Indoor Propagation Models

Transcription

1 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 3-Sep-15 1(3)

2 Variations (fading) are accounted for via log-normal distribution: ( ) ( ) L d [ db] = LA d [ db] + X σ (4.) where X σ is a log-normal random variable (in db) of standard deviation σ [ db]. Variations on the order of σ [ db] should be expected in practice. "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 3-Sep-15 (3)

3 T.S. Rappaport, Wireless Communications, Prentice Hall, 00 Note: n = ν is the path loss exponent; f = 914 MHz. Lecture 4 3-Sep-15 3(3)

4 T.S. Rappaport, Wireless Communications, Prentice Hall, 00 Lecture 4 3-Sep-15 4(3)

5 T.S. Rappaport, Wireless Communications, Prentice Hall, 00 ELG4179: Wireless Communication Fundamentals S.Loyka Lecture 4 3-Sep-15 5(3)

6 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] = 0lg f + 10ν lg d + L ( n) 8 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 500MHz 1 n 3 d > 1m 1 Recommendation ITU-R P Lecture 4 5-Sep-13 6(3)

7 Table 4.1: path loss exponent factor 10ν in various environments Table 4.: 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.) Lecture 4 5-Sep-13 7(3)

8 T.S. Rappaport, Wireless Communications, Prentice Hall, 00 Lecture 4 5-Sep-13 8(3)

9 T.S. Rappaport, Wireless Communications, Prentice Hall, 00 Lecture 4 5-Sep-13 9(3)

10 Ericsson s Indoor Path Loss Model (900 MHz) T.S. Rappaport, Wireless Communications, Prentice Hall, 00 Lecture 4 5-Oct-1 10(3)

11 Log-Normal Fading (Shadowing) Reminder: 3 factors in the total path loss, LP = LALLF LSF (.5) Siwiak, Radiowave Propagation and Antennas for Personal Communications, Artech House, 1998 Lecture 4 5-Oct-1 11(3)

12 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 πσ ( L ) p Lp σ (4.6) L p are in db, and σ is the standard deviation (in Physical explanation: multiple reflection/diffraction + the central limit theorem. 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 Lecture 4 5-Oct-1 1(3)

13 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 t t = 0 Π Γ i t = 0Π Γi i= 1 i= 1 N P E E or P P P = P + 0 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 reflections /scatterings ( N 5) and is used in practice to model large-scale fading (shadowing). Lecture 4 5-Oct-1 13(3)

14 Small-Scale (multipath) Fading Model E 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 cos ωt E sin ϕ sin ωt i i i i i= 1 i= 1 N This is in-phase (I) and quadrature (Q) representation E ( t) = E cosωt E sinω t = E cos( ω t + ϕ) t x y (4.9) Assume that i N x y where E = E + E envelope By central limit theorem, 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, ] i E, E N (0, σ ) x y N π are i.i.d. (4.10) Lecture 4 5-Oct-1 14(3)

15 E is Rayleigh distributed with pdf where σ is the variance of E x (or E y ), x x ρ ( x) = exp, x 0 σ σ N 1 Ex Ei i= 1 (4.11) σ = = (4.1) 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 x / σ Lecture 4 5-Oct-1 15(3)

16 Outage Probability and CDF Outage probability = CDF is Rx operates well if E E x F( x) = ρ ( t) dt = Pr( E < x) (4.13) th 0 the threshold effect. If E < E, the link is lost -> this is an outage. th Lecture 4 5-Oct-1 16(3)

17 Rayleigh Fading For Rayleigh distribution, the outage probability is x x 0 σ (4.14) F( x) = Pr( E < x) = ρ ( t) dt = 1 exp Introduce the instantaneous signal power P = P = σ = the average power, then P out { } = Pr SNR < γ P = x /, 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) P = (4.16) γ γ = γ, where γ is the SNR. Note in (4.16) the 10db/decade law. Lecture 4 5-Oct-1 17(3)

18 Example: ELG4179: Wireless Communication Fundamentals S.Loyka 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 γ / γ, db. Complex-valued model: N E ( t) = E e e (4.17) t i= 1 results in complex Gaussian variables. i jϕ i jωt Propagation channel gain simply normalized received signal, has the same distribution. Lecture 4 5-Oct-1 18(3)

19 Rayleigh Fading Channel 10 SISO 1x Received power (SNR) norm. to the average [db] vs distance (location, time) Lecture 4 5-Oct-1 19(3)

20 T.S. Rappaport, Wireless Communications, Prentice Hall, 00 ELG4179: Wireless Communication Fundamentals S.Loyka Lecture 4 5-Oct-1 0(3)

