- Email: [email protected]
The analysis of a transmission system employing SONET-like frames
computer communications
ELSEVIER
Computer
Communications
The analysis of a transmission
21 (1998) 530-537
system employing SONET-like frames H.N. Kundaeli*
Department
ofPhysics, University of Dar-es-Salaam, P.O. Box 35063, Dar-es-Salaam, Tanzania Received 26 September
1997; accepted 9 January
1998
Abstract Frame synchronization systems that employ multiple-code markers have been shown to give better synchronization performance than systems that employ single-code markers. It has been shown, however, that only a few of the codewords in the marker need to be used in order to attain optimum synchronization performance. Among the multiple-code marker systems which have gained wide usage recently is the SONET scheme for information transmissions at high data rates. In this report, a model for the analysis of the synchronization performance of systems which employ SONET-like frames is developed. The model is then applied to the SONET scheme to study how its performance varies with the choice of the parameters in its frame structure. 0 1998 Elsevier Science B.V. Keywords:
SONET; Frame sychronization;
Multiple code markers; Synchronization
1. Introduction Digital information is often transmitted serially as a stream of blocks of information bits separated by synchronization bits called markers. In order to retrieve the information, the receiver first locates the markers and takes the bits between them as the information. The bits forming an information block and marker are referred to as a frame, and the scheme of synchronizing the receiver to the frames using the markers is referred to as frame synchronization. In systems employing frame synchronization, the markers may consist of single bits or multiple bits. The North American DS 1 basic frame scheme of 193 bits is a typical 1 bit marker scheme [ 11. Networks employing this scheme, however, do multiplex several basic frames into extended frames, and several extended frames into superframes. The markers for the extended frames then come from the markers of the basic frames, and are therefore composed of multiple bits. In some cases, only some of the marker bits of the extended frames are used for frame synchronization, a technique known as windowing [l]. On the other hand, the synchronous optical network (SONET) scheme uses basic frames consisting of 810 words, two of which are for the marker [2]. This scheme is destined for the transmission of data at high speeds in optical fibres, and is supposed to replace conventional systems at those speeds. For example, a DS3 PCM superframe consisting of 12 DS 1
0140-3664/98/$19.00
0 1998 Elsevier
basic frames can be accommodated in a single STS-1 SONET basic frame. When higher transmission capacities are desired, several SONET basic frames are multiplexed by word interleaving. The resulting frame then has twice as many codewords as the number of multiplexed basic frames. If the multiplexed frames are long, and all the codewords of the markers are used for synchronization, then the receivers turn out to be quite complex, and the performance of the systems tends to degrade. In order to avoid this, ‘windowing’ is applied so that only a few of the words in the marker are used for synchronization [3]. In this report, the work reported by the author on systems employing single-code markers is extended to the case of systems employing SONET-like multiple-code markers. The model is then used to study how the performance of the system varies with the choice of its parameters-namely frame length, number of codewords used for synchronization, position of the synchronization codewords in the marker, recovery verify number and loss verify number. The synchronization efficiency, which has been used in the determination of design parameters for systems employing single-code markers [4], is used as the measure of performance.
2. Model of the receiver The receiver analyzed in this report is destined for frames consisting of J multiplexed basic frames, and its transition
* E-mail: [email protected] PII SO140-3664(98)00125-X
efficiency
Science
B.V. All rights reserved
H.N. KundueliLomputer
Communications
531
21 (199X) 530-537
false path is exactly the same as that in the true path. Note, however, that any data which the receiver collects while in the false path is invalid. 2.1. Derivation
Fig.
I A receiver for frames having J SONET-like basic frames.
diagram is given in Fig. 1. The markers of the multiplexed frames are assumed to consist of only two types of specially chosen codewords, designated as Al and A2. In the SONET scheme, the bit sequences of the codewords are 11110110 for Al and 00101000 for A2. The markers, therefore, consist of J codewords of type Al and J codewords of type A2. It is, however, assumed that the receiver uses only I codewords (I = 2...25) for synchronization, B of which (B = I...J) are A2 codewords. The operation of the receiver is therefore as follows. When it has lost synchronization, or is brought up, the receiver starts to search for the first Al codeword on a bit-by-bit basis in state 1. If it locates the A 1 codeword, and finds it error free, it moves to state 2 1 and stays in this state as long as the received words are error-free Al codewords. If it receives a word matching the error-free A2 codeword, the receiver moves to state 22, otherwise it returns to state 1 if the received word matches neither the error-free Al codeword nor the errorfree A2 codeword. In state 22, the receiver checks the next B - 1 words if they match the error free A2 codewords before going to state 23. If it finds any of these B - 1 words in mismatch however, the receiver returns to state 1 and