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

2. 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

3. 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

4. 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

5. 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

6. 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

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

8. 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

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

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

11. 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

12. 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

13. 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

14. The synchronization part transmits a modified likelihood function
Symbol detection

15. 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)

16. 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

17. 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

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

19. 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

20. 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

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

22. 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

23. 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

24. 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…

25. 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

26. 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

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

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

29. 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

30. 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

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

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

33. 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

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

35. 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,…)

36. Thank you for your attention !

37. BER for BICM transmission