Indoor Propagation Models Outdoor models are not accurate for indoor scenarios. Examples of indoor scenario: home, shopping mall, office building, factory. Ceiling structure, walls, furniture and people effect the EM wave propagation. Large/small number of obstacles, material of the walls etc. Modeling approach: classify various environments into few types and model each type individually. Generic model is very difficult to build. Key Model The average path loss is or in db: d ν LA ( d ) = L0 = const d ~ d d 0 d LA ( d )[ db] = L0[ db] + 10νlg d 0 ν ν (4.1) where L 0 is path loss at reference distance d 0. Lecture 4 28-Sep-16 1(27)
ITU Indoor Path Loss Model 1 The model is used to predict propagation path loss inside buildings. The average path loss in db is L ( d )[ db] = 20lg f + 10ν lg d + L ( n) 28 A f where: Limits: f is the frequency in MHz; d is the distance in m; d > 1 m; ν is the path loss exponent (found from measurements); L ( n ) is the floor penetration loss (measurements); f n is the number of floors (penetrated); 900MHz f 5200MHz 1 n 3 d > 1m 1 Recommendation ITU-R P.1238-8. Lecture 4 28-Sep-16 2(27)
Variations (fading) around the average are accounted for via lognormal distribution: ( ) ( ) L d [ db] = LA d [ db] + X σ (4.2) where X σ is a log-normal random variable (in db) of standard deviation σ [ db]. Variations on the order of (2...3) σ [ db] should be expected in practice. (2..3)σ Rule (adopted from "Empirical Rule" by Dan Kernler) Site-specific models will follow this generic model. Additional factors are included (floors, partitions, indoor-outdoor penetration etc.). Lecture 4 28-Sep-16 3(27)
T.S. Rappaport, Wireless Communications, Prentice Hall, 2002 Note: n = ν is the path loss exponent; f = 914 MHz. Lecture 4 28-Sep-16 4(27)
T.S. Rappaport, Wireless Communications, Prentice Hall, 2002 Lecture 4 28-Sep-16 5(27)
T.S. Rappaport, Wireless Communications, Prentice Hall, 2002 Lecture 4 28-Sep-16 6(27)
Table 4.1: path loss exponent factor 10ν in various environments Table 4.2: Floor penetration loss L ( n ) in various environments f Log-normal fading should be added as well, ( ) ( ) L d [ db] = LA d [ db] + X σ (4.2) Lecture 4 25-Sep-13 7(27)
T.S. Rappaport, Wireless Communications, Prentice Hall, 2002 Lecture 4 25-Sep-13 8(27)
T.S. Rappaport, Wireless Communications, Prentice Hall, 2002 Lecture 4 25-Sep-13 9(27)
Ericsson s Indoor Path Loss Model (900 MHz) T.S. Rappaport, Wireless Communications, Prentice Hall, 2002 Lecture 4 4-Oct-17 10(27)
Log-Normal Fading (Shadowing) Reminder: 3 factors in the total path loss, LP = LALLF LSF (2.5) Siwiak, Radiowave Propagation and Antennas for Personal Communications, Artech House, 1998 Lecture 4 4-Oct-17 11(27)
Log-Normal Fading (shadowing): this is a long-term (or largescale) fading since characteristic distance is a few hundreds wavelengths. Due to various terrain effects, the actual path loss varies about the average value predicted by the models above, p p [ db] L = L + L (4.5) where L p is the average path loss, L - its variation, which can be described by log-normal distribution. Overall, L p becomes a log-normal RV, where L p and db as well). ( L p ) ρ = 1 e 2πσ ( L ) 2 p Lp 2σ 2 (4.6) L p are in db, and σ is the standard deviation (in Physical explanation: multiple diffractions + the central limit theorem (in db). Shadowing: due to the obstruction of LOS path. The semi-empirical models above can be used together with the log-normal distribution. Reasonable physical assumptions result in statistical models for the PC. This approach is very popular and extensively used in practice. Lecture 4 4-Oct-17 12(27)
