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203Capacity Evaluation of Cellular CDMA
Q an g Wan gDepartment of Electrical and Com pute r Engineering,
IJnilTersity of Victoria
Victoria, B.C. , Canada V8W 3P6andJ . Gregory AcresM P R
Teltech. L t d . S999 Nelson Way, Burnaby, B.C., Canada V5A 4B5
ABSTRACTTh is paper is concerned with th e capacity of a
cellular codedivision multiple access (CDMA) system i n Rayleigh
fadingwith log-normal shadowing. Th e capacit ,y in terms of
thenumber of users per cell per Hz is shown to be very sensitiveto
assumed values of various parameters. CDMA capacit .yis shown to be
possib ly g reater than the p roposed TD MAcapacity unde r certain
assumpt>ions and vice versa undero thers . An asymmetry between
the up l ink and downl inkcapacit ies in CDMA is shown based on the
finding that t .heuplink intracell interference power is only hal f
of t h a t i n thedown 1 n k
1 IntroductionRecently, there ha s been renewed interest in comm
ercial ap-plic,ations of spread spectrum (SS) communicat ions .
Twotypical examples are QUALCOMM's proposal for digitalcellular
telephony [l] an d SCS- Mill iconi 's proposal for per-sonal
comniunication networks (PCN) [2]. I n both propos-als: DS-CDMA has
been proposed i n conjunction with acellular structure. In th is
paper , we es t imate the capaci tyof a cel lu lar CDMA system.A
log-normal Rayleigh fading channel is assumed here.Unlike [5],
however, we assum e t h at , in a digital CDMA cel-lular system
employing SS diversity, error correction codingand interleaving,
most of the Rayleigh fading can be com-pensated for and the
remaining effect will be reflected inthe f inal average decoded
BER. The log-normal var iat ion(shadowing) is the cause for
outage.
2 Downlink Capacity2.1 Capacity in Rayleigh FadingThi s section
presents results on th e capa city of the downlinkof a CDMA
cellular system where coherent coded BPSK isassumed. T he key
performance parameter is SN R defined as2A2/No, where 2A2 is the m
ean of the ch i -square d is t r ibu tedsignal level at the sa mpli
ng instan t of the matc hed fi l terbefore conibining in the
decoder which is assumed to be
'This research w as performed for M P R Teltech, Ltd., under
Con-tract MPR-H24 -A260 ( 9 1 / 0 4 )
a Viterbi decoder for a convolutional code, and No is one-sided
spec tral density of AWGN . Values of 2A2/N0 are takenfrom [3]
where BER performance of a coded system i n acorrelative fading
channel is studied in detail .As usual ly done wi th a CD MA system
(see, e .g . , [5 ] ) ,we as su me t h a t i t is interference l
imited so t h a t t h e sys-tem thermal noise is negligible. In
this case, No is due tomultiple users only. Here we have implied
that multiple ac-cess interference can be modeled as AWGN which has
beencarefully addressed in, e. g. , [6 , 71.In D S/BPSI< over a
fading channel, the received signalIS
s ( t ) =m d j c o ( t ) o s ( 2 s f c f ) , ( j - )T35 t
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aoBcli is uniformly distributed i n [ 0 , T,). ot,e t.1iat i n
th edownlink of a cel lular system, C; or E,, is th e s a n ~ eor
allusers withiri t h e s a m e cel l for a given j . T h e n ,
based 011the random sequence assumption, the variance of the
inter-ference component at the matched f i l ter output is given
by[31
[ A4 = 4 3 4Pb + ( d B ) r =0 4332"
10-2 20 I 2910 - 3 23 25 I I 7
The above result, is different from t.he corresponding resultin
[12] which has &+& n place of A. he resul t in[ la ] is
found to be inco rrect. Specifically, th e lower limit forthe
second integrat ion in the first, equation in Equation ( 1 2 )in
[12] should be -y, no t y.Since decoding is a combining process in
regard t o boththe signal and the interference: the combined effect
of in-terference can be characterized approximately by the aver-age
variance with respect to the chi-square fading distr ibu-t ion.
