Coverage Probability

20

IEEE TRANSACTIONS ON BROADCASTING, VOL 43, NO. 1, MARCH 1997

THEORETICAL COMPUTATION OF THE COVERAGE PROBABILITY OF A DIGITAL TV TRANSMITTER IN AN ANALOGUE TV NETWORK USING A STOCHASTIC APPROACH Hatem MOKHTARI TDF-C2R (Broadcasting and Radiocommunications Research Center) 1, rue Marconi 57070 Metz, FRANCE Phone : +33 387 20 75 61 e-mail : [email protected] Abstract In this paper, a theoretical computation method for a digital TV transmitter coverage probability is proposed. The computation make use of the assumption that the studied digital TV transmitter is sharing a co-channel frequency band with six surrounding analogue TV transmitters whose arriving signals are considered as co-channel uncorrelated interferers. Coverage probability expressions have been derived considering the digital TV signal as the wanted power and the remaining analogue TV signals as co-channel interferers. After several Monte Carlo sirnulation-based comparisons the utilized theoretical approach has shown to be in good agreement, with a much less important computation time. 1. Introduction

The introduction of digital terrestrial TV will have to take into account the existing analogue networks, especially in the co-channel situations. Subsequently, there is a need in developing a comprehensive planning method dedicated to the future hybrid TV broadcasting network (i.e digital TV and analogue TV services). By evidence, this would lead to a reliable deployment of digital TV networks bearing in mind the superimposed constraints of the existing traditional TV networks, namely protection ratios and transmitting powers. Publisher Item Identifier S 0018-9316(97)02607-3.

In this paper, a digital TV transmitter is assumed to share a cochannel frequency band with six analogue TV transmitters. For the sake of simplicity, the considered hybrid TV network is supposed to have a regular network configuration which allows the use of the well-known honey comb geometrical structure. Rather than using a deterministic computation method for the determination of digital TV coverage, a more accurate approach based on the statistical description of both wanted and interfering powers is proposed. Indeed, the multidimensional phenomenon can be simplified into a two dimensional equivalent problem. Thus, assuming a lognormal shadowing of each individual interfering analogue power [ 11, a single and equivalent probability density function (PDF) for the interfering apportionment can be calculated [2] using an exact formulation. Accordingly, a two-by-two procedure, for the computation of the equivalent mean and variance of the corresponding equivalent Gaussian PDF of the set of six interferers, can easily be used provided that each statistical parameter is assumed to be known by means of propagation models (mean value) and measurements (standard deviation). Regardless to terrain information considerations, especially in the VHF and UHF studied bands, the ITU-R 370 propagation model has been used only for testing. However, one could use any other home-made propagation model provided that its reliability, with comparison to measurements, is proved in these two frequency bands.

0018-9316/97$10.00 0 1997 IEEE

21

The key-feature of our method is the fact of using a double analytical integration instead of computing seven inter-dependent multiple integrals which require the use of numerical timeconsuming integrations. The idea has been previously used [3] in view of reducing the complexity of the problem which obviously leads to a less important computation time. This method is indeed an easy-to-use and versatile procedure especially when the equivalent Gaussian PDF is analytically given [2]. Afterwards, coverage probability for digital TV transmission is computed by integration taking account of the minimum required power for digital TV reception and the co-channel protection ratio.

2. Theory In their theoretical investigation, Schwartz and Yeh [2] have showed that a set of N log-normal variates can be represented accurately by a single lognormal variate. The authors have also derived the correspondent mean and variance exprcssions for the equivalent Gaussian PDF. This idea is simply applied in this paper where the six uncorrelated analogue TV signals are represented by a single random signal whose PDF is calculated using their analytical results. The honey comb structure in figure 1 illustrates the positioning of each digital and analogue TV transmitters.

Although digital TV and analogue TV are supposed to follow a log-normal shadowing variability law, the statistical parameters (namely, mean and variance) are different. Indeed, digital TV standard deviation is less than analogue TV signal one [4]. In addition, digital TV coverage requirements approach 90 % (for signal abrupt failure reasons) which is not necessarily the case of analogue TV receiving conditions. That is the reason why we need to model accurately the coverage probability, unlike the ITU-R methods [5] whose accuracy tends to be limited for higher probability values. Before computing the coverage probability (or outage probability), it is of great interest to focus the efforts on the equivalent PDF statistical parameters for the set of six interferers. According to the procedure given by Schwartz and Yeh, i.e achieving the computation of mean and variance using a two-by-two procedure, we have computed the Gaussian PDF statistical parameters for the interference part. However, the wanted signal PDF remains log-normal (or Gaussian in dBm) which leads to the use of the integral below

~31:

Where the variables in (1) are expressed in dBm which leads to the use of Gaussian PDF instead of log-normal PDF and R is nothing but the co-channel protection ratio. The minimum required received power is noted So and is also expressed in dBm. Consequently, the PDFs in (1) are given as follows :

Figure 1 : Geometrical Configuration of the Hybrid TV Network

8

...

Analogue TV transmitter Dipital TV transmitter

and, with htequrv, = I , one could easily write

22

0,= 5.2 dB. In our simulations we have taken R = 0 dB and So= - 100 dBm.

where the statistical parameters in (3), namely m, and are calculated by means of the abovementioned procedure [2] (i.e two-by-two calculations). The analytical computation of the integrals in (1) is detailed in the appendix. Accordingly, equation (1) could be recasted in an extended form given as follows :

where the constants in (4) are also given in appendix.

