EE6604 Personal & Mobile Communications Week 8 Path Loss Models Shadowing Co-Channel Interference 1
Okumura-Hata Model L p = A+Blog 10 (d) A+Blog 10 (d) C A+Blog 10 (d) D for urban area for suburban area for open area where A = 69.55+26.16log 10 (f c ) 13.82log 10 (h b ) a(h m ) B = 44.9 6.55log 10 (h b ) C = 5.4+2[log 10 (f c /28)] 2 D = 40.94+4.78[log 10 (f c )] 2 19.33log 10 (f c ) Okumura and Hata s model is in terms of carrier frequency 150 f c 1000 (MHz) BS antenna height 30 h b 200 (m) MS antenna height 1 h m 10 (m) distance 1 d 20 (km) between the BS and MS. The model is known to be accurate to within 1 db for distances ranging from 1 to 20 km. 2
The parameter a(h m ) is a correction factor (1.1log 10 (f c ) 0.7)h m (1.56log 10 (f c ) 0.8) a(h m ) = for medium or small city 8.28(log 10 (1.54h m )) 2 1.1 for f c 200 MHz 3.2(log 10 (11.75h m )) 2 4.97 for f c 400 MHz for large city 3
190 170 Large City Suburban Open Area Path Loss (db) 150 130 110 90 70 0.0 0.2 0.5 0.8 1.0 1.2 1.5 Log(distance in km) Path loss predicted by the Okumura-Hata model. Large city, f c = 900 MHz, h b = 70 m, h m = 1.5 m. 4
CCIR Model To account for varying degrees of urbanization, the CCIR (Comité International des Radio-Communication, now ITU-R) developed an empirical model for the path loss as: L p (db) = A+Blog 10 (d) E where A and B are defined in the Okumura-Hata model with a(h m ) being the medium or small city value. The parameter E accounts for the degree of urbanization and is given by E = 30 25log 10 (% of area covered by buildings) where E = 0 when the area is covered by approximately 16% buildings. 5
Lee s Area-to-area Model Lee s area-to-area model is used to predict a path loss over flat terrain. If the actual terrain is not flat, e.g., hilly, there will be large prediction errors. Two parameters are required for Lee s area-to-area model; the power at a 1 mile (1.6 km) point of interception, µ Ωp (d o ), and the path-loss exponent, β. The received signal power at distance d can be expressed as or in decibel units µ Ωp (d) = µ Ωp (d o ) d β f d o f o n α 0 µ Ωp (dbm)(d) = µ Ωp (dbm)(d o ) 10βlog 10 d d o 10nlog 10 f f o +10log 10 α 0, where d is in units of kilometers and d o = 1.6 km. The parameter α 0 is a correction factor used to account for different BS and MS antenna heights, transmit powers, and antenna gains. 6
Lee s Area-to-area Model The following set of nominal conditions are assumed in Lee s area-to-area model: frequency f o = 900 MHz BS antenna height = 30.48 m BS transmit power = 10 watts BS antenna gain = 6 db above dipole gain MS antenna height = 3 m MS antenna gain = 0 db above dipole gain If the actual conditions are different from those listed above, then we compute the following parameters: α 1 = α 2 = α 3 = BS antenna height (m) 30.48 m MS antenna height (m) 3 m transmitter power 10 watts α 4 = BS antenna gain with respect to λ c/2 dipole 4 α 5 = different antenna-gain correction factor at the MS 2 κ 7
Lee s Area-to-area Model The parameters β and µ Ωp (d o ) have been found from empirical measurements, and are listed in the Table below. Terrain µ Ωp (d o ) (dbm) β Free Space -45 2 Open Area -49 4.35 North American Suburban -61.7 3.84 North American Urban (Philadelphia) -70 3.68 North American Urban (Newark) -64 4.31 Japanese Urban (Tokyo) -84 3.05 For f c < 450 MHz in a suburban or open area, n = 2 is recommended. In an urban area with f c > 450MHz, n = 3 is recommended. The value of κ in is also determined from empirical data as κ = 2 for a MS antenna height > 10 m 3 for a MS antenna height < 3 m. 8