starts another search. On reaching state 23, the receiver performs checks on framing patterns consisting of the I codewords instead of single codewords. The receiver must receive N - 1 consecutive error-free framing patterns before going to state 4, otherwise it returns to state 1 in case of a single failure. While in state 4, the receiver collects data from each received frame while monitoring the framing pattern. If it receives an erroneous framing pattern, it moves to state 6 and subsequently to state 1 after M - 1 consecutive errors. If it receives a single error-free framing pattern while in state 6, however, the receiver returns to state 4. Note that while in state 6, the receiver continues to regard all received data as valid data. The path through states 2,4 and 6 is called the true path because it is the path which the receiver takes when it encounters an error free Al codeword while in state 1. On the other hand, if the receiver encounters any other bit sequence resembling an error free Al codeword while in state 1, it moves to state 31 and takes the false path through states 5 and 7. Owing to the symmetry of the transition diagram, the operation of the receiver in the
of transition probabilities
To define the data characteristics of the frame, we assume that the frame consists only of data and synchronization words. Any other words in the frame apart from data and marker words are also assumed to be data words in order to simplify the mathematical derivations. The data characteristics of the frame can, therefore, be defined as follows: J is the number of multiplexed basic frames and K is the length of the resulting multiplexed frame, K is equal to (K-2J) data + JAl + JA2 marker (sync) words; S is the marker length which is equal to JAI + JA2 words; W is the word length; P, = Pr (received bit is in error}; (Yis (duration of one bit)/ (duration of a frame); and, fi is (duration of one word)/ (duration of a frame). From the data characteristics, and noting that L is the frame length in bits (KW bits), the following relationships can be defined: 1 is the bit position in the frame (I = 1.. .L); k is the word position in the frame (k = 1. . .K); s is the sync word position in the marker (s = 1.. .S); k(1) is the word in which the tth bit occurs; and s(l) is the sync word in which the Ith bit occurs if 1 is a marker bit. Fig. 2 shows a snapshot of a received bit stream stretching from A to B and consisting of a portion of frame i - 1, the whole of frame i and a portion of frame i + 1. The indicated frames are composed of two basic frames, four marker words and 4320 bits. The reference point is in frame i at bit 1 = 4299, which gives k(1) = 538 and s(l) = 2. In order to derive the transition probabilities, first note that when the receiver is attempting to gain or regain synchronization, it tests each received W-bit sequence in the received bitstream to see if it matches the corresponding sync word. As a general case, a received W-bit sequence can overlap two words in the frame, thereby consisting of the last W - r bits of word k(l) - 1 followed by the first r bits of word k(1). In Fig. 2 for example, the W-bit sequence includes bits 4292 to 4299, spanning sync words 1 and 2, and r = 1 - (k(l) - 1)W = 3 bits. Next, note that for each sync word, the W-bit sequences which the receiver can test in the frame include W - 1 words with mixed (data + marker) bits at the beginning of the frame, followed by L - (S + l)W + 1 words with data bits only, then W - 1 words with mixed bits, and finally (S - 1) W + 1 words with I=4299 A -_+.----I: i-l
, ,/ , I
i
y
, 1
I
B
i+1
k(1) = 538 S(l) = 2 Fig. 2. A snapshot W = 8.
of a transmitted
bit-stream
with K = 270, J = 2 and
H.N. KundaeWCompufer Communications 21 (1998) 530-537
532
marker bits only. The positions of the Al sync words 1= L - SW + jW and the positions of the A2 sync words 1 = L - JW + jW where j = 1. _..I. The probability that received W-bit sequence under test resembles sync word given by P(b) where:
are are the s is
P(l,s) = Pr(received W-bit sequence resembles sync word s]; = Pr( last W - r bits of word k(l) - 1 resemble first W - r bits of sync word s].Pr{first r bits of word k(l) resemble last r bits of sync word s). If we denote the Hamming distance between the first y bits of sync word u and the last y bits of sync word v by H(u,v,y), represent s by 1 when sync word = Al and by 2 when sync word = A2, and use r = 1 - (k(1) - l)W, we obtain P(l, 1) as f
(1 -p,>
W-r-H(l,2,W~r)pH(l,2,W-T).~ e
and l=L-(S-j)W
P(L l),
(4)
P21(0=
otherwise
0,
l=L-(S-j)W
0, P, I(4 =
I P(l, 1)
otherwise I=L-(S-j)W
P(L 2)7 Pz(l)
(5)
=
otherwise
0,
P3(1) =
0,
l=L-(S-j)W
P(l,2)
otherwise
l
2” -1 2W’
W515L-SW L-SW
P(1, 1) = 1 W-r-Hil,l,W-r)p~~l.I.w-r~
x
(1 -PC?)
(1 _pe)'~H~I,I,r)P~~l,I,r),
l=L-(S-j)W
(1 - PtJW> (1 -P,) \ (1 -P,)
L-(S-l)W
W-r-H(l,l,W-r)~(l,l,W-~)
x (1 _p,)‘-H(2,I,r)p~(2,I,r),
W-r-H(1,2.W-r)~(1,2,W-r)
x
(1
_pe)‘-H(2,1,r)p~(2,1,r),
L_JWL_(J_
1)W
l)W (1)
and P(1,2) as
I
(1 -P,)
W-r-H(2,2,W-r)PH(2,2,W-r).l e
l
2” 1 2%
&.(I
WalsL-SW
- pe)‘-H(l,2,r)~(l.2’I),
L-SW
P(1,2) = 1
I
W-r-H(2,I,W-r)~(I.I,W-r)
x cl _p,)~-H(l,2,r)~(l,2,r),
L_ts_
l)w
W~r~Hi2.1,W-r)p~(2,1.W-r)
x t1 _pe)r-H(2,2,r)~(2,2,r),
L_Jw
< /
W-r-H(‘,‘,W-r)p~(‘,‘,W-r)
x (1 _ pe)‘-H(‘,‘,r)~(‘,‘,r),
1 ~ L_ (J _ l)~
(1 - pe.1 (1
- p,>
(1 -P,)
(2) L_Jw l)w 1 + L _ (J _j)w
l=L-(J-j)W
(1 - PdW,
Using this result, and assuming that the receiver encounters the first Al word at position 1, the dynamic transition probabilities in the recovery path are obtained as
Q, (1)= 1 - P(l, 11, P(L l), Pl2(1)
l=L-(S-j)W PI,(l)
= 0,
otherwise
=
0,
l=L-(S-j)W
P(1, 1)
otherwise
(3)
H.N. Kundaeli/Cotnputer
For the other transition p32
=
Communications
we have Pz2 = P2 and
probabilities,
Q,(l) =
p3.