This approximation becomes accurate for calculat ingthe pairwise
error probabil i ty corresponding to a large pair-wise distance
between the correct codeword and an incor-rect codeword which means
a t ime average over m any codesymbols. For exam ple, a pairwise
distance for the opti mu mconst,raint length 7 rate l / 2 code is
at least, 10 which is con-sidered to be sufficiently large. From
Equa t i ons 1 and 2 ,respective average variances are given by
M =434 A t = 1024 M = 1024r =2 4 1 0 r =0 433 r = 2 4 1 0
16 6 1 3 33 2 111
a nd
Pb10-210-3
- 1 1U; = - E ( q j ) = -8A43A f 3A4
M =434 I h4 =43 4 A4 = 1024 M = 10247 =2 4 1 0* ( , E ) r =0 133
1 r =2 4 1 0 r =0 433
6 6 7 500 I 1 7 6 1 1 7 2 6478 5 318 I 18 2 7 7 1 425
( 3 )
- -Com par ing Equat ions 3 an d 3 and let t ing U; =r i , we
haveequivalent(5)
Note th at the interferelice spectral denszty 211 the aboue
equa-tzon doubles that used zn [5], which will significantly
affectthe system capaci ty as will be seen later . This factor of
2is due to the fact that we are interested in a coded systemwhere
th e MR C principle is used in decoding. In an uncodedsys tem, a
symbol by symbol decision can be made with thesam e error probabil
i ty ei ther with the MRC matched fil-ter operat ion as used here
or mu lt iplying the incoming sig-nal with co(t)cos(27rfct) (note
the omission of the factorm) efore the integrat ion. The lat ter
operat ion resul tsin No =&4A2 which is used in [5].If there
are L O active interfering users within the samecell , then the
equivalent No in the above equation is in-creased by L O t imes .
Suppose tha t there a re A equallyloaded interfering cells . Note t
ha t th e interference from anadjacent cel l arr ives through a
fading path independent ofthe signal path. Then the corresponding
contribution to NOis E:==A: where 2A2 represents the power of
theinterference to the reference user from the k-th
interferingcell. In th e presence of Rayleigh fading only (no
log-normalfading), the average signal power an d th e interference
powerfrom th e k-th interfering cell at the reference user a re
pro-port ional to the l / r z a nd l / r z , respect ively, and
therefore,
where y i s the propagat ion exponent , and T O a n d rk a r e
thedistances from the reference user to its own cell-site andk-th
interfering cell-site, respectively.Taking into account both
intracel l and intercel l intrr-ference. we have the SNR as
follows.
where1c 10,L o = - -
" C C t
with L , to be the number of users per cell. lbrl to be th
evoice act ivity factor ( th e percentage of the t im e when a
useris ac tual ly t ransm i t t ing voice s ign al ) , and A',,ci
to be thenumber of sectors per cel l .Let r =E::($)-'. Then. f rom
Equat ion 6 , we liavt.
For any given position in a cell, we can find the corre-s pond i
ng r value.Suppose we have an infini te cel lular array, and a
userassumes any locat ion in a cel l with a n equal chance. A s s u
mey =4 . Th en, t he average va lue of r , f , can be found to bef
=0.433 a nd t he m a x i m um va l ue r = r,,, = 2 . 4 10
whichcorresponds to the corner at th e cell edge tha t is the
farthestfrom t he cell center .By t h e use of Equat ion 8 , we can
obta in the numberof users corresponding to r = f a n d r = rm = 2
. 410 , re-spect ively. T h e resul ts are shown in Tables 1 and 2
whichcorrespond to two syste ms employing two different
inter-leaving sizes in conjunction with the optimum
constraintlength 7 ra te 1 /2 c ode . fm denotes the maximum
Dopplerfrequency. Here we assume Von =0 .35 a nd h',,,, =3.From
Tables 1 a n d 2 , we can see tha t in order to obtainthe error
probabil i ty Pb = for the average case of r =
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F* U T1 0 - 1 1 4 1 5
5 x 1 0 - 2 I635
Table 4 : Th e numb er o f users per cell (L c ) or a given
outag eprobability fo r in ter leav ing dep th=60 , span=40 and
fmTs=0.001
# ( d B ) 1 A4 =I 3 4 A4 = 31 2 A r =1 0 2 420 1 21 24 4 320 I
20 22 $0
1 0 - 2
0.443, we have 17 and 318 users per cell for each of
twointerleaving sizes and M = 434 which is a v a l u e used in[5].
