Chapter 2: Statistical Analysis of Fading Channels

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# Chapter 2: Statistical Analysis of Fading Channels - PowerPoint PPT Presentation

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

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Presentation Transcript
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 13th
• Channel autocorrelation functions and power spectra
Chapter 2: Shot-Noise Channel Simulations
• Channel Simulations Experimental Data (Pahlavan p. 52)
Chapter 2: Shot-Noise Effect

ti

ti

• Channel viewed as a shot-noise effect [Rice 1944]

Linear

system

Counting process

Response

Shot-Noise Process: Superposition of i.i.d. impulse responses occuring at times obeying a counting process, N(t).

Chapter 2: Shot-Noise Effect
• Measured power delay profile
Chapter 2: Shot-Noise Definition
• Shot noise processess and Campbell’s theorem
Chapter 2: Wireless Fading Channels as a Shot-Noise
• Shot-Noise Representation of Wireless Fading Channel
Chapter 2: Shot-Noise Assumption
• Counting process N(t): Doubly-Stochastic Poisson Process with random rate
Chapter 2: Joint Characteristic Function
• Conditional Joint Characteristic Functional of y(t)
Chapter 2: Joint Moment Generating Function
• Conditional moment generating function of y(t)
• Conditional mean and variance of y(t)
Chapter 2: Joint Characteristic Function
• Conditional Joint Characteristic Functional of yl(t)
Chapter 2: Joint Moment Generating Function
• Conditional moment generating function of yl(t)
• Conditional mean and variance of yl(t)
Chapter 2: Correlation and Covariance
• Conditional correlation and covariance of yl(t)
Chapter 2: Central-Limit Theorem
• Central Limit Theorem
• yc(t)is a multi-dimensional zero-mean Gaussian process with covariance function identified
Chapter 2: Edgeworth Series Expansion
• Channel density through Edgeworth’s series expansion
• First term: Multidimensional Gaussian
• Remaining terms: deviation from Gaussian density
Chapter 2: Edgeworth Series Simulation
• Channel density through Edgeworth’s series expansion
• Constant-rate, quasi-static channel, narrow-band transmitted signal
Chapter 2: Edgeworth Series vs Gaussianity
• Channel density through Edgeworth’s series expansion
• Parameters influencing the density and variance of received signal depend on
• Propagation environment Transmitted signal
• l(t) l(t) TsTs(signal. interv.)
• s (var. I(t),Q(t)) K
• rs
Chapter 2: Channel Autocorrelation Functions

Fc(t )

Power Delay

Profile

|Fc(Df)|

t

Tm

Ft

Bc

Df

Dt=0

Power Delay

Spectrum

Fc( Dt;t )

Ft

FDt

|Fc(Dt;Df)|

Dt=0

Scattering

Function

Df

WSSUS Channel

Sc( l;t )

Dt

Sc(l; Df)

Ft

FDt

Df=0

Df=0

Sc(l;t)

|Fc(Dt)|

Sc( l )

t

Doppler Power

Spectrum

Tc

Dt

FDt

l

Bd

l

Chapter 2: Channel Autocorrelations and Power-Spectra
• Consider a Wide-Sense Stationary Uncorrelated Scattering (WSSUS) channel with moving scatters
• Non-Homogeneous Poisson rate: l(t)
• ri(t,t) = ri(t): quasi-static channel
• pf(f)=1/2p , pq(q)=1/2p
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
Chapter 2: Channel Autocorrelations and Power-Spectra
• Time-spreading: Multipath characteristics of channel
• Autocorrelation in Frequency Domain, (space-frequency, space-time)
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
• Time variations of channel: Frequency-spreading
Chapter 2: Shot-Noise Simulations
• Temporal simulations of received signal
Chapter 2: References
• 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
• 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 41stIEEE Conference on Decision and Control, to appear, Las Vegas, NV, December 2002.