3. Results As it has been previously pointed out, the aim is to propose a calculation method whose accuracy and computation time are optimized. This means that our method ought to be compared to Monte Carlo simulations which is considered as a reference method despite its very important time-consuming aspect. Only for the sake of evaluating the reliability of our analytical method, we have taken as an example a set of six uncorrelated interferers accounting for the analogue TV apportionment, and a unique value for the digital TV part. Results for received powers at 40 km from the digital TV transmitter are reported in table 1 given below :

Table 1 I Individual co-channel mean values for the 6 interferers in dBm

These six signals are supposed to be statistically independent with equal standard deviations of 9.5 dB in the UHF band. Which yields, using the Schwartz and Yeh procedure, m,=-62.1 dBm and

It is well known that the coverage area depends on the transmitting parameters (ERP, antenna heights, polarization, frequency, etc.). All of these considerations are neglected since we suppose that, a priori, one should have calculated the received powers from both wanted and unwanted contributions. The above mentioned mean values for analogue TV are supposed to be computed by means of any propagation model. In our case we have used the ITU-R 370 Recommendation [6] for a smooth terrain (Le for Ah = 50 m) at a given point within the digital TV transmitter coverage area. This propagation model has given a mean value for the digital TV (wanted) power of m, = -46.14 dBm at 40 km (with 8 = (Ox,OM)= 90’) from the digital TV transmitter with a standard deviation Os = 6.0 dB as it has been measured recently according to ITU-R Contributions [6].

ATV4

ATV5

A M

ATV6

Figure 2 : Geometrical Representation of the Coverage Probrhilq Cnlsulation Point

Digital TV Transmitter

0 Analogue TV Transmitters

Preliminarily, a comparison between the studied theoretical investigation and the Monte Carlo procedure is in fact of great interest. It means that the accuracy of this method have to be estimated using the whole six

23

interferers. Figure 3 given below shows the cumulative distribution hnction (probability that the sum of the six interferers is less than the power on the absicssa axis) using both Monte Carlo (taken as a reference) and the above described theoretical method of Schwartz and Yeh. Figure 3 :Cumulative Distribution Function Using the Theoretical Method Compared to Monte Carlo Procedure

4

-time

m,,m,and m3 Computed Probability with Monte Carlo Computed Probability with Schwartz & Yeh

98.60 %

98.68 %

97.63 %

97.78 %

137

172

T(Monte

Carlo)/T(Theor

Y)

2 r;

Since our method is intended to improve the computation time compared to Monte Carlo procedure, two different configurations have been considered. First we consider only the nearest three interferers according to figure 1 and in a second step we introduce the three remaining far contributions, which obviously increases the computation time especially with the Monte Carlo method. Noting that one could add the farest contributions one-by-one in view of studying the run-time variation law and any eventual non-linear considerations when the number of transmitters is getting higher. Table 2 below reports the results for a distance of 40 km from the digital TV transmitter on the Y axis as mentioned above.

We also compared graphically the coverage probability for the hybrid TV network computed by the two methods of interest. Figure 4 illustrates the variations of this probability with the distance. Thus, starting from 40 km along the Y axis to 66 km allows the computation, for each point, of the coverage probability taking account of interferences. Figure 4 : Comparison of Monte Carlo and Theoretical Model for Coverage Probability Calculation

According to the results obtained in figure 4, comparisons have yield an average error of 3.1 1 % between the Monte Carlo method, considered as a reference, and our theoretical method. These comparisons

24

concerned the 11 points within the 40 to 66

km range with a step of 2 kilometers. Indeed, the Monte Carlo method, because of its very important computation time, has restricted our sampling which lead us to consider only few points for the sake of comparing the accuracy of our method. 4. Conclusion As a preliminary work on planning aspects concerning the deployment of digital TV transmitters within the existing analogue TV networks, the obtained results with an analytical computation method showed few discrepancies when compared to the Monte Carlo simulations, which is well recognized as a reference method. Also, for further studies dealing with multiple interferences, especially in the case of digital single frequency networks (SFN) such as DAB for example, this method can easily be implemented since the multidimensional phenomenon can be treated as a two dimensional problem (including the wanted received power). Furthermore, especially for TV reception in general, the calculation of good reception areas can be improved by taking into account the receiving antenna pattern, which is not considered in our investigation. This could be included in link-budget calculation and precisely for received mean power values.

Moreover, it seems that there is no limitation concerning the number of interferers except that the previously described two-by-two procedure will have to be repeated systematically each time it is necessary depending obviously on the number of preponderant interferers.

Components ' I , Bell Syst. Tech. J., 61, pp. 1441-1462, 1982.

[3] K. W. Sowerby, A. C. Williamson, " Outage Probability Calculations for a Mobile Radio System Having two LogNormal Interferers ", Electronics Letters, Vol. 23, No. 25, pp, 1345-1346, 3rd Dec. 1987. [4] DOC,ITU-R TG 11-3WK5, 'I OFDM Digital Television Location Variation Statistics ' I , October 1994.

[ 5 ] ITU-R Report 945-2, Method for the Assessment of Multiple Interference " I'

Appendix

Computation of Coverage

Probability Starting with the expression given above by equation (1) one could simply recast it in the following :

where the inner integral, say F( S ), can be calculated as follows :

Using the probability function given by :

2

(A3)

one could write the coverage probability as References [ l ] W. C. Y. Lee, Mobile Communications Engineering, McGraw-Hill, 1982.

[2] S. C. Schwartz, and Y. S. Yeh, On the Distribution Function and Moments of Power Sums with Log-Normal

which comprises two terms, one depending on the apportionment of interference and

25

the other remaining one is completely independent on interference. Subsequently, we can easily find the following expression and

where the constants and fbnctions in (A5) are given by :

The coverage probability in (A5)is then quite manageable and one could easily use this result for coverage prediction using a stochastic approach taking account of the required minimum power and the COchannel protection ratio.