1Iit.h these two user numbers per cc l l , if we requirePb <
lo-: Iiy Equat, ion 7 ! we have r=2.29 a n d 1 ~ 1 . 8 3which m
eans th at o u tage p robab i li t ies are 0.91%) nd
2.73%,,respectively. Conversely. for a given outage probability ,
wecan obtain the corresponding number of users as show11 i nTables
3 and 4 assuming the ou tage corrcsponds to Pb >10-2 .From the
abo ve results, we have following observations.
2 4 0 0 [ 2 0 1 16 1 8 33
e
e
e
2.2
Po( r 1 w ( d B ) A4 =43 4 I A< =51 2 A< =1 0 2 41 0 - 1 1
4 1 5 I 6 6 1 35 6 I 4 1 9 0 3 4
5 X 1 6 3 5 I 6 67 3 3 s I 3 9 4 7 8 4, 1 0 - 2 I 2 400 1 6 67 \
I 27 6 I 3 2 5 6 4 6
Cod ing and int,erleaving can dra stically affect the sys-tem
capacit.. . For exam ple, fo r any given s and requiredpb = the
number of users per cell is increased byabout 24 times when the
interleaving size is increasedf ro m (d ep t~ h , p an ) = (20 ,
20) t o (GO . 4 0 ) .Th e user capaci ty is proport ional t o the
number o f ch ipsper coded symbol , M which is in turn proportional
tothe processing gain of a C D M A syst , em. I n fact . whenL
O>> 1 , L O sz L O+1, t h e 2 A 2 / N o in Equation 6 is
afunction of L c / M only. In th is case, as M increases,th e numb
er of users per cell will be increased by thesam e facto r . Th is
observat ion is t rue for all capacitiespresen ted in th is repor t
.Under comparable conditions, our results show signifi-cantly fewer
users per cell than those i n [5] d u e to t h edoubly increased
intracell interference spectral densityas discussed earlier.
Capacity in Log-normal Rayleigh Fad-ing
We now presen t the capaci ty resu l ts fo r the downl ~nkof
acellular system in t he presence of correlative Rayleigh fadingand
log-normal shadowing .Now, the quan t i ty E { 2 A : } is a
log-normal randomvariable. Specifically, XO= I O l o g ~ o E { 2 ,
4 ~ }s normallyd is t r ibu ted wi th mean m, = 10log10($)? and
variance g zwhere r-0 is the distance between the user and the
desiredcell-site. yk =1 0 [ 0 g l o E { 2 ~ 4 ~ }s normally
distributed with
T ab le 5 .d ep th =2 0 , sp an =2 0 an d f,,T.- =0 .001 ,
assuming s tanda rddeviation of log-normal fading is 1 d BT h e N u
m b er of users per cell for i n t e r l eav i1 9 5
mean m y , = lOloglo(+-)7 and variance where 7 k is thedistance
between the user and the k-th adjacent interfer-ing cell-site. Y =
lologlo E(2A:) is approximate lynormally distributed with mean my
and var iance U; [13].