Log-Normal Fading: Derivation Tx Rx Assume signal at Rx is a result of many scattering/diffractions: Total Rx power: If E N = E Π Γ, Γ 1 (4.7) t 0 i i i= 1 N N 2 2 2 2 t t = 0 Π Γ i t = 0Π Γi i= 1 i= 1 N P E E or P P P = P + 20 lg Γ db 0, db i i= 1 Γ i are i.i.d., then PdB N P0, dbσdb (, ) (4.8) Log-normal distribution works well for i.i.d multiple diffractions ( N 5) and is used in practice to model large-scale fading (shadowing). In practice: σ db = 5...10 db (can take 8 db as an average). System design: allow for 2σ db margin (for about 95% reliability). Lecture 4 4-Oct-17 13(27)
Small-Scale (multipath) Fading Model E 2 E 3 E 1 E 5 Rx E 4 Many multipath components (plane waves) arriving at Rx at different angles, N E ( t) = E cos( ω t + ϕ ) t i i i= 1 N = E cosϕ cosωt E sinϕ sinωt i i i i i= 1 i= 1 This is in-phase (I) and quadrature (Q) representation N E ( t) = E cos ωt E sin ω t = E cos( ω t + ϕ) t x y (4.9) Assume that i N 2 2 x y where E = E + E envelope I : E = E cos ϕ, Q: E = E sin ϕ x i i y i i i= 1 i= 1 E are i.i.d., and that ϕ [ 0,2 ] i N π are i.i.d. (4.10) Lecture 4 4-Oct-17 14(27)
By central limit theorem, E is Rayleigh distributed with pdf where 2 E, E N (0, σ ) 2 σ is the variance of E x (or E y ), x y x x ρ ( x) = exp, x 0 2 2 σ 2σ N 2 2 1 2 Ex Ei 2 i= 1 2 (4.11) σ = = (4.12) which is the total received power (for isotropic antennas). For this result to hold, N must be large ( N 5 10). 0.8 Rayleigh PDF 0.6 PDF 0.4 0.2 0 0 1 2 3 4 5. xσ / Lecture 4 4-Oct-17 15(27)
Outage Probability and CDF Importance of the CDF in wireless system design. Rx operates well if E E If E th th the threshold effect. < E, the link is lost --> this is an outage. Outage probability = CDF is x F( x) = ρ ( t) dt = Pr( E < x) (4.13) 0 Lecture 4 4-Oct-17 16(27)
Rayleigh Fading For Rayleigh distribution, the outage probability is x 2 x 2 0 2σ (4.14) F( x) = Pr( E < x) = ρ ( t) dt = 1 exp Introduce the instantaneous signal power P 2 = P = σ = the average power, then P out { } = Pr SNR < γ P = x 2 / 2, γ P, (4.15) = 1 exp = 1 exp = F( P) γ P so that normalized SNR ( γ / γ ) or power ( P / P) PDF and CDF: and asymptotically, Note that P P x ( ) ( ) x f x = e, F x = 1 e (4.15a) P γ P P Pout = F( P) = (4.16) P γ γ = γ, where γ is the SNR. Note in (4.16) the 10db/decade law. Lecture 4 4-Oct-17 17(27)
Example: 3 3 P = 10 P = 10 P, or 30 db w.r.t. P out 3 i.e. if P out = 10 is desired, the Rx threshold is 30dB below one without fading (the average). 1 Rayleigh CDF Outage probability 1. 10 3 0.1 0.01 1. 10 4 40 30 20 10 0 10 20 γ / γ, db. Complex-valued model: N E ( t) = E e i e (4.17) t i= 1 results in complex Gaussian variables. i jϕ jωt Propagation channel gain simply normalized received signal, has the same distribution. Lecture 4 4-Oct-17 18(27)
Rayleigh Fading Channel 10 SISO 1x1 0 10 20 30 0 100 200 300 400 500 Received power (SNR) norm. to the average [db] vs distance (location, time) Lecture 4 4-Oct-17 19(27)
