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 Short-term Fading Varying environment Obstacles on/off Area 2 Area 1 Transmitter Log-normal Shadowing Mobiles move

Chapter 3: Motivation for Dynamical Channel Models Complex low-pass representation of impulse response:

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

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

Chapter 3: Static Log-Normal Model

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

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

Chapter 3: Dynamical Spatial Log-Normal Model

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

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

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

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

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.

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

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

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)

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

Chapter 3: Dynamical Temporal Log-Normal Model

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

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

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

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

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

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

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

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

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

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.

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 XXVIIth URSI General Assembly, Maastricht, August 2002.