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Chapter 5: Applications using S.D.E.’s

Chapter 5: Applications using S.D.E.’s. Channel state-estimation State- s pace c hannel e stimation using Kalman f iltering Channel parameter identification a Nonlinear f iltering Power control for flat fading channels Convex o ptimization and p redictable s trategies

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Chapter 5: Applications using S.D.E.’s

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  1. Chapter 5: Applications using S.D.E.’s • Channel state-estimation • State-space channel estimation using Kalman filtering • Channel parameter identificationa • Nonlinear filtering • Power control for flat fading channels • Convex optimization and predictable strategies • Channel capacity • Optimal encoding and decoding

  2. Chapter 5: Linear Channel State-Estimation • The various terms of the state-space description are: • Note that the parameters depend on the propagation environment represented by t

  3. Chapter 5: Channel Simulations • First must find model parameters for a given structure • Method 1: Approximate the power spectral density (see short-term fading model) • Method 2: From explicit equations and data we have • Obtain {k,z,wn} parameters

  4. Chapter 5: Channel Simulations cos wct sin wct ABCD ABCD dWQ dWI + + - Flat-fading channel X X • Flat-fading channel state-space realization in state-space

  5. Chapter 5: Linear Channel State-Estimation • State-Space Channel Estimation using Kalman filtering • Considering flat-fading

  6. Chapter 5: Linear Channel State-Estimation • State-Space Channel Estimation using Kalman Filtering

  7. Chapter 5: Channel State-Estimation: Simulations • Flat Fast Rayleigh Fading Channel, SNR = 10 dB, v = 60 km/h

  8. Chapter 5: Channel State-Estimation: Simulations • Frequency-Selective Slow Fading, SNR=20dB, v=60km/h

  9. Chapter 5: Channel state-estimation: Conclusions • For flat slow fading, I(t), Q(t), r2(t) show very good tracking at received SNR = -3 dB. • For flat fast fading, I(t), Q(t), r2(t) show very good tracking when the received SNR = 10 dB. • For frequency-selective slow fading, I(t), Q(t), r2(t) of each path show very good tracking, w.r.t. MSE, when the received SNR = 20 dB.

  10. Chapter 5: Channel state-estimation: References • J.F. Ossanna. A model for mobile radio fading due to building reflections: Theoretical and experimental waveform power spectra. Bell Systems Technical Journal, 43:2935-2971, 1964. • R.H. Clarke. A statistical theory of mobile radio reception. Bell Systems Technical Journal, 47:957-1000, 1968. • M.J Gans. A power-spectral theory of propagation in the mobile-radio environment. IEEE Transactions on Vehicular Technology, VT-21(1):27-38, 1972. • T. Aulin. A modified model for the fading signal at a mobile radio channel. IEEE Transactions on Vehicular Technology, VT-28(3):182-203, 1979. • C.D. Charalambous, A. Logothetis, R.J. Elliott. Maximum likelihood parameter estimation from incomplete data via the sensitivity equations. IEEE Transactions on AC, vol. 5, no. 5, pp. 928-934, May 2000. • C.D. Charalambous, N. Menemenlis. A state-state approach in modeling multi-path fading channels: Stochastic differential equations and Ornstein-Uhlenbeck Processes. IEEE International Conference on Communications, Helsinki, Finland, June 11-15, 2001.

  11. Chapter 5: Channel state-estimation: References • K. Miller. Multidimensional Gaussian Distributions. John Wiley & Sons, 1963. • M.S. Grewal, A.P. Andrews. Kalman filtering – Theory and Practice, Prentice Hall, Englewood Cliffs, New Jersey 07632, 1993. • D. Parsons. The mobile radio Propagation channel. John Wiley & Sons, 1995. • R.G. Brown, P.Y.C. Hwang. Introduction to random signals and applied Kalman filtering: with MATLAB exercises and solutions, 3rd ed. John Wiley, 1996. • G. L. Stuber. Principles of Mobile Communication. Kluwer Academic Publishers, 1997. • P. E. Kloeden, E. Platen. Numerical Solution of Stochastic Differential Equations. Springer-Verlag, New York, 1999.

