Local Coverage Probability Estimation in Single ... - Semantic Scholar

Local Coverage Probability Estimation in Single Frequency Networks

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Agnes Ligeti and Slimane Ben Slimane, Member, IEEE

Abstract

In OFDM based Single Frequency Networks (SFNs) such as Digital Audio Broadcasting (DAB) and Digital Video Broadcasting (DVB), the mobile receiver is provided by the same signal from several synchronous transmitters at the same carrier frequency. This transmitter diversity structure makes SFNs very ecient in fading channels with an excellent spectrum saving, and very good power utilization over the service area. However, depending on the position of the mobile within the service area, these signals arrive with dierent delays causing possible echoes at the receiver. These echoes contribute to both the useful and the interfering parts of the signal. As the average received power in radio channels is a stochastic process, such contribution creates correlation between the dierent components of the Signal-to-Interference Ratio (SIR). In broadcasting practice, correlation was not taken into account when estimating the distribution of the SIR and the local coverage probability. This paper considers the eect of correlation on the estimation of the SIR and the coverage probability in OFDM based SFNs. Analyses are provided to show that in some situations the estimation error can be considerable when neglecting this correlation. To obtain a better estimation, the correlation coecient between the total useful and total interfering parts of the received signal is rst evaluated and then used as a correction factor in estimating the mean and standard deviation of the SIR. The obtained results show that such method can signi cantly reduce the estimation error of local coverage probability without increasing the computation complexity. 1

Keywords

OFDM, single frequency network, digital audio broadcasting, digital TV, log-normal random variables, correlation I. Introduction

Both Digital Audio Broadcasting (DAB) [2] and Digital Video Broadcasting (DVB) [3] systems use Orthogonal Frequency Division Multiplexing (OFDM) to facilitate Single Frequency Networks (SFNs). This modulation scheme allows all the transmitters in the radio network to transmit the same information in the same frequency block simultaneously. The various incoming signals from the dierent transmitters are seen as replicas (echoes) of the same signal and can combine positively if their temporal spread is compatible with the selected duration of the guard interval of the OFDM modulation. However, these signal replicas also create arti cial multi-path propagation which might not be properly handled by the receiver. For instance, signals from transmitters very far away from the receiver will cause interference. This interference, referred to as self interference, consists not only of Inter-Symbol Interference (ISI), but also of Inter-Channel Interference (ICI) due to losses in orthogonality between the OFDM sub-carriers. Thus, a mobile receiver in an SFN will experience, besides the regular co-channel interference from neighboring SFNs that use the same frequency block, self interference from its own transmitters. In order to successfully design any SFN system, it is important A. Ligeti and S. B. Slimane are with the Radio Communication Systems Laboratory at the Dept. of Signals, Sensors and Systems, Royal Institute of Technology, 100 44 Stockholm, Sweden. E-mail: fagnesl, [email protected] 1 Part of the paper has been presented in 50th Vehicular Technology Conference, 1999, Amsterdam [1]

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to understand and analyse the eects of self interference as well as of the co-channel interference on the system performance. The eect of self interference on OFDM based SFNs have been investigated in the literature [4], [5], [6]. When dealing with coverage estimation, the shadowing of the desired signals is one of the main sources of performance degradation. To model the location (and time) variation of the shadow faded signal components, they are commonly assumed to be random variables (RVs) with log-normal distributions. In all studies on SFN statistical independence between the desired signal power and that of the interfering power have been assumed [7], [8]. This assumption may be reasonable in the presence of co-channel interference alone but not when self interference is present. As the late echoes arriving at the mobile receiver contribute to both desired (useful) and interfering parts of the received signal, a strong correlation will exist between their average received powers. This correlation should not be ignored and need to be considered when estimating the local coverage probability of SFNs. In order to estimate the local coverage probability, the distribution function of the Signal-toInterference Ratio (SIR) has to be calculated. This requires the evaluation of the total useful power, the total interfering power, and their ratio. Since we are dealing with random variables, these three steps are far from being trivial. The situation is even more complicated in SFNs since both useful and interfering powers are linear combinations of several log-normal components, and due to self interference correlation exists between them. Even by ignoring such correlation, the exact statistical treatment of the distribution of a sum of several log-normal components is very dicult to carry out numerically. So far no exact closed form solution for this distribution has been found. Instead, a number of approximation methods have been developed during the past years. These methods commonly assume that such sum can be well approximated by a log-normal distribution. In the analog broadcasting practice, the power sum method, the simpli ed multiplication method, and the log-normal method (LNM) are commonly used [9]. The comparison studies in [10] and [11] have found that among these three methods the LNM gives, in general, the best estimation accuracy. However, the obtained estimation accuracy was not satisfactory in the high coverage probability range (i.e., above 90%). Two extended versions of the LNM with the introduction of some correction factor, the k-LNM and t-LNM methods, were presented in [8]. The problem of evaluating the distribution of a sum of several log-normal random variables has also received a lot of attention in cellular radio systems. For that matter, several approximation methods have been proposed and studied in the literature. Among these methods we nd the Wilkinson's method [12], the Schwartz and Yeh's method [12], the cumulant matching method [13], and Farley's method [13]. A detailed comparison of these methods for uncorrelated random variables can be found in [13]. When correlations among the individual log-normal components exist, the above techniques may

