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Chapter: Wireless Channels
Section: Shadowing


Multiple log-normal signals

In cellular networks, interference does not come from only one source but from many co-channel transmitters. In a hexagonal reuse pattern the number of interferers typically is six.

At least two different methods are used to estimate the probability distribution of the joint interference power accumulated from several log-normal signals. Such methods are relevant to estimate the joint effect of multiple interfering signals with shadowing. Fenton and Schwartz and Yeh both proposed to approximate the pdf of the joint interference power by a log-normal pdf, yet neither could determine it exactly.

Fenton

The method by Fenton assesses the logarithmic mean and variance of the joint interference signal directly as a function of the logarithmic means and variances of the individual interference signals. This method is most accurate for small standard deviations of the shadowing, say, for less than 4 dB.

Schwartz and Yeh

The technique proposed by Schwartz and Yeh is more accurate in the range of 4 to 12 dB shadowing, which corresponds to the case of land-mobile radio in the VHF and UHF bands. Their method first assesses the logarithmic mean and variance of a joint signal produced by cumulation of two signals. Recurrence is then used in the event of more than two interfering signals. A disadvantage of the latter method is that numerical computations become time consuming.

Table: Mean mt and standard deviation st (both in dB) of the joint power of n signals with uncorrelated shadowing, each with mean 0 dB and with identical standard deviation. Networks, with 0, 6, 8.3 and 12 dB of shadowing of individual signals. 

         0 dB               6 dB              8.3 dB             12 dB
n     mt     st         mt       st        mt        st        mt       st
 
1    0.00 0.00       0.00     6.00       0.00     8.30       0.00     12.00
2    3.00 0.00       4.58     4.58       5.61     6.49       7.45      9.58
3    4.50 0.00       6.90     3.93       8.45     5.62      11.20      8.40
4    6.00 0.00       8.43     3.54      10.29     5.08      13.62      7.66
5    7.00 0.00       9.57     3.26      11.64     4.70      15.37      7.13
6    7.50 0.00      10.48     3.04      12.69     4.41      16.74      6.74
Besides these methods, by Fenton and Schwartz and Yeh, a number of alternative (and often more simplified) techniques are used. For instance in VHF radio broadcasting, signals fluctuate with location and with time according to log-normal distributions. Techniques to compute the coverage of broadcast transmitters are in CCIR recommendations.

Outage probabilities for systems with multiple Rayleigh fading and shadowed signals can however be computed easily without explicitly estimating the joint effect of multiple shadowed signals.

PC Computer Software

PC Windows executable to compute the mean and standard deviation of the signal power due to multiple i.i.d. lognormal signals. The user can set the the standard deviation.

Disclaimer: Executable software programmes are provided with no guarantee whatsoever. See License.



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