Effects of correlated shadowing signals on channel reuse in mobile radio systems

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708 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 40, NO. 4, NOVEMBER 1991

Effects

of Correlated Shadowing Signals on

Channel Reuse in Mobile Radio Systems

Aysel Safak and Ramjee Prasad, Senior Member, IEEE

Abstract-This paper presents a study to evaluate the cochan- ne1 interference probability for the desired and interference signals which are correlated due to shadowing. The effects of correlation on the normalized reuse distance is investigated. A generalized expression for the cochannel interference probability is derived by combining uncorrelated (fast) Rayleigh fading and correlated (slow) log-normal signals. Cochannel interference probability for sectorized cell layouts is compared with the lower bound of cochannel interference probability for omnidi- rectional antenna systems. The results are useful for evaluating the performance of cellular telephony and packet radio net- works.

I. INTRODUCTION

NE of the most important design objectives for mobile

0

radio systems is to conserve the available spectrum by reusing allocated frequency channels in the areas that are located geographically as close to each other as possible. The limitation in distance for reusing frequency channel can be determined by the amount of cochannel interference. To achieve a satisfactory frequency channel assignment plan, it is necessary to fully understand the effects of cochannel interference on mobile-radio reception.

One needs to calculate the probability of cochannel interference of mobile radio signals which fluctuate due to fading and shadowing in order to be able to establish satisfactory reuse distances between base stations. Cochannel interference with fading and shadowing for the uncorrelated signals has already been studied [ll-[131.

In the presence of shadowing environment only, Daikoku and Ohdate have derived the expression for cochannel inter- ference probability ( [ 5 , eq. (6)]) for the correlated desired and interference signals; however, their results are erro- neous. The error occurred in deriving the joint probability density function (pdf) of signal-to-interference ratio for the local mean where they apparently considered that the correlation effect is equivalent to decreasing the standard deviation

( a ) from a to a(1 - p ) ' I 2 (where p is the correlation coefficient) [5]. This assumption is true if the standard deviation is same for the desired and interference signals [2]. But in their paper [5], Daikoku and Ohdate derived the equations for a general case considering different standard deviations ad and ai with respect to the desired and interference signals, respectively, which lead to error in the final expressions.

Manuscript received June 1, 1990; revised September 21, 1990. The authors are with the Telecommunications and Traffic-Control Systems IEEE Log Number 9103230.

Group, Delft University of Technology, the Netherlands.

The purpose of this paper is first to derive the correct expression for the cochannel interference probability in presence of correlated desired and interference signals, and then to determine the normalized reuse distance by using the calculated cochannel interference probabilities. Finally, sectorized cell layouts are also investigated and compared with the omnidirectional cell patterns in term of cochannel interference probability.

The cochannel interference probability is defined as F (

cz) = Prob

{

Pd / P i

<

CY} ₍₁₎

where Pd is the instantaneous power of the desired signal and Pi is the interference power from the cochannel and CY is the necessary protection ratio.

The paper is organized as follows. Section I1 describes the analysis of cochannel interference probability in presence of correlated desired and interference signals. Section I11 defines the (normalized) reuse distance and presents computational results. In Section IV, sectorized cell layouts are introduced and compared with omnidirectional cell layouts using computational results. Section V contains the concluding remarks.

11. PROBABILITY OF COCHANNEL INTERFERENCE WITH SHADOWING ONLY

Shadowing of radio signals by buildings and hills leads to gradual changes in the local mean signal level, which can be represented by log-normal pdf

f/'od(Pod) =

[

l / (

fi

' ad

' exp

{

-

[

1/(2 [In ( p o d / t d ) ] 2 ] ( 2 )

where ad, Pod, and

t d

are the standard deviation, local mean, and area median of the desired signal, respectively.

