Instrumentation & Measurement Magazine 23-5 - 36

Time Estimate

Weight Estimate

According to the clock difference and clock frequency of
time t, the estimated clock difference of time (t+τ) can be
obtained, such as equation (1). Xij(t) is the clock difference
between clock i and clock j measured at time t, τ is the time
interval of measurement, and d is the frequency drift parameter of atomic clock. According to the weighted average
principle, the optimal estimation of clock difference of the
reference clock relative to the paper clock is calculated, as
shown in (2):

Estimation error is used to estimate the weight, that is, the difference between the clock relative to the paper clock and the
calculated value of the current moment of each clock is used to
determine the weight, shown in (6):

	

	

Xˆ i  t  
 Xi  t   Yi  t   12 d 2	(1)

Xj t   

n

w   Xˆ  t     X  t   	(2)
i 1

i

i

ij

Xi(t), Yi(t) is the time difference and frequency difference of
clock i at time t relative to a reference time scale, respectively.
Xˆ i  t  is the prediction of clock difference at time t of clock i; wi
is the weight of atomic clock i.

Frequency Estimate
Generally speaking, the mathematical models of the instability of most frequency sources can be established by five
independent random noise combinations. The five kinds of
noise are white phase noise, flicker phase noise, white frequency noise, flicker frequency noise and random walk
frequency noise. When flicker frequency noise or random
walk frequency noise occurs, it needs to be suppressed. The
frequency estimation of the atomic time algorithm does not
record the true value of atomic clock's past frequency but
only pays attention to the change of frequency. An exponential filter is introduced to change the frequency into a time
function with time constant. The average frequency estimation of clock i at time (t+τ) is based on the first order difference
of clock difference data, which is calculated by exponential
filter, as in (3) and (4):

	

	

Xi  t     Xi  t 
	(3)
Yˆi  t    



Y
t  
i

1 ˆ
Y  t     miYi  t  	(4)

mi  1  i

ˆ  t  is the frequency prediction of clock i on the time interval
Y
i
from time t to t+τ; mi is the average time constant of exponential frequency, which is the minimum error introduced in the
time difference estimation of (1), and it is estimated according to (5):

	

mi


 1 4 2
1
1    mini
3
2
3 2







1/2


	(5)



	

ε
Xˆ i  t     Xi  t     Ki	(6)
i

Ki is the deviation of the error estimate.
The mean square time error of each clock is determined by
the exponential filter of (7):
	


 i2  τ 
t 

εi(τ) is the cumulative error of time difference estimation of
clock i over time interval τ; <ε2x(τ)> is the paper clock mean
square deviation of time t with time interval τ, and usually the
initial value is τ 2 σ 2y  τ . Nτ is the time constant of the exponential filter, which is used to estimate the current mean square
error.
The paper clock error estimation is calculated according to
(8). Every participating clock can improve the value, and the
poor performance of the clock will not affect the stability of paper clock:
	

 τ 
2
x

36	

 n
1
 
 i1  2  τ 
i


1


 	(8)



where n is the number of atomic clocks. The weight is applied
in (2) and its calculation is shown in (9):

	

wi 

	

Ki 

 x2  τ 

 i2  τ 
8  x2

 i2

1/2

	(9)

	(10)

Ki estimates the deviation of the first item on the right side of
(6), which is usually very small. This is because each clock belongs to the timekeeping atomic clock ensemble and shows the
performance of the atomic clock itself through the weight factor. The bigger the weight of an atomic clock is, the bigger the
deviation in the calculation is.

Estimation of Frequency Drift of Hydrogen Clock
Based on Minimum Error Theory
The principle of frequency drift estimation of hydrogen masers based on minimum error theory is as follows: We usually
define the time difference process as an integral of the frequency process, shown in (11):
t

	
τmini is the most stable time interval of clock i, which is the τ
when σ 2y  τ  is the minimum on the Allan variance curve.

1  2
  τ   N  i2  τ  	(7)
t
N  1  i

Xi  t    Yi  u  du	(11)


We define the causal first increment operator as (12):

IEEE Instrumentation & Measurement Magazine	

August 2020
Instrumentation & Measurement Magazine 23-5

Table of Contents for the Digital Edition of Instrumentation & Measurement Magazine 23-5
