Shadowing. Log-normal Shadowing. Local Average Power Measurements. Scanning in 3D. Same-Distance Measurements. Repeat for Multiple Distances

Transcription

1 Shadowing Log-nomal Shadowing Instucto: MA Ingam ECE483 Also called slow-fading Accounts fo andom vaiations in eceived powe obseved ove distances compaable to the widths of buildings Exta tansmit powe (a fading magin) must be povided to compensate fo these fades Local Aveage Powe Measuements Take powe measuements in Watts as the antenna is moved in a on the ode of a few wavelengths Aveage these measuements to give a local aveage powe measuement > λ/ Points whee powe measuements ae made Velocity of antenna Scanning in 3D Afte the cat is stabilized, the XYZ actuatos move an antenna in 3 D to make a set local powe measuements One local aveage powe is computed fo each cat location Y The antenna Z X Z Y X Scan volume Same-Distance Measuements Local aveages ae made fo many diffeent locations, keeping the same tansmitte-eceive distance These local aveages will vay andomly with location Repeat fo Multiple Distances Simila collections of aveage powes ae made fo othe Tx-Rx distances Example Tx-Rx locations within a floo of a building

2 Likelihood of Coveage At a cetain distance, d, what is the pobability that the local aveage eceived powe is below a cetain theshold γ? P( P ( < γ ) PG t tg P ( = L L t distance dependence shown explicitly Likelihood of Coveage, cont d Since only the path loss is andom, the pobability can be expessed as a pobability involving path loss: P( P ( < γ ) = P( > β ) maximum toleable path loss Typical Macocell Chaacteistics Path Loss Assumptions Total Path Loss [-db] [not eal data] The set of aveage losses measued at about 4km [1log 1 (4)=6] The mean loss in db follows the powe law : d = d o) + 1nlog1 do The measued loss in db vaies about this mean accoding to a zeo-mean Gaussian RV, X σ, with standad deviation σ d = do ) + 1nlog1 + X σ d o 1log 1 (Distance fom Base Station [km]) Typical Data Chaacteistics Pobability Calculation Total Path Loss [-db] [not eal data] Best Fit Path Loss Exponent: n=4 σ is usually 5-1 db fo mobile comm Since is Gaussian, we need to know how to calculate pobability involving Gaussian RVs P( > β ) 1log 1 (Distance fom Base Station [km])

3 Q Function If X is a Gaussian RV with mean α and standad deviation σ, then b α P ( X > b) σ whee Q is a function defined as The Poblem with Q The integand of Q has no antideivative Q is found tabulated in books Q can be calculated using numeical integation + 1 x Q( z) = exp dx π z What is Log-nomal Shadowing? If Y is a Gaussian RV and Z is defined such that Y=logZ, then Z is a lognomal RV Shadowing is log-nomal shadowing when the path loss in db is Gaussian; this means that the path loss expessed as a atio is log-nomal Invese Q Poblems Sometimes, the pobability is specified and we must find one of the paametes in the agument of Q b α P ( X > b) σ Suppose the value of P ( X > b) is given, along with values of b and α Solve fo σ Must look up the agument of Q that gives the specified value Example Invese Q Poblem Suppose the mean of the local aveage eceived powes at a cetain distance is -3dBm, that the standad deviation of shadow fading is 9 db, and that the obseved eceived powe is above the theshold 95% of the time What is the theshold powe? b ( 3) P( P > b) = 95 9 Q is usually tabulated fo aguments of 5 and less, so we must use the fact that Q z = 1 Q( z ( ) ) The agument of Q that yields 5 is about 165 b + 3 = 165, and b = 4485 dbm 9 Bounday Coveage Suppose that a cell has adius R and γ is the minimum acceptable eceived powe level Then P( P > γ ) is the likelihood of coveage at the bounday of the cell P( P > γ ) is also the faction of time that a mobile s signal is acceptable at a distance R fom the tansmitte, assuming the ca moves aound that cicle

4 Pecentage of Useful Sevice Aea By integating these pobabilities ove all the cicles within a disk, one can compute the faction of the aea within the cell that will have acceptable powe levels π R 1 U ( γ ) = P P γ ddθ πr ( ( ) > ) Integal Evaluated Assuming log-nomal shadowing and the powe path loss model, the faction of useful sevice aea is 1 1 ab 1 ab U ( γ ) = 1 ef ( a) + exp 1 ef b b whee γ P a = σ and 1nlog1 e b = σ The Eo Function ef(x) is anothe fom of the Gaussian integal (like Q(x)) ef(x) has odd symmety, with exteme values ±1 ef 1 (x) ef ( x) = π x t e dt Note that some authos may define ef diffeently 1-1 x ef and Q The ef function and Q ae elated: ef ( ( z) z) = 1 Q When the Aveage Bounday Powe is Acceptable Suppose P (R) = γ Then we may use this gaph fom [Rappapot 96] to figue the pecent useful sevice aea Summay Local aveages in db of eceived powe (o path loss) tend to be Gaussian when the ensemble is all Tx-Rx locations with the same distance in the same type of envionment The mean local aveage path loss follows the standad powe model (popotional to 1logd n ) Can use Q o ef to calculate the likelihood of bounday coveage o the pecent of useful sevice aea

5 Refeences [Rapp, 96] TS Rappapot, Wieless Communications, Pentice Hall, 1996 [Saundes,`99] Simon R Saundes, Antennas and Popagation fo Wieless Communication Systems, John Wiley and Sons, LTD, 1999