RADIO PROPAGATION MODELS

Transcription

1 RADIO PROPAGATION MODELS 1

2 Radio Propagation Models 1 Path Loss Free Space Loss Ground Reflections Surface Waves Diffraction Channelization Shadowing 3 Multipath Reception and Scattering Dispersion Time Variations

3 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

4 Free Space Loss Isotropic antenna: power is distributed homogeneously over surface area of a sphere. The power density w at distance d is w PT = 4 πd where P T is the transmit power. The received power is w = A 4 πd P T with A the `antenna aperture' or the effective receiving surface area. 4

5 FREE SPACE LOSS, continued The antenna gain G R is related to the aperture A according to G A R = 4 π λ Thus the received signal power is 1 P R = PTGR λ 4 π 4πd The received power decreases with distance, P R :: d - The received power decreases with frequency, P R :: f - Cellular radio planning Path Loss in db: L fs = log (f /1 MHz) + 0 log (d / 1 km) Broadcast planning (CCIR) Field strength and received power: E 0 = (10 π P R ) In free space: E 0 = 30 PTG 4πd T 5

6 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 cos cosθ G( θφ, ) = 1.64 sinθ 6

7 Law of Conservation of Energy Total power at distance d is equal to P T 4π G( φθ, ) da = 1 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

8 Groundwave loss: Waves travelling over land interact with the earth's surface. Norton: For propagation over a plane earth, where R c j j ( ) E =E 1+ Re + (1-R ) F( )e +, i 0 c c i 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 groundreflected 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

9 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 t - h r ) +d } h r {(h t + h r ) +d } The received signal power is P = R λ j 4πd 1 + Re P G G T T R The phase difference is found from Pythagoras. Distance between TX and RX antenna = {( h t - h t ) + d } Distance between TX and mirrored RX antenna = {( h t + h t ) + d }} 9

10 Space-wave approximation The phase difference is = π d +(h +h ) - d +(h -h ) λ ( ) t r t r At large a distance, d >> 5h t h r, 4π hh r λd so, the received signal power is t P = R λ 4πd 1+ Rexp 4πjhh r λd t P G G T T R 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

11 Two-ray model (space-wave approximation) Received Power [db] d - d -4 ln(distance) For R c = -1 and approximate, the received power is P R = λ 4πd 4sin πhh λd r t G G P R T T 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

12 Macro-cellular groundwave propagation For dλ >> 4 h r h t, we approximate sin(x) x: P R _ r t 4 PTGRGT hh d Egli [1957]: semi-empirical model for path loss f c L = 40 logd + 0 log -0loghh r t. 40MHz Loss per distance: log d Antenna height gain:... 6 db per octave Empirical factor:... 0 log f Error: standard deviation... 1 db 1

13 Generic path-loss models p is normalized power r is normalized distance Free Space Loss: "0 log d" models p = r - Groundwave propagation: "40 log d" models p = r -4 Empirical model: p = r -β, β... 5 β 3. Micro-cellular models VHF/UHF propagation for low antenna height (h t = 5 10 m) - p = r 1 + r β1 r g -β 13

14 Diffraction loss The diffraction parameter v is defined as v = h m 1 d + 1 λ 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 L d = 6+ 9v 17. v 0< v< log v v>. 4 14

15 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 K = L fs+ L d+ L R, Blomquist: L K = L fs+ L d+l R, 15

16 Statistical Fluctuation: Location Averages Received Power [db] 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 ln(distance) 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

17 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: ( ) exp plog Log - plog f p = 1 πσ 1 σ 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

18 The log-normal distribution Convert 'nepers' to 'watts'. Use and p = Log p ln p ( ) ( ) f pdp= f p dp p p Log Log p p Log = ln p The log-normal distribution of received (local-mean) power is p( ) f p= 1 πσ p s 1 p exp - ln σ, p 18

19 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 localmean power are E m σ p _ p f p dp = p exp m. 0 [ ] m m p( ) N.B. With shadowing, the interference power accumulates rapidly!! Average of sum of 6 interferers is larger than sum of area means. 19

20 Depth of shadowing: sigma = db "Large-area Shadowing": Egli: Average terrain: 8.3 db for VHF and 1 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: 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 100 m 0

21 Rician multipath reception line of sight reflections TX RX Narrowband propagation model: Transmitted carrier s(t) = t cosω c Received carrier where C ρ n φ n v(t)= Ccosω t + ρ cos( ω t + φ ), c N n=1 n c n is the amplitude of the line-of-sight component is the amplitude of the n-th reflected wave is the phase of the n-th reflected wave Rayleigh fading: C = 0 1

22 Rician fading: I-Q Phasor diagram Received carrier: v(t)= Ccosω t + ρ cos( ω t + φ ), c N n=1 n c n where ζ is the in-phase component of the reflections ξ is the quadrature component of the reflections. I is the total in-phase component (I = C + ζ) Q is the total quadrature component (Q = ξ)

23 Central Limit Theorem ζ and ξ are zero-mean independently identically distributed (i.i.d.) jointly Gaussian random variables PDF: f (i,q) = 1 πσ I,Q - i +(q-c ) exp σ Conversion to polar coordinates: Received amplitude ρ: ρ = i + q. i = ρ cos φ; q = ρ sin φ, f ( ρφ, ) = ρ πσ ΡΦ, ρ +C -ρccosφ exp - σ 3

24 Rician Amplitude Integrate joint PDF over φ from 0 to π: Rician PDF of ρ ρ( ρ ρ ρ +C ρ ) exp I0 f = q - q C q, where I 0 ( ) is the modified Bessel function of the first kind and zero order is the total scattered power ( = σ ). Rician K-ratio K = direct power C / over scattered power Measured values K = (6 to 30 db) for micro-cellular systems 4

25 Light fading (K ) Very strong dominant component Rician PDF Gaussian PDF Severe Fading: Rayleigh Fading Direct line-of-sight component is small (C 0, K 0). The variances of ζ and ξ are equal to local-mean power PDF of amplitude ρ is Rayleigh ρ ρ f Ρ( ρ) = exp -. p p The instantaneous power p (p = 1 /ρ = 1 /ζ + 1 /ξ ) is exponential p( ) ρ( ρ ) f p =f dρ dp = 1 exp - p p p. 5

26 Nakagami fading The sum of m exponentially distributed powers is Gamma distributed. t f p ( p ) t t = 1 p p Γ(m) p where m-1 exp t - p p. Γ(m) is the gamma function; Γ(m+1) = m! m is the 'shape' factor The local-mean power E [p t ] = m. The amplitude is Nakagami m-distributed m-1 ρ( ρ ρ ρ ) exp f = Γ (m) p m-1 m - p Application of this model: - Joint interference signal (not constant envelope!!) - Dispersive fading; self interference N.B. The sum of m Rayleigh phasors is again a Rayleigh phasor. 6