1. Chapter 2: Statistical Analysis of Fading Channels Channel output viewed as a shot-noise process Point processes in general; distributions, moments Double-stochastic Poisson process with fixed realization of its rate Characteristic and moment generating functions Example of moments Central-limit theorem Edgeworth series of received signal density Details in presentation of friday the 13 th Channel autocorrelation functions and power spectra

2. Channel Simulations Experimental Data (Pahlavan p. 52) Chapter 2: Shot-Noise Channel Simulations

3. Chapter 2: Shot-Noise Channel Model

4. Channel viewed as a shot-noise effect [Rice 1944] Chapter 2: Shot-Noise Effect titi titi Counting process Response Linear system Shot-Noise Process: Superposition of i.i.d. impulse responses occuring at times obeying a counting process, N ( t ).

5. Measured power delay profile Chapter 2: Shot-Noise Effect

6. Shot noise processess and Campbell’s theorem Chapter 2: Shot-Noise Definition

7. Shot-Noise Representation of Wireless Fading Channel Chapter 2: Wireless Fading Channels as a Shot-Noise

8. Counting process N ( t ) : Doubly-Stochastic Poisson Process with random rate Chapter 2: Shot-Noise Assumption

9. Conditional Joint Characteristic Functional of y ( t ) Chapter 2: Joint Characteristic Function

10. Conditional moment generating function of y ( t ) Conditional mean and variance of y ( t ) Chapter 2: Joint Moment Generating Function

11. Conditional Joint Characteristic Functional of y l ( t ) Chapter 2: Joint Characteristic Function

12. Chapter 2: Joint Moment Generating Function Conditional moment generating function of y l ( t ) Conditional mean and variance of y l ( t )

13. Conditional correlation and covariance of y l ( t ) Chapter 2: Correlation and Covariance

14. Central Limit Theorem y c (t) is a multi-dimensional zero-mean Gaussian process with covariance function identified Chapter 2: Central-Limit Theorem

15. Channel density through Edgeworth’s series expansion First term: Multidimensional Gaussian Remaining terms: deviation from Gaussian density Chapter 2: Edgeworth Series Expansion

16. Channel density through Edgeworth’s series expansion Constant-rate, quasi-static channel, narrow-band transmitted signal Chapter 2: Edgeworth Series Simulation

17. Channel density through Edgeworth’s series expansion Parameters influencing the density and variance of received signal depend on Propagation environmentTransmitted signal (t) (t) T s T s (signal. interv.)  var. I(t),Q(t  r s Chapter 2: Edgeworth Series vs Gaussianity

18. Chapter 2: Channel Autocorrelation Functions  c (  t;  ) S c ( ;  ) S c ( ;  f) Scattering Function FF FtFt FF FtFt WSSUS Channel Power Delay Profile Power Delay Spectrum   c (  ) TmTm ff BcBc |  c (  f)| FF  t=0 tt TcTc |  c (  t)|  f=0  t=0 BdBd S c ( )  f=0 FtFt Doppler Power Spectrum tt |  c (  t;  f)| ff S c (  ) 

19. Consider a Wide-Sense Stationary Uncorrelated Scattering (WSSUS) channel with moving scatters  Non-Homogeneous Poisson rate: (  ) r i (t,  ) = r i (  ): quasi-static channel p  (  )=1/2 , p  (  )=1/2  Chapter 2: Channel Autocorrelations and Power-Spectra

20. Time-spreading: Multipath characteristics of channel Chapter 2: Channel Autocorrelations and Power-Spectra

21. Time-spreading: Multipath characteristics of channel Chapter 2: Channel Autocorrelations Power-Spectra

22. Time-spreading: Multipath characteristics of channel Autocorrelation in Frequency Domain, (space-frequency, space-time) Chapter 2: Channel Autocorrelations and Power-Spectra

23. Time variations of channel: Frequency-spreading: Chapter 2: Channel Autocorrelations and Power-Spectra Double Fourrier transform

24. Time variations of channel: Frequency-spreading Chapter 2: Channel Autocorrelations and Power-Spectra

25. Time variations of channel: Frequency-spreading Chapter 2: Channel Autocorrelations and Power-Spectra

26. Temporal simulations of received signal Chapter 2: Shot-Noise Simulations

27. K.S. Miller. Multidimentional Gaussian Distributions. John Wiley&Sons, 1964. S. Karlin. A first course in Stochastic Processes. Academic Press, New York 1969. A. Papoulis. Probability, Random Variables and Stochastic Processes. McGraw Hill, 1984. D.L. Snyder, M.I. Miller. Random Point Processes in Time and Space. Springer Verlag, 1991. E. Parzen. Stochastic Processes. SIAM, Classics in Applied Mathematics, 1999. P.L. Rice. Mathematical Analysis of random noise. Bell Systems Technical Journal, 24:46-156, 1944. W.F. McGee. Complex Gaussian noise moments. IEEE Transactions on Information Theory, 17:151-157, 1971. Chapter 2: References

28. R. Ganesh, K. Pahlavan. On arrival of paths in fading multipath indoor radio channels. Electronics Letters, 25(12):763-765, 1989. C.D. Charalambous, N. Menemenlis, O.H. Karbanov, D. Makrakis. Statistical analysis of multipath fading channels using shot-noise analysis: An introduction. ICC-2001 International Conference on Communications, 7:2246-2250, June 2001. C.D. Charalambous, N. Menemenlis. Statistical analysis of the received signal over fading channels via generalization of shot-noise. ICC-2001 International Conference on Communications, 4:1101-1015, June 2001. N. Menemenlis, C.D. Charalambous. An Edgeworth series expansion for multipath fading channel densities. Proceedings of 41 st IEEE Conference on Decision and Control, to appear, Las Vegas, NV, December 2002. Chapter 2: References