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Chapter 3: Log-Normal Shadowing Models

Chapter 3: Log-Normal Shadowing Models. Motivation for dynamical channel models Log-Normal dynamical models Short-term dynamical models The Models are Used in the Shot-Noise Representation of Wireless Channels. Chapter 3: Motivation for Dynamical Channel Models. Mobiles move. Area 1.

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Chapter 3: Log-Normal Shadowing Models

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  1. Chapter 3: Log-Normal Shadowing Models • Motivation for dynamical channel models • Log-Normal dynamical models • Short-term dynamical models • The Models are Used in the Shot-Noise Representation of Wireless Channels

  2. Chapter 3: Motivation for Dynamical Channel Models Mobiles move Area 1 Area 2 Short-term Fading Varying environment Obstacles on/off Log-normal Shadowing Varying environment Obstacles on/off Transmitter

  3. Chapter 3: S.D.E.’s in Modeling Log-Normal Shadowing • Dynamical spatial log-normal channel model • Geometric Brownian motion model • Spatial correlation • Dynamical temporal channel model • Mean-reverting log-normal model • Space-time mean-reverting log-normal model • Mean-reverting log-normal model

  4. Chapter 3: Log-Normal Shadowing Model Receiver tn,3 path n tn,2 qn Transmitter tn,1 t ord LOS vmR(t) tk,4 one subpath tk,3 tk,1 tk,2 path k d(t)

  5. Chapter 3: Static Log-Normal Model

  6. Chapter 3: Dynamical Log-Normal Model • t d :time-delay equivalent to distance d=vct • speed of light • S(t,t)andX (t,t) : random processes • modeled using S.D.E.’s • with respect to t and t

  7. Chapter 3: Dynamical Spatial Log-Normal Model • Time, t: fixed (snap shot at propagation • environment) • {S(t,t)}|t=fixed S.D.E. w.r.t. t * S(t, t5) Receiver S(t, t4) * S(t, t2) * S(t, t3) * * S(t, t1) Transmitter

  8. Chapter 3: Dynamical Spatial Log-Normal Model

  9. Chapter 3: Dynamical Spatial Log-Normal Model • Need specific S.D.E.s for {X(t,t), S(t,t)} where • {X(t,t)}|t=fixed => At every t, B.M. with non-zero drift • {S(t,t)}| t=fixed => At every t, G.B.M. • a :models loss characteristics of propagation environment

  10. Chapter 3: Dynamical Spatial Log-Normal Model • Properties of spatial log-normal model • S(t,t) = ekX(t,t) : Geometric Brownian Motion w.r.t. t

  11. Chapter 3: Dynamical Spatial Log-Normal Model • Properties of spatial log-normal model • S(t,t) = ekX(t,t) => Using Ito’s differential rule • S(t,t) = ekX(t,t) : Geometric Brownian Motion w.r.t. t

  12. Chapter 3: Spatial Log-Normal Model Simulations Experimental Data (Pahlavan) • Time t: fixed • Snap-shot at propagation environment • {X(t,t)}|t=fixed : increases logarithmically with d or t • S(t,t) = ekX(t,t) : Log-Normal

  13. Chapter 3: Spatial Correlation of Log-Normal Model • Spatial correlation characteristics: • Indicate what proportion of the environment remains the same from one observation instant or location to the next, separated by the sampling interval. • Consider • Since the mobile is in motion it implies that the above correlation corresponds to the spatial correlation.

  14. Chapter 3: Experimental Correlation • Reported spatial correlation decreases exponentially with d • sX2: covariance of power-loss process • Dd, Dt : distance, time between consecutive samples • v: velocity of mobile • Xc: density of propagation environment

  15. Chapter 3: Spatial Correlation of Log-Normal Model • Consider the following linear process

  16. Chapter 3: Spatial Correlation of Log-Normal Model • Since the mobile is in motion, covariance with respect to t spatial covariance • Identification of parameters {b(t), d(t)} • Use experimental correlation data  identify b(t), • From variance of initial condition and b(t)  identify d(t), • Note: variance of initial condition of power loss process increase with distance. • equivalent to: • d(t) increases or b(t) decreases (denser environment)

  17. Chapter 3: Dynamical Temporal Log-Normal Models Sn(tm-1,t) * * Sn(tm ,t) Receiver Transmitter • T-R separation distance d or t fixed • {S(t,t)}|t=fixed S.D.E. w.r.t. t

  18. Chapter 3: Dynamical Temporal Log-Normal Model

  19. Chapter 3: Dynamical Temporal Log-Normal Model • Need specific S.D.E.s for {X(t,t), S(t,t)} where • {X(t,t)}|t=fixed => At every instant of time t, is Gaussian • {S(t,t)}|t=fixed => At every instant of time t, is Log-Normal • {b(t,t), g (t,t), d (t,t)}: model propagation environment

  20. Chapter 3: Dynamical Temporal Log-Normal Model • Properties of mean-reverting process

  21. Chapter 3: Dynamical Temporal Log-Normal Model • Properties of mean-reverting process

  22. Chapter 3: Dynamical Temporal Log-Normal Model • S(t,t) = ekX(t,t) => Using Ito’s differential rule

  23. Chapter 3: Temporal Log-Normal Model Simulations high low • Illustration of mean reverting model • b(t,t) high: not-dense environment • b(t,t) low: dense environment

  24. Chapter 3: Dynamical Temporal-Spatial Log-Normal Model y Receiver x(t) x Transmitter d (0,0) qn vmT(t) vmR(t) d(t) • Propagation environment varies x(t) • Transmitter-Receiver relative motion d(t)

  25. Chapter 3: Temporal-Spatial Log-Normal Model Sim.

  26. Chapter 3: Temporal-Spatial Log-Normal Model Sim. Receiver Transmitter d qn (t) d(t) d1 d2 d3 vm (t)

  27. Chapter 3: Spatial Correlation of Log-Normal Model • b(t) : inversely proportional to the density of the propagation environment

  28. Chapter 3: References • M. Gudamson. Correlation model for shadow fading in mobile radio systems. Electronics Letters, 27(23):2145-2146, 1991. • D. Giancristofaro. Correlation model for shadow fading in mobile radio channels. Electronics Letters, 32(11):956-958, 1996. • F. Graziosi, R. Tafazolli. Correlation model for shadow fading in land-mobile satellite systems. Electronics Letters, 33(15):1287-1288, 1997. • A.J. Coulson, G. Williamson, R.G. Vaughan. A statistical basis for log-normal shadowing effects in multipath fading channels. IEEE Transactions in Communications, 46(4):494-502, 1998. • R.S. Kennedy. Fading Dispersive Communication Channels. Wiley Interscience, 1969. • S.R. Seshardi. Fundamentals of Transmission Lines and Electromagnetic Fields. Addison-Wesley, 1971. • L. Arnold. Stochastic Differential Applications: Theory and Applications. Wiley Interscience, New York 1971. • D. Parsons. The mobile radio propagation channel. John Wiley & Sons, New York, 1992.

  29. Chapter 3: References • C.D. Charalambous, N. Menemenlis. Stochastic models for long-term multipath fading channels. Proceedings of 38th IEEE Conference on Decision and Control, 5:4947-4952, December 1999. • C.D. Charalambous, N. Menemenlis. General non-stationary models for short-term and long-term fading channels. EUROCOMM 2000, pp 142-149, April 2000. • C.D. Charalambous, N. Menemenlis. Dynamical spatial log-normal shadowing models for mobile communications. Proceedings of XXVIIthURSI General Assembly, Maastricht, August 2002.

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