Recent Advances in Iterative Parameter Estimation

Cédric Herzet and Luc Vandendorpe

Université catholique de Louvain, Belgium

Most estimators rely on the maximum-likelihood criterion

UnbiasedEstimation mean is equal to the actual parameter

EfficientIt reaches the smallest mean square error

Classical estimators operate in non-data-aided (NDA) mode

Suboptimal if the sequence is coded

All sequences are assumed equiprobable

BER for BICM transmission with phase estimation

NDA

Perf. Sync

The estimation quality leads to BER degradation

We resort to the iterative methods to solve the ML problem

Good performanceWe expect the method to converge to the ML solution

Low complexityEach iteration must have a low computational load

The EM algorithm enables an efficient search of the ML solution

It is robustConverges under mild conditions to the ML solution

It might converge slowlyDepends on the quantity of missing information

Easy to maximize

BER for BICM transmission with phase estimation

NDA

Perf. Sync

BER for BICM transmission with phase estimation

NDA

Perf. Sync

EM

BER for BICM transmission with phase estimation

NDA

Perf. Sync

EMIncrease the number of iterations required to achieve converge

Synchronization based on the factor graph framework

We apply the SP algorithm to the ML estimation problem

The likelihood function may be viewed as the marginal of this probability

The considered factor graph has two main parts

SynchronizationOnly depends on synchronization parameters

Only depends on transmitted symbols

Symbol detection

Symbol detection part transmits symbol extrinsic probabilities

Symbol extrinsic probabilities

(= turbo receiver, BCJR decoder…)

Synchronization

Symbol detection

The synchronization part transmits a modified likelihood function

Modified likelihood function

Synchronization

Symbol detection

The extrinsic probabilities are used as a priori informationTransmitted message :

extrinsic probability

(from detection part)

where

The synchronization message is approximated by a delta function

We compute a « well-chosen » point of the likelihood function

Parameter estimate

Synchronization

Symbol detection

We solve a ML problemat each SP iteration

Easier to compute due to the particular factorization of the a priori information:

BER for BICM transmission with phase estimation

NDA

Perf. Sync

EM

BER for BICM transmission with phase estimation

NDA

Perf. Sync

EMDo not increase significantly the receiver complexity

SP

The EM approach drops some information about the parameter

maximized by the SP approach

maximized by the EM approach

Theoretical lower bounds for soft synchronizer performance

Soft synchronizers consider a modified statistical model

The symbol a priori knowledge is assumed to come from 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

We derive a lower bound valid for a soft information distribution

Soft Modified Cramer-Rao Bound: easy to compute in practice…

MSE for BICM transmission with phase estimation

MCRB

SMCRB

The soft synchronizers can reach the MCRB after only a few iterations…

MSE for BICM transmission with phase estimation

MCRB

SMCRB

NDA

Do not take the code structure into account !

MSE for BICM transmission with phase estimation

MCRB

NDA

EM

Do not take fully benefit from the available soft information

MSE for BICM transmission with phase estimation

MCRB

NDA

SP

The SP approach enables to operate very close to the SMCRB

EM

Semi-analytical performance analysis of turbo-equalization schemes

The considered receiver is made up with three blocks

MMSE/ICequalizer

MAPdecoder

Channelestimator

Turbo equalizer

Assumptions : BPSK, one user

Received samplesBER

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 simulationsvery well

4 dB5-tap Porat channel

simulations

calculations

The MAP decoder behavior is simulated

MAP

decoder

Finally, the BER may be expressed as a function of theequalizer inputs, notably the estimation error variance

f is simulated

BER vs. estimation error variance

The BER degradation isaccurately predictedby calculations

4 dB5-tap Porat channel

simulations

calculations

Cooperations andprospective 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