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
• 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

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Chapter 2 Shot-Noise Channel Simulations

• Channel Simulations Experimental Data
  (Pahlavan p. 52)

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Chapter 2 Shot-Noise Channel Model

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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).
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Chapter 2 Shot-Noise Effect

• Measured power delay profile

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Chapter 2 Shot-Noise Definition

• Shot noise processess and Campbells theorem

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Chapter 2 Wireless Fading Channels as a
Shot-Noise

• Shot-Noise Representation of Wireless Fading
  Channel

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Chapter 2 Shot-Noise Assumption

• Counting process N(t) Doubly-Stochastic Poisson
  Process with random rate

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Chapter 2 Joint Characteristic Function

• Conditional Joint Characteristic Functional of
  y(t)

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Chapter 2 Joint Moment Generating Function

• Conditional moment generating function of y(t)
• Conditional mean and variance of y(t)

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Chapter 2 Joint Characteristic Function

• Conditional Joint Characteristic Functional of
  yl(t)

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Chapter 2 Joint Moment Generating Function

• Conditional moment generating function of yl(t)
• Conditional mean and variance of yl(t)

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Chapter 2 Correlation and Covariance

• Conditional correlation and covariance of yl(t)

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Chapter 2 Central-Limit Theorem

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

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Chapter 2 Edgeworth Series Expansion

• Channel density through Edgeworths series
  expansion
• First term Multidimensional Gaussian
• Remaining terms deviation from Gaussian density

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Chapter 2 Edgeworth Series Simulation

• Channel density through Edgeworths series
  expansion
• Constant-rate, quasi-static channel, narrow-band
  transmitted signal

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Chapter 2 Edgeworth Series vs Gaussianity

• Channel density through Edgeworths series
  expansion
• Parameters influencing the density and variance
  of received signal depend on
• Propagation environment Transmitted signal
• l(t) l(t) Ts Ts (signal. interv.)
• s (var. I(t),Q(t)) K
• rs

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Chapter 2 Channel Autocorrelation Functions
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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

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Chapter 2 Channel Autocorrelations and
Power-Spectra

• Time-spreading Multipath characteristics of
  channel

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Chapter 2 Channel Autocorrelations Power-Spectra

• Time-spreading Multipath characteristics of
  channel

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Chapter 2 Channel Autocorrelations and
Power-Spectra

• Time-spreading Multipath characteristics of
  channel
• Autocorrelation in Frequency Domain,
  (space-frequency, space-time)

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Chapter 2 Channel Autocorrelations and
Power-Spectra

• Time variations of channel Frequency-spreading

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Chapter 2 Channel Autocorrelations and
Power-Spectra

• Time variations of channel Frequency-spreading

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Chapter 2 Channel Autocorrelations and
Power-Spectra

• Time variations of channel Frequency-spreading

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Chapter 2 Shot-Noise Simulations

• Temporal simulations of received signal

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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.

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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, 72246-2250, 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, 41101-1015, June 2001.
• N. Menemenlis, C.D. Charalambous. An Edgeworth
  series expansion for multipath fading channel
  densities. Proceedings of 41st IEEE Conference on
  Decision and Control, to appear, Las Vegas, NV,
  December 2002.