1. 1 3.4 Basic Propagation Mechanisms & Transmission Impairments (1) Reflection: propagating wave impinges on object with size >> examples include ground, buildings, walls (2) Diffraction: transmission path obstructed by objects with edges 2 nd ry waves are present throughout space (even behind object) gives rise to bending around obstacle (NLOS transmission path) (3) Scattering propagating wave impinges on object with size < number of obstacles per unit volume is large (dense) examples include rough surfaces, foliage, street signs, lamp posts

2. 2 Models are used to predict received power or path loss (reciprocal) based on refraction, reflection, scattering Large Scale Models Fading Models at high frequencies  diffraction & reflections depend on geometry of objects EM wave’s, amplitude, phase, & polarization at point of intersection

3. 3 3.5 Reflection: EM wave in 1 st medium impinges on 2 nd medium part of the wave is transmitted part of the wave is reflected (1) plane-wave incident on a perfect dielectric (non-conductor) part of energy is transmitted (refracted) into 2 nd medium part of energy is transmitted (reflected) back into 1 st medium assumes no loss of energy from absorption (not practically) (2) plane-wave incident on a perfect conductor all energy is reflected back into the medium assumes no loss of energy from absorption (not practically)

4. 4 (3)  = Fersnel reflection coefficient relates Electric Field intensity of reflected & refracted waves to incident wave as a function of: material properties, polarization of wave angle of incidence signal frequency boundary between dielectrics (reflecting surface) reflected wave refracted wave incident wave

5. 5 (4) Polarization: EM waves are generally polarized instantaneous electric field components are in orthogonal directions in space represented as either: (i) sum of 2 spatially orthogonal components (e.g. vertical & horizontal) (ii) left-handed or right handed circularly polarized components reflected fields from a reflecting surface can be computed using superposition for any arbitrary polarization E || EE

6. 6 3.5.1 Reflection from Dielectrics assume no loss of energy from absorption EM wave with E-field incident at  i with boundary between 2 dielectric media some energy is reflected into 1 st media at  r remaining energy is refracted into 2 nd media at  t reflections vary with the polarization of the E-field plane of incidence reflecting surface= boundary between dielectrics ii rr tt plane of incidence = plane containing incident, reflected, & refracted rays

7. 7 Two distinct cases are used to study arbitrary directions of polarization (1) Vertical Polarization: (E vi ) E-field polarization is parallel to the plane of incidence normal component to reflecting surface (2) Horizontal Polarization: (E hi ) E-field polarization is perpendicular to the plane of incidence parallel component to reflecting surface plane of incidence ii rr tt E vi E hi boundary between dielectrics (reflecting surface)

8. 8 E i & H i = Incident electric and magnetic fields E r & H r = Reflected electric and magnetic fields E t = Transmitted (penetrating) electric field HiHi HrHr EiEi ErEr ii rr tt  1,  1,  1  2,  2,  2 EtEt Vertical Polarization: E-field in the plane of incidence HiHi HrHr EiEi ErEr ii rr tt  1,  1,  1  2,  2,  2 EtEt Horizontal Polarization: E-field normal to plane of incidence

9. 9 (1) EM Parameters of Materials  = permittivity (dielectric constant): measure of a materials ability to resist current flow  = permeability: ratio of magnetic induction to magnetic field intensity  = conductance: ability of a material to conduct electricity, measured in Ω -1 dielectric constant for perfect dielectric (e.g. perfect reflector of lossless material) given by  0 = 8.85  10 -12 F/m

10. 10 often permittivity of a material,  is related to relative permittivity  r  =  0  r lossy dielectric materials will absorb power  permittivity described with complex dielectric constant (3.18) where  ’ = (3.17)  =  0  r -j  ’ highly conductive materials   r &  are generally insensitive to operating frequency  0 and  r are generally constant  may be sensitive to operating frequency

11. 11 Material rr /r0/r0 f (Hz) Poor Ground40.001 2.82  10 7 10 8 Typical Ground150.005 3.77  10 7 10 8 Good Ground250.02 9.04  10 7 10 8 Sea Water815 6.97  10 9 10 8 Fresh Water810.001 1.39  10 6 10 8 Brick4.440.001 2.54  10 7 41094109 Limestone7.510.028 4.21  10 8 41094109 Glass, Corning 70740.00000018 5.08  10 3 10 6 Glass, Corning 70740.000027 7.62  10 5 10 8 Glass, Corning 70740.005 1.41  10 810

