Lower bounds - Sites personnels de TELECOM ParisTech

Estimation Lower Bounds and
Synchronization Issue in Single Carrier System
Alexandre Renaux and Pascal Larzabal
Ecole Normale Supérieure de Cachan, France
Alex - 1
Thurday, 27th October 2005
NEWCOM Automn School 2005
Estimation Lower Bounds and
Synchronization Issue in Single Carrier System
Outline
Estimation Lower Bounds
Estimator Perfomance Breakdown
Relative Insufficiency of Deterministic Bounds
Bayesian Bounds of the Weiss-Weinstein Family
Synchronization Issue in Single Carrier System
Problem Setup
Deterministic Bounds
Weiss & Weinstein Family Bounds
Alex - 2
Thurday, 27th October 2005
NEWCOM Automn School 2005
No Information
ld
esho
Thr
Mean Square Error (dB)
Estimator Perfomance Breakdown
Asy
mpt
otic
SNRTH
Alex - 3
Thurday, 27th October 2005
NEWCOM Automn School 2005
Signal to Noise Ratio (dB)
Mean Square Error (dB)
Estimator Perfomance Breakdown
Signal to Noise Ratio (dB)
Alex - 4
Thurday, 27th October 2005
NEWCOM Automn School 2005
ou
tli
er
Mean Square Error (dB)
Estimator Perfomance Breakdown
Huge MSE
Signal to Noise Ratio (dB)
Alex - 5
Thurday, 27th October 2005
NEWCOM Automn School 2005
Goals
- Approximation of the MSE curve over all
the SNR range
No Information
ld
esho
Thr
Mean Square Error (dB)
Estimator Perfomance Breakdown
- Good estimation of threshold SNRTH
Asy
mpt
otic
SNRTH
Alex - 6
Thurday, 27th October 2005
NEWCOM Automn School 2005
Signal to Noise Ratio (dB)
Mean Square Error (dB)
Insufficiency of the Cramér-Rao Bound
Asymptotic Area
Cram
ér -R
ao B
ound
Efficient estimator
(asymptotically !!!)
Signal to Noise Ratio (dB)
Alex - 7
Thurday, 27th October 2005
NEWCOM Automn School 2005
Insufficiency of the Cramér-Rao Bound
Mean Square Error (dB)
Non-asymptotic Area
What’s happen
- the CRB is too optimistic
Cram
ér -R
ao B
ound
- the CRB doesn’t exhibit threshold phenomenon
Signal to Noise Ratio (dB)
Alex - 8
Thurday, 27th October 2005
NEWCOM Automn School 2005
Relative Insufficiency of Deterministic Bounds
The parameters are assumed to be deterministic
Unification of deterministic lower bounds
Bounds = Minimum MSE of an estimator in
with contraints on the bias
(in possible other points)
MSE
Alex - 9
Thurday, 27th October 2005
NEWCOM Automn School 2005
Bias constraints
Estimator Bias
Relative Insufficiency of Deterministic Bounds
Stronger Bias
Constraint
Parameter
Range
True value of the parameter
Solution of the constraints
optimization problem
Alex - 10
Thurday, 27th October 2005
NEWCOM Automn School 2005
Cramér-Rao bound
Bhattacharyya bound
Estimator Bias
Relative Insufficiency of Deterministic Bounds
Parameter
Range
True value of the parameter
N test points: Barankin bound
1 test point: Chapman-Robins bound
Alex - 11
Thurday, 27th October 2005
NEWCOM Automn School 2005
Relative Insufficiency of Deterministic Bounds
Test points ???
Huge computational cost
Alex - 12
Thurday, 27th October 2005
NEWCOM Automn School 2005
Mean Square Error (dB)
Bayesian Bounds of the
Weiss-Weinstein Family
?
