1. Chapter 4: Motivation for Dynamic Channel Models Short-term Fading Varying environment Obstacles on/off Area 2 Area 1 Transmitter Log-normal Shadowing Varying environment Obstacles on/off Mobiles move

2. Complex low-pass representation of impulse response: Chapter 4: Motivation for Dynamic Channel Models

3. Chapter 3: S.D.E.’s for Short-Term Fading Dynamics represent time-variations of environment Captured by Doppler power spectrum From power spectral densities to S.D.E.’s State-space realizations of In-phase and Quadrature components Time-domain simulations flat-fading, frequency select. Distributions of short-term dynamical channel Summary of distributions

4. Chapter 3: 3-Dimensional Scattering Model nn nn x z y nth incoming wave E n =:{r n,  n,  n,  n }; n=1,…, N O O’(x 0,y 0,z 0 )  direction of motion of mobile on x-y plane v x0x0 z0z0 y0y0 O’’ 3-Dimensional Model [Clarke 68, Aulin 79]

5. Chapter 3: Autocorrelation and Power Spectral Density

6. Doppler Power Spectral Density Factorization (Normalization, Approximation) State-Space Model Chapter 3: Doppler P.S.D. is Band-limited Doppler Power Spectral Density Factorization (Normalization, Approximation) State-Space Model

7. Power Spectral Density: S yy (  )=F   R yy (  t)] Chapter 4: Power Spectral Densities and S.D.E.’s Linear system h(t) Gaussian process S xx (  )S yy (  )|H(  )| 2 x = x(t) y(t) WSS R xx (  t) R yy (  t) LTI

8. Chapter 4: Paley-Wiener Factorization Condition

9. Chapter 4: Factorization of Approximated P.S.D. Factorization

10. Chapter 4: Approximate P.S.D.

11. Approximation

12. Chapter 4: State-space realizations of Fading Process Nominal state-space model

13. Chapter 4: State-Space Realizations of Fading Process Nominal state-space model: mean and variance

14. Chapter 4: T.-D. Simulations of Fading Process Nominal state-space model s(t)delay dW I cos  c t ABCD X dW Q sin  c t ABCD X + X y(t) + - Flat-Fading Channel

15. Chapter 4: T.-D. Simulations of Fading Process Simulation of Flat-Fading Channel using Matlab Experimental Data (Pahlavan)

16. Time-Domain simulations of short-term fading model Simulation of received signal through a flat-fading channel using Matlab

17. Chapter 4: T.-D. Simulations of Fading Process Simulation of frequency-selective Channel using Matlab

18. Chapter 4: T.-D. Simulations of Fading Process Simulation of received signal through a frequency- selective channel using Matlab

19. Temporal simulations of received signal for a multipath channel: From PSD obtain parameters of state-space model. Dynamics of channel gain obtained through solving state-space model (generate independent brownian motions for in-phase and quadrature components). Identify the parameters of the non-homogeneous Poisson process (t). This characterizes the obstacles in the environment. Generate points of non-homogeneous poisson process. This corresponds to generating the path arrival times. Associate each path with a gain computed using the state-space model. Chapter 4: T.-D. Simulations of Fading Process

20. Temporal simulations of received signal: Chapter 4: Shot-Noise Model Simulations

21. Chapter 4: Probability Distributions of Attenuations Probability Distribution – Rayleigh:  n (t)=0.

22. Chapter 4: Probability Distributions of Attenuations X j (s)= 0,  j = 0 j =1,…,n X 1 (s)= X 2 (s)= 0  1 =  2 =0 n=2 Non-Stationary Rayleigh Stationary Rayleigh n=2  Non-Stationary Nakagami Stationary Nakagami t large  1 =  2 =0 Non-Stationary Rician Stationary Rician  and/or t large n=2

23. Envelope Imaginary Real Distance Along Measurement Trajectory (m) CRC-TU/e-TU/d-U/Ottawa Typical CW Time Series (Perpendicular NLOS St.) Distance Along Measurement Trajectory (m) Received Power (dBW) Intersection with LOS street Tx L c1 L c2 -50 -100 -150 20 0 -20 0 20 -20 20 0 10 40 42 44 46 48 50 52 54 0 100 200 300 400 500 600 700 800 900 Re: Paper of R. Bultitude et al. from G.A.

24. CRC-TU/e-TU/d-U/Ottawa Signature Functions & Parameters AutocorrelationPower Spectral Density Lag (S) Correlation Relative Power Frequency (Hz) 1 0 -0.10 0.1 -80 - 60 -40 -20 0 20 40 60 80 50 25 0 Re: Paper of R. Bultitude et al. from URSI G.A.

25. CRC-TU/e-TU/d-U/Ottawa Experimental Observations for a Perpendicular NLOS street Calibrated Received Power (dBW) Distance Along Measurement Trajectory (m) LcsLcs -70 -80 -90 -100 420 440 460 480 500 520 nickie: Check data with Dr. Bultitude nickie: Check data with Dr. Bultitude Re: Paper of R. Bultitude et al. from G.A.

26. E. Wong, B. Hajek. Stochastic Processes in Engineering Systems. Springer- Verlag, New York, 1985. M.C. Jeruchim, P. Balaban, S. Shanmugan. Simulation of Communication Systems. Plenum, New York, 1994. P. E. Kloeden, E. Platen. Numerical Solution of Stochastic Differential Equations. Springer-Verlag, New York, 1999. C.D. Charalambous, N. Menemenlis. Stochastic models for short-term multipath fading: Chi-Square and Ornstein-Uhlenbeck processes. Proceedings of 38 th IEEE Conference on Decision and Control, 5:4959- 4964, December 1999. C.D. Charalambous, N. Menemenlis. Multipath channel models for short- term fading. 1999 International Workshop on mobile communications, pp 163-172, Creta, Greece, June 1999. C.D. Charalambous, N. Menemenlis. A state-space approach in modeling multipath fading channels via stochastic differntial equations. ICC-2001 International Conference on Communications, 7:2251-2255, June 2001. Chapter 4: References