K
We have the SNR3 A 2 1
Using a method in [13]. we can find t he me an and th evariance
of 1 . . Then, we have th e probability density f u n c -t i011 of
2,42/A: as follows:
1 0f 2 A 2 / . \ I O ( t ) = ( l n 1 0 ) 2 ( -*,JW
We assume that each user in a cell is uniformly dis-tributed. At
any point of the desired cell , the outage prob-ability (assuming
is the highest, tolerable Pb)
pout(pb 2 o-?) = fzA2/N0(t)t (11)1where b is t .he SNR
corresponding to Pb = T h e a v -erage outage probability (Pout )
ver the desired cell can beob tained as follows. First we divide
the entire area ( S )ofthe cel l into many smal l areas (dS) and
accum ulate the out ,-age probability of these smal l areas . Then
we average thesum over the en t i re area. That is,
(1 2 )1pout =3 p o u i ( p b 2 10-)dS.Using the BER results
presented in [3], and Equation12 for the outage probability , we
can calculate the numberof users per cell as shown in Table 5 an d
T ab le 6 for a: =T h e results in Table 6 look quite favourable
providedth e sufficient interleaving is avail able . However, if we
as-
sume th e s tand ard dev iat ion o f the log-normal fad ing to
be8 d B as suggested in [I], i . e . , U, = u y , = 8 d B , then
thecapacity will be drastically reduced as shown i n T ab le 7
.
U& =1
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%!?le 6: T h e n u m b er o f us(rs per ce l l for
interleavingdcp,th=GO. s p a n = 4 0 and f , , lT .=0 .001 .
assuming s tan darddeviation of log-normal fadiiig is 1 dF3
T ab le 8: The number of users per cell with 3-cell-site
diver-sity for interleaving deptll=20, span=20 and f,,,T;
=0.001.assuming s tandard dev iat ion of log-normal fading is I d
BPoutl o - l o - ?
F ( d B ) A4 =434 A { =512 A4 =1024 Pout g ( d B ) 124 = 4 3 4 M
=51 2 M =10246.67 3.58 422 839 lo- 20 22 25 456.67 282 332 66 0
10-2 20 17 20 37
Table 7 : T h e n u m b er o f users pe r cell for
interleavingdepth=GO, span=40 and fmTs=0.001, assuming s
tandarddeviation of log-normal fading is 8 d BTab le 9: Th e number
of users per cell with 3-cell-site diver-s i ty fo r inter leav ing
dep th=60 , span=40 and fmTs=0.001.assuming st anda rd deviat, ion
of log-normal fading is 1 d B
Poutlo-IO-
s ( d B ) M =434 .b= 51 2 M =1024 Pour s ( d B ) M =43 4 M = 5 1
2 M =1024 6 .67 68 80 16 0 l o - 6.67 370 435 8636 .67 6 8 16 lo- 6
.67 29 5 347 6902.3 Capacity in Log-normal Rayleigh Fad- From
Eq.14, we have.I n the d o ~ rn l in k o f a CDlZIX cel lular
system , the 2A2cell-site diversity is usual ly employed to improve
t,hesignal quality to a mobile at th e vdge of a cell.In this case,
a mobile receivcs I i d 2 , copies of the We assume th at the
location of a user is unifor~iilysame s ignal f rom ]
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10-1 I 20 II 11 12 23
Table 11: The number of users per cell with 3-cell-site diver-s
i ty for inte r leaving depth=60, span=40 and f,,T, =0.001assuming
standard deviat ion of log-normal fading is 8 d BPout I g ( d B ) I
A4 =434 I M =51 2 I M =1024lo- ' I 6.67 11 180 I 21 1 I 420
10-2 I 20 11 2 3 5
3 Uplink Capacity
lo-' I 6.67 I ] 40
We assume tha t there 1s ideal power control i n the uplinkso
that average arr ival signal powers are identical f r o n i
allusers within a cel l at the c~l l -si te
47 95
3.1 Capac ity in Rayleigh FadingTh is subsect ion presents th e
capaci ty for the uplink of cel-lular system in the presence of
correlative Rayleigh fading.We use the same terminology as th e
last section with t,heund erstan ding tha t now the receiver is at
a cell-site andinterference is from oth er mobiles.S a m e as the
downlink case, now the BDPSIi deniod-ulat ion operat ion consists
of mult iplying s ( t ) plus signalsfrom other users by s ( t )
(recall ou r assu mpt ion that , adja -cent symbols are so correlat
,ed that they correspond to (.hesame fading ampl i tude) and in
tegra t ing the product f roni0 t o Ts in t ime. Th e signal
component a t the demodula-tor output i s d,E,,, where E, , i s the
