Impact of Shadowing Correlation on
Coverage of Multihop Cellular Systems
Koji Yamamoto, Atsushi Kusuda, and Susumu Yoshida
Graduate School of Informatics, Kyoto University
Yoshida-honmachi, Sakyo-ku, Kyoto, 606-8501 Japan
Email: {kyamamot,kusuda,yoshida}@hanase.kuee.kyoto-u.ac.jp
Abstract— The impact of spatial correlation of shadowing on
the coverage of TDMA multihop cellular systems is investigated in
single-cell environments. The introduction of relaying capabilities
to cellular systems may enhance the cell coverage as follows.
Changing single-hop transmission to multihop transmission reduces per-hop path loss. In addition, since a multihop route
is selected among a number of alternatives, the use of mobile
stations suffering from severe shadowing can be decreased. In
most studies on multihop cellular systems, shadowing is modeled
as a location-independent log-normal random variable; however,
adjacent shadowing values are spatially correlated because shadowing is caused by terrain configuration or obstacles between
the transmitter and receiver. Thus, coverage enhancement due
to relaying capabilities may not be achieved as expected.
In this paper, first, according to the commonality between
multihop transmission and symbol rate control, a similar methodology to formulate the spectral efficiency (SE) and outage
probability of rate-adaptive cellular systems is used to estimate
these performances of multihop cellular systems. Second, we
investigate the impact of decorrelation distance, whose typical
value for the urban environment is 20 meters, on the coverage
of multihop cellular systems. By using a model for spatially
correlated shadowing, numerical results reveal that when the
coverage of single-hop cellular systems is ten times larger than
the decorrelation distance, the introduction of relaying capability
may enhance the coverage without significant degradation due
to the spatial correlation of shadowing.
I. I NTRODUCTION
In recent years, there has been lots of interest in applying
a relaying function to conventional wireless cellular systems,
in which all mobile stations (MS’s) are directly connected
to base stations (BS’s), because of its various advantages
such as power reduction or coverage enhancement. There are
some works related to cellular systems with multihop relaying,
referred to as multihop cellular systems. For example, in [1],
Aggélou et al. proposed to integrate relaying functions with
Global System for Mobile Communications (GSM) cellular
systems. In [2], Wu et al. introduced stations dedicated to
relaying for high bandwidth usage. In [3], opportunity driven
multiple access (ODMA) is proposed to enhance the coverage
by allowing MS’s beyond the reach of the cell coverage to
reach the BS. Moreover in [4]–[6] multihop CDMA (code
division multiple access) cellular systems have been analyzed.
This multihop transmission has the same ability as symbol
rate control to enhance the end-to-end communication range
at the loss of bandwidth efficiency (BE), which is defined to
be the maximum end-to-end bit rate through multiple hops
per unit bandwidth [7]. According to this commonality, the
similar methodology for the performance formulation of rateadaptive cellular systems [8] has been used to formulate the
outage probability and SE, which is defined to be the average
of BE in a given cell, of multihop cellular systems in both
single-cell [9] and multi-cell environments [10].
In most studies on multihop cellular systems, shadowing
is modeled as a location-independent log-normal random
variable. However, adjacent shadowing values are spatially
correlated due to the shadowing process versus distance,
because shadowing is caused by terrain configuration or obstacles between the transmitter and receiver. The shadowing
autocorrelation can be described with sufficient accuracy by
an exponential function [11].
In this paper, we evaluate the impact of shadowing correlation on the coverage of multihop cellular systems by using
spatially correlated shadowing models [11], [12]. Because the
main purpose of this paper is to conduct a tradeoff analysis
between multihop cellular systems under correlated shadowing
and that under uncorrelated shadowing, TDMA (time division
multiple access) is used for multiple access for the ease of
evaluation. The remainder of this paper is organized as follows.
In Section II, we illustrate the system model of TDMA cellular
systems and correlated shadowing models. In Section III, we
describe the formulation of SE and outage probability in rateadaptive cellular systems as shown in [8]. In Section IV, we
present the formulation of performances in multihop cellular
systems. In Section V, we report the performance of multihop
cellular systems. Section VI concludes the paper with a
summary and some final remarks.
