IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. VT-36. NO. 4. NOVEMBER 1987
`
`149
`
`Antenna Diversity in Mobile Communications
`
`RODNEY G. VAUGHAN, MEMBER, IEEE, AND J. BACH ANDERSEN, SENIOR MEMBER,
`
`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 thefarfield of the diversity antenna should obey(or nearly obey) an
`orthogonality relationship. The role of mutual coupling is central, andit
`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 someexisting designs.
`
`I.
`
`INTRODUCTION
`
`in
`spectrum efficiency
`HE DEMAND for better
`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 componentis associated with the deeper fades. 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 bystrategically 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-
`trum efficiency is
`reduced because the distance between
`frequency reuse transmitters must be greater to maintain
`acceptable cochannel interference levels. Moreover, the ran-
`dom FM cannot be suppressed by simply increasing the
`
`Manuscript received May, 9, 1986: revised May 10. 1987.
`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 somekind 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 combinerso that existing
`systems can be improved without the need for implementing a
`signaling diversity scheme. The random FM is suppressed
`according to the order of diversity and the combining
`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-
`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.
`
`it is worth adding that adaptive
`Regarding the first point,
`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 diversityis 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 knowledge of the
`channel. Specifically,
`the Rayleigh-like fading gi es rise to
`bursts of errors during the deeper fades. The channelis often
`treated as having ‘‘good’* and ‘‘bad”’ states of transmission in
`a scheme known as the Gilbert-Elliot model (e.g., Ahlin, [1]).
`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
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`© 1988 IEEE
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`IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL, VT-36, NO. 4, NOVEMBER 1987
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`482.0
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`GROUPOELAY(AMICROSECONDS?
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`GROUPDELAY(HICROSECONDS)
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`Fig. 1.
`Illustration that antenna diversity can also work for wide-band (frequency hopping) systems. The three figures are group
`delays from simulation of three diversity antenna elements. Dispersive (bad) channels are independent for each element. Average
`group delay is about 17 ys, which is exaggerated for clarity on the scale (it is typically less than 0.5 ys).
`
`performance may well be quite different from actual perform-
`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).
`Mucheffort has also been expended on wide-band systems.
`The spread spectrum approach scems to be necessary for
`implementation of optimum combining, which is discussed by
`Winters [34]. Frequency hopping schemes (often referred to as
`frequencydiversity) 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 schemeisillustrated by simulation 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-
`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
`random FM. The spikes of high dispersion in Fig.
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`VAUGHAN AND ANDERSEN: ANTENNADIVERSITY [IN COMMUNICATIONS
`
`151
`
`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-
`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 I for both polarization
`matrices (e.g.,
`(4)) and signal-to-noise-ratio (e.g.,
`(17)).
`Someinconsistencies also arise from historical ‘‘misuse.’” For
`example, covariances and (complex) correlations are consid-
`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 CNRis 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 SNRis 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: 7 means
`transpose,
`the asterisk means complex
`conjugate, and H means Hermitian transpose. When discus-
`sing the mobile communications scenario (see Section II), 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 Il. Not a great deal has been
`reported about the scenario of sources incident on an urban-
`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-
`ed combining (or when the diversity antenna elements are
`always terminated), more care than that displayed in the
`literature is
`required to interpret
`the diversity gain. A
`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 viewpcints
`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 techniquesare 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 remainderof the article) concentrates on
`antenna diversity at the mobile. An element figure of merit
`(the element directivity toward the distributed sources sce-
`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
`byconsidering 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 monopolesis also discussed. The advantageis 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 methodsolution
`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-
`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-
`element array.
`
`Tl. Antenna 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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`152
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`IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, 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 lowerlimits 6,, 9.3 62, ¢, where the
`origin is the base station. The incident electric field is denoted
`
`lated. The polarization matrix at the mobile is thus
`
`T’'(6;, G15 62, G2) =S(6, 6)d(G; — 02)6( 1 — o2)
`
`1
`
`0
`
`A(O, b, t)=Ag(O, o, t)O+h6(8, 4, 1
`
`Q)
`
`where
`
`wherethe units of A, As, and A, are volts/meter/steradian. The
`polarization matrix for the incident fields is defined as
`
`S(6, =|
`
`S;
`0,
`
`60° < 6 < 90°, 0°<o< 360°
`elsewhere
`
`8)
`
`T'’(@1, $13 92, 2)= |
`
`Te |Pos Ts
`
`(2)
`
`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-
`
`T4(81, O13 92, $2) = (he (81, Gis HAG, O15 t)).
`
`3)
`
`tions scenario (MCS),
`
`If the polarizations are considered uncorrelated and each
`polarization considered spatially uncorrelated, then
`
`T’ (61, $13 92, G2)=P(8, 6)6(8; — 8:)6(o) — 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 "]
`
`4)
`
`where
`
`P(6, O)=P, 6,86: 6,, O15 OX by
`
`=0,
`
`elsewhere
`
`(5)
`
`is the (constant) power density per steradic square distribution
`and
`
`T f66
`XPD =——
`3o
`
`(6)
`
`the cross polar discrimination (XPD). For vertically
`is
`polarized antennas in urban areas,
`the XPD is given by
`Kozonoetal, [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
`station,
`the value is = —6 dB [21]. Most existing mobile
`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 (Kozonoef 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° < @ < 360°,
`60° < @ < 90°, where @ and ¢ are now the spherical
`coordinates with the mobile at the origin. Both polarizations
`are uncorrelated and equally likely,
`the latter property
`implying that the base station receives equal powers in both
`polarizations. Each polarization is assumed spatially uncorre-
`
`energy =|E|*+|ZoH|?