For the sync phase (state 4 and 6 or 5 and 7) we assume that the receiver uses only I codewords of the marker as the framing pattern. We next assume that the receiver can synchronize on any of the A2 words (Bth A2 word) as the header codeword of the framing pattern, but require that the framing pattern must have at least one Al and one A2 words. We next note that if the receiver synchronizes on the Bth A2 word, then the length of the framing pattern can vary from one Al + BA2 to JAI + BA2 words. We now define a sequence as any sequence of IW bits in the frame, and let the sequence which the receiver is testing be positioned at the Zth bit position in the frame, and be represented by x(t). If we denote l* as the position of the framing pattern, then x(l*) represents the framing pattern. Next, assume that x(l) consists of r marker bits and IW - r data bits, where r varies from 0 to ZW. In Fig. 2 for example, if I = S, then r = 1 (L-SW) = 11 bits. Finally, let h(l,r) represent the Hamming distance between the r bits of the framing pattern and the r marker bits of x(l) in the absence of errors. The corresponding probability of no error for the sequence under test is given by T(I) where:
For the equalities
I
(1 -P,)
T(1) =
of P(l,s), T(1) is obtained
IW - i-h(L/W
- /)ph(l.IW e
- I)
(1 - Py,
1 # 1*
other hold
transition
=
following
-
p,,
(10) z~)X,,(z)
)
(6)
L-sw
where 1=1* X,,(z)
= 1 - Q,?
-
P,2Q21~‘+~
1 - P,,zP B-2
1=1*
(8)
=
otherwise
P12P2Q22z 2BfcY ;z
(P22zPY
-
and the dynamic transition therefore, given by 1 -(l
the
efficiency
hp2w&ZN+BP+ol
(7)
T(l)>
probabilities,
IWSlSL-SW
otherwise
0
(9)
l
IW- h(l,IW)ph(l,IW) e 9
= 0,
P32(0
1 - T(l),
(1
1
We therefore obtain P,,(l)
9,,(z)
y
~+sw~L-h(l,/+SW-Llph(l.I+SW-L) e
(1 -P,)
1=1*
as
(1 -pdW,
i
0,
In order to derive the synchronization efficiency, we first note that the parameters of the synchronization efficiency are computed using the dynamic transition probabilities given in Eqs. (3)-(5), (7)-(9). The derivation of transfer functions from transition diagrams can be carried out by inspection or by elementary graph reduction [5]. The dynamic recovery transfer function through the true path is therefore obtained from Fig. 1 as
1
1 2L-r-(.Y-/)W (1 -P,)
533
3. Analysis for synchronization
T(I) = Pr{x(l) resembles the framing pattern}; = Pr{ ZW - r data bits of x(l) resemble the IW - r bits of x(l*)}.Pr( r marker bits of x(l) resemble the remaining r bits of x(l*)}. Using the notations
21 (1998) 530-537
probabilities
1 - Pz,zP N-l
-PJW,
l=i”
0,
1 # 1*
Q4(0 =
in the loss path are,
P,2P2P;2-‘Q,g1+Ba+a 1 - P,,zP
1 (p& ‘I=o
(11)
H.N. KundaeWComputer
Communications
21 (1998) 530-537
from which we obtain the transition
time as
B-2
N-l
p12 Lls=a+
(1
-
P+W2
-P,,)
x j=O
pi,,+pzw
P
(
w+v-
1v+aY)+
(1 _p2,)
x pn23 n=O )
1
p,2p2&
x3
(1 -P21)
(17)
Fig. 3. Modified efficiency.
transition
diagram
for the analysis
of synchronization
Likewise, when the word encountered in state 1 is not the first Al codeword, the transfer function from state 1 to 8 is given by B-2
The recovery time is then given by
2P+ol 1
(12)
L,8cz) = Q,za + P;3G!;;rzia
+ “3’3”“_
(P,,zfif
p j;O 31z
which evaluates b,(z)
to
N-l P,,P3@2-‘Q33~‘+“~+~
= B-2 P+PP2
a(1 -P2l)+P,2
x
N-1 pi,,+P,P;2-’
n=O
J=o
[P,2P2Pf2-
x
‘P&l
I
+
1 (P33z)” n=O
1 - P3,zP
Pn23
P,,P&
‘f13Q5QY~M+N+
+
’ +BP+or M-l
[I - P3,zpl
1 - Pgz - QsP,z2
m,.
(13) The dynamic as
holding time through the true path is obtained M M+I Q4Qs z M-l
@4,(z) = 1 - P4z -
Q4P,z2 1
(14)
and the equivallent
transition
1(18)
(Q,z)”
time is given by
(Q6d”
m=O
from which we obtain the holding time as M-l 1 Q,Q: m=O
1+ L41 =
(15)
Q4Qf
For the other parameters, we first split state 1 into states 1 and 8 so that for the case when the receiver fails to reach the (true) sync phase from state 1, it ends up in state 8. We, therefore, obtain the modified transition diagram shown in Fig. 3. When the word encountered in state 1 is the first Al codeword, the transfer function from state 1 to 8 is given by L,R(~) = Q,za + p;2ffz;” B-2 P,2P2Q22~~‘+~ +
(P2279
1 - Pz,zP P,,P2Pf2-
+
jz
‘Q23z
I
+BP+a
1 - P2,zD
N-l
x (P23z)” n=O
(16)
Now, assuming that while in state 1 the receiver encounters a word resembling the Al codeword at bit 1 and fails to reach the sync phase, the next bit at which it starts the search phase is obtained as l,{ = ceil(L,s*L) where ceil(x) = smallest integer greater than or equal to X. When the receiver first suffers a synchronization failure, it starts the search phase at bit 1 = L - (J - B)W + 1. It then goes through a number of false synchronizations with each new I, until it encounters bit I, = L - (S - l)W, which is the position of the first Al codeword. Let T, denote the time needed to process bit 1 = L - (J - B)W + 1 until the first encounter of bit I, = L - (S - l)W. Also, let T, denote the time needed to process bit 1, = L - (S - l)W until the next encounter of bit 1, = L - (S - 1)W if the receiver fails to reach the sync phase. Using the notations of the previous reports [6], T, (mean time spent
535
H.N. KundaeWComputer Communications 21 (1998) 530-537 Table I True (L14) and false (L,,) recovery, and true (L,,) and false (LsI) holding times at J = 48, I = 3, I2 = 2, N = 3, M = 1 and E = 3.
