Time Synchronization for Wireless Sensor Networks - Adaptive Signal Processing in Wireless Communications - page 383

where I (θ) is the Fisher information matrix. After some mathematical manipulations,
the joint clock offset and skew estimator can be expressed as [19]

 

N N N N

∑∑ ∑

[] ∑








    







2

Dxi − D Dxi ⋅ []

i i i

 ˆ

ˆ



(BP)

θ
θ

1




o




i = 1 i = 1 i = 1 i = 1

= , (13. 22)

2

N N N

(BP)
s


∑∑



 

N N

∑∑∑


 
 


 
 

    −

NDxi ⋅ [] D x []

ND

2

− D

i i

i i

 

i = 1 i = 1 i = 1

i = 1 i = 1

 

(A) - T 1,1 (A) . The Cramer-Rao lower bound (CRB) can be obtained by invert-

ing the Fisher information matrix I (θ). From (13.21), the Fisher information matrix is
given by

where Di  T 1, i

 

N

 
  
 
 


∑

 
 
  
 
 

N D

i

1
2

I () θ =

i = 1

.

σ

N N

∑∑

D D

2

i i



 i = 1 i = 1 

Then, inverting I (θ) yields

 

N

N

∑ ∑















2

D − D

i i

σ

2

I −

1 i 1 =

() θ =

i = 1

. (13. 23)

2

N

 

N N

∑

∑∑


 
 


 
 

− DN

ND

2

D

i

i i

 

i = 1

i = 1 i = 1

 

Hence, from (13.23), the CRBs for the relative clock offset and skew become

N

∑

σ

2

D

2

i

( ˆ

var θ

(BP)

) ≥ (13. 24)

i = 1

o

2

 

N N

∑∑

 
  

 
  

2

ND − D

i i

i = 1  i = 1 

and

σ

2

N

( ˆ

(BP)

var θ ) ≥ . (13. 25)

s

2

 

N N

∑∑


 
 


 
 

ND

2

− D

i i

i = 1 i = 1

 