CH 9: Shadow Fading models of macrocellul macrocellular ar • The models path loss described in Chapter 8 assume that path loss is a function only of parameters such as antenna heights, environment and distance.
•In practice, the particular clutter (buildings, trees) along a path at a given distance will be different for every path. •Some paths will suffer increased loss, whereas others will be less obstructed and have an increased signal strength, as illustrated in Figure 9.1. •This phenomenon is called shadowing or slow fading. It is crucial to account for this in order to predict the reliability of coverage provided by any mobile cellular system.
Figure 9.1: Variation of path profiles encountered at a fixed range from a base station.
9.2 STATISTICAL CHARACTERIZATION •If a mobile is driven around a base station (BS) at a constant distance, then the local mean signal level will typically appear similar to Figure 9.2, after subtracting the median (50%) level in decibels. •A typical probability density function of the signal is shown in Figure 9.3. The distribution of the signal powers is log-normal; that is, the signal measured in decibels has a normal distribution. The process is known as shadowing or slow fading. The variation occurs over distances comparable to the widths of buildings and hills in the region of the mobile, usually tens or hundreds of meters.
Figure 9.2: Typical variation of shadowing with mobile position at fixed BS distance.
Figure 9.3: Probability density function
The standard deviation of the shadowing distribution (in decibels) is known as the location variability, L . The location variability varies with frequency, antenna heights and the environment; it is greatest in suburban areas and smallest in open areas. It is usually in the range 5–12 dB.
9.3 PHYSICAL BASIS FOR SHADOWING If contributions to the signal attenuation along the propagation path are considered to act independently, then the total attenuation A, as a power ratio, due to N individual contributions A1 , . . ., A N will be simply the product of the contributions:
A = A1
A2 . . .
A N
If this is expressed in decibels, the result is the sum of the individual losses in decibels:
L = L1 + L2 + . . . + L N If all of the Li contributions are taken as random variables, then the central limit theorem holds (Appendix A) and L is a Gaussian random variable. Hence A must be log-normal. In practice, not all of the losses will contribute equally, with those nearest the mobile end being most likely to have an effect in macrocells. Moreover, the contributions of individual diffracting obstacles cannot simply be added, so the assumption of independence is not strictly valid. Nevertheless, when the different building heights, spacing and construction methods are taken into account, along with the attenuation due to trees, the resultant distribution function is very close to lognormal.
9.4 IMPACT ON COVERAGE 9.4.1 Edge of Cell When shadowing is included, the total path loss becomes a random variable, given by
L = L50 + L S • L50 is the level not exceeded at 50% of locations at a given distance, as predicted by any standard path loss model (the local median path loss). • LS is the shadowing component, a zero-mean Gaussian random variable with standard deviation L . The probability density function of L S is therefore given by the standard Gaussian formula:
• In order to provide reliable communications at a given distance, an extra fade margin has to be added into the link budget according to the reliability required from the system.
In Figure 9.4, the cell range would be around 9.5 km if shadowing were neglected, then only 50% of locations at the edge of the cell would be properly covered. By adding the fade margin, the cell radius is reduced to around 5.5 km but the reliability is greatly increased, as a much smaller proportion of points exceed the maximum acceptable path loss.
Figure 9.4: Effect of shadowing margin on cell range
•The probability that the shadowing increases the median path loss by at least z [dB] is then given by:
• By change of variables, (9.5) becomes:
• Q(.) function is the complementary cumulative normal distribution. Values for Q are tabulated in Appendix B (Table B.1), or they can be calculated from erfc(.), the standard cumulative error function, using
• Q(t) is plotted in Figure 9.5 and can be used to evaluate the shadowing margin needed for any location variability by putting t = z/ L: 10
10
10
10
) t ( Q
10
10
10
10
0
-1
-2
-3
-4
-5
-6
-7
0
0.5
1
1.5
2
2.5
3
3.5
t Figure 9.5: The Q function.
4
4.5
5
Example 9.1 A mobile communications system is to provide 90% successful communications at the fringe of coverage. The system operates in an environment where propagation can be described by a plane earth model plus a 20 dB clutter factor, with shadowing of location variability 6 dB. The maximum acceptable path loss for the system is 140 dB. Antenna heights for the system are h m = 1.5 m and hb = 30 m. Determine the range of the system. How is this range modified if the location variability increases to 8 dB? Solution
• The total path loss is given by the sum of the plane earth loss, the clutter factor and the shadowing loss: Ltotal = L PEL + Lclutter + L S = 40 log r – 20 log hb –20 log hm + 20 + L S • To find L S , we need the value of t = z/ L for which the path loss is less than the maximum acceptable value at 90% of locations, or when Q(t) = 10% = 0.1. From Figure 9.5, this occurs when t 1.25. Multiplying this by the location variability gives:
L S = z = t
L
= 1.25
6 = 7.5 dB r = 4.4 km.
•If L rises to 8 dB, the shadowing margin L S = 10 dB and r = 3.8 km. Thus, shadowing has a decisive effect on system range.
• Outage Probability: the probability that the predicted path loss will exceed the maximum acceptable path loss. In the above example, the outage probability is:
• The probability of coverage at a range r is:
Coverage Probability = pe(r ) = 1 – pout •In general, the coverage probability is given by:
where Lm is the maximum acceptable path loss and L(r ) is the median path loss model, evaluated at a distance r , and M = Lm – L(r ) is the fade margin chosen for the system.
• This coverage probability is shown in Figure 9.6, using the same values as Example 9.1.
Figure 9.6: Variation of coverage percentage with distance for example 9.1
9.4.2 Whole Cell It is more appropriate to design the system in terms of the coverage probability experienced over the whole cell, rather than coverage at the edge of the cell. The geometry is shown below. The coverage probability within the shaded area is pe(r ). The coverage probability of the whole cell is pcell and is calculated as follows:
r=0 where erf( x ) = 1 – erfc( x ). This may be solved numerically for any desired path loss model L(r ). In the special case of a power law path loss model, the result may be obtained analytically. If the path loss model is expressed as:
The result will be:
where
Figure 9.8: Probability of availability over whole cell area, with pe as a parameter.
Figure 9.9: Probability of availability over whole cell area, with fade margin as a parameter, varying from 1–10 dB in steps of 1 dB.
9.5 LOCATION VARIABILITY Figure 9.10 shows the variation of the location variability L with frequency, as measured by several studies. It is clear there is a tendency for L to increase with frequency and that it depends upon the environment. Figure 9.10 also includes plots of an empirical relationship fitted to the Okumura curves and chosen to vary smoothly up to 20 GHz. This is given by: L
= 0.65 (log f c)2 – 1.3 log f c + A
where A = 5.2 in the urban case and 6.6 in the suburban case.
Figure 9.10: Location variability versus frequency.
