`
`MICROWAVE
`MOBILE
`COMMUNICATIONS
`
`Edited by William C. Jakes
`FORMERLY DIRECTOR, RADIO
`TRANSMISSION. LABORATORY
`BELL TELEPHONE LABORATORIES
`NORTH ANDOVER, MASSACHUSETTS
`
`IEEE
`PRESS
`
`IEEE COMMUNICATIONS SOCIETY, SPONSOR
`
`THE INSTITUTE OF ELECTRICAL AND ELECTRONICS ENGINEERS, INC., NEW YORK
`
`1
`
`SAMSUNG 1028
`
`SAMSUNG 1028
`
`1
`
`

`

`IEEE PRESS
`445 Hoes Lane, P. O. Box 1331
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`
`IEEE PRESSEditorial Board
`William Perkins, Editor in Chief
`
`R. 8. Blicq
`M. Eden
`D. M.Etter
`J. J. Farrell I
`G. F. Hoffnagle
`
`R. F. Hoyt
`J. D. Irwin
`S. V. Kartalopoulos
`P. Laplante
`E. K. Miller
`
`J. M. F. Moura
`I. Peden
`L. Shaw
`M.Simaan
`
`Dudley R. Kay, Director ofBook Publishing
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`Lisa S. Mizrahi, Review and Publicity Coordinator
`
`Reissued in cooperation with the
`IEEE Communications Society
`
`IEEE Communications Society Liaison to IEEE PRESS
`Jack M. Holtzman
`
`Copyright © 1974, AT&T IMP Corp. reprinted by permission.
`
`Printed in the United States of America
`
`1098 765 43 2
`
`ISBN 0-7803-1069-1
`
`IEEE Order Number: PC4324
`
`2
`
`

`

`xi
`
`xiii
`
`11
`
`11
`13
`19
`
`3 4
`
`5
`
`65
`
`79
`
`79
`80
`
`112
`
`119
`
`123
`
`contents
`
`preface to the IEEE edition
`preface to the first edition
`foreword
`introduction
`
`Wm. C. Jakes
`
`PART | MOBILE RADIO PROPAGATION
`
`chapter 1 multipath interference
`Wm. C. Jakes
`
`Synopsis of Chapter
`1.1 Spatial Distribution of the Field
`1.2 Power Spectra of the Fading Signal
`1.3. Power Spectrum and Other Properties of the Signal Envelope
`1.4 Random Frequency Modulation
`1.5 Coherence Bandwidth
`1.6 Spatial Correlations at the Base Station
`1.7 Laboratory Simulation of Multipath Interference
`
`chapter 2 large-scale variations of the
`average signal
`D. O. Reudink
`
`Synopsis of Chapter
`2.1. Factors Affecting Transmission
`2.2 Observed Attenuation on Mobile Radio Paths over Smooth
`Terrain
`2.3 Effects of Irregular Terrain
`2.4 Statistical Distribution of the Local Mean Signal
`2.5 Prediction of Field Strength
`
`3
`
`

`

`vi
`
`Contents
`
`chapter 3 antennas and polarization effects
`Y. S. Yeh
`
`Synopsis of Chapter
`3.1 Mobile Antennas
`3.2 Base Station Antennas
`3.3 Polarization Effects
`
`PART Il MOBILE RADIO SYSTEMS
`
`chapter 4 modulation, noise, and
`interference
`
`M. J. Gans and Y. S. Yeh
`
`Synopsis of Chapter
`Frequency Modulation
`4.1
`4.2 Digital Modulation
`4.3 Channel Multiplexing
`4.4 Man-Made Noise
`
`chapter 5 fundamentals of diversity systems
`Wm. C. Jakes, Y. S. Yeh, M. J. Gans,
`and D. O. Reudink
`
`Synopsis of Chapter
`5.1 Basic Diversity Classifications
`5.2 Combining Methods
`5.3 Antenna Arrays for Space Diversity
`5.4 Effect of Diversity on FM Noise and Interference
`5.5 Diversity Against Shadowing
`
`chapter 6 diversity techniques
`D. O. Reudink, Y. S. Yeh, and
`Wm. C. Jakes
`
`Synopsis of Chapter
`Postdetection Diversity
`
`6.1
`
`133
`
`133
`134
`
`150
`152
`
`161
`
`161
`162
`218
`240
`295
`
`309
`
`309
`310
`
`313
`329
`341
`377
`
`389
`
`389
`390
`
`4
`
`

