Time Synchronization for Wireless Sensor Networks - Adaptive Signal Processing in Wireless Communications - page 379

where

∑ +

N

( )

Q  T TTTTTd + − ( ) .

12 , , i i 34 , , i i 2 , i 3 , i

i = 1

Note that the joint MLE depends on the value of the fixed portion of delays d , which is
assumed to be known in this section. Although estimating d is an achievable task, we
do not consider d another unknown (nuisance) parameter due to the inherent highly
nonlinear and complex operations required for estimating d .
The Cramer-Rao lower bound (CRB) for the vector parameter θ = [θ o , θ s ] T can be
derived from the 2 × 2 Fisher information matrix I (θ) by taking its inverse. From (13.6),
the second-order derivatives of the log-likelihood function with respect to θ o and θ′ s are
found as


  


  

∂

2

ln L θθσ

θ

, ′,

2

2

4 N θ ′ ,

os

=−

s

∂

2

σ

2

o







2 2

∂ ln L θ , ′, θσ

N

   

2

∑

oos


 


 

2 2

=− ( −+ θ ) ( T −, θ )

2 , i o 3 , i o

2 2

∂ ′ θ σ

s

i = 1







2 2 N

∂ ln L θθσ

θ

, ′,

   

2

∑

os
o ′′


 


  .

=− 2 θθ θ ′ − ′ +−′ − TT T θ

so s 2 , i 1 , i s 3 , i 4, i

∂ θ

2

σ

s

i = 1

Taking the negative expectations yields

 


  


  

∂

2

ln θθσ
θ

L , ′,

2

2

4 N θ ′




os



− E

 =

,

s

∂

2

σ

2



o



 







2 2

∂ ln L θθσ

θ

, ′,

N

   

∑


 


 

2




os



2 2

( XT dYTd +++−+

′

) ( )

− E

 =

E i

i 1 , i i 4 , i

XY ,

2 2

2

∂ ′ σ

i

θ

s

s

 

i = 1







N 2 2

= ∑ =

 ( ) + ( ) +

2

2 TdTd 2 σ

 

( a

 1 , i 4 , i

i 1

,

2 2

σ θ

′

s

 







2 2

∂ ln L , ′,

∂ ′

θθσ
θθ

N

   

2

∑






os


 


 

 

  ,

− E

 =−

E 22 θθ ′ −− T T ++ TT

 

XY , s o 2 , i 3 , i 1 i 4 , i

2

σ

i i

os

 

i = 1

() b 






= TT +

2 N

   

1 4

2

σ

where ( a ) and ( b ) are due to X i = θ′ s ( T 2, i - θ o ) - ( T 1, i + d ) and Y i = θ′ s (θ o - T 3, i ) + ( T 4, i - d ),
and