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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
    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

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
  • Time-spreading Multipath characteristics of
    channel

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

22
Chapter 2 Channel Autocorrelations and
Power-Spectra
  • Time-spreading Multipath characteristics of
    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
    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.
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