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• How to terminate the numerical integration without changing the computed field– Abrupt termination like an absorbing screen above the termination point.
– Make the field go smoothly to zero above the significant region
• Discretize the integral in step size of at least /2
Shadow Fading for Propagation PastSuccessive Rows of Different Height
From base station
Because the width of the Fresnel zone is on the order of the width of the buildings, the random embodiments of buildings along the propagation path for mobiles located between different rows have the same statistical distribution as the embodiments along the propagation path for different mobile locations along a row.
Cumulative Distribution Functions for Receive Power at Rooftop and Street Level
Plane wave incidence ( f = 900 MHz, = 0.5º, d = 50 m )HB uniformly distributed 8 - 14 m
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0-50 -40 -30 -20 -10 0
Field (dB)
CD
F
(At mobile)Mean = -34.68Std = 4.22
(At rooftop)Mean = -11.49Std = 5.77
Because the distributions are nearly a straight line for a linear vertical scale, the CDF’s are nearly those of a uniform distribution. Addition sources of variation are needed to get a lognormal distribution.
Missing Buildings, Roof Shape and BuildingMaterials Also Cause Signal Variation
Additional sources of variability that influence diffraction down to the mobile are roof characteristics and construction, and the absence of buildings in a row, such as at and intersection. For simulations we assume: 50% peaked, 50% flat 50% conducting, 50% absorbing boundary conditions 10% of buildings are missing
Cumulative Distribution Function for Combinationof Random Height and Other Random Factor
CDF of the received power at Street levelfor:
f = 900 MHz = 0.5°d = 40 mHB distribution is Uniform Rayleigh Nearly straight line for the distorted vertical scale indicates a Normal distribution of power in dB.
• Sequence of random processes, each of which multiply the signal by a random number: - Random building height - Random diffraction down to mobile due to roof shape, construction, missing buildings
• On dB scale, multiplication of random numbers is equal to addition of their logs
• By central limit theorem of random statistics, a sum of random numbers has normal (Gaussian) distribution
• Adding just two random numbers gives normal distribution, except in tails