`
`Switched antenna diversity within a DECT system
`
`J-P van Deursen and M.G.Jansen
`Telecommunications and Traffic Control Systems Group
`Departmentof Electrical Engineering
`Delft University of Technology
`P.O.Box 5031, 2600 GA Delft, The Netherlands
`Tel.: +31 15 782417, Fax: +31 15 781774
`E-mail: M.G.Jansen@ET.TUDelft.NL
`
`Abstract
`
`the signal-degrading effects of multipath
`To combat
`propagation in a DECT system, diversity techniques
`can be employed. The performance of a switched an-
`tenna diversity scheme based on received power
`is
`analyzed and compared with the performance ofselec-
`tion diversity for a frequency non-selective radio chan-
`ne] with either Rayleigh or Ricean fading characteris-
`tics. Received output power statistics are chosen as
`performance measure. Expressions are derived for the
`performance of
`two-branch switched diversity with
`unequal branch statistics, antenna (space) correlation
`and (partial) time correlation between succeeding time
`samples. Results show that switched diversity perfor-
`mance is always inferior to selection diversity perfor-
`mance and that the Rice factor influences the diversity
`gain non-linearly. Both diversity schemes are sensitive
`to unequal branch statistics, while switched diversity
`gain deteriorates fastest. Antenna correlation does not
`have a large influence on the performance of switched
`diversity. Finally, it is also shown that the diversity gain
`with switched diversity is reduced considerably if there
`is only partial time correlation between succeeding time
`samples.
`
`1. Introduction
`
`The last few years show a rapidly increasing demand for
`wireless communication systems. The growing trend is
`to supply different services, such as speech, video or
`data
`transmission, with
`the
`same
`network and
`peripheral equipments. The DECT (Digital European
`Cordless Telecommunications) standard [1] provides a
`platform for the wireless digital transmission of speech
`and data.
`DECT will most likely mainly be used for indoor
`communications, where it has to deal with a very hos-
`tile radio environment. Time dependent multipath
`propagation can cause recurring deep signal
`fades,
`severely distorting reception quality. An effective means
`to combat multipath fading is diversity, which implies
`that several copies of the transmitted signal are ac-
`quired at the receiver and subsequently combined to
`improve performance.
`This paper presents a performance evaluation of
`such a diversity technique, namely switched antenna
`diversity, in a DECT system. This evaluation is based
`
`on a prestudy performed by Kopmeiners [2]. Perfor-
`mance is compared with that of selection diversity.
`Only two-branch diversity is
`considered,
`i.e.
`the
`receiving end acquires
`only
`two
`copies of
`the
`transmitted signal. The switching scheme is based on
`the received power level on both branches [3]. The
`radio channel is assumed to be frequency non-selective,
`with either Rayleigh or Ricean fading statistics. Results
`are given in terms of the cumulative density function
`(CDF) of
`the received power after diversity. The
`influence of several different effects that can degrade
`switched diversity performance is examined,
`including
`unequal branch statistics, correlated signals on both
`branches and incomplete correlation between suc-
`ceeding time samples.
`
`Il. The DECT radio link
`
`DECT is based on the cellular radio concept with very
`small ceil sizes (micro- or pico-cells, cell radii ranging
`from 25 m upward)
`to obtain a very high traffic
`capacity (typically 10000 Erlang/km?).
`CEPT, the standardization organization of European
`PTT’s, has appointed the 1880-1900 MHz frequency
`band for DECT applications. Traffic between the base
`Station and the handhelds in each cell is regulated by
`an FDMA/TDMA/TDD (Frequency Division Multiple
`Access/Time Division Multiple Access/Time Division
`Duplex) access protocol. Within the assigned frequency
`band 10 carrier frequencies are used, evenly spaced by
`1.728 MHz. Each of the 10 carriers (FDMA) is sub-
`divided in time into 10 ms frames, each frame consis-
`ting of 24 timeslots (TDMA). Thefirst 12 timeslots are
`used for downlink communication (from base station to
`handheld), the last 12 for uplink communication (from
`handheld to base station). The framestructure is shown
`in Fig. 1.
`A pair of timeslots with corresponding sequence
`numbers in the first and second half of a frame, e.g. 0
`and 12, form one duplex channel (TDD). Note that
`these time slots are separated by half a frame-iength,
`ie. by approx. 5 ms.
