Recent Advances in Iterative Parameter Estimation
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Transcript of Recent Advances in Iterative Parameter Estimation
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