`
`(9)
`
`which is a six-component sum in the MCS (noearth plane is
`assumed present). The envelopes of the |Z,|*? 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 channelis 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 assumethat 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 energyat a point at the
`base station has a theoretical
`limit of only two combined
`branches.
`
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`154
`
`IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. VT-36, NO. 4, NOVEMBER 1987
`
`Signal Combination
`In this section, predetection maximum ratio combiningis of
`principal interest. There is little difference in diversity gains
`between equal gain, selection, and maximum ratio combina-
`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 meanlevel
`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
`performanceis 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-
`tion case.
`the
`While switched schemes offer practical advantages,
`maximum ratio combining is mainly of theoretical use and as a
`performance benchmark. More recently,
`the more compli-
`cated optimum combining (Bogachevand Kiselev [6], Winters
`[34]) has been discussed, although implementation details are
`lacking. The advantage of optimum combiningis the possibil-
`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 amountof interference suppres-
`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 outto be the sum of
`the branch SNR’s. The latter property makes maximum ratio
`combining very attractive for finding theoretical characteris-
`tics of the combined signal.
`If uncorrelated Rayleigh distributions and identical mean
`SNR’s are assumed for each input channel, then the cumula-
`tive probability of the SNR of the maximum ratio combined
`signal is (e.g., Jakes [13, p. 319])
`
`M
`Pu(y=l—-e-v? Y
`k=1
`
`k-1
`
`Y
`
`(¢)
`
`(k-1)!
`
`(10)
`
`is the mean
`where M is the numberof input channels and T'
`SNRofeach channel. Setting the number of branches M to 1
`in (10) leads to the Rayleigh distribution.
`
`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 Py(y). For example, two-branch antenna
`diversity with maximum ratio combining gives a diversity gain
`of about 16 dB for P,(y)
`0.001. After
`three-branch
`diversity, diminishing returns from adding extra branchessets
`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-
`cifically, when branches becomecorrelated, 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-
`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
`
`|p|? = pe
`
`(11)
`
`It
`where p, is the correlation coefficient of the envelopes.
`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-
`surements by the authors of envelope correlations obtained in
`urban environments have often been negative. Kozonoet 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
`
`r,=|sin x|
`
`r,=|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 that 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 elementsignals, as far as curing the fadingis
`concerned, is zero,
`it becomes
`When the branch signals become correlated,
`very difficult
`to find Py(y) for combinations other than
`maximum ratio. P2(y) for a finite branch correlation is well
`
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`
`6
`
`6
`
`

`

`VAUGHAN AND ANDERSEN: ANTENNADIVERSITY IN COMMUNICATIONS
`
`155
`
`known and P;() for a circular array (identical correlations for
`all branches in the MCS oranyrotationally symmetric source
`scenario) is established in the Appendix. The Rayleigh curves
`for P(y) and P;(y) are displayed in Figs. 3(a), 3(b). The curves
`for P(y) are well known(e.g., Jakes [13, p. 327]). Note the
`SNRis 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 onlyif the mutual impedance has no
`effect. At the base station, this is not completely unreasonable
`because the mutual impedance decreases much morerapidly
`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 meansthat 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 Jp (Kx) and sinc
`(kx) (Vaughan [30]) showsthatfor finite correlations (appre-
`ciable values, greater than, say 0.5),
`the antennas must be
`closer than a fraction of a wavelength. (In space diversity at
`the mobile. there is seldom interest in having a larger spacing
`than the first zero of the correlation function.) Now,
`in the
`limit as p > 1, the spacing approaches zero and the elements
`merge into one. Nevertheless,
`the curves of Fig. 3(a),
`(6)
`indicate a 3-dB and 4.77 dB (power factors of 2 and 3,
`respectively) diversity gain for
`this case! Evidently,
`the
`diversity gain has to be defined in these cases as having a
`reference (SNR) from a single element in the presence of the
`other elements of the diversity antenna whileit is operating
`as a diversity antenna. This definition can only be properly
`corrected by accounting for the mutual coupling. In Section
`III,
`it
`is shown that, for certain high-gain mobile antenna
`elements, the open circuit signal correlation coefficient po is
`closely related to the normalized mutual resistancer,
`
`po®l.
`
`(14)
`
`For many antennas, the open circuit and terminated circuit
`correlation coefficients are reasonably close (cf. for example,
`Figs. 12 and 13 for sloping monopoles discussed below) and
`so to a reasonable approximation,
`2 me
`r? = pe.
`
`(15)
`
`the approximate effect of mutual
`With these results,
`coupling can be included in the Rayleigh diagrams. The
`abscissa is modified by the multiplicative factor (additive, for
`dB quantities)
`
`(SNR(1 branch, mutual coupling ignored))
`(SNR(1 branch, mutual coupling accounted for)) ’
`
`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 p, = 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 p, (the loaded circuit correla-
`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 importantrole for this defintion of diversity gain (the
`unused elements are assumedto 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 p,, as long as
`peis less than about 0.7. Indeed, p, = 0.7 is quoted almost
`universally to be acceptable for diverstiy considerations. For
`maximum ratio combining at the mobile, this figure corres-
`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 pp < ~0.7 at the base station or p, < ~0.5 at the mobile.
`
`Mean SNR Differences
`It has been assumed that all branches have the same mean
`SNR’s. When these become different, a combiner will, of
`course, favor the branch with the highest mean SNR, and the
`diversity returns will be reduced. In terms of the diversity
`gain,
`the degradation is similar
`to that caused by finite
`correlations. In the case of two branches, it is clear that the
`condition of one branch having much higher mean SNR than
`the other will result in the combined signal having the fading
`characteristics of a single channel independent of the branch
`signal correlation. This same effect occurs for correlated
`branchs (9 > 1), where the combined signal fades as a single
`channel,
`independent of the difference in the branch mean
`SNR’s. The trade-off in