`.
`The use of TDMA/TDD makes it possible for DECT
`base stations to use only one transceiver, instead of one
`for each carrier (as in an FDMA system). The guard
`space between time slots allows for a base station to
`Switch between carriers, so adjacent slots can be on
`different carriers. This leads to a possible 120 duplex
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`channels per base station, of which 12 can be used
`simultaneously.
`
`‘frame = ifims
`
` *————. 12 downsiots
`
`2 up skits
`two corresponding time
`slots form one duplex
`channel
`
`Fig. 1 DECT frame structure
`
`Il. Switched and selection diversity principles
`
`several versions of the transmitted
`With diversity,
`signal are obtained at the receiver. If these signals are
`not strongly correlated, the probability that all of them
`are simultaneously in a fade is much lower than the
`probability that a single signal is in a fade.
`There are numerous ways of obtaining several ver-
`sions of the transmitted signal at the receiver. Switched
`and selection diversity are two forms of antenna (or
`Spacé) diversity, where multiple spatially separated
`antennas are used at the receiver, the distance between
`the antennas being large enough to ensure that signals
`on different antennas are not correlated.
`
`With selection diversity a branch consists of an an-
`tenna plus a receiver, see also Fig. 2. On the basis of a
`certain signal quality measure the ‘best’ branch at any
`instant
`is chosen as output branch, so the signal
`received on that branch becomes the output signal.
`Contrary to more complex schemes such as equal-gain
`combining or maximum-ratio combining, which can
`achieve a higher output SNR, the SNR of the output
`signal with selection diversity is only as good as the
`SNR of the chosen branch. Due to the fact that each
`branch has its own separate receiver, there is no delay
`in the selection of the optimum branch. In this paper
`the instantaneous received signal power is chosen as
`the. selection criterion. The disadvantage of selection
`diversity is the fact that two radios are needed, which
`
`increases the size and cost of the receiver.
`
`Fig. 2 Selection diversity
`
`With switched diversity a branch consists of just an
`antenna. A switch can connect only one branch at a
`
`time to a single recciver, see also Fig. 3. As with selec-
`tion diversity, instantaneous received signal power on a
`branch is
`chosen as
`the switching criterion,
`the
`switching algorithm is as follows:
`the power on the
`chosen (first) branch is compared with a certain preset
`threshold value A. If the power on this branch drops
`below A, the other (second) branch is chosen. If the
`power on this second branch is above A, it becomes the
`output branch. If however,
`the power on the second
`branch is also below A,
`two different
`follow-up
`Strategies can be discerned:
`- a Switch back to the first branch is made, which may
`lead to a period of continuous switching. between
`branches until the power on one of them exceeds A,
`this strategy is called switch-and-examine[4], or
`the second branch stays the chosen branch to avoid
`needless switching, a strategy called switch-and-stay.
`A Switch to the other branch is only made if power
`on the chosen branch crosses A in negative direction.
`All calculations involving switched diversity done in
`‘this paper are based on the switch-and-examine
`
`-
`
`Strategy.
`
`Fig. 3.
`
`Switched antenna diversity
`
`IV. Performance analysis
`
`AS was mentioned before, only two-branch switched
`and selection diversity are considered here. Fading has
`either Rayleigh or Ricean characteristics and is as-
`sumed to be frequency non-selective. The instantaneous
`received power on both branches is represented by the
`random variables p, and p». The output power after
`the diversity operation is denoted by p, = c(p,,p2),
`where c(.) is the relevant combining function.
`
`IV.A Independently faded branches with equal statistics
`If the signals on both branches are uncorrelated and
`have equal
`statistics,
`the largest diversity gain is
`achieved, ie. we have optimum diversity. To derive the
`performance expressions, we start with the probability
`density functions
`(PDF's) of the faded signal on a
`single branch. In case of Rayleigh fading, the PDF of
`instantaneous received power on each branch is given
`by
`
`Fray,p(Pi) = —,
`
`G;
`
`exp Fi PB = 9,
`
`G;
`
`(1)
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`

`
`
`examine can be written as [2]
`
`where 0? is the average received power, i = 1 or 2. The
`PDFfor Ricean fading is
`1 sel. A p(k-1)>A
`2
`
`1 2p,+8°||y2pis
`
`Fric,p(Pi) = —xP|-—5— |fo| ——|, P; = 0,
`(2)
`P,(k)
`V
`o:i
`26;
`2 sel. A p,(k-1) <A
`
`where o7 is the average scattered component power,
`s*/2 is the average dominant component power and
`Ip(.) is the modified Bessel function of the first kind
`and zero order. An important factor in the case of
`Ricean fading is the Rice factor K, defined as the ratio
`of average dominant component power and average
`scattered component power
`(3)
`K as*/207.