p,
L l4 (frames)
L 15(years)
&I
LSI (frames)
10-j IO-6
2.03 2.00
I .28E + 7 I .25E + 22
6.7342 min 1.195E + 7 years
5.00 5.00
0996
processing lost frames), T,, (mean time out of synchronization) and T, (mean time in synchronization) are obtained as r,,, = T, + ]L,b - (CY+ P(J + B -
l))l.T,,
I
(20)
Ts = 1 + L4, and the synchronization
efficiency
as
(21)
’ = T, + T, 4. Results and discussion
The results are given in Table 1 and Figs. 4-9. The parameters used are J as the number of the multiplexed basic frames, I as the length of the framing pattern in words, I, as the number of Al codewords in the framing pattern, J2( = B) as the number of A2 codewords in the framing pattern, N as the recovery verify number, M as the loss verify number and E as the error rate (P, = 1OPE). In Table 1, values of the recovery and holding times based on the mean transition probabilities [6] are given. These were computed separately and are presented here for comparison only. The parameters used for their computation were the same as those used in Ref. [3] (frames withJ=48,1=3,1,=2,N= l,M=3,E=3orE=6and frame duration = 125 ps). The results given here for the true recovery and holding times were found to agree well with
16
24
Number of frames Fig. 4. Efficiency M=landE=3.
32
40
48
J
versus number of frames J with I = 6, I? = 1. N = 1,
5
12
Iz with J = 48, I = 6,
those given in Ref. [3]. In these results, however, there is the added advantage of seeing that the receiver will take more than 10’ years to lock on non-framing bit patterns as compared with 250 ps needed to lock on the framing pattern. Moreover, at the error rate of 1O-6 for example, the receiver will take only 625 ps to lose synchronization if it locks on the wrong bit pattern, whereas it will take more than 10’ years to lose synchronization if it locks on the framing patterns. The system is therefore quite robust. For the computation of the synchronization efficiency, it was found that framing pattern lengths exceeding six codewords resulted in overflow computational errors. The errors arose because the recovery and holding times tended to increase very rapidly at low error rates. A maximum value of six codewords for the framing pattern was therefore adopted for the computations in this analysis. Moreover, except where stated, the frame length has been fixed at J=48 multiplexed frames. To start with, Fig. 4 shows how the efficiency varies with the frame length J while the other parameters have been fixed at I = 6, I2 = 1, N = A4 = 1, and E = 3. It is seen that the efficiency decreases with the number of frames. This is expected because the receiver has a greater probability of locking on data words when there are more data words. Fig. 5
0995 8
4
3
Fig. 5. Efficiency versus number of A2 codewords 1 andE=3. N=I.M=
T,,=T,+(a+i3(J+B-1))
0
2
Number of A2 codewords
-i
1
I
I
2
3
Number of A I codewords Fig. 6. Efficiency versus number of Al codewords N=l,M= 1 andE=3.
4
I1
I, with J = 48, Iz = 2,
536
H.N. KundaeWComputer Communications 21 (1998) 530-537
\
0.96
0
12
3
4
5
6
7
Recovery verify number
8
9
10
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
Error rate (-E)
N
verify number N with J = 48, I = 3,
Fig. 9. Efficiency versus error rate P, = lo-” with J = 48, I = 3, I2 = 2. N=l andM= I.
shows how the efficiency varies with the number of A2 codewords (Z2) while the other parameters have remained unchanged. It is seen that the system has a critical value of two A2 codewords below which the efficiency drops drastically, and above which it remains almost constant. Variations in the other parameters also produced the same result. In Fig. 6, it is seen that the efficiency decreases with I,, the number of Al codewords while the other parameters have been fixed at I2 = 2, N = 1, M = 1 and E = 3. This decrease of efficiency with I, was found to be consistent even when the other parameters were varied. The explanation is that the important detection during synchronization recovery is the change from the Al sequence of codewords to the A2 sequence. Increasing the number of Al codewords in the framing pattern, therefore, only lowers the performance because the Al to A2 detection is more decisive than the mere detection of Al codewords. Also, only a few cheeks on the A2 codewords are necessary to verify correct synchronization after the Al to A2 change. Therefore, increasing the number of A2 codewords beyond a certain value does not improve the performance very much. Following the results of Figs. 5 and 6, I has been fixed at three codewords with I2 = 2 codewords in the next results. In Fig. 7, it is seen that the efficiency decreases
with the recovery verify number, whereas in Fig. 8 it increases with the loss verify number. The other parameters have been fixed at E = 3 and either N = 1 or M = 1. Finally, Fig. 9 shows the characteristic decrease of the efficiency to very low values at very high error rates, and the increase towards unity at low error rates. The results of Figs. 4, 7-9 show the same trend as those obtained for systems employing single-code markers [6,4]. Also, the synchronization efficiency which was used to obtain the optimum codewords for single-code systems in Ref. [4], has been decisive in establishing the minimum codeword combination of one Al and two A2 codewords in this analysis. In the analysis of Luan et al. [7], the minimum framing pattern of six codewords with I2 = 3 has been proposed. On the other hand, in a prototype receiver developed for SONETsystems, the framing pattern of three codewords with f2 = 2 was found satisfactory [3].