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fail and provide unacceptable estimations. To handle correlated log-normal components, extended versions of some of the above techniques were introduced [14]. The Schwartz and Yeh's method was extended in [15]. The Wilkinson's method, the extended Schwartz and Yeh's method, and the cumulant matching method were used to compute the distribution function of a sum of several correlated lognormal random variables and their estimation accuracy were compared in [14]. However, very little has been done in the literature concerning the computation of the distribution function of the SIR when possible correlation exists between the total useful and total interfering signals. A brief analysis in [14] showed that such correlation has strong impact on the distribution function of the SIR and thus the coverage probability. To the best knowledge of the authors, no method has been suggested to determine such distribution for the general case where arbitrary correlation exists among useful and interfering components, which as mentioned earlier very typical in SFNs. In this paper the cumulative distribution function (CDF) of the SIR and the local coverage probability are analyzed in dierent signal environments. For determining the total useful and total interfering components we apply existing methods. Then we propose a method to approximate the correlation coecient between the total useful and total interfering power as the function of the correlation between the individual components. Assuming that the SIR is log-normally distributed the derived correlation coecient can be used as a correction factor to calculate its mean and standard deviation. We refer to this assumption as the Gaussian assumption. We present an analysis of the validity interval of the Gaussian assumption, and provide an alternative estimation method for the cases in which the estimation error is expected to be high. We use Monte-Carlo simulation to validate our methods. We show that for signal constellations in which signals outside of the guard interval have signi cant contribution to the total useful and/or to the total interfering signals, our method yields signi cant improvement in the accuracy of estimating the local coverage probability. The rest of the paper is organized as follows. We develop a model for the received OFDM signal in SFN in Section II. In Section III we discuss the distribution of the individual power components in the SIR and provide models for the two most important sources of correlation among these components. In Section IV we present the conventional coverage probability estimation method and develop a method to estimate the correlation coecient between the total useful and total interfering power, which is then used as a correction factor in the local coverage probability estimation. Similar to the conventional method, the extended method is based on the Gaussian assumption, i.e., it assumes that the SIR can be well approximated by a log-normal distribution. We provide several numerical examples in Section V, including cases in which the Gaussian assumption holds and also cases in which it does not hold. A further analysis of the validity conditions of the Gaussian assumption is provided in Section VI, and an improved approximation method is developed to provide accurate coverage probability estimates for those (practically interesting) cases in which the Gaussian assumption is not

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τ’2

τ1

τ2 τ’2’

τ’1

τ3

τ4

Fig. 1. Single frequency network: multi-path propagation with natural and arti cial echoes

valid. Concluding remarks are given in Section VII. II. System Model

A. Received OFDM Signal

The transmitters in a Single Frequency Network (SFN) are fed with synchronous data and transmit identical signals occupying the same frequency block using OFDM modulation. The equivalent lowpass of the transmitted OFDM signal is

s(t) =

NX c ?1 k=0

j 2k Tu ; a(0) k e t

? Tg  t  T u ;

(1)

where Nc denotes the total number of sub-carriers, a(0) k is the information symbol on sub-carrier k during the 0th OFDM block interval, Tu is the duration of useful part of the OFDM symbol, and Tg is the duration of the guard interval. The multiple copies of the signal arrive at the receiver antenna with dierent delays. The time dispersion is caused by two main mechanisms (see Fig. 1): The natural dispersion caused by re ections in the vicinity of the receiver; and the arti cial delay caused by the reception of the same signal from the dierent transmitters at slightly dierent delays. Since the block duration, Tt = Tu + Tg , in SFN applications is very large compared to the natural time dispersion, the eect of the natural dispersion can be neglected. Thus, the received signal at the receiver can be written as:

r(t) =

M X i=1

his(t ? i) + z(t) + n(t);

(2)

where M is the total number of transmitters in the SFN, i is the propagation delay from transmitter i to the receiver, and hi is a tap gain which is modeled as a zero mean complex Gaussian process with average power n

o

Pr;i = E jhi j2 = Pt;i Lp (di )10i =10 :

(3)

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In (3) Pt;i denotes the eective radiated power of transmitter i, and i is a Gaussian random variable with zero mean and variance 2i describing the local shadow fading environment. The function Lp(di ) represents the path gain which is a decreasing function of the distance between transmitter i and the receiver, di . In the simulation results given in Section V we calculate Lp(di ) based on the existing model provided by ITU-R [16]. Such model takes into account the positions of the transmitters beyond the radio horizon. The additive term z (t) in (2) represents the external interference (co-channel interference) coming from other SFNs using the same frequency block, and n(t) is the background noise. The received signal r(t) is then passed through the OFDM demodulator. We assume that the receiver synchronizes to the rst arriving copy of the signal, and that out of the M copies K copies arrive within the guard interval; the other M ? K copies arrive after the guard interval, but before the end of the OFDM block. With a slowly varying channel the tap gains stay constant during the OFDM block and the received sample at sub-carrier k is obtained [17] as:   rk = H (k) ? G0 (k; k) a(0) k ?

NX c ?1

l=0;l6=k

G0(k; l)a(0) l +

NX c ?1

where H (k) is the channel transfer function at sub-carrier k:

H (k) = and2

M X i=1

l=0

G0(k; l)a(l ?1) + zk + nk ;

hi e?j2k Tui ; 

e?j2(l?k) Tu G(k) ? G(l) j 2(l ? k) with G(k) representing the transfer function of the M ? K late paths: Tg

G0(k; l) =

G(k) =

M X

i=K +1

hie?j2k Tui : 

(4)

(5)

(6)

(7)

The received sample rk consists of two main parts: The useful part of the signal which carries the information symbol a(0) k , and the interfering part which is a linear combination of background noise, external interference, and self interference (Inter-channel Interference (ICI) and Inter-Symbol Interference (ISI)). Notice that all the paths that arrive within the guard interval contribute only to the useful part of the signal, but the other late paths contribute both to the useful and to the interfering parts of the signal. B. Signal-to-Interference Ratio

The ratio between the average powers of the useful and interfering parts of the signal represents the Signal-to-Interference Ratio (SIR) denoted by ?, and is used in this paper as a measure of the 2

Notice that 0 ( G

k; k

) = ? j?21 dGdk(k) .