The local mean, P o i , of the interference signal is also described by the density function of the form (2) with ai and

t i ,

respectively, the standard deviation and area median of the interferer. Since some of the signals are shadowed by the same obstacles, the local means of the signals in the same cell are partially correlated [ 11, [2], [5]. Assuming a correlation between Pod and Poi, the joint pdf of P o d / P o i is defined as

r

141 m

**f*(4**

=

1

f(Pod9 Poi) *

1

a ( p o d , w >

1

dw ( 3 ) where A = P o d / P o i , w = P o i ,

I

aPod, P o , ) / a ( A , w )

I

= w 0018-9545/91$01.00 0 1991 IEEE

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SAFAK AND PRASAD: EFFECTS OF CORRELATED SHADOWING SIGNALS

is Jacobian and

f(

Pod, P o i ) is the joint density function [ 111:

where ud = ( l / u d ) * In ( P o d / E d ) and

ui

= ( l / a i ) *

The cochannel interference probability for shadowing only, ln(Poj/Ei)*

F'(CI), is obtained using ( 1 ) and

(4):

F'(CZ) = l a [ 1 / ( 2 & a,, A)]

0

* exp

{

-

[In ( A / A , ) / ~ . U , , ] ~ ] d A nm

-

- (l/v'%)

.

exp

{

- u 2 / 2 }

.

du ( 5 ) where A, =

td/E,,

p o = 1n{Ed/Ei* a}, and a,, is the effective standard deviation given as

( 6 )

1 12

a,, = [ (0;

+

- 2 p

.

ad * 0 ; ) /2]

Equation (5) is the correct expression, which should re- place (6) of [ 5 ] . An obvious difference is observed in the expression of U,,. The expression for a,, of [5] should

therefore be replaced by (6). A correct graphics for determining a,, for given values of ad or a, or both and p as a parameter using ( a ) is shown in Fig. 1 . One may observe that a,, of [5] is correct for p = 0 and ad = a, for p = 0. For ai = 0, a,, of [5] reduces to ad * ( 1

-

p ) l I 2 , which is evidently incorrect. Note that (5) reduces to (17) of [ l ] for the uncorrelated case ( p = 0) and a, = ad.

Fig. 2 shows the variation of cochannel interference with p o for various values of the correlation coefficient, U = ad =

ai = 6 dB and 12 dB. It is observed from Fig. 2 that the cochannel interference probability becomes smaller as the signals become more correlated for all values of the p o . This can be explained intuitively by the fact that, as the signals become more correlated, the fluctuations in Pod and Poi would affect the probability that Pod

/

Poi

<

a to a lesser extent. Fig. 2 also confirms that the cochannel interference probability increases with the increase in the shadowing effect, i.e., U , for equal correlation coefficient. For unequal

p , it is not always true, as it can be seen from Fig. 2 that

F(CZ) is lower for U = 12 dB with correlated signals p = 0.8 than F(C1) for U = 6 dB with uncorrelated signals p = 0 for a given value of p o .

In mobile radio systems, the received signal envelope fluctuates rapidly due to multipath propagation and wave interference. These fluctuations are described by Rayleigh statistics, i.e., the signal envelope has Rayleigh pdf condi- tional to the local mean level which fluctuates typically 6- 12 dB due to shadowing. Note that the envelopes of signals, suffering Rayleigh fading, are uncorrelated while their local means are partially correlated as a result of shadowing [ l ] .

In case of combined Rayleigh fading and log-normal shadowing, the cochannel interference probability,

Ffs(

CZ), is

10 1 . 0 0 . 1 0 . 1 1 . 0 01 / a d 709 10 Fig. 1 . a parameter: a: p = 0; b: p = 0.2; c: p = 0.4; d: p = 0.6; e: p = 0.8. Curves for determining uefl for given values of U, and uj and p as

Y 1 -1 10 - 2 10 -3 10 -4 10 -5 10 I 1 1 I I \ I I I , I

,

11 I T 0 4 8 1 2 1 6 20 P

Fig. 2. Probability of cochannel interference with shadowing only for correlation coefficient and standard deviation as parameters. U = 6 dB: a:

p = 0.8; b: p = 0.4; c: p = 0. U = 12 dB: d: p = 0.8; e: p = 0.4; f: 0

p = 0.

