Fast Computation of Time Deviation and Modified Allan Deviation for Telecommunications Clock Stability Characterization Mingfu Li: Hsin-Min Peng, Chia-Shu Liao Advanced Tech. Research Lab. Telecommunication Labs., Chunghwa Telecom Co., Ltd. 12, Lane 551, Min-Tsu Road Sec. 5, Yang-Mei, Taoyuan, Taiwan 326, R.O.C. Tel: +886-3-4245 181 Fax:+886-3-4245474 E-Mail:
[email protected]
Abstract
chronous Digital Hierarchy (SDH) [4] has been widely deployed and acknowledged as a transmission standard replacing the existing PDH. For the introduction of SDH standard, network synchronization and timing becomes more apparent and challenge.
Time and frequency characterization of precision clocks and oscillators is an important task in the maintenance of time and frequency standards. Recently, time domain signal characterization of clocks is generally preferred and several pedormance measures of clocks and oscillators are defined in ITU-T Recommendation G.810. Among these measures, Maximum Time Interval Error (MTIE), Time Deviation (TDEV) and Modified Allan Deviation (MDEV) are frequently used, especially in telecommunications measurement. However, the direct computation of MTIE, TDEV and MDEK defined in ITU-T Recommendation G.810, tends to be unmanageable when the number of samples becomes large. In this paper, we propose a fast computation approach for TDEV and MDEV The approach is based on the recursive algorithm and the computation exactly conforms to the ITU-T G.810 dejinition. As compared with the direct computation approach, the time complexity of the proposed approach is reduced by a factor of N , the number of samples. It reveals that the real-time measurement or monitoring will be feasible by employing the proposed computation approach.
In digital communication systems, the variations of the significant instants of a digital signal from their ideal position in time are called jitter or wander [5]. Short-term variations which are of frequency larger than or equal to 10 Hz are defined as jitter, while long-term variations which are of frequency less than 10 Hz are defined as wander. Jitter is usually induced in the regenerators or multiplexiers where bit stuffing or pointerjustification events can occur, and may increase the bit error rate in the receiving end. Thus, in order to reduce the jitter, many shemes has been proposed, including Stuff Threshold Modulation (STM) [6], jitter attenuator 173, Phase-Lock Loop (PLL) [8] and bit leaking [9] schemes, etc. As to the wander, it usually results from the frequency offset or changes in cable delay due to temperature variation, and can lead to data slips. The data slips can be a severe problem for data services. Hence, the digital switching equipment requires network synchronization to avoid slips in the input elastic store. For guaranteeof network synchronization quality ,ITU-T has enacted the jitter and wander specifications in [ 10, 11, 12, 131for equipments testing and telecommunications measurement.
1 Introduction
In time/frequency standards and measurements, clock The precision clocks and oscillators have become in-
stability characterization, jitter and wander are the main
creasingly important in society, especially in the high-speed digital communication networks. As one knows, network synchronization has played a central role in telecommunication networks since the introduction of digital exchanges [l, 21. The early digital transport standard [3] is called Plesiochronous Digital Hierarchy (PDH). Recently, Syn-
concems. In order to evaluate the quality of precision clocks and oscillators, severalparameterssuch as Time Interval Error (TIE), Maximum Time Interval Error (MTIE), Time Deviation (TDEV), Allan Deviation (ADEV), Modified Allan Deviation (MDEV) and Root Mean Square Time Interval Error (TIErms) are defined [5].The parameters TIE, MTIE and TDEV are employed to measure wander of a timing signal, while ADEV and MDEV are usually applied to charac-
*Corresponding author
1087-4089/00 $10.00 0 2000 IEEE
156
terizing the clock stability. In this paper, we will focus our discussion on the TDEV and MDEV , which characterizes the wander and stability of a Clock Under Test (CUT) respectively. We will first introduce the formal definition of TDEV and MDEV, and then suggest an efficient approach for their computation. The rest of this paper organizes as follows. In Section 2, the definition of TDEV and MDEV will be described. Next, a fast computation approach for TDEV and MDEV is proposed in Section 3. In Section 4, the superiority of the proposed approach is demonstrated in terms of time complexity. Finally, in Section 5 we make a concise remark.
where
A(n)=
MDEV(nT0) =
(1)
1 dd(t) 2 ~ f o dt .
l2
-I xi)
’ (6)
”J” 2n4$ ( N
- 3n
+ I)
(7) ‘
nT0
TDEV(n‘i-0)= -MDEV(nTo).