Fig. 7. Efficiency versus recovery I>=Z,M= 1 andE=3.
1.00
I
1
5. Conclusion The analysis of a communication system employing frames based on the SONET scheme has been performed. It has been found that the performance of the receiver decreases with the number of frames employed, although not too rapidly. It has also been found that the performance of the system decreases with the recovery verify number but increases with the loss verify number. The important result, however, was the possibility to determine a minimum length for the framing pattern and its composition of codewords. This was found to be the combination of one Al codeword and two A2 codewords.
Acknowledgements 0
12
3
4
5
6
Loss verify number Fig. 8. Efficiency N=I andE=3.
7
8
9
IO
M
versus loss verify number M with J = 48, I = 3, 12 = 2,
The author thanks the International Centre for Theoretical Physics, Italy, where this work was carried out, and the Swedish Research Agency, SAREC for providing the financial support.
H.N. KundaeWComputer Communications 21 (1998) 530-537
References
[l] A. Nilsson, M. Perry, M. Sutton, Frame synchronization failure: detection and recovery, IEEE Trans. Commun. 39 (1991) 613-618. [2] R. Ballart, Y.-C. Ching, SONET: It’s now the standard optical network, IEEE Commun. Mag. (March 1989) 8- 15. [3] D.T. Kong, 2.488 Gb/s SONET multiplexer/demultiplexer with frame detection capability, IEEE JSAC 9 (1991) 726-731.
537
[4] H.N. Kundaeli, The effect of synchronization codes on system design parameters, Int. J. Electron. 80 (1996) 693-701. [5] R.A. Howard, Dynamic Probabilistic Systems, vol. I: Markov Models, Wiley, New York, 1971. [6] H.N. Kundaeli, Design parameters for a code-synchronized transmission system, Int. J. Electron. 78 (I 995) 37-53. [7] 2. Luan, J.F. Hayes, M.K. Mehmet Ali, Frame synchronization performance of SONET signals, Computer Networks ISDN Systems 25 (1992) 183-190.
ELSEVIER
Computer
Communications
The analysis of a transmission
21 (1998) 530-537
system employing SONET-like frames H.N. Kundaeli*
Department
ofPhysics, University of Dar-es-Salaam, P.O. Box 35063, Dar-es-Salaam, Tanzania Received 26 September
1997; accepted 9 January
1998
Abstract Frame synchronization systems that employ multiple-code markers have been shown to give better synchronization performance than systems that employ single-code markers. It has been shown, however, that only a few of the codewords in the marker need to be used in order to attain optimum synchronization performance. Among the multiple-code marker systems which have gained wide usage recently is the SONET scheme for information transmissions at high data rates. In this report, a model for the analysis of the synchronization performance of systems which employ SONET-like frames is developed. The model is then applied to the SONET scheme to study how its performance varies with the choice of the parameters in its frame structure. 0 1998 Elsevier Science B.V. Keywords:
SONET; Frame sychronization;
Multiple code markers; Synchronization
1. Introduction Digital information is often transmitted serially as a stream of blocks of information bits separated by synchronization bits called markers. In order to retrieve the information, the receiver first locates the markers and takes the bits between them as the information. The bits forming an information block and marker are referred to as a frame, and the scheme of synchronizing the receiver to the frames using the markers is referred to as frame synchronization. In systems employing frame synchronization, the markers may consist of single bits or multiple bits. The North American DS 1 basic frame scheme of 193 bits is a typical 1 bit marker scheme [ 11. Networks employing this scheme, however, do multiplex several basic frames into extended frames, and several extended frames into superframes. The markers for the extended frames then come from the markers of the basic frames, and are therefore composed of multiple bits. In some cases, only some of the marker bits of the extended frames are used for frame synchronization, a technique known as windowing [l]. On the other hand, the synchronous optical network (SONET) scheme uses basic frames consisting of 810 words, two of which are for the marker [2]. This scheme is destined for the transmission of data at high speeds in optical fibres, and is supposed to replace conventional systems at those speeds. For example, a DS3 PCM superframe consisting of 12 DS 1
0140-3664/98/$19.00
0 1998 Elsevier
basic frames can be accommodated in a single STS-1 SONET basic frame. When higher transmission capacities are desired, several SONET basic frames are multiplexed by word interleaving. The resulting frame then has twice as many codewords as the number of multiplexed basic frames. If the multiplexed frames are long, and all the codewords of the markers are used for synchronization, then the receivers turn out to be quite complex, and the performance of the systems tends to degrade. In order to avoid this, ‘windowing’ is applied so that only a few of the words in the marker are used for synchronization [3]. In this report, the work reported by the author on systems employing single-code markers is extended to the case of systems employing SONET-like multiple-code markers. The model is then used to study how the performance of the system varies with the choice of its parameters-namely frame length, number of codewords used for synchronization, position of the synchronization codewords in the marker, recovery verify number and loss verify number. The synchronization efficiency, which has been used in the determination of design parameters for systems employing single-code markers [4], is used as the measure of performance.