derived using (1)-(3) and considering the joint pdf's of P o d / P o i and P d / P , as independent occurrence. Here Pd and Pi are the instantaneous power of the desired and interference signals, respectively, with their pdf s as expo- nential distributions:

F ~ ~ ( c z )

=

( I / & )

lm

exp {-U')/

- m

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710 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 40, NO. 4, NOVEMBER 1991 1 -1 10 - 2 1 0 - 3 10 -4 10 -5 10 0 4 8 1 2 16 2 0 P

Fig. 3. Probability of cochannel interference with fading and shadowing for correlation coefficient and standard deviation as parameters. U = 6 dB: a: p = 0.8; b: p = 0.4; c: p = 0. U = 12 dB: d: p = 0.8; e: p = 0.4; f:

0

p = 0.

Again, (7) of [5] should be correctly used with aefi given by (6). Note that (7) reduced to (12) of [l] for the special case of p = 0 and a; = ad. The above integral has been evaluated numerically and is plotted in Fig. 3 as a function of p o for U = U, = a; = 6 dB and 12 dB, (Y = 8 dB and with correlation coefficient as a parameter. As in Fig. 2 which deals only with shadowing, the increasing correlation between the shadowed signals decreases the cochannel interference probability. However, the presence of Rayleigh fading may be observed to increase the probability of cochannel interference compared to the absence of fading, due to the uncorrelated fluctuations of signal envelopes in the presence of fading.

111. REUSE DISTANCE

The cell size and geographic location of cells should be chosen so that the cochannel interference probability does not exceed a given value. Let the distances from the mobile to the desired and interference stations be d, and d ; , respectively. The ratio of area median levels,

5 ,

/ t i ,

is given by

( di

/

d d ) g , assuming identical base station parameters. The parameter g is the ground wave propagation path-loss slope (here g = 4).

The (normalized) reuse distance is defined as the ratio of the distance between the centres of the nearest neighboring cochannel cells to the cell radius of the desired station [l]. From geometrical considerations, the reuse distance R , may simply be written as

Expressing p o in terms of the reuse distance and inserting in

(3,

one can calculate the cochannel interference probabil-

1 -1 10 -2 10 -3 10 -4 10 -5 10 0 6 1 2 1 8 2 4 30 REUSE DISTANCE

Fig. 4. Probability of cochannel interference with shadowing only versus reuse distance for correlation coefficient and standard deviation as parameters. U = 12 dB: a: p = 0, CY = 18 dB; b: p = 0.4, a = 18 dB; c: p = 0,

a = 8 dB; d: p = 0.4, CY = 8 dB. U = 6 dB: e: p = 0, a = 18 dB; f: p = 0.4, a = 18 dB; g: p = 0, CY = 8 dB; h: p = 0.4, CY = 8 dB. ity in case of shadowing only as a function of the reuse distance. The results are shown in Fig. 4 for a = 8 and 18 dB, a = 6 and 12 dB.

Apparently, the cochannel interference probability decreases with increasing reuse distance, increasing correlation between the signals, and decreasing standard deviation of signals due to shadowing.

Again, expressing p o in terms of R , and inserting into (7), one can find the variation of cochannel interference probability as a function of the reuse distance in the presence of fading and shadowing. This is shown in Fig. 5 for (Y = 8 and 18 dB and a = 6 and 12 dB. Here again the correlation helps reduce the cochannel interference probability. Increase in the standard deviation of the signal causes a considerable increase in the cochannel interference probability for a particular correlation factor and reuse distance as shown in Fig. 5.

Reduction in the cochannel interference probability due to correlation between signals is quite obvious because the correlation effectively decreases the standard deviation.

Both Figs. 4 and 5 show that a cellular system can be designed with lower reuse distance in presence of correlated signals for a given cochannel interference probability.