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performance of the receiver at a given location. Reconsidering the expression of the received OFDM sample given in (4), the SIR is obtained as follows: o

n

E jH (k) ? G0(k; k)j2 o o n ?= n ; P E jG0(k; k)j2 + 2 lN=0c ?;l16=k E jG0(k; l)j2 + z2 + n2

(8)

o

n

where for simplicity we assume that E jal j2 = 1; 8 l. The symbol n2 denotes the variance of the background noise, and z2 is the co-channel interference (from transmitters of other SFNs using the same frequency block), which { assuming L co-channel interferers { can be written as

2 = z

MX +L i=M +1

Pr;i:

Replacing H (k) and G0 (k; l) by their expressions in (8) and taking their averages, the SIR becomes P

M P w ( ?  ) i 0 i=1 r;i ? = UI = PM PM +L 2; i=1 Pr;i [1 ? w (i ? 0 )] + i=M +1 Pr;i + n

(9)

where U and I denote the total useful and the total interfering power, respectively. The function w(i ? 0 ) is a weighting function which is dependent on the receiver window type and the detection scheme ([4], [5], [6]), and is a function of the delay of the signal, i , relative to the starting point of the receiver detection window, 0 . In this paper we apply the following quadratic function for w(
 ): 8 1; > < 

w(
 ) = > :

 Tu ?
 +Tg 2 ; Tu

0;

0 
 < Tg Tg 
 < Tf otherwise;

(10)

where Tu and Tg are as de ned earlier. The length of the quadratic part of w(
) is set by Tf , where Tf = Tg + Tu in DAB [4] and Tf = 2Tg in DVB [5]. This quadratic function is illustrated in Fig. 2 where we can see that depending on the relative propagation delay, the received signal from transmitter i may contribute to the useful part, to the interfering part, or to both parts of the received signal. Fig. 2 also shows that paths arriving within the guard interval do not introduce any interference. C. Local Coverage Probability

The accuracy of the predicted received power is limited by the resolution of the terrain database, which typically ranges from 50 m  50 m to 1000 m  1000 m. The coverage estimation is con ned to these small square areas of the terrain, which we call area elements throughout the paper. Consider the expression of the local received power from a given transmitter i at some receiver location as de ned in (3). This power usually changes much faster with the random variable i as compared to its variation with respect to the distance di . Thus, within a small area element the local received power Pr;i can be modeled as a log-normal random variable, i.e., its logarithm Qi = 10 log10 Pr;i is a

7 w(∆τ) 1

DAB

DVB

∆τ Tg

Tu+Tg

Fig. 2. Weighting function for DAB and DVB

Gaussian RV with parameters mqi and qi where

mqi = 10 log10 Pt;i + 10 log10 Lp(di ) and

qi = i :

Since the received power is a random variable, the received useful and interfering power from transmitter i, denoted by Ui and Vi , respectively, and the interfering power, Wj , of the j th external transmitter are also random variables:

Ui = Pr;i w (i ? 0 ) ; 1  i  M Vi = Pr;i [1 ? w (i ? 0 )] ; 1  i  M Wi = Pr;i+M ; 1  i  L:

(11) (12) (13)

Using the notation introduced in (11){(13), the SIR can be rewritten in the following form


+ UM ? = UI = V +


+ VU1 + : (14) 1 M + W1 +


+ WL + n2 The above expression indicates that, both useful and interfering parts of the received signal have an average power which is a linear combination of several random variables and the SIR is also a RV. An important performance measure commonly used in broadcasting practice is the so-called local coverage probability de ned as the probability that the SIR exceeds a system speci c threshold 0 , also known as protection ratio in an area element: 4 pc = Prf?  0 g:

The total coverage probability within the target service area can then be derived from the local coverage probability of all area elements contained within the service area. To guarantee high homogeneous coverage within the service area, usually requirements set for the local coverage probability

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of each area elements. In digital radio systems very high requirements are set for such probability due to the rapid degradation of the reception quality with unsucient SIR. For example, for a good reception quality pc should be above 90 ? 99% in DAB networks and above 70 ? 95% in DVB networks [8]. To estimate the local coverage probability the probability distribution function of the SIR needs to be known, which requires that sucient information about the distribution and cross correlation of the individual components in (14) is available. III. Modeling the Distribution and the correlation of the Components in SIR

If Pr;i is assumed to be log-normally distributed, Wi is also a log-normal random variable. However, such a log-normal assumption may not apply to the random variables Ui and Vi . In order to determine the probability distribution function of these two RVs and that of the SIR, one needs to investigate the variation of the weighting function w(
 ) over the area element and the cross-correlation between Ui and Vi . An additional source of correlation is the correlation of the shadow fading of the signals, which can be considerable between signals coming from the same direction. These issues are further discussed in the next two subsections. A. Distribution of the Individual Signals and their Correlation Due to Self-Interference

Let us consider the received signal from a given transmitter i at some location of the receiver within the service area. Depending on its relative delay
i = i ? 0 with reference to the rst received signal path, this signal can contribute to both useful and interfering parts of the signal at the receiver. These two contributions are random variables given by Ui and Vi . One can easily see from (10) that if
i  Tg this signal contributes to the useful part only, thus the interference Vi is zero and Ui = Pr;i is log-normally distributed. Similarly when
i  Tf , the signal contributes to the interfering part only, yielding a useful part Ui equals zero and log-normally distributed interfering part. However, when
i falls within the critical region Tg <
i < Tf , both Ui and Vi are non zero with a strong correlation between them and unknown probability distribution functions. The above critical region and the weighting function w(
i ) can be rearranged in such a way that they contain distances instead of delays. We therefore can write the following instead

dg 
di = di ? d0 = c
i  df ; 8 1; > < 

w(
d) = > :

 du +dg ?
d 2 ; du

0;

0 
d < dg dg 
d < df otherwise;

(15)

(16)

9 TABLE I

Parameters for DAB/DVB transmission modes.

Transmission Tu Tg Ts dg df Mode [s] [s] [s] [km] [km] DAB Mode I 1000 246 1246 73.8 373.8 DAB Mode II 250 62 312 18.6 93.6 DAB Mode IV 500 123 623 36.9 186.9 DVB 8k-1/4 896 224 1120 67.2 134.4 DVB 2k-1/4 224 56 280 16.8 33.6 with

dg = cTg ;

df = cTf ;

du = cTu;

(17)

where c is the speed of light, d0 is the distance from the closest transmitter to the mobile receiver, and di is the distance between the ith transmitter and the mobile receiver. The two limit distances are dependent on the length of the guard interval and the OFDM block length. Values of dg and df for some representative DAB and DVB transmission modes are shown in Table I. Within the (open) interval (dg ; df ), the weighting function w(
di ) has a location variation. How fast this varies within the observation area element is quite important as it has direct eect on the distribution function of the random variables Ui and Vi , and on the cross-correlation between them. The variation of the weighting function w (
di ) over the area element depends on the DAB/DVB transmission mode, the size of the area element, and the relative position of the area element with respect to the transmitters. Let us consider the geometry of Fig. 3 and de ne 0 as the azimuth angle of transmitter 0 (the closest transmitter to the mobile receiver) and i the azimuth angle of transmitter i. As the transmitter positions within the SFN are xed, the distance between every pair of transmitters is constant. De ning by Di;0 the distance between transmitter i and transmitter 0, the relative distance
di can be written as
di =

q

Di;2 0 ? 2di d0 [1 ? cos (i ? 0 )]:

(18)

We notice that the relative distance is dependent on the azimuth dierence i ? 0 . This relation controls the variation of the weighting function w(
di ), yielding no variation if i = 0 , and the highest variation when k = 0 + , i.e., when transmitter i, transmitter 0, and the mobile receiver are on a straight line with the mobile receiver in between. The position of the receiver within the area element A (shaded in Fig. 3) is a random variable and can be considered as uniformly distributed. Thus, the weighting function w (
di ) is also a random variable. As a representative case, in Figure 4 we plotted the standard deviation of w(
di ) and 1 ? w(
di ) in dB within the area element, as a function of the relative position of the area element

10 Transmitter i Reference transmitter

α0 Pi

d0

A

a

di

αi

a

U = P
w ((d ? d0 ) =c) V = P
[1 ? w ((d ? d0 ) =c)] i

i

r;i

i

r;i

i

Fig. 3. Reference gure to evaluate the eect of w(
) on the distribution of the useful and interfering power components U and V from remote transmitter i, and on their correlation coecient r . i

xi ;yi

Correlation Coefficient, rx ,y i i

1

2.0

rx , y i i

0.9

1.8

0.8

1.6

0.7

1.4

a = 100m a = 500m

0.6

1.2

0.5

1.0

0.4

0.8

0.3

σ(1-w)

0.6

σw

0.2

0.4

0.1

0.2

0 0

Fig. 4. Correlation coecients r 0 = 0,  = .

10

20

30

40

50

60

70

80

90

Relative distance of the area element from transmitter i, ∆d i [km]

xi ;yi

Standard deviation, σ [dB]

i

100

and standard deviations of w(
d ) and 1 ? w(
d ) for DAB Mode II, i

i

i

between the two transmitters. The position of the area element is de ned by its center coordinates. For a xed i ? 0 , the relative angle of the area element with respect to the transmitters has negligible impact on the magnitude of the variation. We also plotted the correlation coecient rxi ;yi between the logarithmic equivalent of Ui and Vi , i.e., Xi = 10 log10 (Ui ) and Yi = 10 log10 (Vi ), respectively. We assume DAB Mode II, 0 = 0 and the worst case constellation i ? 0 = . The two sets of curves represent the a = 100 m and a = 500 m cases. The results were obtained by Monte Carlo simulation. It can be seen that apart from the borders of the critical region (near
di = dg and
di = df ), the standard deviations are well below 0:2 dB, which is negligible compared to the typical 4:0{8:0 dB standard deviation of Pr;i . Also, the correlation coecient stays very close to 1 over almost the entire dg 
di  df region. We can conclude that assuming w(
di ) to be constant over an area element also within the critical region is reasonable. Consequently, Ui and Vi can be assumed to be log-normally distributed, i.e., Xi is a Gaussian RV with mxi = mqi + 10 log10 (w(
di )) and xi = qi , and Yi is a Gaussian RV with myi = mqi + 10 log10 (1 ? w(
di )) and yi = qi . The correlation coecient between the useful and

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interfering components coming from the transmitter signal can be assumed equal to 1. B. Correlation between Individual Signals due to Shadow Fading

As earlier indicated correlation may also exist between signals coming from dierent transmitters. To describe such correlation we use a simple model where it is assumed that the shadow fading gains are correlated with respects to the dierence in the azimuth angle between the receiver and the two transmitters (see Fig. 5). The model can capture the eects of having several transmitters shadowed by a large obstacle. However, it assumes that if a nearby transmitter is shadowed, all far-away transmitters behind this site will also be shadowed and vice-versa. The correlation coecient between the received powers of any pair of useful transmitters (or interfering, or useful and interfering) can be modeled ([18], [19]) as

rqi qj = r(ji ? j j) = ji ?j j;

(19)

where the parameter 0   1 determines the strength of the correlation. position of the receiver

∆αij

transmitter i

transmitter j

4  ?  ) between two transmitters and the receiver Fig. 5. Angle dierence (
 = ij

i

j

IV. Local Coverage Probability Estimation Under the Gaussian Assumption

As derived in the previous section we can assume, that all components in SIR can be modeled by log-normal RVs. Thus, the total SIR can now be written in a more general form +


+ UM ? = UI = UI1 ++UI 2 + 1 2


+ IN where both Ui and Ii are log-normal random variables. The Xi and Yi are their Gaussian equivalent, i.e., Xi = 10 log10 Ui and Yi = 10 log10 Ii , respectively. We can model the noise as a special log-normal RV with zero variation.

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To estimate the coverage probability the probability density function of both U and I need to be evaluated rst. In the beginning of the paper we have discussed several methods for estimating the sum of log-normal RVs. All methods are based on the assumption that a sum of log-normal RVs is well approximated by a log-normal RV [12]. Therefore, we can describe the total useful power by

S = 10 log10 U; assumed Gaussian with mean ms and standard deviation s , and the total interfering power by

R = 10 log10 I; also assumed Gaussian with mean mr and standard deviation r . The estimation of the parameters ms, s , mr , and r can be done using anyone of the existing summation methods (preferably one that can take possible correlations between the components into account). Once the distributions of U and I are obtained, the distribution of function of ? is then derived. Assuming joint normality between S and R, the random variable ? is also log-normally distributed, or = 10 log10 ? = S ? R p

is a Gaussian random variable with mean m = ms ?mr and standard deviation  = s2 ? 2 rsr s r + r2 , where rsr denotes the correlation coecient between S and R. We refer to the above approximation as the Gaussian assumption (GA). Under the GA, the coverage probability can be estimated as:

p^c = Prf?  0 g = Pr f  10 log10 0g = Q 10 log10 0 ? m !