2. Model of the receiver The receiver analyzed in this report is destined for frames consisting of J multiplexed basic frames, and its transition
* E-mail: [email protected] PII SO140-3664(98)00125-X
efficiency
Science
B.V. All rights reserved
H.N. KundueliLomputer
Communications
531
21 (199X) 530-537
false path is exactly the same as that in the true path. Note, however, that any data which the receiver collects while in the false path is invalid. 2.1. Derivation
Fig.
I A receiver for frames having J SONET-like basic frames.
diagram is given in Fig. 1. The markers of the multiplexed frames are assumed to consist of only two types of specially chosen codewords, designated as Al and A2. In the SONET scheme, the bit sequences of the codewords are 11110110 for Al and 00101000 for A2. The markers, therefore, consist of J codewords of type Al and J codewords of type A2. It is, however, assumed that the receiver uses only I codewords (I = 2...25) for synchronization, B of which (B = I...J) are A2 codewords. The operation of the receiver is therefore as follows. When it has lost synchronization, or is brought up, the receiver starts to search for the first Al codeword on a bit-by-bit basis in state 1. If it locates the A 1 codeword, and finds it error free, it moves to state 2 1 and stays in this state as long as the received words are error-free Al codewords. If it receives a word matching the error-free A2 codeword, the receiver moves to state 22, otherwise it returns to state 1 if the received word matches neither the error-free Al codeword nor the errorfree A2 codeword. In state 22, the receiver checks the next B - 1 words if they match the error free A2 codewords before going to state 23. If it finds any of these B - 1 words in mismatch however, the receiver returns to state 1 and starts another search. On reaching state 23, the receiver performs checks on framing patterns consisting of the I codewords instead of single codewords. The receiver must receive N - 1 consecutive error-free framing patterns before going to state 4, otherwise it returns to state 1 in case of a single failure. While in state 4, the receiver collects data from each received frame while monitoring the framing pattern. If it receives an erroneous framing pattern, it moves to state 6 and subsequently to state 1 after M - 1 consecutive errors. If it receives a single error-free framing pattern while in state 6, however, the receiver returns to state 4. Note that while in state 6, the receiver continues to regard all received data as valid data. The path through states 2,4 and 6 is called the true path because it is the path which the receiver takes when it encounters an error free Al codeword while in state 1. On the other hand, if the receiver encounters any other bit sequence resembling an error free Al codeword while in state 1, it moves to state 31 and takes the false path through states 5 and 7. Owing to the symmetry of the transition diagram, the operation of the receiver in the
of transition probabilities
To define the data characteristics of the frame, we assume that the frame consists only of data and synchronization words. Any other words in the frame apart from data and marker words are also assumed to be data words in order to simplify the mathematical derivations. The data characteristics of the frame can, therefore, be defined as follows: J is the number of multiplexed basic frames and K is the length of the resulting multiplexed frame, K is equal to (K-2J) data + JAl + JA2 marker (sync) words; S is the marker length which is equal to JAI + JA2 words; W is the word length; P, = Pr (received bit is in error}; (Yis (duration of one bit)/ (duration of a frame); and, fi is (duration of one word)/ (duration of a frame). From the data characteristics, and noting that L is the frame length in bits (KW bits), the following relationships can be defined: 1 is the bit position in the frame (I = 1.. .L); k is the word position in the frame (k = 1. . .K); s is the sync word position in the marker (s = 1.. .S); k(1) is the word in which the tth bit occurs; and s(l) is the sync word in which the Ith bit occurs if 1 is a marker bit. Fig. 2 shows a snapshot of a received bit stream stretching from A to B and consisting of a portion of frame i - 1, the whole of frame i and a portion of frame i + 1. The indicated frames are composed of two basic frames, four marker words and 4320 bits. The reference point is in frame i at bit 1 = 4299, which gives k(1) = 538 and s(l) = 2. In order to derive the transition probabilities, first note that when the receiver is attempting to gain or regain synchronization, it tests each received W-bit sequence in the received bitstream to see if it matches the corresponding sync word. As a general case, a received W-bit sequence can overlap two words in the frame, thereby consisting of the last W - r bits of word k(l) - 1 followed by the first r bits of word k(1). In Fig. 2 for example, the W-bit sequence includes bits 4292 to 4299, spanning sync words 1 and 2, and r = 1 - (k(l) - 1)W = 3 bits. Next, note that for each sync word, the W-bit sequences which the receiver can test in the frame include W - 1 words with mixed (data + marker) bits at the beginning of the frame, followed by L - (S + l)W + 1 words with data bits only, then W - 1 words with mixed bits, and finally (S - 1) W + 1 words with I=4299 A -_+.----I: i-l
, ,/ , I
i
y
, 1
I
B
i+1
k(1) = 538 S(l) = 2 Fig. 2. A snapshot W = 8.
of a transmitted
bit-stream
with K = 270, J = 2 and
H.N. KundaeWCompufer Communications 21 (1998) 530-537
532
marker bits only. The positions of the Al sync words 1= L - SW + jW and the positions of the A2 sync words 1 = L - JW + jW where j = 1. _..I. The probability that received W-bit sequence under test resembles sync word given by P(b) where:
are are the s is
P(l,s) = Pr(received W-bit sequence resembles sync word s]; = Pr( last W - r bits of word k(l) - 1 resemble first W - r bits of sync word s].Pr{first r bits of word k(l) resemble last r bits of sync word s). If we denote the Hamming distance between the first y bits of sync word u and the last y bits of sync word v by H(u,v,y), represent s by 1 when sync word = Al and by 2 when sync word = A2, and use r = 1 - (k(1) - l)W, we obtain P(l, 1) as f
(1 -p,>
W-r-H(l,2,W~r)pH(l,2,W-T).~ e
and l=L-(S-j)W
P(L l),
(4)
P21(0=
otherwise
0,
l=L-(S-j)W
0, P, I(4 =
I P(l, 1)
otherwise I=L-(S-j)W
P(L 2)7 Pz(l)
(5)
=
otherwise
0,
P3(1) =
0,
l=L-(S-j)W
P(l,2)
otherwise
l
2” -1 2W’
W515L-SW L-SW
P(1, 1) = 1 W-r-Hil,l,W-r)p~~l.I.w-r~
x
(1 -PC?)