12. 12 because of superposition – only 2 orthogonal polarizations need be considered to solve general reflection problem Maxwell’s Equation boundary conditions used to derive (3.19-3.23) Fresnel reflection coefficients for E-field polarization at reflecting surface boundary  || represents coefficient for || E-field polarization   represents coefficient for  E-field polarization (2) Reflections, Polarized Components & Fresnel Reflection Coefficients

13. 13 Fersnel reflection coefficients given by (i) E-field in plane of incidence (vertical polarization)  || = (3.19) (ii) E-field not in plane of incidence (horizontal polarization)   = (3.20)  i = intrinsic impedance of the i th medium ratio of electric field to magnetic field for uniform plane wave in i th medium given by  i =

14. 14 velocity of an EM wave given by boundary conditions at surface of incidence obey Snell’s Law (3.21)  i =  r (3.22) E r =  E i (3.23a) E t = (1 +  )E i (3.23b)  is either  || or   depending on polarization |  |  1  for a perfect conductor, wave is fully reflected |  |  0  for a lossy material, wave is fully refracted

15. 15 radio wave propagating in free space (1 st medium is free space)  1 =  2  || = (3.24)   = (3.25) Simplification of reflection coefficients for vertical and horizontal polarization assuming: Elliptically Polarized Waves have both vertical & horizontal components waves can be depolarized (broken down) into vertical & horizontal E-field components superposition can be used to determine transmitted & reflected waves

16. 16 (3) General Case of reflection or transmission horizontal & vertical axes of spatial coordinates may not coincide with || &  axes of propagating waves for wave propagating out of the page  define angle  measured counter clock-wise from horizontal axes spatial horizontal axis spatial vertical axis   || orthogonal components of propagating wave

17. 17  vertical & horizontal polarized components components perpendicular & parallel to plane of incidence E i H, E i V E d H, E d V E d H, E d V = depolarized field components along the horizontal & vertical axes E i H, E i V = horizontal & vertical polarized components of incident wave relationship of vertical & horizontal field components at the dielectric boundary E d H, E d V E i H, E i V = Time Varying Components of E-field (3.26) - E-field components may be represented by phasors

18. 18 for case of reflection: D  =   D || || =  || for case of refraction (transmission): D  = 1+   D || || = 1+  || R =,  = angle between two sets of axes DC =DC = R = transformation matrix that maps E-field components D C = depolarization matrix

19. 19 1.0 0.8 0.6 0.4 0.2 0.0 0 10 20 30 40 50 60 70 80 90 |  || |  r =12  r =4 angle of incidence (  i ) Brewster Angle (  B ) for  r =12 vertical polarization (E-field in plane of incidence) for  i <  B : a larger dielectric constant  smaller  || & smaller E r for  i >  B : a larger dielectric constant  larger  || & larger E r Plot of Reflection Coefficients for Parallel Polarization for  r = 12, 4

20. 20  r =12  r =4 |||| 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0 10 20 30 40 50 60 70 80 90 angle of incidence (  i ) horizontal polarization (E-field not in plane of incidence) for given  i : larger dielectric constant  larger   and larger E r Plot of Reflection Coefficients for Perpendicular Polarization for  r = 12, 4

21. 21 e.g. let medium 1 = free space & medium 2 = dielectric if  i  0 o (wave is parallel to ground) then independent of  r, coefficients |   |  1 and |  || |  1  || =   = thus, if incident wave grazes the earth ground may be modeled as a perfect reflector with |  | = 1 regardless of polarization or ground dielectric properties horizontal polarization results in 180  phase shift

22. 22 3.5.2 Brewster Angle =  B Brewster angle only occurs for vertical (parallel) polarization angle at which no reflection occurs in medium of origin occurs when incident angle  i is such that  || = 0   i =  B if 1 st medium = free space & 2 nd medium has relative permittivity  r then (3.27) can be expressed as sin(  B ) = (3.28) sin(  B ) = (3.27)  B satisfies

23. 23 e.g. 1 st medium = free space Let  r = 4  sin(  B ) = = 0.44  B = sin -1 (0.44) = 26.6 o Let  r = 15  sin(  B ) = = 0.25  B = sin -1 (0.25) = 14.5 o