`

`Contents
`
`vii
`
`6.2 Switched Diversity
`6.3 Coherent Combining Using Carrier Recovery
`6.4 Coherent Combining Using a Separate Pilot
`6.5 Retransmission Diversity
`6.6 Multicarrier AM Diversity
`6.7 Digital Modulation-Diversity Systems
`6.8 Comparison df Diversity Systems
`
`chapter 7 layout and control of
`high-capacity systems
`D. C. Cox and D. O. Reudink
`
`Synopsis of Chapter
`7.1 Large Radio Coverage Area Systems
`7.2 Coverage Layout of Small Cell Systems
`7.3 Base Station Assignment in Small Cell Systems
`7.4 Channel Assignment in Small Cell Systems
`
`appendix a computation of the spectra of
`phase-modulated waves by means of
`Poisson’s sum formula
`
`M. J. Gans
`
`appendix b click rate for a
`nonsymmetrical noise spectrum
`M. J. Gans
`
`appendix ¢ median values of transmission
`coef ficient variations
`
`M, J. Gans
`
`index
`
`399
`423
`hod
`489
`512
`517
`531
`
`545
`
`545
`546
`562
`
`572
`
`623
`
`627
`
`631
`
`635
`
`
`
`5
`
`

`

`Fundamentals of Diversity Systems
`
`313
`
`incurred in transmitting M
`thus a delay of M/2f, seconds is
`slot;
`branches. The sampling rate must be at least 8M kHz for voice transmis-
`sion. To keep the transmitted pulse width within the transmission band-
`width of the medium requires that M be limited to about 50 branches, at
`most.
`The transmission delay and upper limit on M are not serious; a much
`more fundamental limitation probably rules out time diversity for mobile
`radio. This is the fact that the minimum time separation between samplesis
`inversely proportional to the speed of the vehicle, since f,, = v/A. In other
`words, for the vehicle stationary, time diversity is essentially useless. This is
`in sharp contrast
`to all of the other diversity types discussed in the
`preceding sections. In these cases the branch separations are not functions
`of the vehicle speed; thus the diversity advantages are realized equally well
`for the vehicle stationary or moving at high speed.
`
`5.2 COMBINING METHODS
`
`Over the years a- number of methods haveevolved to capitalize on the
`uncorrelated fading exhibited by separate antennas in a space-diversity
`array.
`In this section we will divide these methods into four generic
`categories, outline their operating principles, and derive somerelationships
`that describe the improvement in the resulting signal statistics.. Specific
`embodiments appropriate to microwave mobile radio systems will be
`detailed in Chapter 6.
`
`5.2.1 Selection Diversity
`This is perhaps the simplest methodofall. Referring to Figure 5.2-1, that
`one of the M receivers having the highest baseband signal-to-noise ratio
`(SNR) is connected to the output.* As far as the statistics of the output
`signal are concerned,
`it
`is immaterial where the selection is done. The
`antenna signals themselves could be sampled, for example, and the best
`one sent to one receiver. To derive the probability density and distribution
`of the output signal we follow the approach given by Brennan.’
`We assume that the signals in each diversity branch are uncorrelated
`and Rayleigh distributed with mean power bo. The density function of the
`signal envelope is given in Eq. (1.1-14):
`
`i
`
`P(rJ= Fe0
`
`~ 1? /2bo
`
`*Inpractice, the branch with the largest (§+ N) is usually used,since it is difficult to measure
`SNR.
`
`6
`
`

`

`314
`
`Mobile Radio Systems
`
`VOICE
`TRANSMITTER
`
`O OUTPUT
`BEST ONE OF THE
`M RECEIVERS
`
`Figure 5.2-1 Principles ofselection diversity.
`
`wherer; is the signal envelope in the ith branch. We will be interested in
`SNR;thusit is convenient to introduce new variables. The local (averaged
`over one RF cycle) mean signal power per branch is r?/2. Let the mean
`noise power per branch ne be the sameforall branches, ne = N, and let
`4 local meansignal power per branch
`mean noise powerper branch
`
`vi
`
`2
`i
`= aN 5
`
`A meansignal powerper branch
`~ mean noise powerper branch
`
`bo
`= (y= ve
`
`(5.2-1)
`
`(5.2-2)
`
`Then
`
`(5.2-3)
`P(y)= pew.
`The probability that the SNR in one branchis less than or equal to some
`specified value y, is
`
`Plu<ul= ['r(way
`=] —e7%/T,
`
`(5.2-4)
`
`7
`
`