(1 _pe)'~H~I,I,r)P~~l,I,r),
l=L-(S-j)W
(1 - PtJW> (1 -P,) \ (1 -P,)
L-(S-l)W
W-r-H(l,l,W-r)~(l,l,W-~)
x (1 _p,)‘-H(2,I,r)p~(2,I,r),
W-r-H(1,2.W-r)~(1,2,W-r)
x
(1
_pe)‘-H(2,1,r)p~(2,1,r),
L_JW
1)W
l)W (1)
and P(1,2) as
I
(1 -P,)
W-r-H(2,2,W-r)PH(2,2,W-r).l e
l
2” 1 2%
&.(I
WalsL-SW
- pe)‘-H(l,2,r)~(l.2’I),
L-SW
P(1,2) = 1
I
W-r-H(2,I,W-r)~(I.I,W-r)
x cl _p,)~-H(l,2,r)~(l,2,r),
L_ts_
l)w
W~r~Hi2.1,W-r)p~(2,1.W-r)
x t1 _pe)r-H(2,2,r)~(2,2,r),
L_Jw
< /
W-r-H(‘,‘,W-r)p~(‘,‘,W-r)
x (1 _ pe)‘-H(‘,‘,r)~(‘,‘,r),
1 ~ L_ (J _ l)~
(1 - pe.1 (1
- p,>
(1 -P,)
(2) L_Jw l)w 1 + L _ (J _j)w
l=L-(J-j)W
(1 - PdW,
Using this result, and assuming that the receiver encounters the first Al word at position 1, the dynamic transition probabilities in the recovery path are obtained as
Q, (1)= 1 - P(l, 11, P(L l), Pl2(1)
l=L-(S-j)W PI,(l)
= 0,
otherwise
=
0,
l=L-(S-j)W
P(1, 1)
otherwise
(3)
H.N. Kundaeli/Cotnputer
For the other transition p32
=
Communications
we have Pz2 = P2 and
probabilities,
Q,(l) =
p3.
For the sync phase (state 4 and 6 or 5 and 7) we assume that the receiver uses only I codewords of the marker as the framing pattern. We next assume that the receiver can synchronize on any of the A2 words (Bth A2 word) as the header codeword of the framing pattern, but require that the framing pattern must have at least one Al and one A2 words. We next note that if the receiver synchronizes on the Bth A2 word, then the length of the framing pattern can vary from one Al + BA2 to JAI + BA2 words. We now define a sequence as any sequence of IW bits in the frame, and let the sequence which the receiver is testing be positioned at the Zth bit position in the frame, and be represented by x(t). If we denote l* as the position of the framing pattern, then x(l*) represents the framing pattern. Next, assume that x(l) consists of r marker bits and IW - r data bits, where r varies from 0 to ZW. In Fig. 2 for example, if I = S, then r = 1 (L-SW) = 11 bits. Finally, let h(l,r) represent the Hamming distance between the r bits of the framing pattern and the r marker bits of x(l) in the absence of errors. The corresponding probability of no error for the sequence under test is given by T(I) where:
For the equalities
I
(1 -P,)
T(1) =
of P(l,s), T(1) is obtained
IW - i-h(L/W
- /)ph(l.IW e
- I)
(1 - Py,
1 # 1*
other hold
transition
=
following
-
p,,
(10) z~)X,,(z)
)
(6)
L-sw
where 1=1* X,,(z)
= 1 - Q,?
-
P,2Q21~‘+~
1 - P,,zP B-2
1=1*
(8)
=
otherwise
P12P2Q22z 2BfcY ;z
(P22zPY
-
and the dynamic transition therefore, given by 1 -(l
the
efficiency
hp2w&ZN+BP+ol
(7)
T(l)>
probabilities,
IWSlSL-SW
otherwise
0
(9)
l
IW- h(l,IW)ph(l,IW) e 9
= 0,
P32(0
1 - T(l),
(1
1
We therefore obtain P,,(l)
9,,(z)
y
~+sw~L-h(l,/+SW-Llph(l.I+SW-L) e
(1 -P,)
1=1*
as
(1 -pdW,
i
0,
In order to derive the synchronization efficiency, we first note that the parameters of the synchronization efficiency are computed using the dynamic transition probabilities given in Eqs. (3)-(5), (7)-(9). The derivation of transfer functions from transition diagrams can be carried out by inspection or by elementary graph reduction [5]. The dynamic recovery transfer function through the true path is therefore obtained from Fig. 1 as
1
1 2L-r-(.Y-/)W (1 -P,)
533
3. Analysis for synchronization
T(I) = Pr{x(l) resembles the framing pattern}; = Pr{ ZW - r data bits of x(l) resemble the IW - r bits of x(l*)}.Pr( r marker bits of x(l) resemble the remaining r bits of x(l*)}. Using the notations
21 (1998) 530-537
probabilities
1 - Pz,zP N-l
-PJW,
l=i”
0,
1 # 1*
Q4(0 =
in the loss path are,
P,2P2P;2-‘Q,g1+Ba+a 1 - P,,zP
1 (p& ‘I=o
(11)
H.N. KundaeWComputer
Communications
21 (1998) 530-537
from which we obtain the transition
time as
B-2
N-l
p12 Lls=a+
(1
-
P+W2
-P,,)
x j=O
pi,,+pzw
P
(
w+v-
1v+aY)+
(1 _p2,)
x pn23 n=O )
1
p,2p2&
x3
(1 -P21)
(17)
Fig. 3. Modified efficiency.