Deterministic bounds
Do not take into account
an a priori knowledge of
the parameter
Signal to Noise Ratio (dB)
solution
Bayesian Bounds
The parameters are assumed to be random (take into account the a priori pdf)
Alex - 13
Thurday, 27th October 2005
NEWCOM Automn School 2005
Bayesian Bounds of the
Weiss-Weinstein Family
Two Families
M-hypothesis testing
problem
Ziv-Zakaï Family
Alex - 14
Thurday, 27th October 2005
NEWCOM Automn School 2005
Covariance inequality
principle
Weiss-Weinstein
Family
Bayesian Bounds of the
Weiss-Weinstein Family
Two Families
Today
M-hypothesis testing
problem
Ziv-Zakaï Family
Alex - 15
Thurday, 27th October 2005
NEWCOM Automn School 2005
Covariance inequality
principle
Weiss-Weinstein
Family
Bayesian Bounds of the
Weiss-Weinstein Family
Unification of Bayesian lower bounds
In the Bayesian context, the best Bayesian bound is given by the
Conditional Mean Estimator
which is the solution of
Global MSE
Alex - 16
Thurday, 27th October 2005
NEWCOM Automn School 2005
Bayesian Bounds of the
Weiss-Weinstein Family
Unification of Bayesian lower bounds
Alex - 17
Thurday, 27th October 2005
NEWCOM Automn School 2005
Bayesian Bounds of the
Weiss-Weinstein Family
Unification of Bayesian lower bounds
Infinite number of constraints
Something less than the best Bayesian bound
(due to the constraints relaxation)
Minimal bounds on the MSE
Alex - 18
Thurday, 27th October 2005
NEWCOM Automn School 2005
Bayesian Bounds of the
Weiss-Weinstein Family
Unification of Bayesian lower bounds
Infinite number of constraints
Something less than the Conditional Mean
Estimator MSE (due to the constraints
relaxation)
Solution of the constrained optimization problem
Degrees of freedom: choice of h and s
Alex - 19
Thurday, 27th October 2005
NEWCOM Automn School 2005
Bayesian Bounds of the
Weiss-Weinstein Family
Unification of Bayesian lower bounds
Bayesian Bhattacharyya bound
solution
With
Particular case
Bayesian Cramér-Rao bound
solution
Alex - 20
Thurday, 27th October 2005
NEWCOM Automn School 2005
Bayesian Bounds of the
Weiss-Weinstein Family
Unification of Bayesian lower bounds
Reuven-Messer bound (Bayesian Barankin bound)
solution
With
Particular case
Bobrovsky-Zakaï bound
(Bayesian Chapman-Robins bound)
solution
Alex - 21
Thurday, 27th October 2005
NEWCOM Automn School 2005
Bayesian Bounds of the
Weiss-Weinstein Family
Unification of Bayesian lower bounds
Weiss-Weinstein Bound
solution
Alex - 22
Thurday, 27th October 2005
NEWCOM Automn School 2005
Estimation Lower Bounds and
Synchronization Issue in Single Carrier System
Outline
Estimation Lower Bounds
Estimator Perfomance Breakdown
Relative Insufficiency of Deterministic Bounds
Bayesian Bounds of the Weiss-Weinstein Family
Synchronization Issue in Single Carrier System
Problem Setup
Deterministic Bounds
Weiss & Weinstein Family Bounds
Alex - 23
Thurday, 27th October 2005
NEWCOM Automn School 2005
Problem Setup
Data Model
Bayesian case
Alex - 24
Thurday, 27th October 2005
NEWCOM Automn School 2005
Deterministic Bounds
Alex - 25
Thurday, 27th October 2005
NEWCOM Automn School 2005
MSE (dB)
Deterministic Bounds
10 Observations
BPSK
SNR (dB)
Alex - 26
Thurday, 27th October 2005
NEWCOM Automn School 2005
Weiss & Weinstein Family Bounds
Alex - 27
Thurday, 27th October 2005
NEWCOM Automn School 2005
MSE (dB)
Weiss & Weinstein Family Bounds
10 Observations
BPSK
SNR (dB)
Alex - 28
Thurday, 27th October 2005
NEWCOM Automn School 2005