`

`Fundamentals of Diversity Systems
`
`315
`
`The probability that the y; in all M branches are simultaneously less than
`or equalto y, is then
`
`Ply s+ Yae < Yel = (07/7)= Pay(7):
`
`(5.2-5)
`
`This is the distribution of the best signal, that is, largest SNR, selected
`from the M branches. P,,(y,) is plotted in Figure 5.2-2 for diversity
`systems with 1, 2, 3, 4, and 6 branches. The potential savings in power
`offered by diversity are immediately obvious:
`10 dB for two-branch
`diversity at
`the 99% reliability level, for example, and 16 dB for four
`branches.
`
`
`
`
`
`
`
`PERCENTPROBABILITYTHATAMPLITUDE>ABSCISSA
`
`99,99
`-40
`
`-30
`
`-10
`-20
`10 log (y/T ),dB
`Figure 5,2-2 Probability distribution of SNR y, for M-branch selection diversity
`system. [= SNR on one branch.
`
`0
`
`10
`
`
`40.0
`
`
`
`2
`
`
`
` wNo5nnoooooooou
`
`
`
`
`
`8
`
`

`

`316
`
`Mobile Radio Systems
`
`The mean SNR of the selected signal is also of interest. This may be
`conveniently obtained from the probability density function of y, from the
`integration:
`
`(w= f 1sPul¥s) 2¥s5
`
`0
`
`(5.2-6)
`
`where P,,(y,) is obtained from
`
`d
`
`Pra1s) = Ge Pua(Ys)
`
`Fl — en ulylema/P,
`
`Substituting P,,(y,) into Eq. (5.2-6),
`
`M
`
`sla
`CY.) = rs
`
`(5.2-7)
`
`(5.2-8)
`
`The dependenceof <y,> on M is shownin Figure 5.2-3.
`The selection diversity system shown in Figure 5.2-1 is a “receiver”
`diversity type that can be used at either the base station or the mobile, the
`only difference being the somewhatlarger antenna separation required at
`the base station (cf. Section 1.6). It is possible to conceive of a selection
`diversity scheme where the diversity antenna array is at the transmitting
`site, as shown in Figure 5.2-4. The transmitters operate on adjacent
`frequency bands centered at f),f,....fy,. These bands are separated in a
`branching filter at the receiving site; each signal is then separately detected
`and the best one chosen as before. Although more frequency bandwidth is
`required, the transmitter antenna array spacing maybeslightly reduced by
`taking advantage of a certain amount of frequency diversity. Of course, if
`the transmitted bands were separated widely enough (cf. Section 1.5), one
`could completely exchange frequency diversity with space diversity and
`use only one antenna.
`
`5.2.2. Maximal Ratio Combining
`In this method,first proposed by Kahn,® the M signals are weighted
`proportionately to their signal voltage to noise power ratios and then
`summed. Figure 5.2-5 shows the essentials of the method. The individual
`signals must be cophased before combining, in contrast to selection divers-
`ity; a technique described in Chapter 6 does this very simply. Assuming
`
`9
`
`

`

`Fundamentals of Diversity Systems
`
`317
`
`\ f
`
`
`
`10iog<2?dB
`
`NO. OF BRANCHES, M
`Improvement of average SNR from a diversity combiner compared
`Figure 5.2-3
`to one branch, (a) maximal ratio combining, (b) equal gain combining, (c) selection
`diversity.
`
`
`eae==\AAONNVeitiN
`
`
` BB
`
`tm
`fa
`fi
`Figure 5.2-4 Selection diversity scheme with antennaarrayatthe transmitting site.
`
`10
`
`
`
`ey ae
`
`B
`ey
`
`OUTPUT
`
`SPECTRUMari=== BEST ONE OF
`
`10
`
`