II. S YSTEM M ODEL
We consider a single isolated cell as shown in Figure 1
and assume that signals are multiplexed by the TDMA, i.e.
there is no co-channel interference. We consider the situation
where RS’s are uniformly and independently distributed in
the cell. Let m denote the number of candidates for relaying
station (RS) per cell. All stations can either transmit or receive
at a given time and use omnidirectional antennas with the
same transmit power. We assume that the symbol rate and
the multihop route are determined by the local mean received
carrier-to-noise ratio (CNR). In this case, we use the required
BER and the BE for the route selection criteria.
candidates for relaying station (RS): m
In this paper, we particularly assume the spatially correlated
shadowing. We assume ` different propagation paths, the
distance of path i is ri , and the local mean received CNR
of path i is γi . Let ln Γi denote the mean of ln γi and let
σi denote the standard deviation of ln γi . The joint PDF of
γ1 , γ2 , . . . , γ` is given by
base station (BS)
cell radius of
given system: R
coverage of
single-hop system: R0
calling mobile
station (MS)
Fig. 1. Calling MS’s and candidates for RS are uniformly distributed in a
single isolated cell with radius R.
fr1 ,...,r` (γ1 , . . . , γ` )
(
)
1 T −1
1
exp − Z M Z
=
`
2
∏
√
`
(2π) det(M )
γk
(3)
k=1
where · denotes the transposition, det(·) denotes the determinant, and
[ ( )
( )
( )]
γ1
γ2
γ`
Z T = ln
ln
· · · ln
.
(4)
Γ1
Γ2
Γ`
T
receiver
receiver
∆d
receiver
transmitter
transmitter
transmitter
∆d
(a) Station at one end of a radio link moves.
receiver
transmitter
∆dT
(b) Stations at both ends of a radio link
move.
Fig. 2.
σ1 2
µ2,1

M = .
 ..
µ1,2
σ2 2
..
.
...
...
..
.

µ1,`
µ2,` 

.. 
. 
µ`,1
µ`,2
...
σ` 2
(5)
is a covariance matrix where
∆dR
receiver
transmitter

Location model of transmitters and receivers.
A. Wireless Channel
We take into account the correlated log-normal shadowing
as well as the propagation loss with the path loss exponent α
as follows. First, at locations where the distances to a certain
transmitting station are r and r0 , the mean received CNR’s
Γ (r) and Γ (r0 ) satisfy the following relationship:
( )−α
r
Γ (r) = Γ (r0 )
.
(1)
r0
Next, we assume the log-normal shadowing. Under this
assumption, the probability density function (PDF) of the local
mean received CNR γ at location where the distance to a
certain transmitting station is r is then given by
[
(
)2 ]
1
γ
1
exp − 2 ln
fr (γ) = √
(2)
2σ
Γ (r)
2πσγ
where σ is the standard deviation of ln γ.
µi,j = µj,i = ρi,j σi σj ,
(1 ≤ i < j ≤ `).
(6)
ρi,j is the correlation coefficient between ln γi and ln γj . The
correlation coefficient of CNR’s when MS at one end of a
radio link moves as shown in Figure 2(a) can be written as
follows:
(
)
∆d
ρi,j = exp −
ln 2
(7)
dcor
where ∆d represents the movement of the transmitter or
that of the receiver, and dcor represents the decorrelation
distance [11]. The decorrelation distance is dependent on the
environment; 20 meters for the urban environment and 5
meters for the indoor environment [13].
In contrast, when MS’s at both ends of a peer-to-peer
radio link move as shown in Figure 2(b), the joint correlation
function can be written as follows:
(
)
∆dT + ∆dR
ρi,j = exp −
ln 2
(8)
dcor
where ∆dT and ∆dR represent the movement of the transmitter and receiver, respectively [12].
III. R ATE -A DAPTIVE C ELLULAR S YSTEMS
In this section, we formulate outage probability and SE
of TDMA cellular systems with rate adaptation (hereinafter
referred to as “rate-adaptive cellular systems”) as described in
[8]. When QPSK modulation is used, 1/2k -rate QPSK transmission is equivalently obtained from consecutive transmission
of the 2k identical symbols, and the required CNR can be
decreased by 10(log10 2)k dB, where k represents an integer
of zero or more. Let K ( ≥ 1) denote the upper limit of k
[14]. We assume that the symbol rate is set as high as possible
while maintaining the required BER. We assume that when the
BER for the 1/2K−1 -rate does not satisfy the required BER,
the 1/2K -rate is used.
