at 13:14:51 UTC from IEEE Xplore. Restrictions apply.
`
`Ex.1020
`APPLE INC. / Page 1 of 24
`
`

`

`150
`
`IEEE TRANSACTIONS 01' VEHICULAR TECHNOLOGY. VOL. VT-36, NO. 4, NOVEMBER 1987
`
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`Illustration that antenna diversity can also work for wide-band (frequency hopping) systems. The three figures are group
`Fig. I.
`delays from simulation of three diversity antenna elements. Dispersive (bad) channels are independent for each element. Average
`group delay is about 17 µ.s, which is exaggerated for clarity on the scale (it is typically less than 0.5 µ.s).
`
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`performance may well be quite different from actual perform(cid:173)
`ance. To the authors' knowledge, detailed investigation of the
`coding gain from a diversity antenna signal have not been
`reported. This should be a rather straightforward step, since
`the model with diversity would involve only a modification to
`the Rayleigh distribution term (maximum ratio combination
`could be assumed).
`Much effort has also been expended on wide-band systems.
`The spread spectrum approach seems to be necessary for
`implementation of optimum combining, which is discussed by
`Winters [34]. Frequency hopping schemes (often referred to as
`frequency diversity) do not seem to have been implemented in
`
`public systems to date. It is worth noting that antenna diversity
`offers potential channel improvement for wide-band systems
`also. The scheme is illustrated by simulatio11 results in Fig. 1,
`which shows that the group delays are uncorrelated between
`branches, so that a highly dispersive channel in one branch
`will be well behaved in another. The group delay characteris(cid:173)
`tics in a wide-band system are analogous to the random FM in
`the narrow-band case. There is an "irreducible" BER effect
`for wide-band systems with single-port antennas, which is
`caused by the group delay characteristic. This irreducible BER
`is thus analogous to that in narrow-band systems caused by the
`in Fig. 1
`random FM. The spikes of high dispersion
`
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`
`Ex.1020
`APPLE INC. / Page 2 of 24
`
`

