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

Chapter 5: Channel State-Estimation Consider the flat-fading channel model Represented in state-space by Find least squares of the state vector

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 

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, ,  n } parameters

Chapter 5: Channel Simulations Flat-fading channel state-space realization in state- space dW I cos  c t ABCD X dW Q sin  c t ABCD X Flat-fading channel

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

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

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

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

Chapter 5: Channel state-estimation: Conclusions For flat slow fading, I(t), Q(t), r 2 (t) show very good tracking at received SNR = -3 dB. For flat fast fading, I(t), Q(t), r 2 (t) show very good tracking when the received SNR = 10 dB. For frequency-selective slow fading, I(t), Q(t), r 2 (t) of each path show very good tracking, w.r.t. MSE, when the received SNR = 20 dB.

J.F. Ossanna. A model for mobile radio fading due to building reflections: Theoretical and experimental waveform power spectra. Bell Systems Technical Journal, 43: , R.H. Clarke. A statistical theory of mobile radio reception. Bell Systems Technical Journal, 47: , M.J Gans. A power-spectral theory of propagation in the mobile-radio environment. IEEE Transactions on Vehicular Technology, VT-21(1):27-38, T. Aulin. A modified model for the fading signal at a mobile radio channel. IEEE Transactions on Vehicular Technology, VT-28(3): , 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 , May 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, Chapter 5: Channel state-estimation: References

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

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:

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.

Chapter 5: Bayes’ Decision Criteria Detection criteria

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

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

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

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

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

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

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

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

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

Chapter 5: Phase Estimation

Solution of Problem 1: Flat-fading; phase estimation problem

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

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

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

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

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

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

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.

T. Kailath, V. Poor. Detection of stochastic processes. IEEE Transactions on Information theory, vol. IT-15, no. 3, pp , May T. Kailath. A General Likelihood-ration formula for random signals in Gaussian noise. IEEE Transactions on Information theory, vol. 44, no. 6, pp , October 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 , May 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, C.D. Charalambous, A. Nejad. Coherent and noncoherent channel estimation for flat fading wireless channels via ML and EM algorithm. 21 st Biennial symposium on communications, Queen’s University, Kingston, Canada, June, 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: , Chapter 5: Channel parameter estimation: References

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

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

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

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

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

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

Chapter 5: Power Control for Wireless Networks QoS Measures

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

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

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

Power control for short-term flat fading t  t  t-1 t  t  t Base Station calculates Mobile S(t  p m (t-1)  p m (t) p m (t) S m (t  => p m (t+1)  p m (t+1) observe => calculate Send back p m (t-1) S m (t-1  => p m (t) p m (t) p m (t+1) p m (t+1) S m (t  p m (t) S m (t-1  S(t  p m (t) Mobile implements  Pathwise QoS Measures and Predictable Strategies

Power control for short-term flat fading t Base Station Mobile Observe p m (t)S m (t  => calculate p m (t+1) p m (t) S m (t  t  t  p m (t+1) S m (t  p m (t+2) S m (t  p m (t-1) S m (.  p m (t) S m (.  p m (t+1) S m (.  Mobile implements Base Station calculates p m (t+1) Value of signal desired p m (t) Pathwise QoS Measures and Predictable Strategies

Chapter 5: Power Control for Wireless Networks QoS Measures Define

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

Chapter 5: Power Control for Wireless Networks QoS Measures

Chapter 5: Power Control for Wireless Networks QoS Measures

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

Chapter 5: Power Control for Wireless Networks: Conclusions Predictable strategies and dynamic models linear programming Probabilistic QoS measures Stochastic optimal control linear programming

J. Zandler. Performance of optimum transmitter power control in cellular radio systems. IEEE Transactions on Vehicular Technology, vol. 41, no. 1, pp , Feb J. Zandler. Distributed co-channel interference control in cellular radio systems. IEEE Transactions on Vehicular Technology, vol. 41, no. 1, pp , Aug R. Yates. A framework for uplink power control in cellular radio systems. IEEE Journal on Selected Areas in Communications, vol. 13, no. 7, pp , Sept S. Ulukus, R. Yates. Stochastic Power Control for cellular radio systems. IEEE Transaction on Communications, vol. 46, no. 6, pp , Jume P. Ligdas, N. Farvadin. Optimizing the transmit power for slow fading channels. IEEE Transactions on Information Theory, vol. 46, no. 2, pp , March References

C.D. Charalambous, N. Menemenlis. A state-space approach in modeling multipath fading channels via stochastic differntial equations. ICC-2001 International Conference on Communications, 7: , June C.D. Charalambous, N. Menemenlis. Dynamical spatial log-normal shadowing models for mobile communications. Proceedings of XXVII th URSI General Assembly, Maastricht, August C.D. Charalambous, S.Z. Denic, S.M. Djouadi, N. Menemenlis. Stochastic power control for short-term flat fading wireless networks: Almost Sure QoS Measures. Proceedings of 40 th IEEE Conference on Decision and Control, volm. 2, pp , December References