IV. SECTOHZED CELLS LAYOUTS

A sectorized cell layout is defined as a pattern of ( N x S ) sectors with

N

and

S

the number of cell sites per cluster and number of sectors per cell site, respectively. (7 x 6), (7 x 3),

(4 x 6), (4 x 3) sectors are examples among numerous sec- torized cell layouts.

Directional antennas are used to obtain the desired sectorized cell layout. It means that three 120" and six 60" beam directional antennas are needed to obtain (7 x 3) and (7 x 6)

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SAFAK AND PRASAD: EFFECTS OF CORRELATED SHADOWING SIGNALS 1 -1 10 - 2 10 -3 10 - 4 10 -5 1 0 0 6 1 2 18 2 4 3 0 R E U S E D I S T A N C E

Fig. 5. Probability of cochannel interference with fading and shadowing versus reuse distance for correlation coefficient and standard deviation as parameters. U = 12 dB: a: p = 0, a = 18 dB; b: p = 0.4, CY = 18 dB; c: p = 0, CY = 8 dB; d: p = 0.4, a = 8 dB. U = 6 dB: e: p = 0 , a = 18 dB;

f p = 0.4, CY = 18 dB; g: p = 0, a = 8 dB; h: p = 0.4, a = 8 dB.

sectors, respectively [ 111-[13]. As compared with an omni- directional antenna system a directional antenna system, i.e., sectorized cell offers two main advantages: i) reducing the number of cochannel interferers and ii) increasing correlated signals. Therefore, combining both advantages the sectorized cell layouts yield reduced cochannel interference probability thereby enhancing gain in spectrum utilization.

(7 x 6) and (4 X 6) sectorized cell layouts are shown in Figs. 6(a) and 6(b) as the worst case design [13]. In both sectorized cells, the number of interferers is reduced to one. The (normalized) reuse distance for (7

x

6) sectors is given by

R , = - 0.7. (9)

R , = ( [ d / , $ i ) " g - 1 .

(

10)

R , for (4

x

6) sectors can be written as

Computational results are obtained for shadowing only using ( 5 ) , (9) ( R , = 4.6), and (10)

( R ,

= 3.46) and shown in Fig. 7 as cochannel interference probability versus protec- tion ratio. Similar results are also shown in Fig. 8 for combined Rayleigh fading and shadowing environment using (7), (9), and (10). It is seen from Figs. 7 and 8 that for a particular value of protection ratio (7

x

6) sectorized cell layout causes lower cochannel interference probability than due to (4 x 6) sectors.

In case of omnidirectional antenna systems, the lower and upper bounds on probability of cochannel interference can be estimated due to one and six interferers, respectively. For the worst case condition [13], R , can be written as

71 1

A

, le

W

(b)

Fig. 6. Determination of C / I in a directional antenna system for worst case in a 60" directional antenna system. (a) (7 x 6) sectorized cell pattem. (b) (4 x 6) sectorized cell pattern.

1 -1 1 0 - 2 10 -3 10 10 0 6 1 2 1 8 2 4 3 0 PROTECTION R A T I O ( d B )

Fig. 7. Probability of cochannel interference versus protection ratio for shadowing only using sectorized cell layouts for correlation coefficients and reuse distances as parameters. U = 12 dB: a: p = 0, R , = 3.46; b: p = 0.4,

R , = 3.46; e: p = 0, R , = 4.6; f p = 0.4, R , = 4.6. U = 6 dB: c: p = 0 , R , = 3.46; d: p = 0.4, R , = 3.46; g: p = 0 , R , = 4.6; h: p = R , = 1

+

( n * [ d / [ i ) l ' g n = 1 , 2 , .

.

. , 6 . (11) 0.4, R , = 4 . 6 .

