Part IV MULTIPLE ACCESS AND ADVANCED TRANSCEIVER SCHEMES 363
4.3.1 Diffraction by a Single Screen or Wedge
The Diffraction Coefficient
The simplest diffraction problem is the diffraction of a homogeneous plane wave by a semi-infinite screen, as sketched in Figure 4.4. Diffraction can be understood from Huygen’s principle that each point of a wavefront can be considered the source of a spherical wave. For a homogeneous plane wave, the superposition of these spherical waves results in another homogeneous plane wave, see transition from plane A toB. If, however, the screen eliminates parts of the point sources (and their associated spherical waves), the resulting wavefront is not plane anymore, see the transition from planeBtoC. Constructive and destructive interferences occur in different directions.5
B C
P
x y
B'
A' C'
A Screen
Shadow zone
Figure 4.4 Huygen’s principle.
The electric field at any point to the right of the screen (x≥0) can be expressed in a form that involves only a standard integral, theFresnel integral. With the incident field represented as exp(−j k0x), the total field becomes [Vaughan and Andersen 2003]:
Etotal=exp(−j k0x) 1
2−exp(j π/4)
√2 F (νF)
=exp(−j k0x)F (ν˜ F) (4.27)
5For more accurate considerations, it is noteworthy that Huygen’s principle is not exact. A derivation from Maxwell’s equations, which also includes a discussion of the necessary assumptions, is given, e.g., in Marcuse [1991].
whereνF= −2y/√
λx and the Fresnel integralF (νF)is defined as:
F (νF)= νF
0
exp
−j πt2 2
dt (4.28)
Figure 4.5 plots this function. It is interesting thatF (ν˜ F)can become larger than unity for some values ofνF. This implies that the received power at a specific location can actually beincreased by the presence of screen. Huygen’s principle again provides the explanation: some spherical waves that would normally interfere destructively in a specific location are blocked off. However, note that thetotal energy (integrated over the whole wavefront)cannot be increased by the screen.
Consider now the more general geometry of Figure 4.6. The TX is at height hTX, the RX at hRX, and the screen extends from−∞tohs. The diffraction angleθdis thus:
θd=arctan
hs−hTX
dTX
+arctan
hs−hRX
dRX
(4.29)
~
1.5
Imaginary part Fresnel integral F (n)
Real part
Magnitude Phase
1.0
0.5
0.0
Fresnel parameter n
−0.5 1.5
1.0
0.5
0.0
1.5
1.0
0.5
0.0
0 5
−0.5−5
Fresnel parameter n
0 5
−5
Fresnel parameter n
0 5
−5
Fresnel parameter n
0 5
−5
−π/2 π/2 0
−π π
Figure 4.5 Fresnel integral.
Propagation Mechanisms 57
TX RX
hs θd
hTX hRX
dRX dTX
Figure 4.6 Geometry for the computation of the Fresnel parameters.
and the Fresnel parameterνFcan be obtained fromθd as:
νF=θd 2dTXdRX
λ(dTX+dRX) (4.30)
The field strength can again be computed from Eq. (4.27), just using the Fresnel parameter from Eq. (4.30).
Note that the result given above is approximate in the sense that it neglects the polarization of the incident field. More accurate equations for both the TE and the TM case can be found in Bowman et al. [1987].
Example 4.3 Consider diffraction by a screen withdTX=200m, dRX=50m, hTX=20m, hRX=1.5m,hs=40m, at a center frequency of 900 MHz. Compute the diffraction coefficient.
A center frequency of 900 MHz implies a wavelength λ=1/3 m. Computing the diffraction angleθdfrom Eq. (4.29) gives:
θd=arctan
hs−hTX dTX
+arctan
hs−hRX dRX
=arctan
40−20 200
+arctan
40−1.5 50
=0.756 rad (4.31) Then, the Fresnel parameter is given by Eq. (4.30) as:
νF =θd 2dTXdRX
λ(dTX+dRX) =0.756 2ã200ã50
1/3ã(200+50) =11.71 (4.32) Evaluation of Eq. (4.28), with MATLAB or Abramowitz and Stegun [1965], yields
F (11.71)= vF
0
exp
−j πt2 2
dt≈0.527−j0.505 (4.33) Finally, Eq. (4.27) gives the total received field as:
Etotal=exp(−j k0x) 1
2−exp(j π/4)
√2 F (11.71)
=exp(−j k0x) 1
2−exp(j π/4)
√2 (0.527−j0.505)
=(−0.016−j0.011)exp(−j k0x) (4.34)
Fresnel Zones
The impact of an obstacle can also be assessed qualitatively, and intuitively, by the concept of Fresnel zones. Figure 4.7 shows the basic principle. Draw an ellipsoid whose foci are the BS and the MS locations. According to the definition of an ellipsoid, all rays that are reflected at points on this ellipsoid have the same run length (equivalent to runtime). The eccentricity of the ellipsoid determines the extra run length compared with the LOS – i.e., the direct connection between the two foci. Ellipsoids where this extra distance is an integer multiple of λ/2 are called “Fresnel ellipsoids.” Now extra run length also leads to an additional phase shift, so that the ellipsoids can be described by the phase shift that they cause. More specifically, the ith Fresnel ellipsoid is the one that results in a phase shift ofiãπ.
TX dTX dRX
dRX dTX
dRX dTX
TX
nF < 0
nF = 0
nF > 0
RX RX
RX TX
Figure 4.7 The principle of Fresnel ellipsoids.
Fresnel zones can also be used for explanation of thed−4law. The propagation follows a free space law up to the distance where the first Fresnel ellipsoid touches the ground. At this distance, which is the breakpoint distance, the phase difference between the direct and the reflected ray becomesπ.
Diffraction by a Wedge
The semi-infinite absorbing screen is a useful tool for the explanation of diffraction, since it is the simplest possible configuration. However, many obstacles especially in urban environments are much better represented by a wedge structure, as sketched in Figure 4.8. The problem of diffraction by a wedge has been treated for some 100 years, and is still an area of active research.
Depending on the boundary conditions, solutions can be derived that are either valid at arbitrary observation points or approximate solutions that are only valid in the far field (i.e., far away from the wedge). These latter solutions are usually much simpler, and will thus be the only ones considered here.
Propagation Mechanisms 59
The part of the field that is created by diffraction can be written as the product of the incident field with a phase factor exp(−j k0dRX), a geometry factorA(dTX,dRX)that depends only on the distance of TX and RX from the wedge, and the diffraction coefficientD(φTX,φRX)that depends on the diffraction angles [Vaughan and Andersen 2003]:
Ediff=Einc,0D(φTX, φRX)A(dTX, dRX)exp(−j k0dRX) (4.35) The diffracted field has to be added to the field as computed by geometrical optics.6
The definition of the geometry parameters is shown in Figure 4.8. The geometry factor is given by:
A(dTX, dRX)= dTX
dRX(dTX+dRX) (4.36)
The diffraction coefficientD depends on the boundary conditions – namely, the reflection coeffi- cientsρTXandρRX. Explicit equations are given in Appendix 4.B (seewww.wiley.com/go/molisch).
TX
rRX
fTX fRX
py
rTX
dRX dTX
RX
Figure 4.8 Geometry for wedge diffraction.