RADIO PROPAGATION MODELS 1
Radio Propagation Models 1 Path Loss • Free Space Loss • Ground Reflections • Surface Waves • Diffraction • Channelization 2 Shadowing 3 Multipath Reception and Scattering • Dispersion • Time Variations 2
Key Questions about Propagation • Why may radio reception vanish while waiting for a traffic light? • How does path loss depend on propagation distance? • What are the consequences for cell planning? • Why has the received amplitude a ‘Rician’ amplitude? • What can we do to improve the receiver? Key Terms • Antenna Gain; Free-Space Loss; Ground Reflections; Two-Ray Model; "40 Log d"; Shadowing; Rician Fading; Bessel Function I 0 (.); Rician K -Ratio; Rayleigh Fading 3
Free Space Loss Isotropic antenna: power is distributed homogeneously over surface area of a sphere. The power density w at distance d is P = T w π 2 4 d where P T is the transmit power. The received power is A = w P T π 2 4 d with A the `antenna aperture' or the effective receiving surface area. 4
FREE SPACE LOSS, continued The antenna gain G R is related to the aperture A according to π = 4 A GR 2 λ Thus the received signal power is λ 2 1 • • P = P G R T R π π 2 4 4 d The received power decreases with distance, P R :: d -2 • The received power decreases with frequency, P R :: f -2 • Cellular radio planning Path Loss in dB: L fs = 32.44 + 20 log ( f /1 MHz) + 20 log ( d / 1 km) Broadcast planning (CCIR) Field strength and received power: E 0 = √ (120 π P R ) 30 P G T T In free space: E = 0 π 4 d 5
Antenna Gain A theorem about cats: An isotropic antenna can not exist. Antenna Gain G T ( φ , θ ) is the amount of power radiated in direction ( φ , θ ), relative to an isotropic antenna. Definition: Effective Radiated Power (ERP) is P T G T Half-Wave Dipole: A half-wave dipole has antenna gain π 2 θ cos cos 2 θ φ G( , ) = 1.64 θ sin 6
Law of Conservation of Energy Total power at distance d is equal to P T φ θ ∫ G( , ) dA = 1 π 4 ⇒ A directional antenna can amplify signals from one direction { G R ( φ , θ ) >> 1}, but must attenuate signals from other directions { G R ( φ , θ ) < 1}. 7
Groundwave loss: Waves travelling over land interact with the earth's surface. Norton: For propagation over a plane earth, ( ) ∆ ∆ • ••• j j E = E 1+ R e + (1- R ) F( )e + , i c c 0 i where R c is the reflection coefficient, E 0i is the theoretical field strength for free space F ( ⋅ ) is the (complex) surface wave attenuation ∆ is the phase difference between direct and ground- reflected wave Bullington: Received Electric Field = direct line-of-sight wave + wave reflected from the earth's surface + a surface wave. Space wave The (phasor) sum of the direct wave and the ground-reflected wave is called 'space wave' 8
Space-wave approximation for UHF land-mobile communication: • Received field strength ≈ LOS + Ground-reflected wave. Surface wave is negligible, i.e., F ( ⋅ ) << 1, for the usual values of h t and h r . {( h t - h r ) 2 + d 2 } h r h t {( h t + h r ) 2 + d 2 } The received signal power is 2 λ Re ∆ + j P = 4 d 1 P G G R T T R π The phase difference ∆ is found from Pythagoras. Distance between TX and RX antenna = √ {( h t - h t ) 2 + d 2 } Distance between TX and mirrored RX antenna = √ {( h t + h t ) 2 + d 2 }} 9
Space-wave approximation The phase difference ∆ is ( ) π ∆ = 2 2 2 2 2 d +(h +h ) - d +(h -h ) λ t r t r At large a distance, d >> 5 h t h r , ∆ ≈ 4 π h h r t λ d so, the received signal power is 2 π λ 4 jh h + r t P = 4 d 1 R exp P G G R T T R π λ d The reflection coefficient approaches R c → -1 for • large propagation distances • low antenna heights For large distances d → ∞ : ∆ → 0 and R c → -1. In this case, LOS and ground-reflected wave cancel!! 10
Two-ray model (space-wave approximation) Received Power [dB] d -2 d -4 ln( Distance ) For R c = -1 and approximate ∆ , the received power is λ 2 π = 2 h h r t 2 P 4 d 4 G G P sin R π λ R T T d N.B. At short range, R c may not be close to -1. Therefor, nulls are less prominent as predicted by the above formula. 11