`

`VAUGHAN AKD ANDERSEN: ANTENNA DIVERSITY IN COM~UNICATIO.\'S
`
`15 I
`
`correspond to the deep fades of the Rayleigh-like envelope. By
`avoiding the deep fades, depicted here in the frequency
`domain, the highly dispersive channels (where low channel
`bandwidths occur) are also avoided.
`A note on terminology is in order, since the multidiscipli(cid:173)
`nary nature of mobile communcations results inevitably in
`inconsistent nomenclature. Most terms used here follow from
`original articles or by convention according to the pertinent
`discipline. An example is the use of r for both polarization
`matrices (e.g., (4)) and signal-to-noise-ratio (e.g., (17)).
`Some inconsistencies also arise from historical ''misuse.'' For
`example, covariances and (complex) correlations are consid(cid:173)
`ered the same, despite their mathematical distinction, and the
`terms carrier-to-noise ratio (CNR) and signal-to-noise ratio
`are also interchangeable, although this is not generally true.
`Strictly speaking, the CNR is the quantity of interest since the
`signals under consideration are RF (or IF) carriers, yet to be
`demodulated (predetection combining is assumed). SNR
`should only be applied to a signal after detection and will not,
`in general, be the same as the CNR. From here on, however,
`the term SNR is used, following Jakes' principal convention
`[ 13]. The time average is denoted ( ·) and is interchangeable
`with the ensemble expectation since all processes are assumed
`ergodic. For matrix operations, the following superscripts
`apply: T means transpose,
`the asterisk means complex
`conjugate, and H means Hermitian transpose. When discus(cid:173)
`sing the mobile communications scenario (see Section 11), the
`word source refers to each point in space that can be
`considered to supply energy to the mobile antenna. The word
`signal refers to the intelligence conveyed by the energy from
`the sources. (Many sources convey the same signal.) When
`discussing antenna diversity, the diversity gain differs from
`the diversity return in that the latter includes the effects of
`mutual coupling. Strictly speaking, the diversity gain should
`include mutual coupling effects, but traditionally, this has not
`been the case. In referring to mobile antennas, the term high
`gain is used for antennas whose receiving patterns are
`confined ( or almost so) to the directions of the sources.
`Section II covers some basic aspects of antenna diversity
`and gives a fleeting mention of other methods for improving
`the mobile channel. Stein [28] and Jakes [13] discuss diversity
`in great depth, and the basics are indeed well covered. Some
`aspects are clarified in Section II. Not a great deal has been
`reported about the scenario of sources incident on an urban(cid:173)
`based mobile or base station. For diversity antenna pattern
`considerations, a convenient distributed souce model is used to
`describe the (ensemble) average scenario, despite the fact that
`the instantaneous scenario may contain only a few sources.
`Energy considerations demonstrate the potential of multiple
`port antennas without resorting to space diversity. A figure of
`merit for a diversity antenna, the diversity gain, and its
`behavior in the presence of mutual coupling receives attention.
`It is shown that when correlated branches undergo nonswitch(cid:173)
`ed combining (or when the diversity antenna elements are
`always terminated), more care than that displayed in: the
`interpret the diversity gain. A
`literature is required to
`fundamental difference exists between high-gain antennas at
`the mobile and base station antennas in this regard. A short
`
`discussion on the effect of different levels of branch mean
`SNR's concludes the section.
`Section III presents several new ideas and viewpoints
`regarding antenna diversity. The conditions for diversity
`action are investigated. It is shown that under certain idealized
`conditions, the correlation coefficient between branch signals
`of a diversity antenna for the mobile can be equated with the
`mutual resistance between the antenna elements. This result is
`new, fundamental, and useful. It means that the performance
`of a class of diversity antenna designs for urban applications
`can be ascertained in the laboratory. The alternative is to
`measure correlations between branch signals in the field,
`normally an expensive and time-consuming exercise. The
`textbooks (see Stein [28], Jakes [13], Lee [22]) divide antenna
`diversity techniques into classes such as angle, polarization,
`space, field component, etc. These techniques are unified into
`pattern diversity. The condition for diversity action is found to
`be orthogonal element patterns over the sources. This is also a
`new and rather fundamental result. The formulation is given,
`and the situations at both the base station and the mobile are
`discussed.
`Section IV ( and the remainder of the article) concentrates on
`antenna diversity at the mobile. An element figure of merit
`(the element directivity toward the distributed sources sce(cid:173)
`nario) is used to find useful design information. An array
`figure of merit (the diversity return) can also be applied to find
`useful and optimum diversity antenna configurations. The role
`of mutual coupling is investigated in detail, and ideas are fixed
`by considering rotationally symmetric two- and three-element
`array designs.
`Section V discusses specific examples of diversity antennas
`for the mobile in terms of the pattern orthogonality. Both
`existing and new designs are included. It is noted that space
`diversity from concentric horizontal ring elements will not
`work well at the mobile. A circular array of three outward
`sloping monopoles is also discussed. The advantage is that the
`feedpoint spacings can be arbitrarily close. A sinusoidal
`current distribution is assumed for all configurations. As the
`antennas become closely spaced, a moment method solution
`would be better. However, is seems unnecessary to solve the
`problem exactly since both the infinite ground plane and source
`distribution are only approximations. Experimental values of
`the envelope correlation are in excellent agreement with the
`theory for a three-element example. The two-element case is
`mentioned and some remarks are offered for the many(cid:173)
`branched circular array. Section VI concludes the paper, and
`the Appendix details the cumulative probability distribution of
`the combined signal from a circularly symmetric three(cid:173)
`element array.
`
`II. A1'TEN;-;A DIVERSITY : SOME BASIC ASPECTS
`
`Source Scenario at the Base Station [30]
`
`Models are required for the scenario of sources producing
`the fields at the mobile and base station. At the base station,
`the incident fields due to a single mobile in an urban area
`occupy a very small portion of the base station field-of-
`
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`
`Ex.1020
`APPLE INC. / Page 3 of 24
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`