Chapter 5: Capacity, Optimal Encoding, Decoding The channel capacity is the most important concept of any communication channel because it gives the maximal theoretical data rate at which reliable data communication is possible We show an efficient method for computing the channel capacity of a single user time-varying wireless fading channels by means of stochastic calculus. We consider an encoding, and decoding strategy with feedback that is optimal in the sense that it achieves the channel capacity. Although the feedback does not increase the channel capacity, it is a tool for achieving the channel capacity

Chapter 5: Channel Model and Mutual Information in Presence of Feedback

The received signal can be modeled as

Chapter 5: Channel Model and Mutual Information in Presence of Feedback Also, we define the following measurable spaces associated with stochastic processes

Chapter 5: Channel Model and Mutual Information in Presence of Feedback The following assumptions are made

Chapter 5: Channel Model and Mutual Information in Presence of Feedback Theorem 1. Consider the model (1). The mutual information between the source signal X and received signal Y over the interval [0,T], conditional on the channel state , I T (X,Y|F Q ), is given by the following equivalent expressions

Chapter 5: Channel Model and Mutual Information in Presence of Feedback Definition 2. Consider the model (1). The Shannon capacity of (1) is defined by

Chapter 5: Upper Bound on Mutual Information Theorem 2. Consider the model (1). Suppose the channel is flat fading. The conditional mutual information between the source signal X, and the received signal Y is bounded above by It can be proved that this upper bound is indeed the channel capacity, by observing that there exists a source signal with Gaussian distribution such that the mutual information between that signal X and received signal Y is equal to the upper bound (3).

Chapter 5: Upper Bound on Mutual Information The capacity is

Chapter 5: Optimal Encoding/Decoding Strategies for Non-Stationary Gaussian Sources We assume that the channel is flat fading (M=1), that it is known to both transmitter and receiver, that a source is Gaussian nonstationary, and can be described by the following differential F t and G t are Borel measurable and bounded functions, Integrable and square integrable, respectively, G t G t tr >0,  t  [0,T], W is a Wiener process independent of Gaussian random variable X 0 ~

Chapter 5: Optimal Encoding/Decoding Strategies for Non-Stationary Gaussian Sources Decoding. The optimal decoder in the case of mean square error criteria is the conditional expectation while the error covariance is Encoding. The optimal encoder is derived by using equation for optimal decoder, and equation for power constraint (6).

Chapter 5: Optimal Encoding/Decoding Strategies for Non-Stationary Gaussian Sources Definition 3. The set of linear admissible encoders L ad, where L ad  A ad, is the set of linear non-anticipative functionals A with respect to source signal X, which have the form The received signal is then The processes W, and N are independent, and the power constraint (2) becomes

Chapter 5: Optimal Encoding/Decoding Strategies for Non-Stationary Gaussian Sources Theorem 3 (Coding theorem for stochastic source). If the received signal is defined by the equation (5), the source by (4), then the encoding reaching the upper bound, optimal decoder, and corresponding error covariance are respectively given by

Chapter 5: Optimal Encoding/Decoding Strategies for Random Variable Sources Theorem 4 (Coding theorem for random variable source). If a source signal X, which is Gaussian random variable is transmitted over a flat fading wireless channel, then the optimal encoding and decoding with feedback reaching the channel capacity are

On Channel capacity: Conclusions We can use the new stochastic dynamical models developed to compute new results and get better insight on various computations of channel capacity which is a very important measure for transmission of information More information in the session, friday, nov. 13 th.

C. Shannon. Channel with side information at the transmitter. IBM Journal, pp , Oct A. Goldsmith, P. Varaia. Capacity of Fading Channels with channel side information. IEEE Transactions on Information Theory, vol. 43, no. 6, pp , Nov G. Caire, S. Shamai. On the capacity of some channels with channel state information. IEEE Transactions on on Information Theory, vol. 45, no. 6, pp , Sept E. Bigliery, J. Proakis, S. Shamai. Fading Channels: Information theoretic and communication aspects. IEEE Transactions on Information Theory, vol. 44, no. 6, pp , Oct T. Cover. Elements of information theory. Chapter 5: Capacity, Optimal Encoding-Decoding: References

Future Work Robust Modeling Receiver Design Optimal Coding Decoding Joint Source and Channel Coding for Wireless Channels Computation of the Channel Capacity for MIMO Channels and Joint Source and Channel Coding Power Control for Wireless Networks