Lee s Area-to-area Model The path loss L p (db) is the difference between the transmitted and received field strengths, L p (db) = µ Ωp (dbm) (d) µ Ωt (dbm). To compare directly with the Okumura-Hata model, we assume an isotropic BS antenna with 0 db gain, so that α 4 = 6 db. Then by using the same parameters as before, h b = 70 m, h m = 1.5m, f c = 900 MHz, a nominal BS transmitter power of 40 dbm (10 watts), and the parameters in the Table for µ Ωp (dbm) (d o ) and β, the following path losses are obtained: 85.74+20.0log 10 d Free Space 84.94+43.5log 10 d Open Area 98.68+38.4log L p (db) = 10 d Suburban 107.31+36.8log 10 d Philadelphia 100.02+43.1log 10 d Newark 122.59+30.5log 10 d Tokyo 9
190 170 Tokyo Newark Philadephia Suburban Open Area Free Space Path Loss (db) 150 130 110 90 70 0.0 0.2 0.5 0.8 1.0 1.2 1.5 Log(distance in km) Path loss obtained by using Lee s method; h b = 70 m, h m = 1.5 m, f c = 900 Mhz, and an isotropic BS antenna. 10
COST231-Hata Model COST231 models are for propagation in the PCS band. Path losses experienced at 1845 MHz are about 10 db larger than those experienced at 955 MHz. The COST-231 Hata model for NLOS propagation is where L p = A+Blog 10 (d)+c A = 46.3+33.9log 10 (f c ) 13.82log 10 (h b ) a(h m ) B = 44.9 6.55log 10 (h b ) 0 medium city and suburban areas C = with moderate tree density 3 for metropolitan centers 11
COST231-Walfish-Ikegami LOS Model For LOS propagation in a street canyon, the path loss is L p = 42.6+26log 10 (d)+20log 10 (f c ), d 20 m where the first constant is chosen so that L p is equal to the free-space path loss at a distance of 20 m. Themodel parametersare thedistanced(km) and carrierfrequencyf c (MHz). 12
COST231-Walfish-Ikegami NLOS Model h b h b h Roof BS d MS h h Roof m w b φ MS incident wave direction of travel Definition of parameters used in the COST231-Walfish-Ikegami model. 13
For NLOS propagation, the path loss is composed of three terms, viz., L p = The free-space loss is L o +L rts +L msd for L rts +L msd 0 L o for L rts +L msd < 0 L o = 32.4+20log 10 (d)+20log 10 (f c ) The roof-top-to-street diffraction and scatter loss is L rts = 16.9 10log 10 (w)+10log 10 (f c )+20log 10 h m +L ori where 10+0.354(φ), 0 φ 35 o L ori = 2.5+0.075(φ 35), 35 φ 55 o 4.0 0.114(φ 55), 55 φ 90 o h m = h Roof h m 14
The multi-screen diffraction loss is L msd = L bsh +k a +k d log 10 (d)+k f log 10 (f c ) 9log 10 (b) where and L bsh = k a = k d = k f = 4+ 18log 10 (1+ h b ) h b > h Roof 0 h b h Roof 54, h b > h Roof 54 0.8 h b, d 0.5km and h b h Roof 54 0.8 h b d/0.5, d < 0.5km and h b h Roof 18, h b > h Roof 18 15 h b /h Roof, h b h Roof 0.7(f c /925 1), medium city and suburban 1.5(f c /925 1), metropolitan area h b = h b h Roof. 15
k a is the increase in path loss for BS antennas below the roof tops of adjacent buildings. k d and k f control the dependency of the multi-screen diffraction loss on the distance and frequency, respectively. The model is valid for the following ranges of parameters, 800 f c 2000 (MHz), 4 h b 50 (m), 1 h m 3 (m), and 0.02 d 5 (km). The following default values are recommended, b = 20... 50 (m), w = b/2, φ = 90 o, and h Roof = 3 number of floors+roof (m), where roof = 3 (m) pitched and 0 (m) flat. 16