`Next, we determine the cumulative density function
`(CDF) of the instantaneous power of the signal after
`diversity p,, denoted by F,,: It is easily seen that for
`selection diversity the combining function is the non-
`linear function p, = c(p,,p2) = max(p,,p2). According
`to [5,pp.141], F,.(P) in that case is equal to
`F,(p) = Fp,(p)F,,(?):
`For Rayleigh faded branches this is
`
`(4)
`
`FRay,p,P) = FRay,p,P)F’Ray,p,P)
`(5)
`= 1-222]-e|-22) 2 0.
`
`o
`o
`Substituting x = (2p)/a, in (2), the separate cdf of
`P, and p, for Ricean faded branches are given by
`
`-
`
`+
`
`| 2
`
`Frigp(P) =1- fx col20a
`VL
`"7
`=1VIR),
`
`where Q(.,.) is the Q-function, defined [6,p.28] as
`-
`2
`
`Q(a,b) 4 fee|2hana
` 24p2] 2
`= exp] | (27 7,(a0),
`
`2
`\&le
`
`(7)
`
`a>b>0, and K is the Rice factor given by (3). The
`CDF of p, for Ricean faded branches is found by
`squaring eqn. (6) and assuming that co, = o, = o and
`
`Since only one receiver is used with switched diver-
`sity, the optimum branch cannot be determined instan-
`taneously. Therefore the combining function not only
`depends on the current vaiue of p, and p; but also on
`the values of p, and py at
`the previous sampling
`instant. If we denote the value of p, at sampling instant
`t, as p,(k)
`the combining function for switch-and-
`
`Po= C(P)P3) =
`
`(8)
`
`Pk)\
`
`2 sel. A p,(k-1)>A
`Vv
`1 sel. A p,(k-1) <A,
`
`the switching threshold. The
`for
`where A stands
`probability that p,(k-1) is equal to A is infinitely small
`and therefore does not appear in (8).
`In terms of
`probabilities, (8) can be written as
`
`6)
`
`Pr{p,>p} = (P{p,uo>e A py(k-1)>A}
`+ Prip,(k)>p Apy(k-1)<A})Pr{1 sel.}
`+ l{p,0o>p A p,{k-1)<A}
`+ Pr{p,(k)>p A py(ke-1)>A\}Pr{2 sel.}
`Because of the assumption that
`the average received
`power in both branches is equal and they have the
`same statistics, the probability of either branch 1 or 2
`being selected is equal, ie. Pr{1 sel.} = Pr{2 sel.} =
`1/2. Given this and the fact that p, and p, are inter-
`changeable, (9) can be reduced to
`
`Pr{pa>P} = Pripk)>e Apyk-1)>4} ay
`+ Pr{p,(k)>p A p,(k-1)<A}.
`If successive time samples have a correlation of 1 so
`they have equalstatistics, the CDF of p,is
`
`Fp) =1-f ff,(rifp,(P2)4P14P2
`-emax(p,A)
`2A
`~ fffrr Vp{P2)4P,4p.
`
`P --
`
`ap
`
`For independently Rayleigh faded branches,this results
`in
`
`1-en|-4]-e|-4veo£4 o
`
`’
`
`Fruyp(P) =
`
`0 <p <A;
`
`1-2 Pls
`
`on| 5] aa }e 412)
`
`
`Prd
`
`and for Ricean faded branches
`
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`
`
`Fric,pP) =
`
`1 - O,(4) - O,)[1 - 9,(4))
`0 <p <A;
`1 - Q,(p)|2 - 2,(4)], p >A,
`
`(13)
`
`where Q,(x) represents Q(v(2K;), ¥(2x)/a)).