`

`152
`
`IEEE TRA NSACTIONS ON VEHICULAR TECH NOLOGY . VOL. VT-36 , NO . 4 , NOVEMBER 1987
`
`coverage. In fact, the incident signal is often well represented
`by a single direction when the antenna is clear of obstructions.
`Define the directions and extent of the sources ( from a single
`mobile) by upper and lower limits 0u, </>u; 0L, <l>L where the
`origin is the base station. The incident electric field is denoted
`
`lated. The polarization matrix at the mobile is thus
`
`r'(01, <1>1; 02, <1>2)=5(0, <1>)0(01-02)0(</>1-</>2) . [~ n (7)
`
`(1) where
`
`where the units of h, h8, and h4i are volts/meter/steradian. The
`polarization matrix for the incident fields is defined as
`
`S ( 0, </>) = [ ;,·
`
`60° ~ 0 ~ 90°, 0° < ¢ < 360° CS)
`elsewhere
`
`where the elements are of the form
`
`is the constant power density per steradic square distribution
`around the mobile . It is emphasized that the XPD at the mobile
`has been assumed to be unity, a case corresponding to equal
`powers in the vertical and horizontal polarizations at the base
`station. This scenario is referred to as the mobile communica(cid:173)
`tions scenario (MCS).
`
`(2)
`
`(3)
`
`If the polarizations are considered uncorrelated and each
`polarization considered spatially uncorrelated, then
`
`r ' (01, </>1; 02, </>2)=P(0, </>)0(01-02)0(</>1-</>2)
`
`Energy Considerations at the Mobile and Base Station
`The energy density at a point (or in a small volume , strictly
`speaking) in space is proportional to
`
`. [ XP~(D) ~]
`
`(4)
`
`where
`
`P(0, </>)=P,
`
`{JL ~ 01 ~ Ou,
`
`¢L ~ q> ~ <Pu
`
`=0,
`
`elsewhere
`
`(5)
`
`is the (constant) power density per steradic square distribution
`and
`
`(6)
`
`is the cross polar discrimination (XPD). For vertically
`polarized antennas in urban areas, the XPD is given by
`Kozono et al. [ 17) as a weak empirical function of the distance
`D between the mobile and base station. However, it is also a
`function of the polarization of the mobile antenna and the type
`of terrain along the path. For a vertically polarized base station
`and a vertically polarized urban based mobile antenna, XPD
`,== 6 dB (Lee and Yeh (21)). For a horizontally polarized base
`- 6 dB [21). Most existing mobile
`station, the value is ,==
`antennas are principally vertically polarized. At the base
`station, then, we choose an average value XPD = 6 dB , but
`note that "instantaneous" values between - 6 dB and 18 dB
`can occur (Kozono et al. [17)).
`
`Source Scenario at the Urban Based Mobile [30]
`At the mobile, the model is that the distributed sources
`occupy the far field evenly in the directions 0° ~ <f> < 360°,
`60° ~ 0 ~ 90°, where 0 and </> are now the spherical
`coordinates with the mobile at the origin. Both polarizations
`latter property
`the
`likely,
`are uncorrelated and equally
`implying that the base station receives equal powers in both
`polarizations. Each polarization is assumed spatially uncorre-
`
`(9)
`
`which is a six-component sum in the MCS (no earth plane is
`assumed present) . The envelopes of the IEzl 2 component and
`the total energy are plotted as a function of position in Fig. 2
`along with their Rayleigh curves. Very little fading of the total
`energy occurs, and in principle , if an antenna could be
`designed to gather the energy coherently, there would be no
`need to resort to space diversity . Obviously , this antenna
`cannot have just a single port (a combiner is required as in
`space diversity). The presence of an earth plane close to the
`antenna reduces the number of field components to three.
`Pierce's energy density antenna (Gilbert [9)) was designed to
`receive these three components, and the technique is often
`called field component diversity. The antenna is mentioned in
`Section V. The reason it works well is that the three field
`components are uncorrelated at a point in an omnidirectional
`scenario (see Jakes [13, p. 38)).
`One interpretation of Fig . 2 is that the Rayleigh-like fading
`of the mobile channel is a result of using a single port antenna.
`At (or rather above) the mobile, the total energy is relatively
`constant so that compact diversity antennas are possible, at
`least in principle .
`At the base station, it is not unreasonable to assume that the
`incident signal from a single mobile is from a single direction.
`This means that the incident energy is restricted to the two
`orthogonal polarizations in this direction. The maximum
`theoretical performance without resorting to space diversity
`(as far as the fading is concered) can thus be realized by
`polarization diversity (Vaughan and Bach Andersen [31]).
`There is an important difference between the fading of
`energy at the mobile and at the base station. The energy at a
`point above a mobile in the MCS corresponds closely to a
`maximum ratio combination of five uncorrelated branches of
`equal mean SNR's (cf. Fig. 2(c)). The energy at a point at the
`base station has a theoretical limit of only two combined
`branches.
`
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`
`Ex.1020
`APPLE INC. / Page 4 of 24
`
`

`

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`plane is assumed present. Sec [30] for the dB definition of standard deviation used here. (c) Rayleigh curves for fading of (a) and (b).
`(a) Energy in i component of electric field along path within MCS. (b) Total energy at point moving within MCS. No ground
`Fig. 2.
`
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`
`STANDARD DEVIATION 108 I= 1.94
`I08Ml = -32.76
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`
`Ex.1020
`APPLE INC. / Page 5 of 24
`
`

