giovanellimethod

8/3/2019 giovanellimethod

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2 Diffraction by Multiple Knife Edges 361

Giovanelli Method

Like the Deygout method, this method is based on the assumption that one of the

edges is dominant and is primarily responsible for the attenuation due to

diffraction, the other edges having but a secondary influence (Giovanelli 1984).

Two edge case. Let us here consider the case of two knife edges M1 and M2. The

determination of the main edge proceeds from the consideration of the parameterq

= h/r defined with respect to the radius of the first Fresnel ellipsoid. The edge

corresponding to the highest value of the parameterq is defined as the main edge,

while the other edges are considered as secondary edges.

1. We first consider the case, represented in Fig. 12, where M1 is defined as themain edge.

The projection of the ray originating at the apex point of the edge M1 and

grazing the edge M2 creates a fictitious receiverR' within the plane Pr. The

Fresnel-Kirchhoff parameter is a function of the diffraction angle 1, and does

not depend on the angle '1. The effective height h1 of the principal obstacle isdetermined from the obstruction by the edge M1 of the Fresnel ellipsoid with foci

atEandR':

( ) '1 1 11 2 3

R EE

H Hh H H d

d d d

=

+ +

(K.59)

( ) 2 1' 1 2 32

R

H HH H d d

d

= + + (K.60)

E

R0H

1H 2H3H

1h2h

1M

2M1 '

1

2

( )rP

'R

1d 2d 3dFig. 12. Radius of the Fresnel ellipsoid: M1 is the main edge (Giovanelli method)


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362 K Diffraction Models

The secondary obstacle M2 is then considered along the propagation path

M1M2R, with a diffraction angle 2 and an effective height h2 given by theequation:

( ) 12 2 1 22 3

RH Hh H H d d d

=

+

(K.61)

2. Let us now consider the case, represented in Fig. 13, where M2has been defined

as the main edge.

The projection of the ray originating from the apex point of the edge M1 andgrazing the edge M2 creates a fictitious transmitterE' within the plane Ep. The

Fresnel-Kirchhoff parameter is a function of the diffraction angle 2, and does

not depend on the angle '2. The effective height h2 of the main obstacle isdetermined from the obstrcution by the edge M2 of the Fresnel ellipsoid with loi

located atE'andR:

( ) ( ) 3 '2 2 ' 1 21 2 3

EE

H Hh H H d d

d d d

= +

+ +

, (K.62)

( ) 1 2' 1 12

E

H HH H d

d

= + . (K.63)

The secondary obstacle M1 is then considered along the propagation path

EM2R, with a diffraction angle 1 and an effective height h1 given by the equation:

( )2

1 1 1

1 2

EE

H Hh H H d d d

= +. (K.64)

E

R0H

1H 2H3H

1h

2h

1M2M

1

'

22

( )EP

'E

1d 2d

2d

Fig. 13. Radius of the Fresnel ellipsoid: M2is the main edge (Giovanelli method)


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2 Diffraction by Multiple Knife Edges 363

E

R

( )EP ( )RP

'E

'R

kh

0H 1kH kH 1kH+1N

H+

kM

1kM 1kM +

kd 1kd +

Fig. 14. Radius of the Fresnel ellipsoid : generalisation toN edges (Giovanelli method)

Generalisation to N edges. The determination of the attenuation due to the

diffraction by N successive edges is based on the repetition of the two edge

procedure. Different cases are to be considered, depending on which edge is the

main obstacle betweenEandR:

- if the main obstacle is an edge Mk located between the first edge M1 and the

last edge MN, two observation planesPEandPR are introduced at the level of

the transmitterEand receiverR. The projection of the light ray which grazes

the edgesMk

andMk-1

is unaffected by the obstaclesMk-2, Mk-3, M2, M1

andthus defines a fictitious transmitterE' within the plane PE. Likewise the

projection of the light ray which grazes the edges Mkand Mk+1 is unaffected

by the obstacles Mk+2, Mk+3, MN-1, MNand thus defines a fictitious receiverR'

within the plane PR. The effective height hk of the main obstacle Mk is

determined from the obstruction by this edge of the Fresnel ellipsoid withfoci located at E' and R'. This construction is then extended to the

propagation paths located left and right of the main obstacle, by introducing

a new observation plane at the level ofMk, with the same function asPR for

determining a new main edge at the left of Mk and as PE for determining a

new main edge at the right of Mk respectively. The procedure is thenrepeated. The equations for the heights of the fictitious transmitters and of

the receivers are:

( ) 11' 1

... k kk

H H

E k k dH H d d +

+

= + + + , (K.65)


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364 K Diffraction Models

( ) 1' 1 11

... k kR k k N k

H HH H d d d

+

+ +

+

= + + + , (K.66)

( ) ( ) ' '' 11 1

......

R Ek k E k

k

H Hh H H d d

d d+

= + +

+ +. (K.67)

if the main obstacle is the first edge M1 located immediately after thetransmitter, an observation planePR is introduced at the level of the receiverR:

the projection of the light ray originating from the secondary source M1 and

grazing the edge M2, is unaffected by the obstacles M3, M4 , , MN-1, MN andthus defines a fictitious receiverR'within the planePR. The effective height h1of the main obstacle is determined from the obstruction by M1 of the Fresnel

ellipsoid with foci locatedR andR'.

if the main obstacle is the last edge MNlocated immediately before the receiver,an observation plane PE is introduced at the level of the transmitterE: the

projection of the light ray originating from the secondary source MN and

grazing the edge MN-1 is unaffected by the obstacles MN-2, MN-3, , M2, M1 and

thus defines a fictitious transmitterE'within the planePE. The effective heighth1 of the main obstacle is determined from the obstruction by MN of the Fresnel

ellipsoid with foci located atE'andR..

3 Diffraction by a Single Rounded Obstacle

In general, the diffraction caused by an actual obstacle located between thetransmitter and the receiver is more adequately represented by a rounded obstacle

than by a simple knife-edge obstacle. The vicinity of the obstacle will be

represented here by a cylinder with a radius equal to the radius of curvature of the

relief.

3.1 Wait Method

For a positive and small angle

, and assuming that the transmitter and thereceiver are both at a relatively large distance from the edge, the attenuation

caused by the diffraction over a rounded obstacle can be expressed as the sum of

three terms (Wait 1959):

( ) ( ) ( ) ( ), ,0 0,dB dB dBA A A U = + + , (K.68)

where:

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