`
`IV.B Independenily faded branches with unequal branch
`Statistics
`If the signals on both branches are independent, but
`have unequalstatistics, diversity gain is reduced. When
`there is
`a difference
`in average power on both
`branches, the branch with the highest average power
`will be favoured most of the time, thereby reducing the
`contribution of the other branch to diversity gain.
`Forselection diversity, using (4) with unequal average
`powers 07, i=1,2, the CDF of p, for Rayleigh faded
`branches (07>04) is simply
`
`1
`2
`Pp
`2
`Fray,p,=1~ 5—3| 81XP]5 1792EXP 2
`
` } (14)
`
`92
`
`9 ~ 92
`
`%
`
`p 2 0. For Ricean faded branches, using (4) and (6),
`
`copes
`{ - ol(aR ca 2 0.
`
`For switched diversity we use (9). Since we assume
`unequal branch statistics, we can no longer take Pr{1
`sel.} = Pr{2 sel.}, so (10) is not valid here. To com-
`pute the branch selection probabilities, we use ihe
`following Markov chain
`
`a,
`
`2
`
`oy
`
`“tee
`
`with state 1: branch one selected and state 2: branch
`two selected. The state transition probabilities qj, are
`equal
`to the probability that power on a branch is
`either above or below the threshold A, i.e.
`
`2
`i
`911 ={fray,pPr)Py =exp(-A/o)); 945=1-9,,
`:
`on {frayp,(P2)4P2 =xP(-AI09); Jai =1-dap
`A
`
`(16)
`
`The steady state solution of this Markov chain gives
`the branch selection probabilities as
`
`Prji
`
`sel.}
`
`I
`!
`1
`= ——-.__ =
`
`Q
`
`1 + (412/423)
`set}
`{1
`Pr{2 sel.} - ra!) =1-q
`1 + (412/41)
`
`(17)
`
`If we still assume that successive time samples are
`completely correlated (and branches
`fade indepen-
`dently!), (9) can be written as
`Pr{p,>P} =
`(i ~exp(-A/?)}exp(~p/03)4
`a ~exp(-A/e?)}exp(-plo?)(l-q), p >0
`+exp(-ploi)q +exp(-p/o3) (I -q),p >A
`+ exp(-A/oi)q +exp(-A/os)(1 ~q), 0 <p <A,
`which reduces to (12) if 6? = 0% = 6”, so q = 1/2. The
`state transition probabilities in case of Ricean fading
`are found as
`
`(18)
`
`A
`
`M1* Ifric,p(Px)4P1 = Q(/2K, WA,
`“1
`Ino. = JFric.p(P2)4P2 = QC(2K WA
`
`A
`
`(49)
`
`and qq = 1-4) qa, = 1-Go9. Inserting (19) in q17)
`results in the branch selection probabilities Pr{1 sel.}
`= q and Pr{2 sel.} = 1-q which give the following
`expression for the probability p,(t)>p for Ricean faded
`branches with an unequal K factor
`Prip,>P} =
`Q,(P)(1 - Q,(4))q
`+ Q1(P)(1 - O,(4)\(1-4), p 2 0,
`+O,A)q + Q(4)(1-9),0<p <A,
`
`(20)
`
`+ OQ, (P)q + Q2(P)(1-q), p > A,
`
`where Qc) is as in (13).
`
`IV.C Rayleigh faded branches, correlated in space
`Diversity gain decreases if the correlation between the
`signals on both branches increases. The correlation
`factor depends on the distance between both receiving
`antennas, the presence of local scatterers around the
`receiver as well as on the angle of an incoming signal
`relative to the imaginary line connecting the antennas
`[7].
`The space correlation factor p, can be defined as
`- (pp, - {p,)(p,)
`94 92
`
`(21)
`
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`
`
`(24)-(25), they have been solved numerically.
`
`time correlation in a
`
`IV.D Rayleigh faded branches,
`switched diversity scheme
`If only the base station in a DECT system is equipped
`with more than one antenna,
`the handheld can still
`profit from this antenna diversity if the base station
`uses the same antenna for downlink uansmission as it
`used for receiving on the uplink, based on the as-
`sumption of
`reciprocity of
`the
`channel.
`In
`a
`TDMA/TDD system such as DECT however, there is a
`time gap of half a framelength between the up- and
`downlink time slots. During this time the channel
`characteristics change, and the correlation between
`these characteristics at the time of the uplink and the
`downlink slot determines the effectiveness of diversity
`on the downlink.
