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## Chapter 2: Statistical Analysis of Fading Channels

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### Chapter 2: Statistical Analysis of Fading Channels Channel output viewed as a shot-noise process Point processes in general; distributions, moments – PowerPoint PPT presentation

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Title: Chapter 2: Statistical Analysis of Fading Channels

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 13th
• Channel autocorrelation functions and power
spectra

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

3
Chapter 2 Shot-Noise Channel Model

4
Chapter 2 Shot-Noise Effect
• 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).
5
Chapter 2 Shot-Noise Effect
• Measured power delay profile

6
Chapter 2 Shot-Noise Definition
• Shot noise processess and Campbells theorem

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

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

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

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

11
Chapter 2 Joint Characteristic Function
• Conditional Joint Characteristic Functional of
yl(t)

12
Chapter 2 Joint Moment Generating Function
• Conditional moment generating function of yl(t)
• Conditional mean and variance of yl(t)

13
Chapter 2 Correlation and Covariance
• Conditional correlation and covariance of yl(t)

14
Chapter 2 Central-Limit Theorem
• Central Limit Theorem
• yc(t) is a multi-dimensional zero-mean Gaussian
process with covariance function identified

15
Chapter 2 Edgeworth Series Expansion
• Channel density through Edgeworths series
expansion
• First term Multidimensional Gaussian
• Remaining terms deviation from Gaussian density

16
Chapter 2 Edgeworth Series Simulation
• Channel density through Edgeworths series
expansion
• Constant-rate, quasi-static channel, narrow-band
transmitted signal

17
Chapter 2 Edgeworth Series vs Gaussianity
• Channel density through Edgeworths series
expansion
• Parameters influencing the density and variance
• Propagation environment Transmitted signal
• l(t) l(t) Ts Ts (signal. interv.)
• s (var. I(t),Q(t)) K
• rs

18
Chapter 2 Channel Autocorrelation Functions
19
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

20
Chapter 2 Channel Autocorrelations and
Power-Spectra
channel

21
Chapter 2 Channel Autocorrelations Power-Spectra
channel

22
Chapter 2 Channel Autocorrelations and
Power-Spectra
channel
• Autocorrelation in Frequency Domain,
(space-frequency, space-time)

23
Chapter 2 Channel Autocorrelations and
Power-Spectra
• Time variations of channel Frequency-spreading

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

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

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

27
Chapter 2 References
• K.S. Miller. Multidimentional Gaussian
Distributions. John WileySons, 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, 2446-156, 1944.
• W.F. McGee. Complex Gaussian noise moments. IEEE
Transactions on Information Theory, 17151-157,
1971.

28
Chapter 2 References
• R. Ganesh, K. Pahlavan. On arrival of paths in
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, 72246-2250, June 2001.
• C.D. Charalambous, N. Menemenlis. Statistical