`

`318
`
`Mobile Radio Systems
`
`
`
`VOICE
`
`TRANSMITTER
`
`Yee al ‘eye “#-VARIABLE GAIN
`
`AMPLIFIERS
`
`
`
`
`
`
`DETECTOR
`
`O OUTPUT
`Figure 5.2-5 Principles of maximal ratio combining (note that
`equal-gain combiningresults),
`
`if
`
`the a,=1,
`
`this cophasing has been accomplished,the envelope of the combinedsignal
`is
`
`r= 3 ar,
`
`(5.2-9)
`
`where the a, are the appropriate branch gains. Likewise the total noise
`power is the sum of the noise powers in each branch, weighted by the
`branch gain factors:
`
`Nr=N D> a?,
`
`(5.2-10)
`
`where, again, it has been assumedthat ne = N for all i. The resulting SNR
`is
`
`2
`r
`YrE aw:
`R
`2Nr
`
`the a, are chosen as stated above,
`if
`It can be shown that
`a,=,/n? =r,/N, then yp will be maximized with a value
`
`(S/n)
`
`M r;
`
`M
`
`Yr
`
`(5.2-11)
`
`that
`
`is,
`
`11
`
`11
`
`

`

`Fundamentals of Diversity Systems
`
`319
`
`Thus the SNR out of the combiner equals the sum of the branch SNRs.
`Now we know that
`2. 1 2} (,24y2
`5.
`
`Ve=ani xn i +y?), (5.2-13)
`
`where x, and y, are independent Gaussian random variables of equal
`variance b, and zero mean (cf. Section 1.1.2). Thus yg is a chi-square
`distribution of 2M Gaussian random variables with variance bo/2N=34I.
`The probability density function of yg can then be immediately written
`down:
`
`M ~lg7ye/T
`or 2.
`(y,)=—————,
`R
`TMM — 1)!
`PXYR)™
`The probability distribution function of yp is given by integrating the
`density function,
`
`(5.2-14)
`
`le YR M-1,-x/T
`
`Pu(Yr)= mucpi xMole-*/Tdx
`M> (g/t
`
`(5.2-15)
`
`The distribution P,,(y,) is plotted in Figure 5.2-6. This kind of combining
`gives the beststatistical reduction of fading of any knownlinear diversity
`combiner.
`In comparison with selection diversity,
`for example,
`two
`*"__usnes give 11.5 dB gain at the 99% reliability level and four branches
`give 19 dB gain, improvementsof 1.5 and 3 dB,respectively, over selection
`diversity.
`The mean SNR of the combined signal may be very simply obtained
`from Eq. (5.2-12):
`
`M
`
`M
`
`(5.2-16)
`(a= B= DT MT.
`Thus <y,> varies linearly with M, whereas for selection diversity it in-
`creases much more slowly, as shown in Figure 5.2-3.
`
`5.2.3. Equal Gain Combining
`It may not always be convenient or desirable to provide the variable
`weighting capability required for true maximal ratio combining. Instead,
`
`12
`
`

`

`Ol
`
`999900Pof!©oo00000004go
`
`
`
`
`
`
`
`320
`
`Mobile Radio Systems
`
`-2
`
`AMPLITUDE>ABSCISSA if,Ti
` PERCENTPROBABILITYTHAT
`
`
`
`-40
`
`-30
`
`-20
`
`-10
`
`oO
`
`10
`
`10 logl(y/T)
`Figure §.2-6 Probability distribution of SNR y, for M-branch maximal ratio
`diversity combiner. T, SNR on one branch.
`
`the gains mayall be set equal to a constant value of unity, and equal-gain
`combining results. The envelope of the combined signal is then given by
`Eq. (5.2-9) with a,;=1:
`
`(5.2-17)
`
`M
`r= > 7,
`
`13
`
`13
`
`