`in suc-
`the signal
`Until now,
`it was assumed that
`ceeding time samples has a correlation factor of 1,
`which results in optimum diversity. In this section we
`look at the effect of a time correlation less than 1. The
`value of the stochastic variable p, at
`instant t, is
`denoted by either p,(k) or p,,(t).
`If we
`introduce the time correlation factor p,,
`defined in a similar fashion as p, as
`
`o, = (p(k)p{k-1)) - (pfk))?
`t
`o
`
`(26)
`
`we can state that diversity gain is highest
`if channel
`characteristics have not changed atall, ie. <Pi(K)Pi(K;
`1)> = <p,(k)”> so <p,(k)p,(k-1)> - <p,(k)>* = o
`and p, = 1. At the other extreme,if p, = 0, succeeding
`time samples are completely uncorrelated and all diver-
`sity gain is lost. The value of p, is determined by two
`factors:
`the time between the succeeding samples k-1
`and k (time between up- and downlink slots) T, and
`the fading frequency or Doppler shift, respectively. In
`[3], the correlation factor for two Rayleigh faded sig-
`nals is given as
`
`p, = Jo(2afpT)
`
`(27)
`
`where fp = 2xf.v/c is the maximum Dopplershift, f, is
`the carrier frequency and v the receiver speed. For a
`DECT system, T is the time between the end of the
`uplink and the beginning of the downlink slot, which is
`approximately 5 ms (half the framelength). Some values
`of p, for different speeds v and T = 5 ms are given in
`table 1.
`
`Table 1 Values of the correlation factor p, for different receiver
`speeds v and corresponding Doppler shifts fp.
`
`Vv (m/s)
`
`fp (Hz)
`
`Pr
`
`0.5
`1.0
`1.5
`2.0
`
`0.91
`= 20
`0.65
`= 40
`0.30
`= 60
`
`= 80 0.05
`
`This table shows
`
`that p, decreases rapidly for an
`
`where <.> denotes statistical averaging. If py and p,
`are independent, <p,p,> = <p,><p> so p, = 0,ie.
`Pp, and pz are uncorrelated and diversity gain will be
`highest. For p, = 1, ie. correlated branches, all diver-
`sity gain is lost. Branches are assumed to have equal
`statistics in this section.
`We start with the joint PDF of two Rayleigh
`distributed random variables a, and a,, which is given
`[8] as
`
`Frayaya
`
`aa |p,|@,4
`(4142) = ———-Ip
`sce) a e)%)
`_ 2
`_
`2
`1742
`474
`a,+a
`.
`-
`1
`2 = » 4; 2 0,
`2(1-pp)o
`which reduces to the product of two separate Rayleigh
`PDF's if p, = 0. Transforming (22) with pj = a;°/2
`(i=0,1) results in the joint PDF for the power of a,
`and ay
`
`ay
`
`Ray,ppp,
`
`1’
`
`4o((1-aa-w° a-palo
`
`(23)
`
`‘exp|
`
`P,* P2
`-——__——_ |, p; 20.
`(1-p5)0”
`Given this joint power PDF, we determine the CDF’s
`of powerafter diversity p,(t) for the different diversity
`techniques. Using the fact that p, and p, in (23) are
`interchangeable, the CDF for selection diversity for cor-
`related branches is given by
`oP;
`
`FRay,p,P)
`
`1-{ JVeospse(P}.P2)4p24p,
`
`wP,
`* j fFray,ppfPvP2)P14P2 }
`po
`
`(24)
`
`P f
`
`0
`
`u ou,9
`
`fRay,p,p}P1P2)apzdp,-
`
`For switched diversity, (11) is changed to
`Fray,pP)=
`
`pA
`SSPeay.pyp,Pr-P2)4P142» 0 <p <A,
`00
`
`(25)
`
`PP
`J[tray,p,PrP2)4P14P2
`00
`‘f[nou(PpP2)4p,dp2,p >A.
`No etosed expression has been found for equations
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`
`increasing speed. Normal walking speed is about 1 mjs,
`a value for which p, has already decreased to 0.65.
