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Clock Offset and Skew Estimation in .NET Create PDF 417 in .NET Clock Offset and Skew Estimation

Clock Offset and Skew Estimation using barcode drawer for visual studio .net control to generate, create pdf417 image in visual studio .net applications. iPhone OS MSE of clock offset f Simplified model Correct model 1 Variance of delay Figure 6.2: MSE of clock o ffset estimate as a function of delay variance. found as 2 ln L , , 2 2 2 ln L , , 2 2 ln L.

= = = . 2N , 2 1 2 1 2. k =1 N k =1 ( T2,k )2 + ( T3,k )2 ,. 2 T2,k + T1,k T3,k T4,k . , . , 2. Taking the negative expectations yields E E 2 ln L , , 2 2 2 ln L , , 2 . ( a). 2N , 2 1 2. k =1. EXk ,Yk ( Xk + T1,k + d)2 + (Yk T4 pdf417 2d barcode for .NET ,k + d)2 2. 2 . 2 2 N k=1 ( T1,k + d) + ( T4,k d) + 2 2 2 ln L , , 2 . 1 2. k =1. EX ,Y 2 (2 T2,k T3,k ) + T1,k + T4,k N T1 + T4 , 2. Exponential Delay Model where ( a) and (b) are due t visual .net PDF 417 o Xk = ( T2,k ) ( T1,k + d) and Yk = ( T3,k ) + ( T4,k d). Therefore, the Fisher information matrix becomes E I ( ) = E 1 = 2 .

2 ln L( , , 2 ) 2 2 ln L( , , 2 ) . E E N k =1 2 ln L( , , 2 ) 2 ln VS .NET PDF 417 L( , , 2 ) . 2N 2 N T1 + T4 1 2. ( T1,k + d)2 + ( T4,k d)2 + 2 2 N T1 + T4 (6.14). Now the CRLB can be obtained PDF-417 2d barcode for .NET by taking the inverse of the [k, k]th element of the Fisher information matrix (i.e.

, var( k ) I 1 ( ) ii ), and the inverse I 1 ( ) is given by I. ( ) = . V 2 2 N 2V N ( T1 + T4 ). ( T1 + T4 ). 2V N ( T1 + T4 ). , (6.15). ( T1 + T4 ). 2V N ( T1 + T4 ). 2 2 2V N ( T1 + T4 ). N where V = k=1 ( T1,k + d) 2 + ( T4,k d)2 + 2 2 . Consequently, from the result in [44], the CRLBs of clock offset and skew for the Gaussian delay model are respectively given by. var( GML ) var( . 2 2 V N 2V N T1 + T4 2 2 2. (6.16). ) =. 2 2 . 2V N T1 + T4 2 2 2 2V N T1 + T4 (6.17). 6.2 Exponential Delay Model A detailed justi cation of m PDF 417 for .NET odeling the network delays as coming from an exponential distribution was presented in 5. Since the MLE for the clock offset under exponential delays has been derived in [41] and already mentioned in (5.

1), we derive the corresponding CRLB for the clock offset in the next section. Afterwards, the joint MLE for both the clock offset and skew is obtained and the corresponding algorithms for nding those estimates are also presented..

70 6.2.1.

Clock Offset and Skew Estimation Cramer Rao Lower Bound (CRLB) for Clock Offset It was proven in [41] that t he MLE of exists when d is unknown and exhibits the same form as the estimator proposed in [71], which is given by = U(1) V(1) , 2 (6.18). where N stands for the numbe .net vs 2010 PDF417 r of observations of delay measurements and the subscript (1) denotes the rst-order statistic of the corresponding data set. In this section, we proceed towards obtaining the CRLB for this clock offset under the exponential delay model.

Note that (6.18) can be rewritten as = U(1) V(1) X(1) Y(1) = + , 2 2. N where X(1) and Y(1) denote the corresponding order statistics of { Xk }k=1 and N X(1) Y(1) , then the pdf of Z {Yk }k=1 , respectively. De ne the new variable Z can be found as follows. Since the order statistics X(1) and Y(1) are independent, the pdf of RV Z, f Z (z), can be found by transforming the joint distribution of RVs X(1) and Y(1) using the dummy variable S = Y(1) .

From the modeling assumptions, the pdfs of the uplink and downlink delays, Xk and Yk , are given respectively by. f Xk ( x ) =. 1 x exp 1 1 1 y exp f Yk (y) = 2 2.
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