where +(t)denotes the phase deviation and f o is a constant nominal frequency. The amplitude fluctuations of the voltage output is Usually a w m d to be negligible around vo. Then the normalized frequency deviation is defined as y(t) = --
+ n
MDEV is constructed by a second difference composed of time deviations so-averaged and will remove the ambiguity through bandwidth variation. The behavior of TDEV and MDEV is substantially dependent on the chosen measurement sampling interval in the observation intervals where WPM or FPM noises dominate. From ( 5 ) and (7), it is not is )related to difficult for one to find out that M D E V ( ~ T O T D E V ( ~ T by O )the following equation
In telecommunications, a clock or oscillator is used to generate the ideally periodic timing signal. However, due to unavoidable noises, actual timing signals are pseudoperiodic ones. Mathematically, a timing signal V ( t )can be represented by
+ +(t)),
-h
and = 1 7 % . * ’ L+1- And the Modified Allan Deviation (MDEV) is defined as
2 Preliminaries
V ( t )= VOsin(2nfot
(xi+2n j=1
(8)
d3
That is, the calculations of k f D E V ( n ~ 0and ) TDEV(n70) are almost the same. Accordingly, hereafter we will only consider the computation of T D E V ( ~ T OHowever, ). the same computation technique can be directly used for MDEV(nT0).
(2)
Integrating y(t) yields the time error function x ( t ) ,which has the dimensions of time and is given by
3 Fast Computation Approach for TDEV 3.1 Recursive Formula for TDEV
(3) From (3), one can observe that the time error function can be written as a function of the phase deviation 4(t). Time domain signal characterization of an oscillator can be done by a Time Interval Counter (TIC). Let the sampling interval of the TIC be T O , then the discrete set of samples xi = x(to 270)is defined as the Time Error (TE), where to represents the measurement starting time. Subsequently, the Time Interval Error (TIE) is defined as
Herein, we will derive a recursive form for T D E V ( ~ T O ) to speed UPthe computation. Refer to (6), one can denote s(n,j ) as follows-
+
n+j-1
s(n,j) =
(Zi+2n
- 2xi+n
+Xi)
(9)
i=j
Consequently, T D E V ( ~ T can O ) be rewritten as follows.
(4)
where T = TO is usually called observation interval. The expected time variation of a signal is measured by the Time Deviation (TDEV). Based on the sequence of Time Error lDEV can be estimated using the following calculation.
TDEV(nT0) =
6n2(N - 3n + 1)
Subsequently, according to the following derivation s ( n ,j) can be simplified to a recursive form. n+i-I
157
n+i-l
n+i-I
n=l
.
Figure 1. The recursive process for computing ~ [ junder ] the special case N = 3k, where k is a positive integer.
3n+j-l
=
2n+j-l
i=2n+j
n+j-l
xi+
xi-2
xi
holds for n 2 m. That is, the inmost C summation of A(n) in (6) can be computed recursively with initial condition s [ j ] = 0 for all j.
i= j
i=n+j
3(n--l)+j--1
\
3.2 Recursive Algorithms for TDEV Computation
\
i=(n-l)+j
In order to employ the recursive form to compute TDEV(n70), additional memory space should be allocated to store the parameter s(n,j). The values s ( n , j ) , for j = 1,... , N - 2, can be stored in the array s [ j ] , j = 1 , . . . ,N - 2. And according to (11),the value sb], where 1 5 j 5 N - 2, can be calculated recursively as shown
1
in Fig. 1. Subsequently, some popular sequential recursive algorithms for computing TDEV(n70)are given. In these algorithms, TDEV(n7-0)is stored in the variable T[n].The first algorithm, Algorithm 1, is given to calculate TDEV(n7-0)for all n. In the telecommunicationsmeasurement, the computztion of TDEV can be done simultaneously with the TIC (Time Interval Counter) data acquisition. This can really save much time to accomplish the TDEV measurement. By using the fast computation approach proposed in this paper, it is possible to fulfill the real time TDEV measurement or monitoring. An example of the real time computation algorithm is given in Algorithm 2. Here, n e w 3 is the number of data samples observed up to the present, and
where
Recursively, one can derive that
158
o l d 3 is the number of data samples observed at the previous computation cycle.