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712 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 40, NO. 4, NOVEMBER 1991 - 3 10 I l i 10 4 _w 2 4 U 0 1 -1 10 - 2 1 0 - 3 10 0 6 1 2 18 24 30 PROTECTION R A T I O ( d B ) PROTECTION R A T I O (dl3)

Fig. 8. Probability of cochannel interference versus protection ratio for fading and shadowing using sectorized cell layouts for correlation coefficients and reuse distances as parameters. U = 12 dB: a: p = 0, R, = 3.46; b: p = 0.4, R, = 3.46; e: p = 0, R, = 4.6; f: p = 0.4, R, = 4.6. U = 6 dB: c: p = 0, R, = 3.46; d: p = 0.4, R, = 3.46; g: p = 0, R , = 4.6; h:

Fig. 10. Probability of cochannel interference versus protection ratio for fading and shadowing using omnidirectional cell layouts for correlation coefficients and reuse distances as parameters. U = 12 dB: a: p = 0, R, =

3.46; b: p = 0.4, R , = 3.46; e: p = 0, R , = 4.6; f: p = 0.4, R, = 4.6; i: p = 0, R, = 6; j: p = 0.4, R , = 6. U = 6 dB: c: p = 0, R, = 3.46; d: p = 0.4, R, = 3.46; g: p = 0, R , = 4.6; h: p = 0.4, R , = 4.6; k: p = 0, R, = 6; 1: P = 0.4, R , = 6. p = 0.4, R , = 4.6. 1 -1 rl 1 0 m m a W U - 2 a LI, W

2

10 -4 10 t I , - : ; ' I ! ! ! ! ! I 0 6 1 2 1 8 2 4 30 PROTECTION R A T I O ( d B )

Fig. 9. Probability of cochannel interference versus protection ratio for shadowing only using omnidirectional cell layouts for correlation coefficients and reuse distances as parameters. U = 12 dB: a: p = 0, R, = 3.46: b: p = 0.4, R, = 3.46; e: p = 0, R, = 4.6; f: p = 0.4, R , = 4.6; i: p = 0, R U = 6 ; j : p = 0 . 4 , R U = 6 . u = 6 d B : c : p = 0 , R U = 3 . 4 6 ; d : p = 0 . 4 ,

R, = 3.46; g: p = 0, R , = 4.6; h: p = 0.4, R , = 4.6; k : p = 0, R , = 6; 1: p = 0.4, R, = 6.

The lower bound ( n = 1) of cochannel interference proba- bility is presented in Figs. 9 and 10 for shadowing only (5) and Rayleigh fading and shadowing (7) using (11) ( R , = 3.46, 4.6, and 6), respectively. Comparing Figs. 7-10 de-

picts that the use of sectorized cell layouts leads to considerable reduction in the cochannel interference probability. In fact, the cochannel interference probability due to the sectorized cell layouts is much reduced comparing with the upper bound ( n = 6) of the cochannel interference probability due to the omnidirectional antenna system which is not presented in this paper. In order to have same level of cochannel interference probability as in case of sectorized cell patterns, higher value of reuse distance is required for omnidirectional cell patterns. Thus, a low value of reuse distance is sufficient for the sectorized cells and hence it gives higher correlation between signals.

The influence of protection ratio CY on cochannel interfer-

ence probability is shown in Figs. 7- 10. It is seen from Figs. 7 - 10 that cochannel interference probability increases with increase in the protection ratio. For the future Pan-European digitally modulated system (GSM, Group Special Mobile) CY

will be approximately 9.5 dB and CY = 18 dB for the UK Total Access Communication System (TACS) with a channel spacing of 25 kHz (a European standard) [16]. Thus, using Figs. 7-10 it is confirmed that the cochannel interference probability for GSM will be less than that for TACS for a given set of parameters.