A. Outage Probability
First, we formulate the outage probability. The outage
probability of rate-adaptive cellular systems is defined as the
location probability that the BER for the minimal rate, i.e. the
1/2K -rate, does not satisfy the required BER in the cell with
certain radius R.
Let βA-k (γ) denote the CNR dependence of BER for 1/2k rate QPSK. The subscript “A” of the BER β denotes “adaptiverate”. When the distance between a calling MS and the BS is
d, the probability that βA-k does not satisfy the required BER
βreq can be expressed as
∫
pA-k (r) =
fr (γ) dγ,
(9)
DA-k
where
DA-k = {γ ; γ > 0 , βA-k (γ) > βreq }.
(10)
k
Note that when the BER for the 1/2 -rate, βA-k , satisfies the
required BER, βA-(k+1) also satisfies the required BER and
therefore, the following relationship holds:
DA-(k+1) ⊂ DA-k .
(11)
Assuming that calling MS’s are uniformly and independently
distributed in the cell, we get the outage probability of rateadaptive cellular systems with radius R as follows:
∫ R
2πr
PA (R) =
p
(r) dr
2 A-K
πR
0
∫ R
2
= 2
r pA-K (r) dr.
(12)
R 0
We assume only full-rate QPSK modulation. Let N ( ≥ 2)
denote the upper limit of the number of hops n. We assume
that a calling MS selects as small number of hops as possible
while maintaining the required BER and N -hop transmission
is used when the end-to-end BER of n-hop transmission does
not satisfy the requirement for all 1 ≤ n ≤ N − 1.
A. Outage Probability
We define the outage probability of multihop cellular systems as the location probability that the end-to-end BER
does not satisfy the required BER in the cell with certain
radius R. Let A0 (r0 , θ0 ) and Am+1 (rm+1 , θm+1 ) denote the
polar coordinates of a calling MS and a BS. In addition, let
A1 (r1 , θ1 ), . . . , Am (rm , θm ) denote the polar coordinates of
RS’s. Let
[
]1/2
ri,j = ri 2 + rj 2 − 2ri rj cos(θi − θj )
(15)
denote the distance between Ai and Aj (0 ≤ i, j ≤ m+1, i 6=
j), and let γi,j denote the local mean received CNR between
Ai and Aj (0 ≤ i, j ≤ m + 1, i 6= j).
We assume regenerative relaying (sometimes referred to as
digital relaying) such that RS’s decode and re-encode before
retransmission. When the BER of every hop is small enough,
the end-to-end BER over multiple hops can be approximated
by the sum of the BER of every hop [7]. The set Dn ⊂
R(m+1)(m+2) of (γ0,1 , . . . , γm+1,m ) such that the end-to-end
BER’s for all routes with n hops does not satisfy the required
BER is expressed as
{
Dn =
(γ0,1 , . . . , γm+1,m );
γi,j > 0,
B. Spectral Efficiency
n
∑
}
∀
βA-0 (γsk ,sk+1 ) > βreq (i, j, sk )
(16)
k=1
Next, we formulate the SE. When the distance between a
calling MS and the BS is r, the average BE can be expressed
as
[
]
K−1
∑ pA-k (d)
2Rsmax Feff
tA (r) =
1−
,
(13)
Bch
2k+1
k=0
where Rsmax is the QPSK symbol rate, Feff is the TDMA
frame efficiency, and Bch is the channel bandwidth. The SE
of rate-adaptive cellular systems can be expressed as follows:
∫ R
2
ηA (R) = 2
r tA (r) dr.
(14)
R 0
IV. M ULTIHOP C ELLULAR S YSTEMS
Multihop relaying enables communications between the BS
and calling MS’s that are relatively far apart. In this section, we
formulate the outage probability and the SE of TDMA cellular
systems with multihop transmission (hereinafter referred to as
“multihop cellular systems”) in a similar way to our derivation
of single-hop rate-adaptive cellular systems in the previous
section.
where sk satisfies 0 ≤ sk ≤ m + 1, sk 6= sk+1 , s1 =
0, sn+1 = m + 1. Therefore, the probability that the endto-end BER’s for any routes with less than n hops do not
satisfy the required BER βreq can be expressed as
pn (r0 , θ0 , . . . , rm , θm )
∫
=
fr0,1 ,...,rm+1,m (γ0,1 , . . . , γm+1,m ) dEn
(17)
En
where
En ≡
n
∩
Dk ,
(18)
dEn ≡ dγ01 · · · dγm+1,m .