Macro-cellular groundwave propagation For d λ >> 4 h r h t , we approximate sin( x ) ≈ x : 2 2 P h h r t _ P G G R T R T 4 d Egli [1957]: semi-empirical model for path loss f c L = 40 log d + 20 log -20 log h h . r t 40 MHz • Loss per distance:................ 40 log d • Antenna height gain:............. 6 dB per octave • Empirical factor:................... 20 log f • Error: standard deviation...... 12 dB 12
Generic path-loss models • p is normalized power r is normalized distance • Free Space Loss: "20 log d" models p = r -2 Groundwave propagation: "40 log d" models p = r -4 Empirical model: p = r - β , β ≈ 2 ... 5 β ≈ 3.2 Micro-cellular models VHF/UHF propagation for low antenna height ( h t = 5 ⋅⋅ 10 m) β - 2 1 + r β - p = r 1 r g 13
Diffraction loss The diffraction parameter v is defined as 2 1 + 1 = v h , m λ d d t r where h m is the height of the obstacle, and d t is distance transmitter - obstacle d r is distance receiver - obstacle The diffraction loss L d , expressed in dB, is approximated by + − < < 2 6 9 v 127 . v 0 v 2 4 . d = L + > 13 20 log v v 2 4 . 14
How to combine ground-reflection and diffraction loss? Obstacle gain: • The attenuation over a path with a knife edge can be smaller than the loss over a path without the obstacle! • "Obstacles mitigate ground-reflection loss" Bullington : "add all theoretical losses" L = L + L + L , K fs d R Blomquist : 2 2 L = L + L + L , K fs d R 15
Statistical Fluctuation: Location Averages Received Power [dB] ln( Distance ) • Area-mean power · is determined by path loss · is an average over 100 m - 5 km • Local-mean power · is caused by local 'shadowing' effects · has slow variations is an average over 40 λ (few meters) · • Instantaneous power · fluctuations are caused by multipath reception · depends on location and frequency · depends on time if antenna is in motion · has fast variations (fades occur about every half a wave length) 16
Shadowing Local obstacles cause random shadow attenuation Model: Normal distribution of the received power P Log in logarithmic units (such as dB or neper), Probability Density: ( ) 1 - 1 2 f = exp p p πσ σ p Log Log 2 2 2 Log where σ is the 'logarithmic standard deviation' in natural units. P Log = ln [local-mean power / area-mean power ] The standard deviation in dB is found from s = 4.34 σ 17
The log-normal distribution Convert 'nepers' to 'watts'. Use p ln = p Log p and ( ) ( ) f p d p = f p d p p Log Log p p p Log = ln p The log-normal distribution of received (local-mean) power is ( ) 1 - 1 p 2 p = p exp , f ln π σ p σ 2 2 2 p s 18
Area-mean and local-mean power • The area-mean power is the logarithmic average of the local-mean power • The linear average and higher-order moments of local- mean power are [ ] ∞ σ ( ) 2 m m m ∫ 2 E _ p f p d p = p exp . p m 2 p 0 N.B. With shadowing, the interference power accumulates rapidly!! Average of sum of 6 interferers is larger than sum of area means. 19
Depth of shadowing: sigma = 3 .. 12 dB "Large-area Shadowing": · Egli: Average terrain: 8.3 dB for VHF and 12 dB for UHF · Marsan, Hess and Gilbert: Semi-circular routes in Chicago: 6.5 dB to 10.5 dB, with a median of 9.3 dB. "Small-area shadowing" · Marsan et al.: 3.7 dB · Preller & Koch: 4 .. 7 dB Combined model by Mawira (PTT Research): Two superimposed Markovian processes: 3 dB with coherence distance over 100 m, plus 4 dB with coherence distance 1200 m 20
Rician multipath reception reflections line of sight TX RX Narrowband propagation model: • Transmitted carrier cos ω s(t) = t c • Received carrier N ρ φ ∑ ω ω v(t)= C cos t + cos ( t + ) , c c n n n=1 where C is the amplitude of the line-of-sight component ρ n is the amplitude of the n -th reflected wave φ n is the phase of the n -th reflected wave Rayleigh fading: C = 0 21
Rician fading: I-Q Phasor diagram Received carrier: N ρ φ ∑ ω ω v(t)= C cos t + cos ( t + ) , c c n n n=1 where ζ is the in-phase component of the reflections ξ is the quadrature component of the reflections. is the total in-phase component ( I = C + ζ ) I is the total quadrature component ( Q = ξ ) Q 22
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