A New Mathematical Method for the Exact Evaluation of the Conditional Cochannei Interference Probability in Cellular Mobile

Radio SystemsG.K. Karagiannidis*, CJ. Georgopoulos* & S.A. Kotsopoulos*

SUMMARY High capacity cellular mobile radiocommunication services in Europe and in North America have set extremely demanding minimization interference targets for the involved operating systems, in order to keep spectrum efficiency in high levels. The evaluation of the conditional cochannei interference probability remains a very interesting scientific area and a major concern in mobile communications. Therefore the evaluation is strongly needed to maintain thr rerquired transmission quality level, to control the spectrum efficiency, to optimize the design criteria and to predict the signai-to-interference ratio in different cellular systems. In this paper, a new mathematical method is presented suitable for the exact exact evaluation o f the outage probability of this type of intereference under specific considerations concerning the existing mobile's propagation environment. Using this prototype method, a high level of probability's accuracy is obtained especially in the worst cases, where the received signai is degraded due to the presence of radio-shadowing phenomena.

1 INTRODUCTION

The decision to implement a cellular systems approach to high capacity mobile radio communication services in Europe and in North America, has set extremely demanding engineering targets for the operating consortia involved. In common with all terrestrial mobile radio transceivers, cellular equipment must operate in environments degraded by a great many types of man made interferences. Of these, the non-coherent forms (ignition noise from vehicles, wide band noise from lighting, power lines, etc) have already received attention. As increasing numbers of equipment, are brought into service in the higher UHF bands, the probabilities of interference from coherent sources such as, unwanted em issions from transmitters and receiver generated spurious, became more significant and a great effort has spent on their evaluation [ 1,2.3]. Moreover due to the major problem facing today’s radio communication industry; that is the limitation of available radio frequency spectrum, a high degree of spectrum efficiency is needed by keeping the cochannei intereference in low ievei.

It is noted that spectrum efficiency is one of the most important factors in designing ceiluiar mobile radio systems. It depends mainly upon the selected cluster scheme, the applied modulation-demoduiation techniques and the set of technical parameters ( i.e. the acceptable interference

University of Patras. Departm ent o f Electrical and Computer Engineering RionGR-261 10 Patras GREECE Submitted to the Institution o f Engineers Australiaon 16 April 1995, rms received 29 February 1996. Paper E 95111R.

probability and the call blocking probability) that affect the service quality. In this case, the Conditional Co-channel

Interference Probability (CCIP) denoted as qc, is a measure to control the Interference's level helping the designers to re-adjust the system's operating parameters in orderto keep this type of interference in low levels. Also when attempting to increase the system's capacity by increasing the cochannei interference sometimes is increased in high levels. In this case the transmission quality cannot be maintained at the specified level and new techniques should be applied in order to increase the system’s capacity by xaving as a

reference point the predicted qc CCIP is defined as the

probability that the undesired signai Local Mean Power (LMP) exceeds the desired signai LMP, by the protection ratio denoted as O' according to the Average Interference Criterion [4], The main problem of calculating the CCIP is to evaluate the probability density function (pdf) of the sum of log-normaily distributed variables. More precisely, in a ceiluiar environment, the received signai is produced by the summation of all the cochannei interfering signals having a total Local Mean Power I. The problem faced by the designers is to evaluate the pdf of I. This evaluation is

very important for the final calculation of qc.

Existing methods approximate the pdf of the I without giving accurate results in the calculation of CCIP. Fenton [5] attempts to approach the above stated problem, assuming that the sum of two (or more) log-normal pdfs is another log-normal pdf having the same variance and the mean value being the sum of the individual means.

Nagata -Akaiwa derive an expression for qr.using a mathematical analysis based on fenton’s method. This expression was corrected later by Prassad-Amback [6]. Finally, Schwartz-Yeh [7] optimize Fenton’s method and derive an expression for the variance which is the sum of two log-normal variables; and for the case, for more than two variables they use an iterative approximation.