j - th symbol energy(PjT,)which, as mentioned earl ier , is a
chi-squared randomvariable with a mea n 2A 2. If we had AWGN with
one-sidedspectral densi ty N O n place of the interfering signals
fromother users, we would have the variance of the noi sy
con-ponent a t the demodula tor output g iven by
which is the same as Equa t i on 1 . Similarly, suppose t
.hat.we have only one interfering user in the sa me cell employ -in
g a spreading waveform q ( t ) a n d a carrier cos(27~f,t+@),where
I$ is uniformly dis tributed in [0, 27r). Suppose thechip
bo:indaries in tim e of c l ( t ) is offset by T which is
uni-formly distr ibuted in [0, Tc) .Note that in the uplink of acel
lular system, the interfering mobile has power Pi or en -ergy E i j
for the j - th symbol which is independent of thatfor the reference
user sim ply because signals from two mo-biles go through
independent , propagation paths to arr iveat th e cell-si te. We
assume tha t the uplink employs ideal
power control Then, the inean of P, or energy E:, S equ&to
tha t of P, or energ) E,,Th en, based on the ran dom sequence
assumpt ion, t h evariance of the interference component at the
demodulatoro u t p u t IS given by [3]
As argued ear l i e r , f rom Equ at ions 18 an d 1 9, we
canhave respective average variances given by
a nd
Compar ing Equat ions 20 and 21 and le t t ing a; = 0 2 , wehave
equivalentNote t ha t . c om pa r i ng Equa t i ons 21 with 4 or
Equations22 with 5 , t h c ziitracell zi i t fr ference p o w e r
zii th e uplink 1sh a l f t h a t i n Ihc d o w n l i n k . T h i s
is due to the independencebetween pat hs from different mobiles to
the cell-si te.If t,here are Lo act.ive interfering users within
t,he samecell . then the equivalent .Yo i n t,he above equation is
in-creased by Lo t i m e s . Suppos e t ha t all of K interfering
cellsare equally loaded. Note that interference from a mobile i nan
adjacent cel l arr ives through a fading path independentof the
signal pat ,h. The mobile in an adjacent interferingcell is power
controlled by i ts own cell-site. Th en th e cor-responding
intercel l cont ,r ibut ion to No is '22% r='=,Aiwhere 2 A i
represents th e power of a user (assuming all usersin a cel l are
located with an identical distr ibut ion) in theI;-th intercell
interfering cell received by the desired cell sitewhich is 2A2(% )'
in the presence of Rayleigh fading only(no log-normal f ading) ,
where rk a nd TO are the distancesfrom the user of the k-th
intercel l interfering cel l to i ts owncell-site and the desired
cell-sit.e, respectively.Taking into account , both intracel l and
intercel l inter-ference, we have the S NR equivalent to th at used
before andgiven as follows,
Note t ha t a cel l-site ante nna on ly receives interference
froml/iVsect of the total number of adjacent cel ls . That is ,
eachcel l-si te antenna only covers 1 out of NSeel ell sector s
Itonly receives th e interference from those adjac ent cel ls
facingthe sec tor covered by the antenn a pa t t e rn . This fac t
isref lec ted in the up per l imi t of the su mm at ion in the
aboveequat ion.Let r =C,"If(2)7.rom Equation 23, we haver o
We assum e th at users are uniformly distr ibuted in a cel la nd
o t he r pa r a m e t e r s a r e t he s a m e as for the downl
ink. We
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%%le 12 T hc numb er of users per cell for uplink I I I
Kayleiglifading w i t h different la1 BPSK and interlea ving J
epth=20,span=20 a n d fm7;=0 001
I Pb 1 * (do) 11 Ai=434 1 A t = 51 2 I A1 = 1024 1lo-?10-3
i 3 . I! I25.8 11 20 23 3829 II 13 15 2 2
Ta b l e 13 : The num be r of users pe r cell for uplink i n
Rayleiglifading with differential BPSK and interleaving depth= 60.s
p a n = 4 0 a n d fmTs=0.001
Po,* I s ( d I ? ) / M =43 4 I M =512lo- I 11.4 1) 345 I 40 6t
10-3 I 13.1 11 25 1 I 29 5 I 53 3 I M =102480 5
have found considering 3 t iers of adjacent cells is
sufficientlyaccurate wliich gives T =0.04Tables 12 and 13 show
capaci ty re sulh .
