System Identification for Acoustic Comms.:

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1 System Identification for Acoustic Comms.: New Insights and Approaches for Tracking Sparse and Rapidly Fluctuating Channels Weichang Li and James Preisig Woods Hole Oceanographic Institution The demodulation of coherent acoustic communications signals requires estimating the state of the communications channel. A Class of Underwater Acoustic Channel (sparse and rapidly timevarying) Adaptive Equalizers for Coherent Communications require estimates of the time-varying channel impulse response. Rapid Time Variation -> need to estimate both channel impulse response and the parameters describing the time variation. Sparseness results in some of the parameters describing the time variation being unobservable in a state-space model sense. Several classes of methods of jointly address the challenges posed by sparse and rapidly time-varying channels. Results with experimental data

2 Surface Scattered Channels Wavefronts II Experiment (40 meter range) Surface Wave Field Impulse Response Estimation Error

3 Acoustic Focusing by Surface Waves Time-Varying Channel Impulse Response Dynamics of the first surface scattered arrival Time (seconds)

4 Channel Dynamics (Scattering Function)

5 Surface wave focusing and the signal prediction error (channel estimation error) Signal Prediction Error using RLS algorithm Channel Estimate using RLS algorithm

6 Impulse Response Estimation Error (surface scattering dominates, 238 m range) Signal Prediction Error is good surrogate for channel estimation error. Time scale of error oscillations is same as that for dominant surface waves. Single surface bounce arrivals contribute most significantly to the channel estimation error.

7 Relevant Aspects of Surface Scattered Channels The surface scattered arrivals can have very high intensities and be rapidly fluctuating. Thus, they can be a major source of error in estimating the channel impulse response. The dynamics of the impulse response containing surface scattered arrivals can change almost as rapidly as the impulse response itself. Algorithms must accommodate rapid time variation of both the impulse response and the parameters describing the dynamics of the impulse response fluctuations. The channel can be sparse and the subset of energetic taps of the impulse response can change rapidly with time. Arrivals appear and disappear as surface conditions evolve in addition to the movement of arrivals in delay.

8 System Equations Time-Varying Channel Impulse Response Vector Transmitted Data Vector Channel State-Space Model Received Signal (Observation Equation)

9 Model Simplifications A(θ,n) and the covariance matrix of the process noise, w[n], are a diagonal matrices. The temporal fluctuations of the elements channel impulse response vector, g[n], are uncorrelated from element to element. The temporal fluctuations of each element of the channel impulse response vector is a first order AR process. (Single pole) The process noise, w[n], and observation noise, v[n], are both white noise processes. Covariance of process and observation noise assumed known (in reality, they are used as algorithm tuning parameters).

10 Algorithm Approaches The Extended Kalman Filter (EKF) (developed in the context of joint parameter/state estimation and must be modified to address issues related to sparse channels) Comments on the Estimate Maximize (EM) and related approaches. Matching Pursuit and related approaches (developed in the context of representation with a sparse set of basis vectors and must be modified to address issues related to the identification of time-varying systems)

11 The Extended Kalman Filter Approach Random Walk Parameter Model -> Augmented State -> Augmented State and Observation Equations Linearized State Equation

12 Parameter Observability and Detectability Intuition: Estimating the state transition coefficient associated with a state variable that equals zero or is very weak is a ill defined problem. Formally: A necessary and sufficient condition for the observability of the parameter vector a[n] is that the sequence of channel estimates is persistently exciting and the underlying channel model is observable. In a sparse channel and the random walk (unstable) assumed parameter model, the elements of the a[n] vector associated with the very weak channel taps are not detectable. Result: The EKF can be unstable and the error covariance can grow without bound.

13 A Dual-Model EKF Approach: Partition the channel taps (elements of the vector g[n]) into two sets: the energetic taps and the non-energetic or quiescent taps. Different models for the time evolution of the elements of the parameter vector a[n] associated with the energetic and quiescent taps. Model for parameters associated with quiescent taps is stable and tends towards a fixed value.

14 Channel and Doppler Estimates

15 Doppler Estimates

16 Effect of Changing β Parameter

17 Estimate Maximize (EM) Based Approaches Intuition: For typical state estimation algorithms, the dynamics of the sequence of state estimates can be close to the dynamics of the sequence of states. This holds even if the algorithm has too long an averaging interval to accurately estimate the state sequence. State Model: Notional Estimate of Dynamics: The EM Algorithm formalizes this in an iterative estimation algorithm which accounts for the errors in the state estimates. Estimate State Sequence Estimate Transition Matrix (A)

18 Comments on EM based approaches The EM algorithm customarily operates on blocks of data. The parameter A is treated as a non-random parameter that is constant over each block. We have developed recursive variants of the traditional EM algorithm and are working on developing methods of accommodating time variability in the parameter A. The EM algorithm suffers from the same unobservabilty problem as the EKF.

19 Matching Pursuit Approaches Sparse system: MP -> sequentially select columns of C[n] (or equivalently, elements of g[n]). When columns of C[n] are not orthogonal, use variants based on orthogonalization and least squares metrics.

20 Modification for time-variability The i th column of C[n] contains the time series of the transmitted data symbols that map the i th channel tap onto the received signal vector y[n]. Scattering Function Representation:

21 Signal Estimation Error (Wavefronts II data) (NOTE: STANDARD EKF WILL BECOME UNSTABLE)

22 Conclusions Surface scattered channels can be both highly dynamic and sparse. Rapid fluctuations require explicitly estimating parameters describing channel dynamics as well as the channel state. Parameters can be unobservable if the channel is sparse (I.e., some taps of the channel impulse response have low energy.) Channel estimation techniques that jointly account for both the channel sparseness and the rapid fluctuations show performance improvements over techniques that do not account for both factors.

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