`If we assume that p, and p, are independent and
`have an identical distribution, we can still use (10) to
`determine the CDF of p, for switched diversity. To
`solve (10), we need to know the joint PDFof p,(k) and
`p,(k-1). This PDF is equal to (23), with p, replaced by
`P, and p, and p, replaced by p,(k) and p,(k-1), respec-
`tively. Since we assume that p, and p, are independent,
`the CDFofp, is given by
`
`Fray,pel?) =1 -JfFray,PyPyPEP-)PuUPrzer
`pa
`A
`=
`
`~ |Fray.p,(Pu)Pr[rap{P21MPogs
`0
`Pp
`
`)
`
`where fRay,p pu) is similar to (23) and fRay.pia(-)
`are similar to (1). The first integral in (28) has been
`solved numerically, the last
`two can simply be solved
`analytically.
`
`V. Computationalresults
`
`Using the expressions derived in the previous sections,
`some results are calculated. All figures show the CDF
`of power after diversity p, as a performance measure.
`Fig. 4 shows the optimum diversity, Rayleigh fading
`
`therefore
`diversity gain with switched diversity is
`achieved near the switching threshold A. Note that all
`’switch-points’ (where p, = A) coincide with the selec-
`tion diversity curve. Fig. 4 also shows that for selection
`diversity, diversity gain increases monotonously for
`decreasing probabilities. For switched diversity this is
`not
`the case. Diversity gain increases for decreasing
`probabilities until
`the threshold is reached and gain
`Settles at a constant value. The higher the threshold A
`is set, the faster the increase in gain and the lower the
`final constant gain becomes.
`Fig. 5 shows the diversity curves for Ricean fading,
`for three different values of the Rice factor K. Logical-
`ly,
`increasing K decreases the probability that p,
`is
`below a certain power level. Note that here also the
`curves for switched diversity coincide with those for
`selection diversity at the threshold level. The influence
`of K on diversity gain may not be obvious from the
`figure. For selection diversity, increasing K results in a
`decreasing gain for the probabilities of interest (down
`to approx. 1E-4). If the probability decreases further,
`the situation reverses
`and gain increases
`for an
`increasing K. For switched diversity,
`two situations
`exist. For high probabilities where neither of the cdf
`curves is below the threshold, gain decreases when K is
`increased. For low probabilities where both cdf curves
`are below the threshold, gain increases when K is
`increased. Between these two situations there is a
`transition region, where one cdf curve has passed the
`threshold while the other has not.
`
`case.
`
`two-branch
`Fig.4 CDF of power after diversity p(t) for
`switched
`diversity and uncorrelated Rayleigh
`faded
`branches with equal average power, for different values of
`the threshold A. Curves are shown for: a) no diversity; b)
`A = 0 dB; c) A = -5 dB; d) A = -10 dB; e) A = -15 dB;
`® selection diversity. For all curves o = 0 dB was taken.
`
`The switched diversity curves consist of two parts, one
`below and one above the threshold A. The part below
`A is parallel to the no-diversity curve. The reason for
`this is that if p, is below the threshold, the power on
`both branches must be below the threshold, so there is
`constant switching between branches.
`In that case,
`behaviour is the same as for no diversity. Above the
`threshold, the curve moves rapidly from the selection
`diversity curve to the no diversity curve. The highest
`
`Fig.5 CDF of powerafter diversity p,(t) for two-branch diver-
`sity and uncorrelated Ricean faded branches with an
`equal Rice factor K. Curves are plotted for no diversity:
`a) K = 0 dB; b) K = 3 dB; c) K = 5 dB;selection diver-
`sity: d) K = 0 dB; e) K = 3 dB; f) K = 5 dB; switched
`diversity: g) K = 3 dB; h) K = 5 dB. For the switched
`diversity curves, A = -5 dB was taken.
`
`Fig. 6 shaws the switched diversity curves for unequally
`powered, Rayleigh faded branches. A large reduction of
`diversity gain can be seen below the threshold A for
`switched diversity. For a probability region where
`several cdf curves are below the threshold, diversity
`gain (in dB’s) is decreased by approximately twice the
`power difference (in dB’s). For a 12 dB power dif-
`ference this means that the performance is even worse
`than for the no diversity case. This is caused by the
`choice made for a? in the calculations. The power on
`
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`one branch is chosen equal to the no-diversity case, the
`power on the other branch is chosen 3, 6 or 12 dB
`
`below this value. a
`
`ss
`
`a
`
`s
`10
`18
`Received Power Po (dB)
`
`Fig. 6 CDF of power after diversity p,(t) for two-branch diver-
`sity and unequally powered, uncorrelated Rayleigh faded
`branches. Curves are plotted for: a) no diversity; switched
`diversity with a power difference between branches of: b)
`12 dB; c) 6 dB; d) 3 dB; e) 0 dB. The threshold value A
`is set at -15 dB.