This paper presents a new method for evaluating the

probability q(. which is defined in terms of log-normal random variables. The method involves six multiple integrals that evaluated using Gauss-Hermite formula. Finally, the formula is derived by the use of characteristic functions. The presented method offers advantage in the accuracy of computation over the use of direct numerical convolution of the relevant pdfs, greater computational speed and reduced complexity.

Moreover, this method offering a prototype calculation of q can be used either as a reference point for testing the ΙένεΙ of accuracy of Fenton and Schwartz - Yeh techniques in any mobile radio system environment, or as an alternative technique for calculating the conditional co-channel interference.

In section 2. some aspects concerning the basic elements of the propagation modelling of Cellular Mobile Systems are given. These elements are taken into account in order

to extract the formula for q( .

Section 3, is reffered to the new proposed method and also corresponding mathematical analysis is given.

Finally, a comparison is made between the proposed method and the existing methods and the results are shown in section 4.

DESCRIPTION OF THE CELLULAR MOBILE RADIO ENVIRONMENT

Pf (jO = — expV 2x,o y

where: X 0 = is the LMP of the signal y = is the signal's amplitude.

However, according to the literature, fast fading rarely follows true Rayleigh behavior [9].

Slow (log-normal) fading is a much slower variation of the median signal, from sector to sector and is a result of the signal blocking due to either large structures, hills, man made structures or mountains. In contrast to the fast fading phenomenon, where the pdf is not always follows a Rayleigh distribution, the corresponding pdf of the slow fading situation can be accurately modeled by the following pdf that follows a log-normal distribution:

where :

4 Ϊ p OX nexp

— ( l / ix 0 — 1 nm) ϊ σ 1"

2 Λ

(2 )

σ = is the standard deviation m = the mean of the LMP, called Area Mean

Power (AMP).

In this case, for a cellular environment, the received signal is produced by the summation of all the incoming co­channel interfering signals. Let us assume that “I” is the LMP of the sum of all the co-channel interfering signals in a cell, therefore the engineers and designers are faced with the problem to evaluate the pdf of I. The solution of this

problem is very important for the final calculation of qc.

As mentioned previously (section 1), the existing methods approximate the pdfs of I without giving accurate results. For this reason, the present authors propose a new method to derive the exact mathematical expression that is suitable to evaluate the Conditional Cochannel Intereference Probability.

Propagation is rarely line-of-sight and generally a number of waves are received at the vehicle's antenna by reflection from hilis, buildings, etc. depending upon the mobile’s urban or rural environment. The interference of these waves produces a standing wave pattern that gives a variation of field strength (known as “fading”) along the street. As the vehicle moves through the fadings pattern, rapid fluctuations in the received signal level occur.

Cellular Mobile Radio signals, in the 900 MHz frequency band, are characterized by a fast fading component superimposed on a Slow fading one. The fast (Rayleigh) fading is produced by the rapid fluctuation in the received signal’s amplitude and phase, due to the multipath phenomenon [8]. The pdf of the fast fading signal follows the Rayleigh distribution and is given by the following formula:

3 M ATHEM ATICAL ANALYSIS OF THE NEW PROPOSED METHOD

For the purpose of the present analysis, the following symbolisms are made:

I, is thes is the

m, is the

is the

σ,· is the

σ* is the

Pu is the

Px is thek is the

Moreover, the present method considers the following setof assumptions:

i. The cellular system zoning is hexagonal and the subscribers are uniform distributed over the examined communication service area.

ii. The co-channel interference, from the first tier is the dominant one [10,11], and the interference from the adjacent channels can be ignored due to the receiver’s high selectivity according to the current mobile radio engineering demands.

iii. The probability of interference is the probability that the ratio of LMP of the desired signal to the LMP of the net interfering signal is less than the protection ratio β.

iv. If the probability of interference for a given call is satisfactory at the Base Station, it is also satisfactory at the Mobile Terminal.

v. There are six cochannel interfering cells (interferers) in the first tier, in a fuily equipped hexagonal-shaped cellular system.