(19)
k=1
The set Ui ⊂ R2 of Ai (ri , θi ) within the cell is expressed as
Ui = {(ri , θi ); rm+1 < ri ≤ R + rm+1 , 0 < θi ≤ 2π}. (20)
Under the assumption that calling MS’s are uniformly and
independently distributed in the cell, we get the outage prob-
TABLE I
2.5
Parameters
Values
Path loss exponent α
Shadowing
3.5
Log-normal distribution
σ 0 = 4 dB, 8 dB
Rayleigh fading
10−2
Channel
Required end-to-end BER βreq
ability of multihop cellular systems with radius R as follows:
∫
1
PN (R) =
pN (r0 , θ0 , . . . , rm , θm ) dVm
(πR2 )m+1 Vm
(21)
Spectral efficiency (×RsmaxFeff/Bch)
PARAMETERS USED IN EVALUATIONS .
full-rate,1-hop
2
1.5
{½,full}-rate
{1,2}-hop
1
{¼,½,full}-rate
0.5
0
0
where
Vm ≡
m
∏
Ui ,
(22)
i=0
dVm ≡ dr0 dθ0 · · · drm dθm .
1
R/R0
2
Fig. 3. Cell-radius dependence of spectral efficiency (α = 3.5, σ 0 = 8 dB).
(23)
0.3
B. Spectral Efficiency
# candidates for
relaying station: m=5
Next, the average BE, tN (r0 , θ0 , . . . , rm , θm ) (bps/Hz), can
be expressed as
Outage probability
full−rate,1-hop
{¼,½,full}-rate
0.1
(25)
where p0 = 1. Therefore, we get the cell-radius dependence
of SE of multihop cellular systems, ηN (bps/Hz), as
∫
1
tN (r0 , θ0 , . . . , rm , θm ) dVm .
ηN (R) =
(πR2 )m+1 Vm
(26)
{½,full}-rate
0.2
{1,2}-hop
R0/dcor=1
R0/dcor=10
}
tN (r0 , θ0 , . . . , rm , θm )
[
]
N
−1
∑
pn−1 (pn−1 − pn )
2Rsmax Feff pN −1
+
=
Bch
N
n
n=1
(24)
pn ≡ pn (r0 , θ0 , . . . , rm , θm )
}
R0/dcor=100
0
0
0.5
1
R/R0
1.5
2
V. N UMERICAL R ESULTS
Fig. 4. Cell-radius dependence of outage probability (α = 3.5, σ 0 = 8 dB).
Table I summarizes the parameters we used in our evaluation. We assume σ1 = σ2 = · · · = σ and σ 0 = 10σ/ ln 10
for shadowing. We also assume Rayleigh fading channels and
two-branch maximal ratio combining diversity reception. The
maximum number of hops is assumed to be limited up to two
(N = 2).
Figure 3 shows the cell-radius dependence of the SE of
rate-adaptive systems (14) and that of multihop systems (26)
for α = 3.5 and σ 0 = 8 dB. When the cell radius increases,
the SE decreases because these systems have a tendency
to admit MS’s with lower BE. We define the coverage of
cellular systems to be the distance from the BS at which the
outage probabilities PA (R) or PN (R) are equal to a certain
required outage probability Preq . In this figure, R0 is set
to be the coverage of full-rate single-hop cellular systems
associated with the outage probability of 10%, which is shown
in Figure 4. As shown in Figure 3, the SE’s of the half-rate
system (K = 2) and the two-hop system (N = 2) are the
same because of their definitions.
Figure 4 shows the outage probabilities of rate-adaptive
systems (12) and multihop systems (21). We see that with
smaller decorrelation distance dcor , multihop cellular systems
have smaller outage probability. We also see that the outage
probability of multihop cellular systems can be decreased
along with the number of candidates for RS m. Note that
the decorrelation distance dcor and m have no effect on the
outage probability of rate-adaptive cellular systems.
We then show the coverage associated with the required outage probability of 10% in Figure 5. The higher decorrelation
distance results in smaller coverage because the attenuation
due to shadowing of a given calling MS tends to be similar
to that of neighboring RS’s. However, when the cell radius of
5
ACKNOWLEDGMENT
m=100
4
Coverage/R0
This work is supported in part by a Grant-in-Aid from
the 21st Century COE Program (no. 14213201) of the
MEXT, Japan, a Grant-in-Aid for Scientific Research (A)
(no. 16206040) from the JSPS.
m=50
R EFERENCES
3
m=20
m=10
2
m=5
m=2
m=1
1
0
1
10
100
R0/dcor
Fig. 5.