= Q 10plog210 0 ? (ms ? m2r ) ; s ? 2rsr s r + r

(20)

where Q(
) is the complementary cumulative distribution function of a zero mean, unit variance Gaussian distribution. The only unknown parameter in (20) is rsr . Conventionally, the correlation between useful signal and interference has always been regarded as very small and was ignored in the estimation of coverage probability, i.e., rsr = 0. This assumption proved to be reasonable in multi-frequency networks. The numerical results in the following section will show that in SFNs this correlation factor can be considerable due to the two major correlation sources discussed in the previous section. It is obvious from (20) that a positive correlation coecient increases the coverage probability; neglecting such a positive correlation yields conservative coverage probability estimates. There is however no known method to carry out the estimation of this correlation coecient. Here we provide a simple estimation method, for an arbitrary M  1 and N  1, as a function

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of the correlation coecients between the individual signal components, their means, and standard deviations. For presentation clarity we use the natural base logarithm in the following formulation. Let us rst introduce the random variable  = S + R: Since S and R are jointly Gaussian,  is also Gaussian. By matching the rst moment of e with the rst moment of the product UI , we have E [UI ] = E [eS eR ] = = E [(eX +


+ eXM )(eY +


+ eYN )] 1

1

= =

N M X X

i=1 j =1 N M X X i=1 j =1

E [e(Xi +Yj ) ]

e(mxi +myj )+ (xi +yj +2rxiyj xi yj )=2 2

2

2

= v

(21)

and E [UI ] = E [eS eR ] = E [e(S +R) ] = E [e ] = em +  =2 : 2 2

(22)

where  = ln(10)=10  0:23026. Utilizing the fact that m = ms + mr and 2 = s2 + r2 + 2 rsr s r , (21) and (22) can be solved for rsr : ?
2 [ln (v) ?  (ms + mr )] ? 2 s2 + r2 rsr = (23) 2 2 s r We demonstrate in Section V that even this simple method can provide rather accurate estimates for rsr , and the estimation accuracy of the local coverage probability is also improved by using rsr (in many practically interesting cases). V. Numerical Results

We consider the following two SFN con gurations, both based on the hexagonal transmitter layout shown in Fig. 6. The rst case is a symmetrical arrangement in which all the 19 transmitter sites are used, with 75 m antenna height and emitted power 500 W. The second case is asymmetrical in which the three encircled sites have 150 m antenna towers and emitted power 5 kW, the sites enclosed in rectangles use 75 m towers and emit 50 W, all other sites are unused. To model the external interference six transmitters with antenna height 150 m and power 10 kW are placed at 60 km from

14 reference path

D=15km

Fig. 6. SFN layout with 19 transmitter sites. In the symmetrical case all 19 sites are used with antenna height 75 m and emitted power 500 W. In the asymmetrical case encircled sites use 150 m and 5 kW, sites in rectangles use 75 m and 50 W, the other sites are not used.

the border of the SFN with a 60 degree angle separation (not shown in the gure). For both cases, the noise level is xed to n2 = ?140 dBW, which practically means that the eect of the noise can be neglected compared to the self- and external interference. The received local mean power from each transmitter is calculated based on the ITU-R 370 propagation model [16] for frequency range 450{ 1000 MHz over land. The standard deviation of each component is assumed be 5:5 dB [8]. The shadow correlation between the signals is calculated by (19) using = 0:15, which is a representative value according to [19]. We assume transmission of DAB mode II, which corresponds to the parameters Tg = 62 s, Tu = 250 s and Tt = 312 s. We estimate the correlation coecient, rsr , and the local coverage probability, pc , along a reference path in the network, also shown in Fig. 6. To evaluate the accuracy of the estimates, reference values were obtained using Monte Carlo (MC) simulations. In the MC simulations 105 samples were generated for each power component Pr;i , 1  i  M + L. The correlation due to self-interference was automatically implemented by deriving Ui and Vi from Pr;i according to (11) and (12). The desired level of shadow correlations among the Qi = 10 log10 Pr;i components were achieved by the method of Scheuer and Stoller ([20], pp.505): We denote the required correlated normal RVs by Q = (Q1 ;


; QM +L ) with mean mq = (mq ;


; mqM L ), standard deviation q = (q ;


; qM L ) and correlation matrix rqq . Since the covariance matrix  = q qT rqq is symmetric and is positive de nite, we can factor it uniquely using Cholesky decomposition as  = HH T where the (M + L)  (M + L) matrix H = [hij ] is lower triangular. We generate Z = (Z1 ;


; ZM +L ) as IID N (0; 1) RVs and obtain Q as Q = mq + H Z . As a method for obtaining ms , s , mr and r we used the extended Schwartz and Yeh method, because it provides higher accuracy in estimating the standard deviations than Wilkinson's method. 1

+

1

+

15

a. Symmetrical SFN 1

b. Asymmetrical SFN 1

MC-SI+SF

MC-SI+SF

MC-SI

MC-SI

0.8

Estimation-SI+SF

Estimation-SI

Correlation Coefficient, rsr

Correlation Coefficient, rsr

Estimation-SI+SF

0.6

0.4

0.8

0.6

0.4

0.2

0.2

0

0

0

5

10

15

20

25

Estimation-SI

30

0

5

Distance from center [km]

10

15

20

25

30

Distance from center [km]

Fig. 7. MC-simulated and estimated r values along the reference path for the symmetrical (a.) and asymmetrical (b.) network con gurations in case of correlation environments SI and SI+SF. sr

10

a. Symmetrical SFN

0

10

b. Asymmetrical SFN

0

Reference point P2

10

-1

10

1-p c

1-p c

10

-2

10

-1

-2

MC 10

-3

MC 10

Conventional (rsr = 0)

-3

Conventional (rsr = 0) Estimation with rsr

Estimation with rsr

10

-4

0

5

10

Reference point P1

15

20

Distance from center [km]

25

30

10

-4

0

5

10

15

20

25

30

Distance from center [km]

Fig. 8. MC simulated and estimated coverage probability along the reference path for the symmetrical (a.) and asymmetrical (b.) network con gurations in case of correlation due to self-interference (SI).