According to the above stated assumptions, q can beexpressed by the following formula:

qc= P roh

f \ f

~Γ— < β = Pr oh

Σ '.V <=ι V

Σ'.V I-I

- β < 0(3)

Let (r ) = <ps (r ), Φ βΙι (r) the characteristic

functions of the variables w, S, /?./, respectively. By taking into account that S, I are statistically independent, then the characteristic function is shown [12] to have the following form:

Φ „(γ) = Μ γ) Π 0 (_r) (9)/ . ( pi,

and by recalling the definition o f the characteristic function, equation (9) gets the following form:

* 7Φ„(γ) = Φ, (r) J exp (-irx,) ρβι (x;) dxt (10) ;=i o

Let

\nXj - 1 ηβ - m, = σ,η (1 1 )

then, (r ) is given by:

* » = *,(/·).(2π) ( j J . J.-t-ee

Jexp ~ ir ·|^τ + L p e

.expk r·2

- Σ γ/ -1 Δ

(12)

If w - s - β ^<=i

then (3) can be written as follows:

q

(4)

(5)Cc = Prob (w , 0)

Considering log-normal pdfs for the /, andS, the followingexpressions are given:

A-0 0 =•Jim ·,m

- { l n y x - \n n } ) \’ exP (---- —---- ). Λ (6)2cr

and

1 -i\ny2 -\ran)A (Λ )=-7=---------exp(—— — 7-------- ) ,y2 > 0 (7)

v 2 p o j 2 ±crs

It can be proved [12] that the pdf of the variable /?./,· is given by:

I y, l 1/ivt -\η β - ιτ \ )2

— s?— L*-° (8)Λ 2 σ ,

From (6) and by definition, the following is obtained:

I) j (> -

qc = \ p w{X)dr=— \ \ (r) exp ( - j r τ) drdr (13)

By substituting (12) to the (13) we get:

- (Hqc = (2π)

■foer /J exp - j . r .

Vr +

.expk r 2-Σγ

(=1 Δ

• M i ·k

Σβ-/si

φχ{τ) .dr ,dx .drx.---.drk

(mf+OiSi)

But considering that:

-feeJ φ,. (r) exp — ir ■ τ + Σ β -e\ /«I

( k λ = 2 π ρ \τ + Σ β . β{η'+αι'') .

V <=ι /

(ιη +«T, ,rf)

(14)

. dr —

(15)

Hence, (14) can be written as follows:

<7 r =* r 2i

- ς 2#*1 ^

( 2 π ) —

] p A T + j 10 ( e ," m ) ) d x r :- d r ,

or, alternatively:

<?< = — r J J e x P h I y J(27Γ) 2 -----

(16)

(17)

environment for realizing the appropriate comparisons withthe already published results Γ61and

b) m, = · · · = mk = rrijthis assumption is generally valid

τ Η « ρ [ - Σ γ :( 2 * ) 2 - -

In β + mi - m, +1 n (e0Γ| Η------ 1- ear>). d r , - - -d r

(22)

the following expression is valid according to the conditions stated at [4,5 and 10]:

m, - ms = 1/2 [(3/2^) 2 ] (23)

where: Ps(x) = is the Cumulative Distribution Function (cdf) of S.

Ps(x) can be written as:

.r ,

Ρ λ*) = J1 r (1 n t - m ^ ,

"7r— e x P t------- - T T - WV 2πω 2 σ .(18)

or

(19)

with F(x) being the cdf of the Normal distribution and having the following form:

F(x) = j exp [ - — ] du (20)

where: γ is the path loss propagation factor (usually γ=4)

ηχ the cluster size.

By substituting (23) to the equation (22), the final formula

for qc is obtained:

Qc — “ Γ J " ‘ J exP i-Σ γ - ί(2π )2f r

‘ r 2i.

tr 2

1 ndr i ■■· drk(24)

the second part of equation (24) can be calculated using the following Gauss-Hermite formula:

oo v

From (16), (19) the following expression of qc is obtained: J ex p [ - λ 2 ] g U ) d x = £ a ,-g U ·)—OO / = 0

(25)

=1

<2*>{jj« p [ - Σ γ ΐ

In β - m, + \n( k \ ηΐί+σ,ηV/=I ■dr, ··■ drk (21)

Equation (21) is simplified to the following form, by considering the case where:

3) σ ? = σ , =··· = σ κ = σthis assumption is considered in order to match the

where: , JC, are constants given from special tables

i is a constant that denotes the accuracy at the f decade digit.