24. 24 3.6 Ground Reflection – 2 Ray Model Free Space Propagation model is inaccurate for most mobile RF channels 2 Ray Ground Reflection model considers both LOS path & ground reflected path based on geometric optics reasonably accurate for predicting large scale signal strength for distances of several km useful for - mobile RF systems which use tall towers (> 50m) - LOS microcell channels in urban environments Assume maximum LOS distances d  10km earth is flat

25. 25 Let E 0 = free space E-field (V/m) at distance d 0 Propagating Free Space E-field at distance d > d 0 is given by E(d,t) =(3.33) E-field’s envelope at distance d from transmitter given by |E(d,t)| = E 0 d 0 /d (1) Determine Total Received E-field (in V/m) E TOT E LOS EiEi E r = E g ii 00 d E TOT is combination of E LOS & E g E LOS = E-field of LOS component E g = E-field of ground reflected component θ i = θ r

26. 26 d’ d” E LOS EiEi EgEg ii 00 d htht h r E-field for LOS and reflected wave relative to E 0 given by: and E TOT = E LOS + E g E LOS (d’,t) =(3.34) E g (d”,t) =(3.35) assumes LOS & reflected waves arrive at the receiver with - d’ = distance of LOS wave - d” = distance of reflected wave

27. 27 From laws of reflection in dielectrics (section 3.5.1)  i =  0 (3.36) E g =  E i (3.37a) E t = (1+  ) E i (3.37b)  = reflection coefficient for ground E g d’ d” E LOS EiEi ii 00 EtEt

28. 28 resultant E-field is vector sum of E LOS and E g total E-field Envelope is given by |E TOT | = |E LOS + E g | (3.38) total electric field given by (3.39)E TOT (d,t) = Assume i. perfect horizontal E-field Polarization ii. perfect ground reflection iii. small  i (grazing incidence)   ≈ -1 & E t ≈ 0 reflected wave & incident wave have equal magnitude reflected wave is 180 o out of phase with incident wave transmitted wave ≈ 0

29. 29 path difference  = d” – d’ determined from method of images  = (3-40) if d >> h r + h t  Taylor series approximations yields (from 3-40)   (3-41) (2) Compute Phase Difference & Delay Between Two Components E LOS d d’ d” ii 00 h t h r h h h t +h r EiEi EgEg

30. 30 once  is known  we can compute phase difference  = = (3-42) time delay  d = (3-43) As d becomes large   = d”-d’ becomes small amplitudes of E LOS & E g are nearly identical & differ only in phase (3.44) if Δ = /n    = 2π/n 0 π 2π Δ

31. 31 (3) Evaluate E-field when reflected path arrives at receiver (3.45)E TOT (d,t)| t=d”/c = t = d”/creflected path arrives at receiver at  = =

32. 32 (3.46) = |E TOT (d)|= = =(3.47) (3.48) E TOT  Use phasor diagram to find resultant E-field from combined direct & ground reflected rays: (4) Determine exact E-field for 2-ray ground model at distance d

33. 33 As d increases  E TOT (d) decreases in oscillatory manner local maxima 6dB > free space value local minima ≈ -  dB (cancellation) once d is large enough  θ Δ < π & E TOT (d) falls off asymtotically with increasing d -50 -60 -70 -80 -90 -100 -110 -120 -130 -140 10 1 10 2 10 3 10 4 m f c = 3GHz f c = 7GHz f c = 11GHz Propagation Loss h t = h r = 1, G t = G r = 0dB

34. 34 if d satisfies 3.50  total E-field can be approximated as: k is a constant related to E 0 h t,h r, and (3.49) d > (3.50)this implies For phase difference,   < 0.6 radians (34 o )  sin(0.5   )    |E TOT (d)|  e.g. at 900MHz  if  < 0.03m  total E-field decays with d 2 (3.51) E TOT (d)  V/m

35. 35 Received Power at d is related to square of E-field by 3.2, 3.15, & 3.51 P r (d) =(3.52b) P r (d) =(3.52a) received power falls off at 40dB/decade receive power & path loss become independent of frequency if d >>

36. 36 Path Loss for 2-ray model with antenna gains is expressed as: for short Tx-Rx distances use (3.39) to compute total E field evaluate (3.42) for   =  (180 o )  d = 4h t h r / is where the ground appears in 1 st Fresnel Zone between Tx & Rx - 1 st Fresnel distance zone is useful parameter in microcell path loss models PL(dB) = 40log d - (10logG t + 10logG r + 20log h t + 20 log h r )(3.53) PL = 3.50 must hold