The estimated correlation coecient rsr is, however, based on parameters estimated by Wilkinson's method, which in our experience yields better accuracy in estimating rsr . In Fig. 7 we plotted the correlation coecient rsr along the reference path obtained by our method and by MC simulation for the symmetrical and asymmetrical networks considering two sub-cases each: 1. Sub-case SI: Only the correlation introduced by the self-interference is considered, the individual Pr;i components are assumed to be uncorrelated. 2. Sub-case SI+SF: Correlations due to both self-interference and correlated shadow fading are considered. The results show that our simple rsr estimation method provides rather accurate estimates in all cases. Note that in the symmetrical network for sub-case SI (correlation only due to self-interference) the correlation coecient is approximately zero. In Figs. 8 and 9 we present coverage probability estimates. Each gure contains estimates obtained

16

1-p c

10

10

a. Symmetrical SFN

0

10

-1

10

1-p c

10

-2

10

b. Asymmetrical SFN

0

-1

-2

MC 10

-3

MC 10

Conventional (rsr = 0)

-3

Conventional (rsr = 0)

Estimation with rsr

10

Estimation with rsr

-4

0

5

10

15

20

Distance from center [km]

25

30

10

-4

0

5

10

15

20

25

30

Distance from center [km]

Fig. 9. MC simulated and estimated coverage probability along the reference path for the symmetrical (a.) and asymmetrical (b.) network con gurations in case of correlation due to self-interference and shadow fading (SI+SF).

via the Gaussian assumption without and with considering rsr , as well as reference estimates obtained by MC simulations. Fig. 8.a and 8.b contain results for sub-case SI in the symmetrical and asymmetrical SFNs, respectively. Fig. 9.a and 9.b provide the same set of estimates for sub-case SI+SF. Fig. 8.a illustrates that the eect of correlation introduced by the self-interference can be neglected in the symmetrical case due to the low signal level of the distant transmitters (i.e., whose signals contribute to both the useful and the interfering components). As expected from Fig. 7.a, neglecting rsr is reasonable in this case (i.e., that is why the two Gaussian estimates coincide). In the asymmetrical case where distant transmitters have high power, the eect of the correlation due to self-interference is not negligible. When the correlated shadow fading is taken into account (Fig. 9) the rsr -corrected p^c estimate is considerably better than the p^c with rsr = 0 for both network con gurations. It can be concluded from the gures that the proposed method can signi cantly improve the accuracy of the estimation compared to the conventional method, albeit the estimation error is still considerable at some parts of the path (see Figs. 8.b and 9.b). The error can be attributed to the fact that the Gaussian assumption of the SIR does not hold in these cases. The validity of the Gaussian assumption requires further investigations. VI. The Validity of the Gaussian Assumption and Possible Estimation Improvements

The results in Section V showed that in certain cases (e.g., as in the asymmetrical SFN example, Figs. 8.b and 9.b) the Gaussian p^c estimator fails to provide accurate estimates, despite the fact that rsr is estimated very accurately (Fig. 7.b). A closer analysis of the empirical distribution of ? in the critical points reveals that in the problematic cases the Gaussian assumption seems to fail. For demonstration purposes, in Fig. 10 we plotted the probability density function of = 10 log10 ? in two

17 0.16

0.14

Probability density of Γ

0.12

At reference point P2 0.1

0.08

At reference point P1

0.06

0.04

0.02

0

0

5

10

15

20

Γ [dB]

25

30

35

40

Fig. 10. The probability density of = 10 log10 ? in two representative reference points.

reference points on the path (they are shown in Fig. 8.b). In point 1 (P1) for which the pc estimation was successful (Fig. 8.b) the shape of the distribution is indeed close to be Gaussian. But in point 2 (P2), in which the estimation failed, the left part of the Gaussian curve is almost entirely missing. To explain the phenomena we introduce the following extended model for ?: c + U i eA + eB ? = UI c + (24) I i = eC + eD where U c and I c \considerably" correlate, while U i and I i are \(almost) independent" of any other terms. The reader should think of (24) as a result of a classi cation of the useful and interfering components, U1 ; : : : ; UM and I1 ; : : : ; IN , according to their correlation properties. Speci cally, U c is the log-normal sum of those Ui components that has strong correlation with one or more Ij components, I c is the log-normal sum of those Ij components that has strong correlation with one or more Ui components, and U i and I i are the log-normal sum of the rest of the Ui and Ij components, respectively. As a result of the classi cation, there is a very high correlation between U c and I c , correlations between any other pairs are close to zero. Let A; B; C and D be the corresponding Gaussian variables as shown in (24), which can be de ned by the means ma , mb , mc, md , and standard deviations a , b , c , d . Let furthermore rac be the correlation coecient between A and C . Using this model a signal constellation can be uniquely described by the above nine parameters. When shadow correlation is neglected, the above classi cation is very straightforward in SFNs: P A: U c = i2f1;::: ;M gjwi0 Pr;i (1 ? wi ), i.e., Group of interfering signals correlated with useful compo-

18 TABLE II

Distribution of SIR ( = 10 log10 ?) and estimation error of p^ in different signal constellations in (24). c

Signal constellation

Distribution of = 10 log10 ? I ma  mb Gaussian II m c  md Gaussian III ma  mb , mc  md Gaussian IV ma  mb , mb > md skewed to the right V mc  md , mb < md skewed to the left O: Over-estimation U: Under-estimation