Shadowing Shadows are very often modeled as being log-normally distributed. Let Then distributions of Ω v and Ω p are where and ξ = ln10/10. p Ωv (x) = p Ωp (x) = Ω v = E[α(t)], µ Ωv = E[Ω v ] Ω p = E[α 2 (t)], µ Ωp = E[Ω p ] 2ξ xσ Ω 2π exp ξ xσ Ω 2π exp ( 10log10 x 2 µ Ωv (dbm) 2σ 2 Ω ( 10log10 x µ Ωp (dbm) 2σ 2 Ω µ Ωv (dbm) = 30+10E[log 10 Ω 2 v ] µ Ωp (dbm) = 30+10E[log 10 Ω p ] ) 2 ) 2 17
Shadowing By using a transformation of random variables, Ω v (dbm) = 30+10log 10 Ω 2 v and Ω p (dbm) = 30+10log 10 Ω p have the Gaussian densities p Ωv (dbm) (x) = p Ωp (dbm) (x) = 1 exp 2πσΩ 1 exp 2πσΩ (x µ Ω v (dbm) ) 2 2σ 2 Ω (x µ Ω p (dbm) ) 2 Note that the standard deviation σ Ω of Ω v (dbm) and Ω p (dbm) are the same. However, for Rician fading channels the means differ by where 2σ 2 Ω µ Ωp (dbm) = µ Ωv (dbm) +10 log 10 C(K) C(K) = 4e2K (K +1) π 1 F 2 1(3/2,1;K) 1F 1 (, ; ) is the confluent hypergeometric function (see Chap. 2, Appendix 3). Note that C(0) = 4/π, C( ) = 1, and 1 C(K) 4/π for 0 K.. 18
Shadow Simulation Shadows can be modelled by low-pass filtering white noise. Here we suggest a first-order low pass digital filter. In a discrete-time simulation, the local mean Ω k+1 (dbm) at step k +1 is generated recursively as follows: k is the step index. Ω k+1 (dbm) = ξω k (dbm) +(1 ξ)v k {v k } is a sequence of independent zero-mean Gaussian random variables with variance σ 2. ξ controls the shadow correlation The autocorrelation function of Ω k (dbm) can be derived as: φ Ω(dBm) Ω (dbm) (n) = 1 ξ 1+ξ σ2 ξ n 19
The variance of log-normal shadowing is σ 2 Ω = φ Ω (dbm) Ω (dbm) (0) = 1 ξ 1+ξ σ2 Consequently, we can express the autocorrelation of Ω k as φ Ω(dBm) Ω (dbm) (n) = σ 2 Ω ξ n Notice that the shadows decorrelated exponentially with the time lag in the autocorrelation function. Suppose we use discrete-time simulation, where each simulation step corresponds to T seconds. For a mobile station traveling at velocity v, the distance traveled in T seconds is vt meters. Let ξ D be the shadow correlation between two points separated by a spatial distance of D meters. Then the time autocorrelation of the shadowing is φ Ω(dBm) Ω (dbm) (n) φ Ω(dBm) Ω (dbm) (nt) = σ 2 Ωξ (vt/d) n D Measurements in Stockholm have shown ξ D = 0.1 for D = 30 meters (roughly). However, this can vary greatly depending on local topography. 20
Co-channel interference on the forward channel d 1 d 6 d 2 mobile subscriber d 0 d 5 d 3 d 4 The mobile station is being served by the center base station. 21
At a particular location, let d = (d 0, d 1,, d NI ) be the vector of distances between a mobile station and the serving base station BS 0 and N I co-channel base stations BS k,k = 1,...,N I. The received signal power power at distance d, Ω p (dbm) (d), is a Gaussian random variable that depends on the distance d through the path loss model, i.e., µ Ωp (dbm) (d) = E[Ω p (dbm) (d)] = µ Ωp (dbm) (d o ) 10βlog 10 (d/d o ) Experiments have verified that co-channel interferers add noncoherently (power addition) rather than coherently (amplitude addition). The C/I a function of the vector d is or in decibel units Λ(d) = Ω p(d 0 ) N I k=1ω p (d k ) Λ(d) (db) = Ω p (dbm) (d 0 ) 10log 10 The outage probability given vector d is N I k=1 O(d) = P r ( Λ(d)(dB) < Λ th(db) ) Ω p (d k ) Although the Ω p (d k ) are log-normal random variables, the sum N I k=1ω p (d k ) is not a log-normal random variable. 22
Multiple Log-normal Interferers Consider the sum of N I log-normal random variables I = N I k=1 Ω k = N I 10 Ω k(dbm) /10 k=1 where the Ω k (dbm) are independent Gaussian random variables with mean µ Ωk (dbm) and variance σ 2 Ω k. The sum I is commonly approximated by another log-normal random variable with appropriately chosen parameters, i.e., I = N I k=1 10 Ω k(dbm) /10 10 Z (dbm) /10 = Î where Z (dbm) is a Gaussian random variable with mean µ Z (dbm) and variance σ 2 Z. The task is to find µ Z (dbm) and σ 2 Z. 23
