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

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

Chapter 2: Shot-Noise Channel Model

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 ).

Measured power delay profile Chapter 2: Shot-Noise Effect

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

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

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

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

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

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

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

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

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

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

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

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

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 (  ) 

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

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

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

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

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

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

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

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

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

R. Ganesh, K. Pahlavan. On arrival of paths in fading multipath indoor radio channels. Electronics Letters, 25(12): , 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: , June 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: , June 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 Chapter 2: References