An Edgeworth Series Expansion for Multipath Fading Channel Densities

1. An Edgeworth Series Expansion forMultipath Fading Channel Densities Nickie Menemenlis C. D. Charalambous McGill University University of Ottawa 41stIEEE 2002 Conference on Decision and ControlDecember 10 - 13, 2002Las Vegas, Nevada

2. Overview • Wireless Communication System • Channel output viewed as a shot-noise process • Double-stochastic Poisson process with fixed realization of its rate • Characteristic and moment generating functions • Central-limit theorem • Edgeworth series of received signal density

3. Wireless Communication Propagation Channels Area 1 Area 2 Short-term fading Log-normal shadowing Transmitter

4. Shannon’s Wireless Communication System Channel code word Message Signal Modulated Transmitted Signal Source Source Encoder Channel Encoder Mod- ulator Wireless Channel User Source Decoder Channel Decoder Demod- ulator Received Signal Estimate of Message signal Estimate of channel code word

5. Impulse Response Characterization Time variations property t2 t(t2) t1 t(t1) Time spreading property t0 t(t0) • Impulse response: Time-spreading : multipath • and time-variations: time-varying environment

6. Impulse Response Multipath Fading Channel

7. Band-pass Representation of Impulse Response • Band-pass representation of impulse response:

8. Shot-Noise Channel Model

9. 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).

10. Shot-Noise Effect • Measured power delay profile

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

12. Shot-Noise Definition • Shot noise processess and Campbell’s theorem

13. Wireless Fading Channels as a Shot-Noise • Shot-Noise Representation of Wireless Fading Channel

14. Shot-Noise Assumption • Counting process N(t): Doubly-Stochastic Poisson Process with random rate

15. Joint Characteristic Function • Conditional Joint Characteristic Functional of y(t)

16. Joint Moment Generating Function • Conditional moment generating function of y(t) • Conditional mean, variance and covariance of y(t)

17. Joint Characteristic Function • Conditional Joint Characteristic Functional of yl(t)

18. Joint Moment Generating Function • Conditional moment generating function of yl(t) • Conditional mean and variance of yl(t)

19. Correlation and Covariance • Conditional correlation and covariance of yl(t)

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

21. Edgeworth Series Expansion • Channel density through Edgeworth’s series expansion • Consider the conditional joint characteristic function • The conditional density of y(t) is given by

22. Edgeworth Series Expansion • Channel density through Edgeworth’s series expansion • First term: Multidimensional Gaussian • Remaining terms: deviation from multidimensional Gaussian density

23. Edgeworth Series Expansion • Channel density through Edgeworth’s series expansion • Consider the received signal y(t) • The conditional density of y(t) is given by

24. Edgeworth Series Expansion • Conditional density of y(t) • Remaining terms: deviation from Gaussian density

25. Received Signal Density: Example

26. Received Signal Density: Example • Conditional density of y(t)

27. Received Signal Density: Example • Conditional density of y(t): Rayleigh channel, Constant rate, Transmitted signal: narrow band • First term: centered Gaussian density • Remaining terms decrease as (lTs) increases • Variance of received signal depends on characteristics of environment (l, s) and transmitted signal (K, Ts) • Oscillatory behaviour due to basis functionsf(n)(x)

28. Received Signal Density: Example; Simulation • Channel density through Edgeworth’s series expansion • Constant-rate, quasi-static channel, narrow-band transmitted signal

29. Received Signal Density: Example • Conditional density of y(t): Dynamic channel, Time-varying Rayleigh, Variable rate, Transmitted signal: wide-band • First term: centered Gaussian density • Remaining terms decrease as (lTs) increases • Variance of received signal depends on characteristics of environment (l(t), s(t)) and transmitted signal (p(t), Ts)

30. 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, sl(t) • rs

31. Conclusions • Received signal density • is not Gaussian • can be computed through Edgeworth’s series expansion • Methodology brings forward the parameters influencing the density and variance of received signal • depend on propagation environment • depend on transmitted signal • Characterization of received signal density is important in the design of transmitters and receivers