Fenton-Wilkinson Method The mean µ Z (dbm) and variance σ 2 Z of Z (dbm) are obtained by matching the first two moments of I and Î. Switching from base 10 to base e: Ω k = 10 Ω k (dbm) /10 = e ξω k (dbm) = eˆω k where ˆΩ k = ξω k (dbm) and ξ = (ln10)/10 = 0.23026. Note that µˆωk = ξµ Ωk (dbm) σ 2ˆΩk = ξ 2 σ 2 Ω k The nth moment of the log-normal random variable Ω k can be obtained from the moment generating function of the Gaussian random variable ˆΩ k as E[Ω n k] = E[e nˆω k ] = e nµˆωk +(1/2)n 2 σ 2ˆΩ Here we have assumed identical shadow variances, σ 2ˆΩk = σ 2ˆΩ, which is a reasonable assumption. 24
Suppose that ˆΩ 1,..., ˆΩNI are independent with means µˆω1,..., µˆωni and identical variances σ 2ˆΩ. The appropriate moments of the log-normal approximation are obtained by equating the means on both sides of where Ẑ = ξz (dbm). This gives µ I = E[I] = N I N I k=1 k=1 e µˆωk Also equate the variances on both sides of E[eˆΩ k ] E[eẐ] = E[Î] = µ Î e (1/2)σ2ˆΩ = e µ Ẑ +(1/2)σ2 Ẑ (1) σ 2 I = E[I 2 ] µ 2 I E[Î2 ] µ 2 Î = σ2 Î This gives N I k=1 e 2µˆΩk e σ2ˆω(e σ2ˆω 1) = e 2µ Ẑe σ2 Ẑ(e σ2 Ẑ 1) (2) 25
To obtain µẑ and σ 2 Ẑ 1. Square Eq. (1) and divide by Eq. (2) to obtain σ 2 Ẑ. 2. Obtain µẑ from Eq. (1) The above procedure yields σ 2 Ẑ = ln µẑ = σ2ˆω σ 2 Ẑ 2 (eσ2ˆω 1) ( NI +ln N I k=1e 2µˆΩk k=1e µˆωk ) 2 +1 N I k=1 e µˆωk Given the means µˆω1,..., µˆωni and variance σ 2ˆΩ, µẑ and σ 2 Ẑ are easily obtained. Finally, we convert back to base 10 by scaling, such that µ Z (dbm) = ξ 1 µẑ σz 2 = ξ 2 σ 2 Ẑ where ξ = 0.23026. 26
Fenton s method breaks down in the prediction of the first and second moments for σ Ω > 4 db. Schwartz and Yeh s method yields the exact first and second moments. However, Fenton s method accurately predicts the tails of the complementary distribution function cdfc F c I (x) = P r(i x) and the cdf F I (x) = 1 F c I (x) = P r (I < x). We are interested in the accuracy of the approximations F I (x) Q lnx µẑ F c I(x) 1 Q when x is small and large, respectively. σẑ lnx µẑ σẑ The cdfc is more important than the cdf for outage calculations and predictions, since outages typically occur when the interference is large. 27
10 0 10 1 10 2 Wilkinson Schwartz and Yeh Farley Simulation P(I<x) 10 3 10 4 N I =2 N I =6 10 5 10 6 10 2 10 1 10 0 10 1 x Comparison of the cdf for the sum of two and six log-normal random variables with various approximations; σ Ω = 6 db. 28
10 1 10 2 P(I>x) 10 3 10 4 Wilkinson Schwartz and Yeh Farley Simulation 10 5 10 6 10 1 10 2 x Comparison of the cdfc for the sum of two log-normal random variables with various approximations; σ Ω = 6 db. 29
10 1 10 2 P(I>x) 10 3 10 4 10 5 Wilkinson Schwartz and Yeh Farley Simulation 10 6 10 1 10 2 x Comparison of the cdfc for the sum of six log-normal random variables with various approximations; σ Ω = 6 db. 30
10 0 10 1 P(I>x) 10 2 10 3 10 4 Wilkinson Schwartz and Yeh Farley Simulation 10 3 10 4 x Comparison of the cdfc for the sum of six log-normal random variables with various approximations; σ Ω = 12 db. 31
Outage with Multiple Interferers 1. First obtain the mean and variance µ Z = µẑ/ξ σ 2 Z = σ 2 Ẑ /ξ2 ξ = 0.23026 2. Treat the average CIR as Gaussian distributed with mean and variance µ Λ(d) = µ Ω(d0 ) µ Z (dbm) σ 2 Λ(d) = σ 2 Ω +σ2 Z. 3. Compute the outage for a given location, described by d O(d) = Q µ Ω(d0 ) µ Z Λ th(db) σ 2 Ω +σ 2 Z 4. Average over all locations d by Monte Carlo integration O = R N O(d)p d (d)dd 32