`

`The SNR of the outputis
`
`Fundamentals of Diversity Systems
`
`321
`
`r
`YE~ 9NM”’
`
`5.2-18
`(5.2-
`
`)
`
`again assuming equal noise in the branches.
`The combined output r is a sum of M Rayleigh variables. The problem
`of finding the distribution of the square of this sum (y,) is an old one,
`going back even to Lord Rayleigh, but has never been solved in terms of
`tabulated functions for M > 3. However, Brennan’ has obtained values by
`computer techniques, and his results for P(y,) are shown in Figure 5,2-7.
`The distribution curvesfall in between the corresponding ones for maximal
`ratio and selection diversity, and generally only a fraction of a decibel
`poorer than maximalratio.
`In contrast to the distribution function, the mean value of y, can be
`simply obtained:
`
`2
`
`(=)=say‘mt>"M
`
`S
`20
`
`(5.2-19)
`
`Now <7r?)=2b,, <7,)= V7bo/2 from Chapter 1. Furthermore, since we
`have assumed that the signals from the various antennas are uncorrelated,
`(rir) =r<r>, i#J. Thus Eq. (5.2-19) can be evaluated:
`
`(Y= nay [2+M(M- |
`
`=r[1+(m-1)F].
`
`(5.2-20)
`
`The dependency of y,; on M is also shownin Figure 5.2-3, andit is seen to
`be only a little poorer
`than maximal
`ratio combining.
`In fact,
`the
`difference is only 1.05 dB in the limit of an infinite number of branches.
`
`5.2.4 Feedback Diversity
`A very elementary type of diversity reception, called “scanning” divers-
`ity,’ is similar to selection diversity except that instead of always using the
`best one of M signals, the M signals are scanned in a fixed sequence until
`
`14
`
`14
`
`

`

`AMPLITUDE>ABSCISSA
`PERCENTPROBABILITYTHAT
`
`
`
`-10
`
`10
`
`322
`
`Mobile Radio Systems
`
`“=40
`
`10 log (y/T), 4B
`Figure 5.2-7 Probability distribution of SNR y for M-branch equal-gain diversity
`combiner. I’, SNR on one branch.
`
`it falls below
`is used until
`one is found above threshold. This signal
`threshold, when the scanning process starts again. The resulting fading
`Statistics are somewhat
`inferior to those from other diversity systems;
`however, a modification of this technique appears promising for mobile
`radio applications. The principles of operation are shown in Figure 5.2-8
`
`15
`
`15
`
`

`

`Fundamentals of Diversity Systems
`
`323
`
`RECEIVER
`
`One
`
`IN
` VOICE
`
`| C
`
`ONTROL
`SIGNAL
`
`VOICE
`
`|
`
`cot YY Baa
`
`OUT
`
`DIVERSITY
`RECEIVER
`
`VOICE IN
`
`BASE STATION
`
`MOBILE
`
`Figure 5.2-8 Principles of feedback diversity.
`
`for two-branch base transmitter diversity. In this system the fact that every
`base-mobile contact is a two-way affair is exploited by using the mobile-to-
`base path as a signaling channel in addition to carrying the voice modula-
`tion. It is assumed that the mobile-to-base path is reliable, using some sort
`of base-receiver diversity. The base transmitter is connected to one of its
`two antennas by a switch, and remainsthere until the received signal at the
`mobile falls below a preset threshold level. It signals this fact over the
`mobile-to-base path, and the transmitter then switches to the other
`antenna and remains there until
`the new signal again falls below the
`threshold. Since the chance of having both transmission paths poor simul-
`taneously is smaller than either one being weak, there should be an average
`improvement in the signal received by the mobile. The performanceis
`affected by the total time delay in actuating the switch, whichis the sum of
`the round-trip propagation time and the time delay corresponding to the
`bandwidth of the control channel. If this delay is too great, the signal at
`the mobile could continue into a fade below the threshold before the
`transmitter switches and the new signal arrives. At UHF and vehicle
`speeds of 60 mi/hr, however, the expected signal drop is only | or 2 dB;
`thus the technique appears promising for use in this frequency range. A
`variation on this scheme can provide very simple mobile receiver diversity,
`with the receiver switching between two antennas using the same logic
`principles previously described. In this case the time delay in the switching
`process can be made very small, with a consequent improvement in the
`signal fadingstatistics. Specific embodiments of these schemes andsignal
`distribution curves are given in Section 6.2.
`
`16
`
`16
`
`