Dependence of coverage on decorrelation distance dcor (α =
3.5, σ 0 = 8 dB).
single-hop cellular system with QPSK R0 is ten times larger
than the decorrelation distance dcor (i.e. R0 /dcor = 10), there
is little decrease from the coverage when no correlation is
assumed.
VI. C ONCLUSION
The impact of spatial correlation of shadowing on the
coverage of multihop cellular systems has been addressed. We
formulated the cell-radius dependence of spectral efficiency
and outage probability of TDMA multihop cellular systems by
using the similar methodology to formulate the performance
of rate-adaptive cellular systems. Numerical results reveal
that when the coverage of single-hop cellular system is ten
times larger than the decorrelation distance, which is about 20
meters for the urban environment and 5 meters for the indoor
environment, the coverage is not significantly reduced from
the coverage under the assumption of no spatial correlation.
[1] G. N. Aggélou and R. Tafazolli, “On the relaying capability of nextgeneration GSM cellular networks,” IEEE Pers. Commun. Mag., vol. 8,
no. 1, pp. 40–47, Feb. 2001.
[2] H. Wu, C. Qiao, S. De, and O. Tonguz, “Integrated cellular and ad
hoc relaying systems: iCAR,” IEEE J. Select. Areas Commun., vol. 19,
no. 10, pp. 2105–2115, Oct. 2001.
[3] T. Rouse, I. Band, and S. McLaughlin, “Capacity and power investigation of opportunity driven multiple access (ODMA),” Proc. IEEE ICC
’02, vol. 5, pp. 3202–3206, Apr. 2002.
[4] A. N. Zadeh and B. Jabbari, “Performance analysis of multihop packet
CDMA cellular networks,” Proc. IEEE GLOBECOM ’01, pp. 2875–
2879, Nov. 2001.
[5] A. Fujiwara, S. Takeda, H. Yoshino, and T. Otsu, “Area coverage
and capacity enhancement by multihop connection of CDMA cellular
network,” Proc. IEEE VTC ’02-Fall, Sept. 2002.
[6] K. Yamamoto and S. Yoshida, “Analysis of reverse link capacity enhancement for CDMA cellular systems using two-hop relaying,” IEICE
Trans. Fundamentals, vol. E87-A, no. 7, pp. 1712–1719, July 2004.
[7] ——, “Tradeoff between area spectral efficiency and end-to-end throughput in rate-adaptive multihop radio networks,” IEICE Trans. Commun.,
vol. E88-B, no. 9, pp. 3532–3540, Sept. 2005.
[8] S. Sampei, Applications of Digital Wireless Technologies to Global
Wireless Communications. Prentice-Hall, 1997.
[9] K. Yamamoto, A. Kusuda, and S. Yoshida, “Performance evaluation
of multihop cellular systems and rate adaptive cellular systems,” Proc.
the 5th International Conference on Information, Communications and
Signal Processing (ICICS 2005), pp. 1–5, Dec. 2005.
[10] A. Kusuda, K. Yamamoto, and S. Yoshida, “Performance comparison
of multihop relaying and rate adaptation in cellular communication
systems,” Proc. the 16th IEEE International Symposium on Personal,
Indoor and Mobile Radio Communications (PIMRC 2005), Sept. 2005.
[11] M. Gudmundson, “Correlation model for shadow fading in mobile radio
systems,” Electron. Lett., vol. 27, no. 23, pp. 2145–2146, Nov. 1991.
[12] Z. Wang, E. K. Tameh, A. R. Nix, and O. Gasparini, “A joint shadowing
process model for multihop/ad-hoc networks in urban environment,”
Proc. 11th WWRF meeting, June 2004.
[13] ETSI TR 101 112 (v3.2.0), “Universal mobile telecommunications system (UMTS); selection procedures for the choice of radio transmission
technologies of the UMTS (UMTS 30.03),” Apr. 1998.
[14] T. Ue, S. Sampei, and N. Morinaga, “Symbol rate controlled adaptive
modulation/TDMA/TDD for wireless personal communication systems,”
IEICE Trans. Commun., vol. E78-B, no. 8, pp. 1117–1124, Aug. 1995.