T.S. Rappaport, Wireless Communications, Prentice Hall, 2002 Lecture 4 4-Oct-17 20(27)
Measured Fading Channels T.S. Rappaport, Wireless Communications, Prentice Hall, 2002. Lecture 4 4-Oct-17 21(27)
LOS and Ricean Model Motivation: a LOS component. The distribution of in-phase and quadrature components is still Gaussian, but non-zero mean: N t ( ) = i cos( ω + ϕ i) + 0 cos( ω + ϕ0) (4.18) i= 1 E t E t E t Both E 0 and ϕ 0 are fixed (non-random constants). E ( t) = ( E + E )cos ωt ( E + E )sin ωt t x x0 y y0 2 2 x = y = 0, = ( x + x0) + ( y + y0) E E and E E E E E (4.19) Pdf of E has a Rice distribution( x = E, x0 = E0) : 2 2 x x xx 0 2 2 0 2 x + ρ = σ 2 σ σ 0 ( x) exp I, Note that if x 0 = 0, it reduces to the Rayleigh pdf. Introduce K-factor: 2 (4.20) K = x 2 2 0 / (2 σ ) (4.21) where x 0 / 2 is the LOS power, it is called LOS ( steady or 2 specular ) component, σ is the scattered (multipath or diffused) power, it is called diffused component. K tells us how strong the LOS is. Total average power = 2 2 2 x 0 / 2 + σ = σ (1 + K ) Lecture 4 4-Oct-17 22(27)
PDF of E becomes 2 2 2 0 x x x ρ ( x) = exp K I 2K σ 2 σ σ (4.22) 0.8 Rayleigh & Rice Densities 0.6 K=0 pdf 0.4 K=1 K=10 K=20 0.2 0 0 2 4 6 8 10. xσ / Note: normalized to the multipath power only. Q: do the same graph if normalized to the total average power. Lecture 4 4-Oct-17 23(27)
10 Rice CDF 1 0.1 Outage probability 0.01 1. 10 3 1.10 4 1.10 5 1.10 6 1.10 7 K = 0 K = 10 K = 1 K = 20 1.10 8 40 30 20 10 0 10 γ / γ, db Q.: find the CDF of Ricean distribution as a function of K and total (LOS + multipath) average power or SNR. Lecture 4 4-Oct-17 24(27)
Applications of Outage Probability 1) Fade margin evaluation for the link budget: P F P 1 out ( γth / γ 0) = ε = γ0 / γ th = 1/ out ( ε ) 2) Average outage time: T = P T out out 2) Average # of users in outage: N = P N out out To be discussed later on: level crossing rates (# of fades per unit time) average fade duration Lecture 4 4-Oct-17 25(27)
Monte-Carlo Method It is a powerful simulation technique to solve many statistical problems numerically in a very efficient way. You should be familiar with it. Detailed description of the method and many examples can be found in numerous references, including the following: [1] M.C. Jeruchim, P. Balaban, K.S. Shanmugan, Simulation of Communication Systems, Kluwer, New York, 2000. [2] W.H. Tranter et al, Principles of Communication System Simulation with Wireless Applications, Prentice Hall, Upper Saddle River, 2004. [3] J.G. Proakis, M. Salehi, Contemporary Communication Systems Using MATLAB, Brooks/Cole, 2000. This is used in labs extensively. Lecture 4 4-Oct-17 26(27)
Summary Indoor propagation path loss models. Log-normal shadowing. Small-scale fading. Rayleigh & Rice distributions. Physical mechanisms. o Rappaport, Ch. 4. Reading: References: o S. Salous, Radio Propagation Measurement and Channel Modelling, Wiley, 2013. (available online) o J.S. Seybold, Introduction to RF propagation, Wiley, 2005. o https://en.wikipedia.org/wiki/itu_model_for_indoor_attenua tion o Other books (see the reference list). Note: Do not forget to do end-of-chapter problems. Remember the learning efficiency pyramid! Lecture 4 4-Oct-17 27(27)