`

`324
`
`Mobile Radio Systems
`
`5.2.5 Impairments due to Branch Correlation
`In the preceding sections the analysis has been based on the assumption
`that the fading signals in the various branches are uncorrelated. It may
`happen in some casesthat this is difficult to achieve, for example, if the
`antennas in the diversity array are improperly positioned, or if
`the
`frequency separation between diversity signals is too small.
`It
`is thus
`important
`to examine the possible deterioration of performance of a
`diversity system when the branchsignals are correlated to a certain extent.
`Intuitively one might expect that a moderate amountof correlation would
`not be too damaging. Since deep fades are relatively rare in any event,it
`would require a very high degree of correlation between two fading signals
`to bring about a higher correspondence between the deep fades.
`The ettect of correiation in diversity branches has been studied by many
`workers. For maximal ratio combining it has been shown possible to derive
`the probability distribution of the combined signal for any number of
`branches. In the case of selection diversity, however,
`it does not appear
`that one can handle more than two branches.In either case the analysis is
`beyond the scope of this book; results described by Stein! will simply be
`presented for M=2.
`In the following the quantity p stands for the magnitude of the complex
`cross-covariance* of the two fading Gaussian signals (assumedalso to be
`jointly Gaussian); p” is very nearly equal
`to the normalized envelope
`covariance of the twosignals.
`
`Selection Diversity
`
`P(y,)=1—e~*/"[1— Q(a,b) + Q(b,a)],
`
`where
`
`Q(a,b)= fe #4(ax)xdx,
`
`‘
`
`eB
`2%
`fom
`“Vra-)
`
`(5.2-21)
`
`(5.2-22)
`
`5.2-23
`ae
`
`For p=0 the distribution reduces to (l1—e~*/")’, using the fact that Q(b,
`0)=1 and Q(0,b)=e~”/*. For y, <I,
`.
`YsPap):
`
`(5.2-24)
`
`2
`
`P,(y,) =
`
`*Defined in Section 1.3. See Eqs. (1.3-1), (1.3-12), (1.3-13), and (1.3-16).
`
`17
`
`17
`
`

`

`Maximal Ratio Combining
`
`Fundamentals of Diversity Systems
`
`325
`
`Py(yg) = 1 Fe[(+phew/e (pe-P], (52-25)
`
`Again for p=0 the distribution reduces to that for uncorrelated fading,
`and for yr<I,
`
`Ye
`(5.2-26)
`P,(YR)= We)"
`The above distributions are shown in Figures 5.2-9 and 5.2-10. We can
`easily see that the intuitive feeling expressed earlier is borne out in the
`numerical results. For example, with selection diversity’ one finds the
`combined signal to be 9.3 dB better than Rayleigh fading for 98% of the
`time with uncorrelated signals. Even for a correlation as high as 80% the
`combinedsignalis 6.3 dB better than nodiversity for 98% of the time.
`
`Impairments due to Combining Errors
`5.2.6
`The description of the various combining methods so far has implied
`that the combining mechanism, whatever it may be, operates perfectly.
`Since the information needed to operate a combineris extracted in some
`way from the signals themselves, there is the possibility of making an error
`and thus not completely achieving the expected performance. This effect
`has been studied in detail by Gans? for the particular case of the maximal
`ratio combiner, and his results will be summarized here. Similar results
`were also obtainedearlier by Bello.'°
`Werecall (Section 5.2.2) that in the maximal ratio combinerthe signals
`are cophased and then summed, with the amplitude of each branch signal
`being weighted by its own SNR. A particular embodiment of this method
`is described in Section 6.4, and involves use of a CW “pilot” signal
`transmitted adjacent to the message band. The phase and amplitude of the
`pilot signal are sensed and used to adjust the complex weighting factors of
`the individual branches so that true maximal ratio combining results. Now
`the fading of the pilot may not be completely correlated with that of the
`message, possibly becausethe pilot frequency is too far removed from the
`message. In this case the complex weighting factors would be somewhatin
`error, and degraded performanceresults. The effects can be completely
`described in terms of the quantity p, defined as the magnitude of the
`complex cross correlation between the transmission coefficients of
`the
`medium associated with the pilot and message frequencies. Approximately,
`
`18
`
`18
`
`

`

`99.98 99.99
`
`
`
`
`
`
` OYPERCENTPROBABILITYTHATAMPLITUDE>ABSCISSA
`
`— 15
`—-10
`10 log (y, /T), dB
`Figure 5.2-9 Probability distribution for a two-branch selection diversity combiner
`with correlated branch signals. ', SNR on one branch; p?= envelopecorrelation.
`
`—5
`
`0
`
`— 25
`
`—20
`
`326
`
`19
`
`

`

`AMPLITUDE>ABSCISSA
`PERCENTPROBABILITYTHAT
`
`
`
`-25
`
`-20
`
`-15
`10 log(yp/l),4B
`Figure 5.2-10 Probability distribution for a two-branch maximalratio diversity
`combiner with correlated branch signals. [, SNR on one branch; p*=envelope
`correlation.
`
`-|
`
`327
`
`20
`
`20
`
`