10 -
4 Log-normal ShadowingNow the quan t i ty E(2A;) i s a
log-normal randoni var iable .Specifically. X O = l O l o g l ~ E (
2 A ~ )s normally distr ibutedwi th mean 772, =1010glo(2A2) (du e
to the ideal power con-t .rol assumed) and variance .:.k =
10/ogloE{2AAi}snormal ly d i s t r ibuted wi th mean my, = I O l
og~~( 2 .4 ( )> )and var iance U; , .
is approximately norinal lydi s t r ibuted wi th mean mt+/ nd
variance U& [5]. J1.e haveth e SN R as follows,
w = lo logloE:==
1.0 .25.8 3 3 625.8 0 0 1
1W
A. --N o Lo& +L,Von&10=Me then have the probability
density function of
2A as follows:
where
We assume tha t users a re uni formly di s t r ibuted in ace ll
. Th en , we f ind the average va lue of the mean and thevariance
of wk = lologlo E { A Z ) corresponding to the userin the k-th
intercell interfering cell which are then used t,ode termine mw a n
d uh us ing the meth od in [13].
Table 14: T h e n u m b er of users per cell fo r the uplink w i
t hdifferential BPS K a nd int .er leaving dep th= 20 , span =20 a
ndfm7;= 0.001, assuming s tandard devia t ion of log-normalfading
is 1 d B
Table 15: T he n um ber of users per cell for the uplink withdi
fferentia l BPSK and in ter leaving depth =60 , sp an=4 0
andfmTs=0.001, assuming s tandard devia t ion of log-normalfading
is 1 dB
The outage probabi l i ty
Tables 14 a n d 1 5 showd B . capaci ty resul ts U* =01; = 1Mhen
the standard deviat ion of the log-normal fadingis increased to 8
dB , then th e capaci ty i s dras t i ca l ly reducedas shown in
Tables 16 and 17.
5 Comparison with FDMA andTDMA
Present analog cel lular mobile system uses 30 k H z FMchannels
with frequency division mult iple access (FDMA).Wi th 12 0 degrees
antenna sectors, the frequency reuse ef-f iciency is limited to 14%
due to the requirem ent of 17 dBcar r ie r - to- in ter ference ra
t io . Th is l eads to th e capaci ty of0 .14/ (30 x lo3)=4 .67 x
users per cell per Hz or 4.67users per cel l per MHz. In the
current ly proposed f irst gen-era t ion d ig i ta l TDM A sys tem
by TIA , each 30 kHz chaiineli s t ime shared by 3 users. Th us ,
the ca paci ty is increasedby a fact,or of 3 t o 14 x users per
cell per Hz or 14
Table 16: T he nu mber of users per cel l for the uplink w ithdi
fferentia l BPSK and in terleaving depth =20 , spa n=20 andfmTs =
0.001, assuming s tandard devia t ion of log-normalfading is 8 d
B
I Pout g ( d B ) 1 M =43 4 I M =51 2 1 M =1024 1
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2h0113 d l 3 bc i k r i n SN H ( I i a i i ( , l ie downlink.
Cotiibiiittigthese t.wo fac tors , we see t ha t th e uplink is
about 1 to 3 d~worsc i n SNR than the downl ink. This difference
suggeststhat the use of a more powerful code for the uplink
thant.liat used for the downlink would he required to assure ai
lequal capaci ty.
1121 C; I, . r l i i r i i i . T h e effects of mul t ipa th
atid fading otithe per formance of di rec t - sequence CDhlA sys
tems ,I E E E Jourital o n Selecled Areas zn Com niunzca tzons,vol.
SAC-2, No.4, pp. 597-603, Ju ly , 1984.[13] S . C. Scliwartz and Y.
S . Yeh, On the Dis t r ibu-t ion Funct ion and Moments of Power S
ums Wi thLog-Normal Co mponent . s , T he Bel l Sys tem
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