Mobile Radio Channel
5.3 Fading
5.3.1 Shadowing/Slow Fading
Referring to Figure 4.3, we may associate shadowing or slow fading with a second com- ponent of received signal strength. As discussed in Section 4.3, this component has a spatial scale of variations in order of several tens wavelengths. The shadowing compo- nent is especially important in radio coverage prediction where we need to ensure that a signal strength received at certain location, normally at the edge of the projected radio cell, will exceed certain threshold level. The statistical characteristics of shadowing can be described in terms of probability density function.
Consider the generic probability distribution plot in Figure 5.9. In this particular plot, the distribution is normalized at specific value ofglobal mean(see Figure 4.3), say this value is the lowest median operating signal level that a radio receiver can operate with to achieve a given degree of performance.
With level of performance corresponding to a global mean, we do not take into account random variations in received signal. The signal would be acceptable for 50%
of samples and not acceptable for the remainder. This is a level of performance that does not meet service requirements. We determine the limit of acceptable performance as being some value higher than the minimum operating level; that is, creating some margin of performance for acceptable signal level. This would have the effect of setting
Median (0 dB) –X dB
(A) –Y dB
(B)
90% exceeded
99% exceeded Median value, 50% exceeded
Received field strength, dB
Figure 5.9 Probability density function of signal variations normalized at the global mean.
the minimum median operating level to some point on the left-hand side of the curve, such as point (A) or (B), see Figure 5.9.
As a simple illustration, if the value X is 10 dB, then shifting the reference value by 10 dB would change the percentage of signals above the reference value from 50 to 90%.
If the median value was−90 dBm for example, and if we set a limit of−80 dB, then we can be sure that 90% of the signals experienced would be above−90 dBm, even taking account of the variability of the signal.
When shadowing is included, path loss L is composed of deterministic (L50) and random (Ls) components
L=L50+Ls (5.15)
at 50% of locations at a given distance, as predicted by any standard path loss model (the median path loss orglobal mean).
TheLsis the shadowing component that follows alognormalstatistical distribution of signal power; that is, thesignal measured in decibels has a normal distribution(i.e. is a zero-mean Gaussian random variable with standard deviation dB). The variation inLs occurs over distances comparable to the widths of buildings and hills in the region of the mobile, usually tens or hundreds of metres.
Application of a lognormal distribution for shadowing models can be seen as follows.
If contributions to the signal attenuation along the propagation path are considered to act independently, then the total attenuationA, as a power ratio due toN individual contributionsA1,…, ANwill be simply the product of the contributions:
A=A1×A2×A3· · ·AN (5.16)
If this is expressed in decibels, the result is the sum of the individual losses in decibels:
Ls=L1+L2+L3+ · · · +LN. (5.17) All of the contributions in (5.17) are taken as random variables, then, withN>>1, the central limit theorem holds andLbehaves as a Gaussian random variable. Hence,A must be lognormal.
Probability density function ofLsis given by the standard Gaussian formula p(Ls) = 1
2√ 𝜋𝜎Lexp
[
− L2s 2𝜎L2
]
, (5.18)
where𝜎Lis a standard deviation of received power in dB. Figure 5.10 represents typical path loss measured along the distance from the base station. The solid curve represents the global mean or signal strength incorporating slow fading/shadowing; that is, theLs component.
In Figure 5.10, the cell range would be around 8 km if the effect of 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 5.3 km, but the relia- bility is greatly increased as a much smaller proportion of points exceed the maximum acceptable path loss. In other words, this margin is defined to maintain coverage with a certainlocation probability.
Mobile Radio Channel 39
–70
–80
–90
–100
–110
–120
2 3 4 5 6 7 8 Distance, km
Fade margin
Cell range RX sensitivity
level Received power
level, dBm
Figure 5.10 Effect of slow fading/shadowing on estimation of the cell range.
Another widely accepted name forlocation probabilityisaccessibility. The common definition for location probability of X% is that the path loss is less than maximum acceptable value in X% of locations at the cell edge. Based on statistic properties of lognormal fading the location probabilityPr(Ls>z)with a given marginzcan be calcu- lated using the standard error function
Pr(Ls>z) =Q ( z
𝜎L )
(5.19) and
Q(t) = 1
√2𝜋∫
∞ x=t
exp (
−x2 2
) dx= 1
2erfc (
√t 2
)
(5.20) The plot of standard error functionQ(t)is provided in Figure 5.11. It can be used to evaluate the shadowing marginzneeded for any location probability with substitute z=t•𝜎L. The lognormal (LNF) margin values versus cell edge probability are presented in Table 5.1.
Illustration of the lognormal fading model is given in Figure 5.12. The drop-outs in signal strength, also called outageA, are considered to occur when the local mean falls below a critical level, see Figure 5.12. The accessibility is then defined as 1−A=Pr(Ls>
z); that is, location probability.
1.00E–08 1.00E–07 1.00E–06 1.00E–05 1.00E–04 1.00E–03 1.00E–02 1.00E–01
1.00E+00 0.0 0.3 0.6 0.8 1.1 1.4 1.7 2.0 2.3 2.5 2.8 3.1 3.4 3.7 4.0 Argument of Q function
Figure 5.11 The Q function.
Table 5.1 Tabularized LNF margin in a function of location probability at cell border.
Location probability at the cell border, %
Lognormal fade margin, dB
50 𝜎×0
60 𝜎×0.25
70 𝜎×0.52
80 𝜎×0.84
90 𝜎×1.28
91 𝜎×1.34
92 𝜎×1.4
93 𝜎×1.48
94 𝜎×1.55
95 𝜎×1.64
96 𝜎×1.75
97 𝜎×1.89
98 𝜎×2.05
99 𝜎×2.33