It is noted that in real time applications the execution time

for calculation of qc, by applying equation (24), is

approximately 3 sec using a Micro-Computer Pentium/ 166MHz.

4 NUM ERICAL RESULTS AND COM PARISON WITH THE EXISTING METHODS

Within the framework of the calculation procedure and the suitability of the proposed technique, equation (24) is used to evaluate qc for several values of a and β, common in the

Cellular Mobile Radio systems.The following considerations are taken into account:

i. k=6, i.e: there are six cochannel interfering cells in the first tier

ii. the propagation path loss slope γ is equal to 4

iii. the system’s cluster size ng is equal to 13iv. the standard deviation o f log-normal fading σ was

varied from 4 to 13 dB.v. the cells are centre illuminated (using omni-directional

antennas)vi. υ=5, i.e. accuracy at the fifth decade digit.

Table 1 The qc for several values of σ and 0 as evaluated using the Fenton's method.

p=4dB P=8dB β= 12dBσ qc a qc a qc4 0.00001 4 I 0.00050 4 0.00842 15 0.00076 5 | 0.00682 5 0.03912 I6 0.00613 6 1 0.02700 6 0.09000 I7 0.02223 7 I 0.06272 7 0.14576 18 0.04819 8 I 0.10259 8 0.19461 I9 0.07608 9 I 0.13998 9 0.23305 !10 0.10473 10 I 0.17228 10 0.26260 I1 1 0.13136 11 ! 0.19965 11 0.28577 |12 0.15550 12 1 0.22296 12 0.30454 !13 0.17723 13 ! 0.24302 13 0.32015 I

Table 2 The qc for several values of σ and β as evaluated usine the Schwartz-Yeh method.

|5=4dB β=8ά B β= l2dB

« ! qt a q< CT qc4 1 0.00001 4 0.00050 i 0.00808 i5 I 0.00069 > ; 0.00649 5 0.03819 |6 | 0.00557 6 0.02572 6 0.08828 !7 I 0.02023 - 0.06037 7 0.14938 !8 | 0.04647 3 0.10786 8 0.21343 I

| 0.08192 9 0.15931 9 0.27201 !10 | 0.12528 10 j 0.21252 10 0.32753 |11 ! 0.16770 11 0.26237 11 0.37526 |12 | 0.21461 12 : 0.30872 12 0.41711 ί13 1 0.25762 13 0.35069 13 0.45583

Table 3 The qc for several values of σ and β as evaluated using the proposed method.

P=4dB 0=8dB β= 12dBσ q< σ i qc <τ qc4 0.00001 4 I 0.00051 4 0.00835 I5 0.00074 5 I 0.00667 5 0.03839 j6 0.00615 6 : 0.02700 6 0.08938 |- 0.02157 - 0.06282 7 0.15057 |8 0.04844 S 1 0.10917 3 0.21314 !0 0.08444 9 ! 0.16027 9 0.27228 j10 0.12599 10 i 0.21 181 10 0.32595 I11 0.16988 1 1 ! 0.261 16 11 1 0.37361 112 0.21369 12 1 0.30696 12 0.41546 !13 0.25585 13 : 0.34868 13 0.45202 |

Figure 1 The CCIP for several values of s. as evaluated using the three methods with β=4 dB.y=4 and ng=l3

Figure 2 The CCIP for several values of s. as evaluated using the three methods with (3=8 dB. γ=4 and ng =13

The results concerning the new method using equation (24), are depicted in Table 3. In tables 1 and 2 the corresponding

values of qcare shown by considering the Fenton’s and Schwartz-Yeh’s methods respectively, under of course the same initial conditions.