Figs. 9 and 10 show that R , = 4.6 is not sufficient for

CY = 9.5 dB and 18 dB to maintain F ( C I ) = lo-'. There- fore, a higher value of R,, i.e., larger cluster size, is

required for omnidirectional cell pattern (e.g.,

R,

= 6 is sufficient, Figs. 9 and 10). Whereas in case of sectorized pattern, even R , = 3.46 is adequate to maintain F ( C I ) =

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SAFAK AND PRASAD: EFFECTS OF CORRELATED SHADOWING SIGNALS 713 V. CONCLUSION

An expression for the effective standard deviation is derived in terms of the standard deviation of the desired and interference signals and correlation factor. A graph is also provided to obtain the effective standard deviation which shows that the correlation effect is equivalent to offering lower standard deviation with uncorrelated shadowing.

This paper also deals with the effect of the correlation between signals on the cochannel interference probability in mobile radio systems. The multipath effects resulting in uncorrelated Rayleigh fading are considered. The influence of correlation on the normalized reuse distance is also investigated.

The results show that the correlation between the signals reduces the cochannel interference probability and its effect becomes more pronounced for higher variances of the shadowed signals, which lead to higher cochannel interference probabilities.

Again it is confirmed that, the presence of fading with shadowing causes considerable increase in the cochannel interference probability. Further, it is noted that the correlation between the signals allow the use of shorter reuse distances compared to the uncorrelated case.

Cochannel interference probability is evaluated as a function of protection ratio for (7 x 6) and (4 x 6) sectorized cell layouts with

R,

= 4.6 and 3.46, respectively. A com- parative analysis is presented between sectorized and omnidirectional cell layouts.

Omnidirectional cell layouts cause the highest level of cochannel interference probability as compared with (7 x 6) and (4 x 6) sectors. Computational results confirm that the digital Pan-European GSM System using MSK modulation will be superior to the UK Total Access Communication System in term of the level of cochannel interference probability. In case of omnidirectional antenna system requires higher cluster size as compared with the directional antenna to maintain acceptable cochannel interference. The influence of two or more correlated log-normal interferers is not considered in this paper.

.

ACKNOWLEDGMENT

The authors are grateful to Prof. Jens C. Arnbak for fruitful discussions. They would like to thank the anonymous reviewers for valuable suggestions.

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Aysel Safak was born in Unzunkopru, Turkey, in 1952. She received the B.Sc. and the M.Sc. (Eng.) degrees from Hacettepe University in 1975 and 1977, respectively.

During 1977-1978, she worked as an assistant in the same department. She was with the Satellite Communications Division of the Turkish PTT Ad- ministration in Ankara between 1979 and 1984, where she was involved with planning, evaluation, and procurement of analogue and digital earth stations. Currently, she is an associate researcher in the Electrical Engineering Department of the Delft University of Technol- ogy, The Netherlands and is working toward the Ph.D. degree. Her research interests include mobile and indoor communications.

Ramjee Prasad (M’88-SM’90) was born in Babh- naur (Gaya), Bihar, India, on July 1, 1946 He received the B.Sc. (Eng.) from Bihar Institute of Technology, Sindri, India, in 1968, and the M.Sc. (Eng.) and Ph.D degrees from Birla Institute of Technology (BIT), Ranchi, India, in 1970 and

1979, respectively

He joined BIT as Senior Research Fellow in 1970 and became Associate Professor in 1980. While he was with BIT, he supervised many research projects m the area of Microwave and Plasma Engineering. During 1983-1988 he was with the University of Dar es Salaam (UDSM), Tanzania, where he became Professor in Telecommuni- cations at the Department of Electrical Engineering in 1986. At UDSM he was responsible for the collaborative project “Satellite Communications for Rural Zones” with Eindhoven University of Technology, The Netherlands. He has published over 80 technical papers. His current research interest lies in packet communications, adaptive equalizers, spread-spectrum systems, and telematics Since February 1988, he has been with the Telecommunica- tions and Traffic Control Systems Group, Delft University of Technology, The Netherlands, where he is actively involved in the area of mobile and indoor radio communications

Dr Prasad is a Fellow of the Institution of Electronics and Telecommuni- cation Engineers, India