  12. Chapter 5: Channel Parameter Identification • Consider the quasi-static multi-path fading channel model • Given the observation process for each path find estimates for the channel parameters:

  13. Chapter 5: Non-Linear Filtering-Sufficient Statistic • Methodology: • Use concept of sufficient statistics in designing non-linear channel parameter estimator. • Sufficient statistic: any quantity that carries the same information as the observed signal, i.e. conditional distribution.

  14. Chapter 5: Bayes’ Decision Criteria • Detection criteria

  15. Chapter 5: Non-Linear Filtering • Sketch of continuous-time non-linear filtering for parameter estimation. • Derive a sufficient statistic and obtain the incomplete data likelihood ratio of multipath fading parameters (for flat and frequency selective channels) • One parameter at a time while keeping others fixed, • All parameters simultanously

  16. Chapter 5: Non-Linear Filtering • Sketch of continuous-time non-linear filtering approach • Non-linear filtering theory relies on successful computation of pN(.,.)

  17. Chapter 5: Non-Linear Filtering • Continuous-time non-linear filtering • Radon-Nikodym derivative (complete data likelihood ratio)

  18. Chapter 5: Non-Linear Filtering • Continuous-time non-linear filtering; Bayes’ rule

  19. Chapter 5: Phase Estimation • Problem 1: Flat-fading; phase estimation • Given the observation process

  20. Chapter 5: Phase Estimation • Defintion: Flat-fading; phase estimation problem

  21. Chapter 5: Phase Estimation • Solution of Problem 1: Flat-fading; phase estimation problem

  22. Chapter 5: Phase Estimation • Solution of Problem 1: Flat-fading; phase estimation problem

  23. Chapter 5: Phase Estimation • Solution of Problem 1: Flat-fading; phase estimation problem

  24. Chapter 5: Phase Estimation

  25. Chapter 5: Phase Estimation • Solution of Problem 1: Flat-fading; phase estimation problem

  26. Chapter 5: Phase Estimation • Solution of Problem 1: Flat-fading; phase estimation • Neglecting double frequency terms

  27. Chapter 5: Phase Estimation • Solution of Problem 1: Flat-fading; phase estimation • Neglecting double frequency terms

  28. Chapter 5: Phase Estimation • Solution of Problem 1: Flat-fading; phase estimation • Neglecting double frequency terms

  29. Chapter 5: Phase Estimation • Solution of Problem 1: Flat-fading; phase estimation • Neglecting double frequency terms

  30. Chapter 5: Channel Estimation • Same procedure for • Gain • Doppler Spread • Joint Estimation of Phase, Gain, Doppler Spread • Frequency Selective Channels

  31. Chapter 5: Simulations of Phase Estimation • Phase estimation in continuous-time

  32. Chapter 5: Nonlinear Filtering Conclusions • Conditional density is a sufficient statistic. • Explicit but complicated expressions can be found for the various parameters of the channel. • These estimations are very useful in subsequent design of various functions of a communications system.

  33. Chapter 5: Channel parameter estimation: References • T. Kailath, V. Poor. Detection of stochastic processes. IEEE Transactions on Information theory, vol. IT-15, no. 3, pp. 350-361, May 1969. • T. Kailath. A General Likelihood-ration formula for random signals in Gaussian noise. IEEE Transactions on Information theory, vol. 44, no. 6, pp. 2230-2259, October 1998. • C.D. Charalambous, A. Logothetis, R.J. Elliott. Maximum likelihood parameter estimation from incomplete data via the sensitivity equations. IEEE Transactions on AC, vol. 5, no. 5, pp. 928-934, May 2000. • S. Dey, C.D. Charalambous. On assymptotic stability of continuous-time risk sensitive filters with respect to initial conditions. Systems and Control Letters, vol. 41, no. 1, pp. 9-18, 2000. • C.D. Charalambous, A. Nejad. Coherent and noncoherent channel estimation for flat fading wireless channels via ML and EM algorithm. 21st Biennial symposium on communications, Queen’s University, Kingston, Canada, June, 2002. • C.D. Charalambous, A. Nejad. Estimation and decision rules for multipath fading wireless channels from noisy measurements: A sufficient statistic approach. Centre for information, communication and Control of Complex Systems, S.I.T.E., University of Ottawa, Technical report: 01-01-2002, 2002.