Single Co-channel Interferer For a single co-channel interferer where p Λ(d)(dB) (x) = The outage for a given d is 1 exp 4πσΩ (x µ Λ(d) (db) ) 2 4σ 2 Ω µ Λ(d)(dB) = µ Ω(d0 ) (db) µ Ω(d1 ) (db) O(d) = Pr(Λ(d) (db) < Λ th(db) ) = Λ th(db) = Q 1 4πσΩ exp µ Λ(d)(dB) Λ th(db) 2σΩ (x µ Λ(d) (db) ) 2 4σ 2 Ω dx 33
R D-R serving MS co-channel BS BS Worst case interference from a single co-channel base-station. In this case d = (R,D R). The worst case outage due to a single co-channel interferer is O(R) = Q µ Ω(R)(dB) µ Ω(D R)(dB) Λ th (db) 2σΩ 34
Using a simple inverse-β path loss characteristic µ Ω(dB) = Ω (db) (d o ) 10βlog 10 (d/d o ) gives O(R) = Q 10log 10 ( D R 1) β Λth (db) 2σΩ The minimum CIR margin on the cell fringe is M Λ = 10log 10 D R 1 For an ideal hexagonal layout D R = 3N, so that N = 1 3 10 β Λ th (db) M Λ +Λ 2 th (db) 10β +1 A small cluster size is achieved by making the margin M Λ and receiver threshold Λ th small. 35
Rician/Multiple Rayleigh Interferers Sometimes propagation conditions exist such that the received signals experience fading, but not shadowing. In this section, we calculate the outage probability for the case of fading only. The received signal may consist of a direct line of sight (LoS) component, or perhaps a specular component, accompanied by a diffuse component. The envelope of the received desired signal experiences Ricean fading. The interfering signals are often assumed to be Rayleigh faded, because a direct LoS is unlikely to exist due to the larger physical distances between the co-channel interferers and the receiver. Let the instantaneous power in the desired signal and the N I interfering signals be denoted by s 0 and s k, k = 1,, N I, respectively. Note that s i = α 2 i, where α2 i is the squared-envelope. The carrier-to-interference ratio is defined as λ = s 0 / N I k=1s k, and for a specified receiver threshold λ th, the outage probability is O I = P(λ < λ th ). 36
Single Interferer For the case of a single interferer, the outage probability reduces to the simple closed form O I = λ th exp λ th +A KA 1 1 λ th +A 1, where K is the Rice factor of the desired signal, A 1 = Ω 0 /(K + 1)Ω 1, and Ω k = E[s k ]. If the desired signal is Rayleigh faded, then the outage probability can be obtained by setting K = 0. 37
Multiple Interferers For the case of multiple interferers, each with mean power Ω k, the outage probability has the closed form O I = 1 N I k=1 1 λ th exp λ th +A KA k k λ th +A k N I j=1 j k A j A j A k, where A k = Ω 0 /(K +1)Ω k. This expression is only valid if Ω i Ω j when i j, i.e., the different interferers have different mean power. If all the interferers have the same mean power, then the outage probability can be derived as λ th O I = exp λ th +A KA 1 1 λ th +A 1 N I 1 k=0 A 1 (λ th +A 1 k k m=0 k m 1 m! Kλ th λ th +A 1 If the desired signal is Rayleigh faded, then the probability of outage with multiple Rayleigh faded interferers can be obtained by setting K = 0. m. 38
Probability of Outage, O I 10 0 10 1 10 2 10 3 K = 0, N I = 1 K = 7, N I = 1 K = 7, N I = 2 K = 7, N I = 6 10 4 10 15 20 25 30 35 Λ (db) Probability of outage with multiple interferers. The desired signal is Ricean faded with various Rice factors, while the interfering signals are Rayleigh faded and of equal power; λ th = 10.0 db. Λ = Ω 0 N I Ω 1 39