Polytechnic University, Brooklyn, NY ©2002 by H.L. Bertoni1 X. Numerical Integration for Propagation Past Rows of Buildings Adapting the physical optics integrals for numerical evaluation Applications –Computed height dependence of the fields –Buildings with flat roofs –Buildings of random height, spacing –Rows of building on hills –Trees located next to buildings –Penetration through buildings at low frequencies

Polytechnic University, Brooklyn, NY ©2002 by H.L. Bertoni2 Numerical Integration for Field Variation in 2D 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  nn nn ynyn x n=1n=2n=3nn+1 Incident wave y n+1 dndn

Polytechnic University, Brooklyn, NY ©2002 by H.L. Bertoni3 Termination Strategy Multiply H n (y n ) by the neutralizer function  (y n ) to smoothly reduce the integration to zero in order to avoid the spurious contribution given by abrupt termination of the integral. 3w3w

Polytechnic University, Brooklyn, NY ©2002 by H.L. Bertoni4 Discretization

Polytechnic University, Brooklyn, NY ©2002 by H.L. Bertoni5 Discretization - cont.

Polytechnic University, Brooklyn, NY ©2002 by H.L. Bertoni6 Height Variation of the Field Above the M=120 Row of Buildings for Plane Wave Incidence (  = 1 o, d = 50 m, f = 900 MHz, M = 120 > N 0 ) Dashed curve for y < 0 is |H(y)| Height in wavelength y/ K.H. UTD Q(gp)Q(gp)

Polytechnic University, Brooklyn, NY ©2002 by H.L. Bertoni7 Standing Wave Behavior for y > 0 (  = 1 o, d = 50 m, f = 900 MHz, M = 120 > N 0 ) |H(y)| Height in wavelength y/ y 

Polytechnic University, Brooklyn, NY ©2002 by H.L. Bertoni8 Side View of Buildings With Different Roofs - Representation by Absorbing Screens - Representation of (a) and (c) by absorbing screen for Tx and Rx at rooftop height (a)(b)(c) w Tx Rx Tx d

Polytechnic University, Brooklyn, NY ©2002 by H.L. Bertoni9 Effect of Roof Shape on Reduction Factor Q Computed Midway Between Rows Constant offset of 3.3 dB between two case (a) and (c), but no change in range index n.

Polytechnic University, Brooklyn, NY ©2002 by H.L. Bertoni10 Shadow Fading Variation from building-to-building along along a row Variation from row-to-row Why the shadow fading is lognormal ?

Polytechnic University, Brooklyn, NY ©2002 by H.L. Bertoni11 Shadow Fading Due to Variations in Building Height Along the Rows As the subscriber moves along street, the received signal passes over buildings of different height, or misses the last row of buildings Street and side walks Subscriber From base station

Polytechnic University, Brooklyn, NY ©2002 by H.L. Bertoni12 Shadow Fading for Propagation Past Successive 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.

Polytechnic University, Brooklyn, NY ©2002 by H.L. Bertoni13 Modeling Shadow Fading for Random Building Height Incident Plane wave x y Building height determined by random number generator Use numerical integration to find fields at successive screen, mobiles

Polytechnic University, Brooklyn, NY ©2002 by H.L. Bertoni14 Row-to-Row Variation of Rooftops Field Due to Random Building Height Plane wave incidence ( f = 900 MHz,  = 0.5º, d = 50 m ) H B uniformly distributed m Screen number H( y )H( y )

Polytechnic University, Brooklyn, NY ©2002 by H.L. Bertoni15 Cumulative Distribution Functions for Receive Power at Rooftop and Street Level Plane wave incidence ( f = 900 MHz,  = 0.5º, d = 50 m ) H B uniformly distributed m 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.

Polytechnic University, Brooklyn, NY ©2002 by H.L. Bertoni16 Missing Buildings, Roof Shape and Building Materials 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 h BS hmhm  HBHB d

Polytechnic University, Brooklyn, NY ©2002 by H.L. Bertoni17 Cumulative Distribution Function for Combination of Random Height and Other Random Factor CDF of the received power at Street level for: f = 900 MHz  = 0.5° d = 40 m H B distribution is Uniform Rayleigh Nearly straight line for the distorted vertical scale indicates a Normal distribution of power in dB.

Polytechnic University, Brooklyn, NY ©2002 by H.L. Bertoni18 Dependence of Standard Deviation of Signal Distribution on  H B for 900 MHz and 1.8 GHz

Polytechnic University, Brooklyn, NY ©2002 by H.L. Bertoni19 Why Shadow Fading is Lognormal Distributed 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