Recent Advances in Iterative Parameter Estimation Cédric Herzet and Luc Vandendorpe Université catholique de Louvain, Belgium Sequel of Marc's presentation Topics

Most estimators rely on the maximum-likelihood criterion Unbiased Estimation mean is equal to the actual parameter Efficient It reaches the smallest mean square error Particular approximation of the optimal receiver Explain what kind of parameter \Theta can include Often used due to its good asymptatical properties Unfortunately impossible to implement in practice since computation of LLF intractable

Classical estimators operate in non-data-aided (NDA) mode All sequences are assumed equiprobable Common assumption:NDA i.e all the transmitted sequences are assumed equiprobable Good approximation at high SNR Large gap between true (CA) solution et NDA solution at low SNR Transition: we would like to get rid of this approximation and compute the actual the ML solution i.e. the ML solution which takes into account the code structure. Since impossible to compute a close form expression, we resort to iterative methods. Suboptimal if the sequence is coded

BER for BICM transmission with phase estimation NDA The estimation quality leads to BER degradation The NDA synchronizer does not able to recover the performance of the perfectly synchronized system. Reason: does not take the code structure into account Perf. Sync

We resort to the iterative methods to solve the ML problem Good performance We expect the method to converge to the ML solution Low complexity Each iteration must have a low computational load Proposed solution : resort to iterative methods

The EM algorithm enables an efficient search of the ML solution Easy to maximize It is robust Converges under mild conditions to the ML solution It might converge slowly Depends on the quantity of missing information Algorithm well-suited to maximization problem when function to maximized is a probability Reasonable complexity Robust Drawback: may converge slowly

BER for BICM transmission with phase estimation NDA Perf. Sync

BER for BICM transmission with phase estimation NDA EM The EM synchronizer enables to improve the parameter estimate at each iteartion. BER improvement at each iteration but required a high number of iterations Perf. Sync

BER for BICM transmission with phase estimation NDA EM Increase the number of iterations required to achieve converge Perf. Sync

Synchronization based on the factor graph framework Explanation of factor graph and SP algorithm principle

We apply the SP algorithm to the ML estimation problem The likelihood function may be viewed as the marginal of this probability Likelihood difficult to handle Likelihood function = marginal function of a more global probability. Idea: Apply the SP algorithm to compute the likelihood function in order to get an expression more easy to handle

The considered factor graph has two main parts Only depends on synchronization parameters Synchronization 2 parts… Cycles : the SP algorithm is iterative (iterative exchange of message between the upper part and the lower part) Transition: talk about the kind of messages exchanged by the two parts. Only depends on transmitted symbols Symbol detection

Symbol detection part transmits symbol extrinsic probabilities Synchronization Symbol extrinsic probabilities Run SP algorithm on lower part = symbol detector (may be itself iterative depending on the kind of transmission) Example : turbo decoder Transmitted messages = symbol extrinsic probabilities (= turbo receiver, BCJR decoder…) Symbol detection

The synchronization part transmits a modified likelihood function Symbol detection

The extrinsic probabilities are used as a priori information Transmitted message : where Same structure as actual likelihood function A priori information modified : extrinsic prob used Transition: difficult to handle : \theta continuous => infinite number of value has to be transmitted Solution : approximate message by canonical distribution => limit the number of values to be passed extrinsic probability (from detection part)

The synchronization message is approximated by a delta function We compute a « well-chosen » point of the likelihood function Synchronization Parameter estimate Build a new LLF and compute a well-chosen point Classical turbo synchronization scheme: use soft information delivered by symbol detector to compute a new estimate Symbol detection

We solve a ML problem at each SP iteration Easier to compute due to the particular factorization of the a priori information: We have to solve an ML problem at each iteration. Easier to solve than the initial ML problem to to the particular factorization of the symbol a priori probability

BER for BICM transmission with phase estimation NDA EM Perf. Sync

BER for BICM transmission with phase estimation NDA Do not increase significantly the receiver complexity EM SP synchronizer only leads to a small degradation in terms of turbo iterations w.r.t. a perfectly synchronized system SP Perf. Sync

The EM approach drops some information about the parameter maximized by the SP approach maximized by the EM approach The modified likelihood function represents the best current statistical model relating observations to the parameter to be estimated. May be rewritten as the sum of two terms whose the first is the function maximized by the

Theoretical lower bounds for soft synchronizer performance Given an amount of soft information, what is the best possible performance achievable by a soft synchronizer

Soft synchronizers consider a modified statistical model The symbol a priori knowledge is assumed to come from a soft information vector e Approach based on the observation that all soft synchronizer are based (explicitly or implicitly) on a modified likelihood function. Modification consists in assuming that symbol a priori information is given by a soft information vector e

We can compute the CRB related to the modified statistical model CRB related to the observation of a particular vector e Using the modified LLF, we may derive a CRB which the estimation error variance given a particluar soft information vector e In practice, we want to bound the synchronizer performance for a distribution of vector e

We derive a lower bound valid for a soft information distribution First expression is intractable to compute in practice We derive a modified CRB much easier to compute in practice. We prove that this bound is not looser than the CRB as long as the frame length is large enough. Soft Modified Cramer-Rao Bound: easy to compute in practice…

MSE for BICM transmission with phase estimation The soft synchronizers can reach the MCRB after only a few iterations… Explanation plot: what vs what Comments: … SMCRB MCRB

MSE for BICM transmission with phase estimation NDA Do not take the code structure into account ! Huge gap at low SNR between the performance achievable by a NDA synchronizer and a CA synchronizer SMCRB MCRB

MSE for BICM transmission with phase estimation NDA Do not take fully benefit from the available soft information EM MCRB

MSE for BICM transmission with phase estimation NDA The SP approach enables to operate very close to the SMCRB SP EM MCRB

Semi-analytical performance analysis of turbo-equalization schemes Likelihood function = marginal function of a more global probability. Idea: Apply the SP algorithm to compute the likelihood function

The considered receiver is made up with three blocks Turbo equalizer Received samples BER Channel estimator MMSE/IC equalizer MAP decoder Assumptions : BPSK, one user

We want to calculate the equalizer outputs as functions of the inputs MMSE/IC equalizer Goal : find analytical expressions of functions f1 and f2

Variance of LLR at equalizer output vs. estimation error variance Calculations fit simulations very well 4 dB 5-tap Porat channel simulations calculations

The MAP decoder behavior is simulated f is simulated Finally, the BER may be expressed as a function of the equalizer inputs, notably the estimation error variance

BER vs. estimation error variance The BER degradation is accurately predicted by calculations 4 dB 5-tap Porat channel simulations calculations

Cooperations and prospective researches Any problems related to parameter estimation: Channel estimation Time-varying parameters … Receiver design based on factor graphs Analytical performance analysis (BER, CRB,…)

Thank you for your attention !

BER for BICM transmission