Error of p^c

? ? ? O U

nents. P P D: I i = i2f1;::: ;M gjwi=0 Pr;i + j 2f1;:::;Lg Pr;j + n2 , i.e., group of interfering signals independent of useful components. Signals from far-away transmitters, whose signal contributes only to the interfering power, and signals from other SFNs operating on the same frequency band belong to this category. In the presence of correlated shadow fading the classi cation requires further research. The value of ma relative to mb and the value of mc relative to md determine the shape of the CDF of ?, and in certain cases this distribution may signi cantly deviate from the log-normal distribution. For example, if rac  1 and mc  md , (24) becomes A B ?  e + e = e(A?C ) + e(B?C )  e(ma ?mc ) + e(B?C ) ; (25)

eC

which is a sum of a constant and a log-normal RV (e(B?C ) can be still approximated by log-normal distribution). If ma and mb is in the same order, the expression of (25) is not log-normal. In cases when rac is high but less then 1, the standard deviation of the rst term in (25) may become much smaller then the standard deviation of the second term; though both terms are log-normal their sum will not be log-normal. Table II summarizes all possible signal constellations in (24) and the outcome of ?. Here we assumed that a  b  c  d , where by x  y we denote that the two terms x and y are in the same order. Estimating pc using the Gaussian assumption when ? is not log-normal results in under- or overesti4 mation. In Fig. 11 we plot the relative error of the estimation, de ned as " = 10 log10 ((1 ? p^c )=(1 ? pc )), as a function of mb and md while keeping ma = 0 dB and mc = 0 dB. The standard deviations were set to a = b = c = d = 3 dB in Fig. 11.a and 5:5 dB in Fig. 11.b. We adjusted 0 to yield pc = 0:99. To isolate the error of the Gaussian assumption from possible summation errors we performed the log-normal summations numerically, i.e., using MC simulation. As can be seen from Fig. 11, the size of the critical region is smaller for the lower standard deviation. We also mapped the ve cases of Table II to the results in Fig. 11.a.

19

a., a = b = c = d = 3 dB

b., a = b = c = d = 5:5 dB

40

40

30

30

I

0 20

20

0

0

0

5

b

IV

0

5

10 m [dB]

b

m [dB]

10

0

-15

0

5

-20

-20

-10

0

V

-30

-10

-25

III

0-5

-10

-5 -10

0

-5

-30

II

-1 5

-20

5

-10

10

-5

-10

-5

-40 -40

-30

-20

-10

0 m [dB]

10

20

30

-40 -40

40

-30

-20

-10

d

0 m [dB]

10

20

30

40

d

Fig. 11. Relative error of the Gaussian p^ estimate expressed in dB (in both cases m = m = 0 dB). a

c

c

Practically, case I corresponds to the typical signal constellation within a homogeneous, dense SFN, i.e., with many similar transmitters (such as in our symmetrical network example). In this case the Gaussian assumption holds because the signals of nearby transmitters dominate over remote contributors, i.e., mb is higher by more than 20 dB than ma , thus ma can be neglected. Case II can occur at the edges of the SFN coverage area where the external interference is dominant. Case III is very unlikely, since it means that the dominant part of both the total useful and the total interfering power comes from the same set of transmitters. This would happen only in abnormal situations, such as when a receiver miss-positions its receiver window. Similarly, case V means that signals outside of the guard interval contribute to the total useful power more than signals within the guard interval, which can be considered abnormal. Finally, case IV can appear inside of the coverage area of a large national SFN, where the eect of external interfering (neighboring) networks is negligible compared to the self-interference and when remote transmitters have comparable contribution to the total useful power as nearby (gap- lling) transmitters. Since this is a practically important case in which the Gaussian assumption fails, an improved estimation method is needed. A. Dominant Contribution of Signals Outside of Guard Interval

Under case IV, i.e., where I  I c , (24) can be modi ed as c i ?  U I+c U = e(A?C ) + e(B?C ) = eG + eG ; 1

(26)

2

where G1 and G2 are Gaussian RVs with mg = ma ? mc, g2 = a2 ? 2 rac a c + c2 and with mg = mb ? mc and g2 = b2 ? 2 rbc b c + c2 , respectively. According to the grouping rac  0 and rbc  0, which results in g  g provided that a , b , c and d are \similar". If g  g , the sum of (26) is no longer log-normal. 1

2

1

2

1

2

1

2

20 40

-5

30

-10 20

-10

-15

-5

0

-20

10

m [dB]

-25

0 0

-1

-10

-5

-1

5

b

0

-25

-20

-20

-20

-30

-40 -40

-30

-20

-10 m

d

0 [dB]

10

20

30

40

Fig. 12. Relative error of the improved p^ estimate expressed in dB (m = m = 0 dB,  =  =  =  = 5:5 dB). I

a

c

c

a

b

c

d

We utilize the fact the pc is only one value of the complementary CDF of ?, namely in point 0 . By approximating the continuous distribution of G1 with a discrete distribution, an improved pc estimator can be obtained by numerical integration (assuming that G1 and G2 are jointly normal): n

p^c = Pr f?  0 g = Pr eG + eG  0 I

= =

Z

1

Z

1

1

?1 10 log ( 0 ?eG1 ) Z 1 Z 1 10

2

o

f (G1; G2 ) dG2 dG1 f (G2jG1 ) dG2 f (G1 )dG1

?1 10 log10 ( 0 ?eG1 ) Z 1  G1 ) ? ml  10 log (

? e 0 10 Q = f (G1 )dG1 l ?1



Np X i=1





G i Q 10 log10 ( 0 ? e ) ? ml pG i ; 1

1

l

where ml = mg ? gg rg g (G1 ? mg ), l = (1 ? rg2 g )g . Values G1i , i = 1; : : : ; Np with corresponding probabilities pG i represent the discrete approximation of G1 . If rg g = 0 then ml = mg and l = g . Based on the de nition we can determine rg g as: 2

2

1

1 2

1

1 2

2

1

1 2

2

2

1 2

2

rg g = rab a b + c ? rac a c ? rbc b c 1 2

g1 g2

In Fig. 12 we plotted the relative error of the improved estimator, "I = 10 log10 ((1 ? p^c )=(1 ? pc )) for the same parameter setting as in Fig. 11.b. We can conclude that p^c provides improved estimation accuracy over the critical region of case IV. For the plot, we approximated G1 with 100 pulses, which can be regarded as a very accurate approximation. However, according to our experience, if I

I

21 10

1-p c

10

10

0

-1

-2

MC 10

-3

Conventional Gaussian(rsr = 0) Gaussian estimation with rsr Improved - 1 pulse Improved - 10 pulses

10

-4

0

5

10

15

20

25

30

Distance from center [km]

Fig. 13. MC simulated and estimated coverage probability along the reference path for the irregular network con guration in case of correlation due to self-interference -90 ms - MC

-95

ma - MC mb - MC

-100

ms - Estimation ma - Estimation

mean in [dB]