Figures 1 to 3 show the obtained results, in a form of histograms, where the numerical deficiencies on the application of the three methods, are observed more clearly.

As it can be seen, the method based on Fenton’s technique gives satisfactory results for small values of σ (σ=7άΒ) and β; moreover, shows a high degree of accuracy deviation for σ >7dB, therefore is not applicable in mobile radio systems where shadow phenomena are existing due to peculiarities of the geographical terrain irregularities (4 dB<a<7 dB). The method based on the Schwartz-Yeh technique is more accurate with an error less than 1%, compared with the new proposed method.

Figure 3 The CCIP for several values of s. as evaluated using the three methods with B-12 dB. γ=4 and n =13

V

Figure 5 The CCIP for several values of a. as evaluated using the proposed new method with a=4. 8. 12 dB. σ=10

dB and ng =13

7 S 13 2! 36

Figure 4 The CCIP for several values of n g , as evaluatedusing the proposed new method with β=4, 8, 12 dB.o=10 dB and a=4

)In Figure 4, qc is histogramed as a function of the

cluster size ng (reuse factor) for β=4. 8 and 12dB.a=10dB

and a=4. The examined cluster size ( n g ) values are 7, 9,

13,25, 36 correspondingly. It is noted that the selected ng

values span a range from heavily cochannel interference

limited to noise-limited. In relative terms, n g =7 gives the highest user capacity with the lowest transmission quality,

while ng =36 has got the highest transmission quality with

the lowest capacity within the framework of the considered

range. M oreover, the n g values 9, 13 and 25 are of

intermediate transmission quality and capacity interests.

Figure 5, depicts qc as a function of the path loss slope a,

for a taking values 4, 8 and 12dB, a=10dB and f lg =13. In

this case, a was varied from 2 to 6. It is noted that a=2

corresponds to free-space path loss, while a=6 corresponds to heavily attenuating environment (i.e: a multistorey office building).

5 CONCLUSIONS

A new mathematical method to evaluate the Conditional Cochannel Interference Probability has been presented. This outage probability has been calculated under specific assumptions for the path loss and shadow fading modelling; and an advantage in the improvement of the accuracy and the computational speed over the existing approximated methods have been shown. The obtained accuracy of computation, strengthens the interference minimization targets that Europe and North america have set for the cellular systems in order to maintain transmission quality in required level and to control the spectrum efficiency.

6 REFERENCES

1. Gardiner J. and Kotsopoulos S.: “Relationship between base station transmitter muiticoupling requirements and frequecyplanning strategies for cellular mobile radio”, Proc. IEE. vol. 132, No. 5. 1985

2. Gardiner J., Kotsopoulos S. and Heathman A.: “Intermodulation Interference probabilities in the cellular radio frequency bands”, IEE International Conference on Electromagnetic Compatibility, Kent, U.K. publ. No. 60, Sept. 1984

3. Kotsopoulos S.:” Minimization of Intermodulation Interference”. Proc. IEE. Part F, Vol. 130, No. 4, p.p 350- 355, 1983

4. Nagata Y. and Akaiwa Y.: "Analysis for Spectrum Efficiency in Single Cell Trunked and Cellular Mobile Radio,” IEEE Trans. On Vehicular Tech.. Vol. VT-35, No. 3, pp. 100-113, August 1987.

5. Fenton L.: "The Sum of Log-Normal Probability Distributions in Scattcr Transmission Systems.” IRE Trans, on Commun. Systems. Vol CS-8. No. I, pp. 57-67, March 1960.

Network Design on the basis of Analytic Traffic Measurements1’, IEE Electronics letters, Vfc>l. 28, No. 15. July 1992

6. Prasad R. and Arnbak J.: “Comments on ‘Analysis for Spectrum Efficiency in Single Cell trunked and Cellular Mobile radio". IEEE Trans, on Vehicular Tech., Vol. 37, No.4. pp.220-222. November 1988.

9. “Special Issue on Mobile Radio Propagation”, IEEE TYans. On Vehicular Tech.. Vol. 37, No. 1, February 1988.