`

`328
`
`Mobile Radio Systems
`
`we can also identify p” as the correlation between the envelopes of two
`single sinusoids sent over the transmission path at the frequencies of the
`pilot and message. Gans? has then shown that the probability density of
`the combiner outputsignal can be written as
`M-1
`
`n=0
`
`a
`
`n
`
`Py(Yr)= =(1 —p?)"te-w/t Z; _ |
`ea nl
`
`2
`l
`YrP
`
`.
`
`x|——|—, 5.2-27
`
`eae
`
`where the binomial coefficientis
`
`-
`
`[“ ‘| Gena
`
`=f
`
`(5.2-28)
`
`The mean SNR may nowbesimply calculated by integrating:
`
`co
`
`<re>= f YrP(Yr) Yr
`
`0
`
`=I[1+(M-1)p?].
`
`(5.2-29)
`
`To get the probability distribution of the output signal, Eq. (5.2-27) is
`integrated:
`
`Pulra)= fu(x)ax.
`
`r
`_
`_—
`Yr/r)
`
`
`=—enm/t S| MOT|pam— pay SS a/—. (5.2-30)
`n
`kt
`n=0
`k=0
`
`M-1
`
`n
`
`
`
`In the worst case (for diversity action), the pilot and messagesignals are
`uncorrelated, p=0, and Py(yz)=1- e~%/T the same as for no diversity
`at all. In this case, also, <yp> =I, the SNR for one branch. On the other
`hand, if p=1 (pilot and message signals perfectly correlated) the distribu-
`tion function and mean SNRreduceto the expressions previously derived:
`
`cat SY (R/T)
`Pu(YrR)=1-e7% »vEe {Yr = MT.:
`
`k=0
`
`.
`
`21
`
`21
`
`

`

`Fundamentals of Diversity Systems
`
`329
`
`As an example of the effect of combinererror, the distribution and mean
`value of the SNR are plotted in Figures 5.2-11 and 5.2-12, respectively, for
`a four-branch maximal ratio combiner. Examining these figures we see that
`combinererrors have significant impact for the deeper fades for relatively
`small decorrelation. For example,
`if p?=1 the combined signal is above
`—7 dB for 99.99% of the time, whereas if p?=0.75 it is above this value
`only 99.6% of the time. On the other hand, the mean value only drops 0.9
`dB if p? changes from 1.0 to 0.75. Thus combinererrors affect the mean
`SNR negligibly in comparison to their effect on deep fades.
`The specific results presented have been for maximalratio combining.It
`appears reasonable, however, that for equal gain or selection diversity, if
`the combiner is controlled by sampling a pilot, decorrelation of the pilot
`and message channel would have similar effects on the probability distri-
`bution and mean SNR.
`
`5.33. ANTENNA ARRAYS FOR SPACE DIVERSITY
`
`In Section 5.2 the diversity combining of Rayleigh fading branches was
`discussed. It has been shown that diversity combining can improve the
`probability distribution af the resultant carrier-to-noise ratio (CNR) over
`that of a single branch system. Since the E, field at the mobile has a spatial
`correlation coefficient p=J,)( Bd), by separating antennas a sufficiently
`large distance d such that p—>0, we should expect to have the required
`independent branches for space-diversity application.
`The problem becomes complicated if we want to place dipoles closer
`together because of space limitations on the mobile. By placing dipoles
`close to each other we musttake into account the mutual coupling among
`antennas and also the finite correlations between the received fading
`signals. These effects can reduce the diversity advantage as mentioned in
`Section 5.2.5, and can also present complicated antenna matching prob-
`lems.
`In this section we shall examine the antenna spacing requirements of the
`linear and planar monopole arrays shown in Figure 5.3-1, assuming
`maximalratio diversity combining. The results presented are based on the
`work of Lee.!!:!* The emphasis will be on the average and the cumulative
`probability distribution (CPD) of the combiner output CNR. The CNRof
`an equal-gain system!*!* is very close to the maximalratio case; therefore
`it is not reported here. The level crossing rates and power spectrum of
`linear and planardiversity arrays can also be foundin Refs. 13 and 14.
`the
`It should be mentioned that
`in the mobile radio environment,
`diversity array gains are obtained by coherent combining of the random
`signals received from each array element. Since the random phase
`
`
`
`22
`
`22
`
`