-105

mb - Estimation

-110

-115

-120

-125

-130

0

5

10 15 20 Distance from center [km]

25

30

Fig. 14. MC simulated and estimated m , m and m along the reference path in the asymmetrical network con guration in case of correlation due to self-interference s

a

b

mb  ma + 10 dB, approximating G1 with a single constant mg provides satisfactory results. 1

In Fig. 13 we have plotted the coverage probability estimates obtained by the Gaussian methods as well as by the above improved estimator for the asymmetrical network of Fig. 8.b. For the p^c estimator we made two curves, one with Np = 1 and one with Np = 10. The results show that p^c matches well with the reference estimates (MC simulation). In Fig. 14 the parameters ms, ma and mb are plotted. It can be seen that the estimation error by p^c is high where the mb is close to ma , and in this interval p^c gives the best estimate. The Fig. 15 justi es our assumption, the correlation coecient rar is close to 1, while rbr is close to zero. I

I

I

VII. Conclusions

In this paper an analysis of the local coverage probability in SFNs in presence of multiple correlated log-normal useful and interfering components is presented. The accuracy of the existing methods

22 1

Correlation Coefficient

0.8 r sr - MC r ar - MC

0.6

r br - MC r sr - Est. 0.4

r ar - Est. r br - Est.

0.2

0 0

5

10 15 20 Distance from center [km]

25

30

Fig. 15. MC simulated and estimated correlation coecients along the reference path in the asymmetrical network con guration in case of correlation due to self-interference

is investigated and corrections are suggested for improvement. The correlation between the total useful and total interfering power is determined as a function of the correlation between the individual components. This is used as a correction factor in estimating the mean and standard deviation of the log-normally modeled SIR. For signal constellations in which the estimation error of pc is high an alternative method is suggested. We show by numerical examples that our methods can signi cantly decrease the estimation error of pc in the critical regions. For the case when the individual components are independent and only the correlation due to self-interference is present, the classical estimate gives good accuracy in a relatively homogeneous network. In presence of large irregularity, where the signals outside the guard interval have dominant contribution, our method p^c yields estimation improvements. When the correlated shadow fading is taken into account the rsr -corrected p^c estimate is considerably better than the p^c estimate with rsr = 0, in both network con gurations. I

[1] [2] [3] [4] [5] [6] [7] [8] [9]

References A. Ligeti, \Coverage Probability Estimation in Single Frequency Networks in the Presence of Correlated Useful and Interfering Components," in 50th Vehicular Technology Conference, VTC-99 Fall, Sept. 1999, pp. 2408{2412. ETSI, \Radio broadcast systems; Digital Audio Broadcasting (DAB) to mobile, portable and xed receivers," ETSI document Draft prevision ETS 300 401, Mar. 1994. ETSI, \Digital Video Broadcasting (DVB); Framing structure channel coding and modulation for digital Terrestrial television (DVB-T)," ETSI document Final Draft pr ETS 300 744, Nov. 1996. D. Castelain, \Analysis of Interfering Eects in a Single Frequency Network," EBU Doc. GT V4/RSM 93, Sept. 1989. V. Mignone, A. Morello, and M. Visintin, \An Advanced Algorithm for Improving DVB-T Coverage in SFN," in Proc. International Broadcasting Convention, Sept. 1997, pp. 534{540. G. Malmgren, Single Frequency Broadcasting Networks, Ph.D. thesis, Dept. of Signals, Sensors and Systems, Royal Institute of Technology, 1997, TRITA-S3-RST-9701, ISSN 1400-9137, ISRN KTH/RST/R-97/01-SE. R. Beutler, \Optimization of Digital Single Frequency Networks," Frequenz, vol. 49, no. 11{12, pp. 245{252, Nov.{Dec. 1995. ETSI, \Digital Video Broadcasting(DVB): Implementation guidelines for DVB terrestrial services; Transmission aspects," ETSI document ETSI TR 101 190, June 1998. ITU-R, \Methods for the Assessment of Multiple Interference," Report ITU-R 945-2, 1990.

23 [10] R. Brugger, \Comparative analysis of standard methods for the combination of multiple interfering eldstrengths," GT RT/DIG 063, GT R4 383, Dec. 1991. [11] R. Brugger, \Treatment of minimum eldstrength in coverage calculations for digital services," CCETT GT R1/DIG 205, Jan. 1994. [12] S. C. Schwartz and Y. S. Yeh, \On the Distribution Function and Moments of Power Sums With Log-Normal Components," The Bell System Technical Journal, vol. 61, no. 7, pp. 1441{1462, Sept. 1982. [13] N. C. Beaulieu, A. A. Abu-Dayya, and P.J. McLane, \Estimating the Distribution of a Sum of Independent Lognormal Random Variables," IEEE Trans. Communications, vol. 43, no. 12, pp. 2869{2873, Dec. 1995. [14] A. A. Abu-Dayya and N. C. Beaulieu, \Outage Probabilities in the Presence of Correlated Lognormal Interferers," IEEE Trans. Vehicular Technology, vol. 43, no. 1, pp. 164{173, Feb. 1994. [15] A. Safak, \Statistical Analysis of the Power Sum of Multiple Correlated Log-Normal Components," IEEE Trans. Vehicular Technology, vol. 42, no. 1, pp. 58{61, Feb. 1993. [16] ITU-R, \VHF and UHF Propagation Curves for Frequency Range from 30 MHz to 1000 MHz," Recommendation ITU-R P.370-7, 1995. [17] S. B. Slimane, \Multicarrier Systems with Time-Limited Waveforms over Mobile Radio Channels," to appear in IEEE Trans. Communications. [18] M. Gudmunson, \Correlation Model for Shadow Fading in Mobile Radio Systems," Electronics Letters, vol. 27, no. 23, pp. 2145{2146, Nov. 1991. [19] G. Malmgren, \On the Performance of Single Frequency Networks in Correlated Shadow Fading," IEEE Trans. Broadcasting, vol. 43, no. 2, pp. 155{165, June 1997. [20] A. M. Law and W. D. Kelton, Simulation Modeling and Analysis, McGraw-Hill, Inc., 1991.