10. Lee W.: “Mobile Cellular Systems Conserve Frequency Resources." MSN & CT, Vol. 15, No. 7, pp. 139-150, June 1985.Cox D.: “Cochannel Interference Considerations in Frequency Reuse Small-Coverage-Area Radio Systems,” IEEE Trans. On Commun., Vbl. COM-30, No. 1, pp. 135- 142. January 1982.

7. Schwartz S. and Yeh Y.: “On the Distribution Function and Moments of Power Sums with Log-normal Components,” The Beil System Tech. Journal. Vol. 61, No. 7, pp. 1441- 1462. September 1982.

8. Lymberopoulos D.. kotsopoulos S. Koukias M. and Kokkinakis G.:"Ceilular Mobile Radio Communication

11. Papoulis A.:" Probability, Random Variables, and Stochastic Processes", McGraw-Hill. 1989.

G.K. KARAGIANNIDIS

Georgios Karagiannidis was bom in Pithagorion-Samos. Greece, on April 17, 1963. he received the diploma in Electrical Engineering from Patras Univeisity, Greece, in 1987. Mr Karagiannidis, is a researcher and postgraduate student in the W ireless Telecommunications Laboratory of the Dept, of Electrical and Computer Engineering in the University of Patras. His field of specialization is the Interference Problems in Ceiluiar Mobile Radio-communication Systems. He is a member of Technical Chamber of Greece.

C J . GEORGOPOULOS

Professor Chris J. Georgopouios has been educated in both Greece and USA and possesses the degrees of BA. BS(EE), MS(EE) and Ph.D (EE). From 1963 to 1972 he worked for three USA companies (Svlvania. Wang Laboratories, and Raytheon Labs) and in 1972 he joined the technical Research Staff of the University of Patras. Greece where he worked on research projects as Senior Scientist. In 1977 he was elected full professor of the Chair of Electronics of Univeisity ofThrace. Greece and in 1989 returned to University of Patras, as full Professor. Professor Georgopouios has received five patents covering his inventions and several awards for his innovative and cost reduction ideas;has wtitten some 50 technical reports on research, design and development projects:has published and presented more than 80 technical papers, and has authored a series of text books in the field of electronics and digital systems. He is also the author of four books of worldwide circulation. Presently, Prof. Georgopouios is heading the Optoelectronics and EMC Research Lab. of university of Patras, and his research and development interests include fiber optics and IR communications. Telematics and EMC sudies. He has been a consultant in the above areas in Europe and USA. Dr. Georgopouios is a Professional Engineer in the State of Massachusetts, a member ofTechnical Chamber Of Greece, a senior member of IEEE, a member of IEEE Optical Communications Committee, a member of the E.A.M.E.C, a member of the IECTC 84/WG 17 (IR Standardization Group) and a member of the CLC/TTF71-3 CENELEC Management Committee (IR free-Air Application Standard).

S.A. KOTSOPOULOS

Stavros Kotsopoulos was bom in Argos-Argolidos, Greece, on November 11, 1952. He received the diploma in Physics from Aristotelian University of Thessaloniki, Greece in 1975 and the diploma in Electrical Engineering from the University of Patras, Greece in 1984. Dr. Kotsopoulos received the M.Phil and Ph.D degrees from the University of Bradford, United Kingdom, in 1978 and 1986 respectively. He joined the Department of Electrical Engineering (post graduate division) of Bradford University, in 1990, as a visiting post-doctoral Research Fellow, working on a structure of new microcellular radio systems. Since 1987, he has been an Ast. Professor in the Dept, of Electrical and Computer Engineering at the University of Patraw, where he gives lectures on wave Propagation and Antennae design. Also, he is currently. Technical Supervisor of the research group working on various research programs supported by the Europpean Community. His current research interests focus on conceptual problems in cellular m obile radio com m unications networking, intermodulation interference, diffraction of Electromagnetic waves by knife-edge obstacles, low profile antennae, image processing and multimedia services. Dr. Kotsopoulos is a member of the Technical Chamber of Greece and also a member of the Union of Greek Physisists.