Global System for Mobile Communications (GSM) cellular systems. In [2], Wu et al. introduced stations dedicated to relaying for high bandwidth usage. In [3] ...
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], Agg´elou 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.
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