`
`With switched diversity, not always the best branch is
`chosen (contrary to selection diversity), so the branch
`having the lowest average power of the two has a finite
`chance of being chosen some of the time. Of course
`performance on average will then be worse than for the
`no diversity case.
`Fig. 7 shows the case of unequally powered, Ricean
`
`faded branches.
`
`Fig. 7 CDF of powerafter diversity p,(t) for two-branch diver-
`sity and unequally powered, uncorrelated Ricean faded
`branches. Curves are plotted for: a) no diversity, K = 5
`dB; selection diversity with a difference between K factors
`of: b) 5 dB; c) 2 dB; d) 0 dB; switched diversity with a
`difference between K factors of: e) 5 dB; f) 2 dB; g) 0
`dB.
`
`For a probability of 1E-3, the loss of diversity gain for
`selection diversity is approximately equal to the diffe-
`rence in K factors. For a decreasing probability this
`loss increases slowly since the cdf curves divert. As with
`Rayleigh fading, for switched diversity the loss of diver-
`sity gain strongly depends on whether p,(t\)>A or
`p,(t)<A. Above the threshold loss is almost negligible,
`below the threshold it increases. As an example: for
`
`probabilities where both switched diversity cdf curves
`are belowthe threshold, diversity gain is reduced by
`approx. 4 dB if the difference in K factors increases
`from 0 to 2 dB.
`Fig. 8 shows the case of selection diversity with
`equally powered, Rayleigh faded branches with cor-
`
`related branch signals. 0
`
`=a
`
`-20
`
`4
`10
`ty
`Received Power Me (dB)
`
`o
`
`s
`
`Fig. 8 CDF of powerafter diversity p,(t) for two-branch selec-
`tion diversity and correlated Rayleigh faded branches.
`Curves are plotted for a correlation factor p, of: a) 1 (no
`diversity); b) 0.95; ¢) 0.7; d) 0.5; e) 0 (uncorrelated,
`optimum diversity gain).
`
`From this plot it is apparent that even for a very high
`correlation factor (p, = 0.95) diversity gain is still sig-
`nificant. The loss of diversity pain decreases rapidly for
`a decreasing correlation factor. Take p, = 0.7 as an
`example, for this value the loss of diversity gain for a
`large range of probabilities is in the order of only 1 dB.
`The loss for even lower correlation factors is negligible.
`
` 2
`
`=o
`
`<«
`10
`18
`Received Power Po (48)
`
`Qo
`
`s
`
`Fig. 9 Cumulative density functions of power after diversity
`Po(t) for two-branch switched diversity and correlated
`Rayleigh faded branches. Curves are plotted for a cor-
`relation factor p, of: a) 1 (no diversity); b) 0.95; ¢} 0.7;
`d) 0.5; e) 0.3; ) © (uncorrelated, optimum diversity gain).
`Forall curves A = -10 dB was taken.
`
`Fig. 9 shows a similar behaviour for switched diver-
`sity. For p,>A, almost no diversity gain is lost up to p,
`~ 0.7. For py<A diversity gain is decreased more than
`for selection diversity, but
`loss is still not very sig-
`nificant up to high values of p,. For p, = 0.7, diversity
`gain.is decreased by approximately 2.5 dB.
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` “ao
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`26
`
`2
`
`<s
`10
`AS
`Received Powsar Po (dB)
`
`eo
`
`&
`
`the cdf curve above the threshold little diversity gain is
`lost, for the part below the threshold approximately
`twice the difference in power level between branches is
`lost (in dB).
`Antenna correlation does not degrade performance
`of any of the diversity techniques very much. Up to an
`antenna correlation factor of 0.7, the loss of diversity
`gain is almost negligible (less than 2 dB for selection
`diversity and switched diversity). Above this value, the
`loss increases only very slowly.