plexity decreases significantly. This is demonstrated by the following illustration. According to Algorithm 1, the complexity of calculating T D E V ( n T 0 )is O ( C ~ ; ' ~ + ' which is equivalent to O ( N - 3n 1). Consequently, for computingallTDEV(nTo),n = l , . . . ,[$J,thecomplex-
+
Algorithm 1: (Recursive TDEV Computation Algorithmfor all n)
1 1 9
N
( N~-~3 n + l ) ) , whichequals O ( N 2 ) . ity s h o u l d b e O ( C ~ The result indicates that the complexity of TDEV computation using (5) directly can be improved by a factor of N when the recursive computation technique is employed.
for (1 5 j 5 N - 2) s [ j ]= 0; n = 1; while (3n 5 N ) {x=O; for (1 5 j 5 N - 3n + 1)
5
TDEV and MDEV are two of the key performance measures in clock stability characterization and telecommunications measurement. The important features and computation issues of TDEV and MDEV have been pointed out. For resolving the troublesome computation, we have presented an efficient and fast approach to compute TDEV and MDEV in this paper. The superiority of the proposed approach has been demonstrated in terms of time complexity. The new approach is found to be the best one at present from the time complexity criterion. For the fast speed computation merit, we believe that it would improve the performance of some commercial timing testing equipment which includes TDEV and MDEV measurements.
T[n]= &; n=n+l;
Algorithm 2: (Real Time Recursive TDEV Computation Algorithm for all n) while (new-N 5 N ) { for ( o l d 3 - 1 5 j 5 n e w 3 - 2) s [ j ]= 0; n = 1; while (3n 5 new-N) o l d 3 - 3n + 1 . { 2 = ~ ~ x[ n ] new-N - 3n + 1' m = m a { 1, o l d 3 - 3n + 2); for (m 5 j 5 n e w 3 - 3n + 1 ) { 4.d = + A(n,j); 5=2+
1
References [ 13 M. J. Klein and R. Urbansky, " Network Synchroniza-
tion -A Challenge for SDH/SONET? ",IEEE Commun. Mag., pp. 42-50, Sept. 1993.
s2['il 6n2(new-N - 3n + 1)'
[2] J. C. Bellamy, " Digital Network Synchronization, " IEEE Commun. Mag., Vol. 33, pp. 70-83, April, 1995.
T [ n ]= L ,E;
[3] ITU-T Rec. G.702, 1988.
n=n+l;
}
//ends a computation cycle here
"
Digital Hierarchy Bit Rates,
"
[4]ITU-T Rec. G.707, Synchronous Digital Hierarchy "
old_N = new-N; update n e w x ;
Bit Rates, " 1993.
}
4
Conclusions
[5] ITU-T Rec. G.810, " Definition and Terminology for SynchronizationNetworks, " Aug. 1996.
[6] W. D. Grover, T. E. Moore, and J. A. McEachern, " Waiting Time Jitter Reduction by Synchronizer Stuff Threshold Modulation, " IEEE GLOBECOM'87, pp. 514-518,1987.
Complexity Analysis
If one computes TDEV(nT0) directly according to (5), the complexity of calculating a certain TDEV(nT0) is ( ~ j N _ ~n), ~ + whichisequivalenttoO((N--3n+l)n). l
o
Thus, for computing all TDEV(nTO), n = 1,
[7] R. F. Bridge, S. Bily, J. Klass, and R. Taylor, " Jitter Attenuation in T1 Networks, " IEEE ICC'90, pp. 685689,1990.
, L+J,
+
the complexity should be O( (N - 3n l)n), which equals O ( N 3 ) . When the new computation technique is used, the com-
[8] Y. Matsuura, " Reducing Jitter in PDH and ATM, IEEE GLOBECOM'9.5,pp. 1323-1326,1995.
159
"
[9] H. Sari and G. Karam, " Cancelation of Pointer Adjustment Jitter in SDH Networks, '' ZEEE Trans. Commun., Vol. 42, No. 12, pp. 3200-3207, Dec. 1994. [ 101 ITU-T Rec. G.823, " The Control of Jitter and Wander
Within Digital Networks Which are Based on the 2048 kbit/s Hierarchy, " March 1993. [ 111 ITU-T Rec. G.824, " The Control of Jitter and Wander
Within Digital Networks Which are Based on the 1544 kbit/s Hierarchy, " March 1993. [12] ITU-T Rec. 0.171, " Timing Jitter and Wander Measuring Equipment for Digital Systems Which are Based on the Plesiochronous Digital Hierarchy (PDH) ," April 1997. [ 131 ITU-T Rec. 0.172,
" Timing Jitter and Wander Measuring Equipment for Digital Systems Which are Based on the SDH ," June 1999.