`

`154
`
`IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. VT-36, NO. 4, NOVEMBER 1987
`
`Signal Combination
`In this section, predetection maximum ratio combining is of
`principal interest. There is little difference in diversity gains
`between equal gain, selection, and maximum ratio combina(cid:173)
`tions. The relative performance returns for each of these
`schemes are well known (e.g . , Jakes, (13, ch. 5]).
`Switched diversity offers economical and practical schemes
`which are usually the type implemented. The local mean level
`of the signal can be measured so the threshhold can be
`floating, but relative to the local signal mean. Arnold and
`Bodtmann [2] give an example with wide-band simulation
`results of this technique. An interesting result is that the
`performance is rather insensitive to the threshhold value, over
`a range of several ( ~ 5) dB [2, fig. 7, p. 159]. Their
`simulation used four uncorrelated signals, and the switching
`rule was just sequential commutation, which surprisingly
`gives significantly better results than the three-branch selec(cid:173)
`tion case.
`While switched schemes offer practical advantages, the
`maximum ratio combining is mainly of theoretical use and as a
`performance benchmark. More recently, the more compli(cid:173)
`cated optimum combining (Bogachev and Kiselev [6], Winters
`[34]) has been discussed , although implementation details are
`lacking. The advantage of optimum combining is the possibil(cid:173)
`ity of improving strong interference suppression ( over other
`combining schemes) , an issue which will also become of
`increasing importance as the demands on spectrum efficiency
`in cellular systems increase. The degree of interference
`suppression is related to the number of branches, so optimum
`combining motivates many-branch systems. For interferers of
`similar or less power than the wanted signal, conventional
`combining gives quite good interference suppression. Miki
`and Hata [21] give some examples for two-branch switched
`combining which include the amount of interference suppres(cid:173)
`sion.
`In maximum ratio combining (Kahn [15]), the weights are
`proportional to the conjugate of the signal voltage and the
`inverse of the branch noise power. Implementation of a
`maximum ratio combiner is expensive since the weights have
`both amplitude and phase, and measurement of the channel
`(instantaneous) SNR is required for each weight update. The
`technique is the best linear combination in the sense that it
`yields the largest output SNR, which turns out to be the sum of
`the branch SNR's. The latter property makes maximum ratio
`combining very attractive for finding theoretical characteris(cid:173)
`tics of the combined signal .
`If uncorrelated Rayleigh distributions and identical mean
`SNR's are assumed for each input channel, then the cumula(cid:173)
`tive probability of the SNR of the maximum ratio combined
`signal is (e.g. , Jakes [13 , p. 319])
`
`where M is the number of input channels and r is the mean
`SNR of each channel. Setting the number of branches M to 1
`in ( 10) leads to the Rayleigh distribution.
`
`(10)
`
`The diversity gain is defined as the decrease in SNR
`compared to a nondiversity receiver for a given performance
`factor. The performance factor usually used with antenna
`diversity is related to PM(-y). For example, two-branch antenna
`diversity with maximum ratio combining gives a diversity gain
`of about 16 dB for P2(-y) = 0.001. After three-branch
`diversity , diminishing returns from adding extra branches sets
`in for this measure of diversity gain.
`Rather lax application of the term diversity gain has led to
`some misconceptions regarding actual diversity returns. Spe(cid:173)
`cifically, when branches become correlated, it is incorrect to
`read the diversity gain off a Rayleigh diagram without taking
`proper account of the mutual coupling. Before elaborating on
`this point, some discussion is in order regarding the correla(cid:173)
`tion coefficient.
`
`Correlated Branch Signals
`The correlation coefficient p of two narrow-band signals
`whose envelopes follow a Rayleigh distribution is known
`(Pierce and Stein [27]) to obey
`
`(11)
`
`where Pe is the correlation coefficient of the envelopes. It
`follows that the square root of the envelope correlation gives
`the signal correlation to within an arbitrary angle. This angle is
`usually considered as zero for practical purposes , and the
`absolute value sign in (11) is correspondingly dropped.
`The property that the correlation coefficient is never
`negative for Rayleigh distributed signals is interesting. Mea(cid:173)
`surements by the authors of envelope correlations obtained in
`urban environments have often been negative. Kozono et al.