`is greatly
`With switched diversity, performance
`reduced if the correlation between succeeding time
`samples is not 1. For example: for a very high cor-
`relation factor of 0.95 diversity gain for a large range of
`probabilities is
`reduced from 10 dB to 3 dB. For
`walking speeds, the correlation has dropped to a level
`where practically no diversity gain is left. This means
`that the examined switched diversity scheme does not
`provide any gain if the receiver is moving.
`
`Acknowledgements
`
`The authors would like to thank Ir. Kiwi Smit and
`Prof. Dr. R. Prasad for the fruitful discussions and their
`contributions to the work presented in this paper.
`
`References
`
`{1] ETS! Technical Report, "Radio Equipment and
`Systems Digital European Cordless Telecom-
`munications (DECT) Reference document", ETR
`O15, March 1991.
`
`[2] RJ. Kopmeiners, “Selection and Switching Diver-
`sity
`in TDMA Systems
`like DECT", Cost
`231TD(93) 108, Limerick, September 1992.
`
`[3] W.C. Jakes, "Microwave Mobile Communications’,
`New York: John Wiley & Sons, 1974.
`
`{(4] AJ. Rustako, Y.S. Yeh and R.R. Murray, "Perfo-
`rmance of Feedback and Switch Space Diversity
`900 MHz FM Mobile Radio Systems with Rayleigh
`Fading", EEE Trans. on Commun., Vol. COM-21,
`No. 11, pp. 1257-1268, November 1973.
`
`[5] A. Papaulis, "Probability, Random Variables and
`Stochastic Processes", McGraw-Hill, 1991.
`
`[6]
`
`"Digital Communications", 24
`Proakis,
`J.G.
`edition, McGraw-Hill, Singapore, 1989.
`
`[7] W.C.-Y. Lee, "Effects on Correlation Between Two
`Mobile Radio Base-Station Antennas",
`JEEE
`Trans. on Commun., Vol. COM-21, No. 11, pp.
`1214-1224, November 1973,
`
`i8] K. Suwa and Y. Kondo, “Transmitter Diversity
`Characteristics
`in Microcellular TDMA/TDD
`Mobile Radio", IEEE proceedings, Boston, 1992.
`
`
`
`Finally, fig. 10 shows the performance of switched
`diversity with Rayleigh faded branches,
`for different
`values of the time correlation factor p,.
`
`two-branch
`for
`Fig. 10 CDF of power after diversity p,(t)
`switched
`diversity
`and
`uncorrelated Rayleigh
`faded
`branches with equal average power. Curves are drawn for
`different values of the time correlation p, between succes-
`sive samples k and k-1: a) p, = 0 (uncorrelated, no diver-
`sity gain); b) 0.65; ¢) 0.9; d) 0.95; ¢) 1 (correlated).
`
`This figure shows the detrimental effect of incomplete
`time correlation. Even for a high correlation factor of
`0.95 diversity gain is reduced considerably. For p, =
`0.65 almost all diversity gain is lost. This indicates that
`the switched diversity scheme, based on instantaneous
`received power, as it is examined here probably will not
`function very well when the receiver is moving around.
`
`VI. Conclusions
`
`As was to be expected, switched diversity performance
`is always inferior to selection diversity performance.
`The smallest gain difference is found for average power
`levels that are near the switching threshold. An op-
`timum switching threshold in general cannot be given,
`this depends on the exact channelstatistics. Diversity
`gain for switched diversity is largely influenced by the
`choice of the threshold value. If the threshold value is
`very low, the probability that power on a chosen branch
`is above the threshold increases, which means that the
`probability of a switch decreases. The behaviour of
`powerafter diversity will then increasingly resemble the
`behaviour for no diversity. If power on both branches is
`below the threshold, diversity gain becomes a constant.
`In case of Rice fading, the Rice factor K has a non-
`linear influence on diversity gain. In the CDFplots this
`means that for an increasing K factor, diversity gain
`decreases for high probabilities but
`increases for low
`probabilities. This holds for both diversity techniques.
`Different branch statistics degrade diversity perfor-
`mance for both diversity techniques. For selection
`diversity the reduction of diversity gain for Rayleigh
`fading is approximately equal
`to the difference in
`average power between branches. For switched diver-
`sity, performance can even get worse than for no diver-
`sity due to the fact that there is always a finite pos-
`sibility that the "bad’ branch is selected. For the part of
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