Polytechnic University, Brooklyn, NY ©2002 by H.L. Bertoni1 X. Numerical Integration for...
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Transcript of Polytechnic University, Brooklyn, NY ©2002 by H.L. Bertoni1 X. Numerical Integration for...
Polytechnic University, Brooklyn, NY
©2002 by H.L. Bertoni 1
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. Bertoni 2
Numerical Integration for Field Variation in 2D
H(xn+1,yn+1) =ejπ / 4
λH(xn,yn)
jke−jkρn
ρndyn
hn
∞
∫ , ρn = (xn+1 −xn)2 +(yn+1 −yn)
2
• 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
yn
x
n=1 n=2 n=3 n n+1
€
E
€
H
Incidentwave
yn+1
dn
Polytechnic University, Brooklyn, NY
©2002 by H.L. Bertoni 3
Termination Strategy• Multiply Hn(yn) by the neutralizer function (yn) to smoothly reduce the integration to
zero in order to avoid the spurious contribution given by abrupt termination of the integral.
€
η yn( )=
1 for yn <yc
exp− yn −yc( )2/w2
[ ] ; yc <yn <yc +3w
0 for yn >yc +3w
⎧
⎨ ⎪
⎩ ⎪
yc >Ndtanα + Ndλ secα3w
exp−yn −yc( )2/w2
[ ]
Hn yn( ) η yn( )
Polytechnic University, Brooklyn, NY
©2002 by H.L. Bertoni 4
Discretization
€
Hn+1 yn+1( )=e jπ /4
λη yn( )Hn yn( )
e−jkρn
ρndyn
0
ym
∫
where
ρn = xn+1 −xn( )2+ yn+1 −yn( )
2
Define
φ yn( ) =∠Hn yn( )−kρn and A yn( )=η yn( )Hn yn( )1λρn
Then
Hn+1 yn+1( )=e jπ /4 A yn( )ejφ yn( )dyn
mΔ
(m+1)Δ
∫m=0
M
∑Using the linear approximations
φ yn( ) ≈φ mΔ( )+ φ mΔ+Δ( )−φ mΔ( )[ ]yn −mΔ
Δ
A yn( ) ≈A mΔ( )+ A mΔ+Δ( )−A mΔ( )[ ]yn −mΔ
Δ
Polytechnic University, Brooklyn, NY
©2002 by H.L. Bertoni 5
Discretization - cont.
€
The integration in each interval can be evaluated in closed form so that
Hn+1 pΔ( ) =Δe jπ /4 A mΔ+Δ( )e jφ mΔ +Δ( ) −A mΔ( )eφ mΔ( )
φ mΔ+Δ( )−φ mΔ( )
⎧ ⎨ ⎩ m=0
M
∑
+jA mΔ+Δ( )−A mΔ( )
φ mΔ+Δ( )−φ mΔ( )[ ]2 e jφ mΔ+Δ( ) −eφ mΔ( )[ ]
⎫ ⎬ ⎭
Polytechnic University, Brooklyn, NY
©2002 by H.L. Bertoni 6
Height Variation of the Field Above the M=120 Row of Buildings for Plane Wave Incidence
(= 1o, d = 50 m, f = 900 MHz, M = 120 > N0 )
Dashed curve for y < 0 is ( ) θTp DgQ )(2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8|H
(y)|
-30 -20 -10 0 10 20 30 40 50 60
Height in wavelength y/
K.H.UTD
Q(gp)
Polytechnic University, Brooklyn, NY
©2002 by H.L. Bertoni 7
Standing Wave Behavior for y > 0 (= 1o, d = 50 m, f = 900 MHz, M = 120 > N0 )
|H(y
)|
Height in wavelength y/
0.00.20.40.60.81.01.21.41.61.8
-30-20-10 0 10 20 30 40 50 60
y
€
Height variation e jκy +Γe−jκy =1+Γe−j2κy where κ =ksinα
Separation Δy between minima is 2κΔy=2π or
Δy=π κ =π ksinα =λ 2sinα
For α =1o, Δy=28.6λ
Also, 1+Γ ≈1.6 and 1- Γ ≈0.4 so that Γ ≈0.6
Polytechnic University, Brooklyn, NY
©2002 by H.L. Bertoni 8
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
TxRx
RxTx
d
Polytechnic University, Brooklyn, NY
©2002 by H.L. Bertoni 9
Effect of Roof Shape on Reduction Factor QComputed Midway Between Rows
Constant offset of 3.3 dBbetween two case (a) and (c),but no change in range index n.
Polytechnic University, Brooklyn, NY
©2002 by H.L. Bertoni 10
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. Bertoni 11
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
Stre
et a
ndsi
de w
alks
Subscriber
From base station
€
Full width of Fresnel
zone near one end of link:
2wF =2 λs
For 900 MHz at mid street
s=20 m, λ =1 3 m
2wF =5.2 m
about the width of a house
Polytechnic University, Brooklyn, NY
©2002 by H.L. Bertoni 12
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.
Polytechnic University, Brooklyn, NY
©2002 by H.L. Bertoni 13
Modeling Shadow Fading for Random Building Height
IncidentPlane 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. Bertoni 14
Row-to-Row Variation of Rooftops FieldDue to Random Building Height
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
0.0100 110 120 130 140 150
Screen number
H( y
)
Polytechnic University, Brooklyn, NY
©2002 by H.L. Bertoni 15
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.
Polytechnic University, Brooklyn, NY
©2002 by H.L. Bertoni 16
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
hBS hm
HB
d
Polytechnic University, Brooklyn, NY
©2002 by H.L. Bertoni 17
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.
Polytechnic University, Brooklyn, NY
©2002 by H.L. Bertoni 18
Dependence of Standard Deviation of Signal Distribution on HB for 900 MHz and 1.8 GHz
Polytechnic University, Brooklyn, NY
©2002 by H.L. Bertoni 19
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