`( 17] also report negative correlation coefficients from their
`base station measurements . This is one way to demonstrate
`that the signal envelope of the mobile channel does not have a
`truly Rayleigh distribution. For diversity considerations ,
`signals with a negative envelope correlation coefficient can
`offer better diversity gain than signals with zero correlation ,
`such as those indicated in Fig. 2. Consider a two-source model
`in which the sources are directly in front of and behind the
`mobile. If two space diversity antennas were mounted such
`that the envelopes were
`
`and
`
`r2= \cos x\
`
`(12)
`
`(13)
`
`then the envelope correlation is readily established to be
`- 0.92. In this case, two-channel diversity is sufficient to
`eliminate the fading almost completely. The reason is thac the
`correlation coefficient is nearly - 1, which represents the
`ideal value . For the scenario which gives rise to Rayleigh
`fading, the best value for envelope correlations between
`diversity antenna element signals, as far as curing the fading is
`concerned , is zero.
`When the branch signals become correlated, it becomes
`very difficult to find PM("() for combinations other than
`maximum ratio. Pi(-y) for a finite branch correlation is well
`
`Authorized licensed use limited to: Fish & Richardson PC. Downloaded on May 16,2022 at 13:14:51 UTC from IEEE Xplore. Restrictions apply.
`
`Ex.1020
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`VAUGHAN AND A."sDERSEN: ANTE?'-iNA DIVERSITY IN COMML:NICATIONS
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`155
`
`the abscissa. For small and medium values of envelope
`correlation, the shift is quite small. For very large values of
`correlatin coefficient , the shift is large . For example , in the
`two-branch case, the curves for Pe = 0, 0.1, 0.5, 0.9, and 1
`are shifted to the left by about 0, 0.2, 0.9 , 2.4 , and 3 dB,
`respectively . For the three-branch case , the corresponding
`shifts are about 0 .2, 0 .9 , 1.6, and 4.77 dB , respectively .
`These shifted curves would then have the effect of mutual
`coupling fully included, albeit approximately , and can be used
`to read off the true diversity gain (now identical to the
`diversity return).
`An explicit relation between PL (the loaded circuit correla(cid:173)
`tion), Po and r is available in Section IV, so that for a given
`antenna, the curves can be derived exactly . The above
`approximations are good for high-gain antennas at the mobile
`and the curves will not change much for all such antennas.
`Note also that the factor of (16) does not affect switched
`antenna diversity systems , where mutual coupling does not
`play an important role for this defintion of diversity gain (the
`unused elements are assumed to be open circuit and to obey the
`approximation of (14)) .
`The diversity gain available from Fig . 3 is not particularly
`sensitive to the envelope correlation coefficient Pe, as long as
`::::: 0. 7 is quoted almost
`Pe is less than about 0. 7. Indeed, Pe
`universally to be acceptable for diverstiy considerations . For
`maximum ratio combining at the mobile, this figure corres(cid:173)
`ponds in a given diversity gain sense, to about 0.5 when the
`mutual coupling is accounted for. A condition for good
`diversity action using maximum ratio combining is that the
`correlation coefficient should be " low ," which can be taken
`as Pe < ~ 0. 7 at the base station or Pe < ~ 0.5 at the mobile .
`
`known and P 3(-y) for a circular array (identical correlations for
`all branches in the MCS or any rotationally symmetric source
`scenario) is established in the Appendix . The Rayleigh curves
`for Pi(-y) and PJ(-y) are displayed in Figs. 3(a), 3(b). The curves
`for Pi(-y) are well known (e.g., Jakes [13, p. 327]). Note the
`SNR is that of the combined signal and the reference (SNR) is
`that of a single branch. It is common practice to read the
`diversity gain off these curves for a given correlation
`coefficient. This is correct only if the mutual impedance has no
`effect . At the base station , this is not completely unreasonable
`because the mutual impedance decreases much more rapidly
`than the signal correlation as similar antennas are spaced
`apart. Space diversity , for example, at the base station requires
`distances of tens of wavelengths between elements (e.g., Lee
`[22, p. 201]), which for conventional antennas means that the
`mutual coupling is very low . Stated in another way , the
`correlation coefficient between base station elements can be
`very close to unity while the mutual coupling is negligible.
`At the mobile , this cannot be the case. Consider again space
`diversity , but now at the mobile. The spatial correlation
`coefficient in the MCS which lies between 10 (kx) and sine
`(kx) (Vaugha