  34. Chapter 5: Channel parameter estimation: References • P.M. Woodward. Probability and Information Theory with Applications to Radar. Oxford, U.K.: Pergamon, 1953. • A.D. Whalen. Detection of signals in noise, Academic Press, New York, 1971. • A. Leon-Garcia. Probability and Random Processes for Electrical Engineering. Addison-Wesley, New York, 1994. • L.A. Wainstein, V.D. Zubakov. Extraction of signals from noise, Englewood Cliffs, Prentice-Hall, New Jersey, 1962. • C.W. Helstrom. Statistical theory of signal detection. Pergamon Press, New York, 1960. • M.S. Grewal, A.P. Andrews. Kalman filtering – Theory and Practice, Prentice Hall, Englewood Cliffs, New Jersey 07632, 1993. • A.H. Jazwinski. Stochastic processes and filtering theory, Academic Press, New York, 1970. • V. Poor. An Introduction to signal detection and estimation, Springer-Verlag, New York, 2000.

  35. Chapter 5: Stochastic power control for wireless networks: Probabilistic QoS measures • Review of the Power Control Problem • Probabilistic QoS Measures • Stochastic Optimal Control • Predictable Strategies • Linear Programming

  36. Chapter 5: Power Control for Wireless Networks • QoS Measures • Review of the Power Control Problem

  37. Chapter 5: Power Control for Wireless Networks • QoS Measures • Vector Form [Yates 1981] • Then

  38. Chapter 5: Power Control for Wireless Networks • QoS Measures • Probabilistic QoS Measures • Define • The Constraints are equivalent to

  39. Chapter 5: Power Control for Wireless Networks • QoS Measures • Decentralized Probabilistic QoS Measures

  40. Chapter 5: Power Control for Wireless Networks • QoS Measures

  41. Chapter 5: Power Control for Wireless Networks • Centralized Probabilistic QoS Measures

  42. Chapter 5: Power Control for Wireless Networks • QoS Measures • Stochastic optimal control • Received signal • State-space representation

  43. Chapter 5: Power Control for Wireless Networks • QoS Measures • Pathwise QoS and Predictable Strategies • define • then • where

  44. Power control for short-term flat fading Base Station calculates Mobile t-1 observe =>calculate Mobile implements Send back t-1 pm(t-1) Sm (t-1) =>pm(t) <=pm(t) S(t-1/2)pm(t-1) pm(t) Sm (t-1) <= t-1/2 pm(t) S(t-1/2)pm(t) t pm(t) Sm (t) =>pm(t+1) <=pm(t+1) pm(t+1) pm(t+1) Sm (t) t+1/2 t+1 • Pathwise QoS Measures and Predictable Strategies

  45. Power control for short-term flat fading • Pathwise QoS Measures and Predictable Strategies Base Station calculates Value of signal Mobile implements Base Station Mobile pm(t) Sm (t-1) pm(t+1) Sm (t) pm(t+2) Sm (t+1) desired pm(t+1) Sm (.) pm(t) Sm (.) pm(t-1) Sm (.) pm(t) pm(t+1) t-1 t+1 t Observe pm(t)Sm (t) =>calculate pm(t+1)

  46. Chapter 5: Power Control for Wireless Networks • QoS Measures • Define

  47. Chapter 5: Power Control for Wireless Networks • QoS Measures • Predictable Strategies over the interval • Predictable Strategies Linear Programming

  48. Chapter 5: Power Control for Wireless Networks • QoS Measures

  49. Chapter 5: Power Control for Wireless Networks • QoS Measures

  50. Chapter 5: Power Control for Wireless Networks • QoS Measures • Generalizations • Linear Programming • Stochastic Optimal Control with Integral/Exponential-of-Integral Constraints

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