21 Measured Fading Channels T.S. Rappaport, Wireless Communications, Prentice Hall, 00. Lecture 4 5-Oct-1 1(3)

22 LOS and Ricean Model There is 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 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) : x x xx I 0 0 x + ρ x = σ σ σ 0 ( ) exp, Note that if x 0 = 0, it reduces to the Rayleigh pdf. Introduce K-factor: (4.0) K = x 0 / ( σ ) (4.1) where x 0 / is the LOS power, it is called LOS ( steady or 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 = x 0 / + σ = σ (1 + K ) Lecture 4 5-Oct-1 (3)

23 PDF of E becomes 0 x x x ρ ( x) = exp K I K σ σ σ (4.) 0.8 Rayleigh & Rice Densities 0.6 K=0 pdf 0.4 K=1 K=10 K= xσ / Note: normalized to the multipath power only. Q: do the same graph if normalized to the total average power. Lecture 4 5-Oct-1 3(3)

24 10 Rice CDF Outage probability K = 0 K = 10 K = 1 K = 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 5-Oct-1 4(3)

25 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 ) ) Average outage time: T = P T out out ) 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 5-Oct-1 5(3)

26 Nakagami Distribution Originally developed to fit measured data (HF channel fading). Provides better fit and is more robust. The PDF of the envelope is m ( m) m 1 mx σ m x ρ ( x) = e, m 1/. (4.3) m Γ σ where σ = x = E, ( ) m 1 t is the Gamma function, ( m) ( m ) Γ m = t e dt (4.4) 0 Γ = 1! for integer m, and ( ) ( ) m = x x x It can model fading that is more or less severe than Rayleigh one: For m = 1 -> Rayleigh distribution. For m = 1/ -> single-sided Gaussian. For m -> delta-function (no fading). Normalized SNR ( γ / γ 0) or power ( P / P 0) PDF and CDF: m 1 m 1 x x x x f ( x) = e, F ( x) = 1 e (4.5) ( m 1)! i! i= 0 i Lecture 4 5-Oct-1 6(3)

27 Nakagami distribution can also approximate the Rice distribution: m m K =, m > 1 m m m ( K + 1) m = K + 1 (4.6) Nakagami distribution is better for analytical analysis since there is no Bessel function in it. Rayleigh distribution requires the same RMS values of various multipath components. Nakagami one does not. Physically, it can be justified by clustering of multipath waves (in time). Lecture 4 5-Oct-1 7(3)

28 1 Nakagami CDF Outage probability m = 1 m = m = 3 Pout = 10 : m F1 = 1 = 30dB, m = F = 13dB, m = 3 F = 7dB. 3 γ / γ 0 Lecture 4 5-Oct-1 8(3)

29 Suzuki Distributions While small-scale fading (Rayleigh) and large-scale fading (lognormal shadowing) are caused by different physical mechanisms, they occur at the same time. Composite distribution: includes both effects. Small scale fading Rayleigh dist., its local mean log-normal RV. Small-scale: x ( x ) x x ρ σ = (4.7) σ e σ Log-normal: ( ) ρ σ = 1 e πασ ( ln σ M ) α (4.8) where M = ln σ, and α = ( ln σ M ). The composite (unconditional) distribution is ρ x = ρ x σ ρ σ dσ x ( ) ( ) ( ) x (4.9) 0 The model is sounder theoretically, but the pdf is not available in closed-form difficult for analytical analysis. Lecture 4 5-Oct-1 9(3)

30 Physical explanation: strong wave with log-normal distribution (due to multiple reflection/diffraction) is composed of individual identically distributed waves. Note: if Lp = LRLLN, where L R is a Rayleigh fading factor, and L LN is a log-norm fading factor then it can be shown that L p has Suzuki distribution. Many other models are available see the reference list. Lecture 4 5-Oct-1 30(3)

31 Monte-Carlo method 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, 000. [] W.H. Tranter et al, Principles of Communication System Simulation with Wireless Applications, Prentice Hall, Upper Saddle River, 004. [3] J.G. Proakis, M. Salehi, Contemporary Communication Systems Using MATLAB, Brooks/Cole, 000. Lecture 4 5-Oct-1 31(3)

32 Summary Indoor propagation path loss models. Log-normal shadowing. Small-scale fading. Rayleigh, Rice, Nakagami and Suzuki distributions. Physical mechanisms. o Rappaport, Ch. 4. Reading: References: o S. Salous, Radio Propagation Measurement and Channel Modelling, Wiley, 013. (available online) o J.S. Seybold, Introduction to RF propagation, Wiley, 005. o 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 5-Oct-1 3(3)