transition
diagram
for the analysis
of synchronization
Likewise, when the word encountered in state 1 is not the first Al codeword, the transfer function from state 1 to 8 is given by B-2
The recovery time is then given by
2P+ol 1
(12)
L,8cz) = Q,za + P;3G!;;rzia
+ “3’3”“_
(P,,zfif
p j;O 31z
which evaluates b,(z)
to
N-l P,,P3@2-‘Q33~‘+“~+~
= B-2 P+PP2
a(1 -P2l)+P,2
x
N-1 pi,,+P,P;2-’
n=O
J=o
[P,2P2Pf2-
x
‘P&l
I
+
1 (P33z)” n=O
1 - P3,zP
Pn23
P,,P&
‘f13Q5QY~M+N+
+
’ +BP+or M-l
[I - P3,zpl
1 - Pgz - QsP,z2
m,.
(13) The dynamic as
holding time through the true path is obtained M M+I Q4Qs z M-l
@4,(z) = 1 - P4z -
Q4P,z2 1
(14)
and the equivallent
transition
1(18)
(Q,z)”
time is given by
(Q6d”
m=O
from which we obtain the holding time as M-l 1 Q,Q: m=O
1+ L41 =
(15)
Q4Qf
For the other parameters, we first split state 1 into states 1 and 8 so that for the case when the receiver fails to reach the (true) sync phase from state 1, it ends up in state 8. We, therefore, obtain the modified transition diagram shown in Fig. 3. When the word encountered in state 1 is the first Al codeword, the transfer function from state 1 to 8 is given by L,R(~) = Q,za + p;2ffz;” B-2 P,2P2Q22~~‘+~ +
(P2279
1 - Pz,zP P,,P2Pf2-
+
jz
‘Q23z
I
+BP+a
1 - P2,zD
N-l
x (P23z)” n=O
(16)
Now, assuming that while in state 1 the receiver encounters a word resembling the Al codeword at bit 1 and fails to reach the sync phase, the next bit at which it starts the search phase is obtained as l,{ = ceil(L,s*L) where ceil(x) = smallest integer greater than or equal to X. When the receiver first suffers a synchronization failure, it starts the search phase at bit 1 = L - (J - B)W + 1. It then goes through a number of false synchronizations with each new I, until it encounters bit I, = L - (S - l)W, which is the position of the first Al codeword. Let T, denote the time needed to process bit 1 = L - (J - B)W + 1 until the first encounter of bit I, = L - (S - l)W. Also, let T, denote the time needed to process bit 1, = L - (S - l)W until the next encounter of bit 1, = L - (S - 1)W if the receiver fails to reach the sync phase. Using the notations of the previous reports [6], T, (mean time spent
535
H.N. KundaeWComputer Communications 21 (1998) 530-537 Table I True (L14) and false (L,,) recovery, and true (L,,) and false (LsI) holding times at J = 48, I = 3, I2 = 2, N = 3, M = 1 and E = 3.
p,
L l4 (frames)
L 15(years)
&I
LSI (frames)
10-j IO-6
2.03 2.00
I .28E + 7 I .25E + 22
6.7342 min 1.195E + 7 years
5.00 5.00
0996
processing lost frames), T,, (mean time out of synchronization) and T, (mean time in synchronization) are obtained as r,,, = T, + ]L,b - (CY+ P(J + B -
l))l.T,,
I
(20)
Ts = 1 + L4, and the synchronization
efficiency
as
(21)
’ = T, + T, 4. Results and discussion
The results are given in Table 1 and Figs. 4-9. The parameters used are J as the number of the multiplexed basic frames, I as the length of the framing pattern in words, I, as the number of Al codewords in the framing pattern, J2( = B) as the number of A2 codewords in the framing pattern, N as the recovery verify number, M as the loss verify number and E as the error rate (P, = 1OPE). In Table 1, values of the recovery and holding times based on the mean transition probabilities [6] are given. These were computed separately and are presented here for comparison only. The parameters used for their computation were the same as those used in Ref. [3] (frames withJ=48,1=3,1,=2,N= l,M=3,E=3orE=6and frame duration = 125 ps). The results given here for the true recovery and holding times were found to agree well with
16
24
Number of frames Fig. 4. Efficiency M=landE=3.
32
40
48
J
versus number of frames J with I = 6, I? = 1. N = 1,
5
12
Iz with J = 48, I = 6,
those given in Ref. [3]. In these results, however, there is the added advantage of seeing that the receiver will take more than 10’ years to lock on non-framing bit patterns as compared with 250 ps needed to lock on the framing pattern. Moreover, at the error rate of 1O-6 for example, the receiver will take only 625 ps to lose synchronization if it locks on the wrong bit pattern, whereas it will take more than 10’ years to lose synchronization if it locks on the framing patterns. The system is therefore quite robust. For the computation of the synchronization efficiency, it was found that framing pattern lengths exceeding six codewords resulted in overflow computational errors. The errors arose because the recovery and holding times tended to increase very rapidly at low error rates. A maximum value of six codewords for the framing pattern was therefore adopted for the computations in this analysis. Moreover, except where stated, the frame length has been fixed at J=48 multiplexed frames. To start with, Fig. 4 shows how the efficiency varies with the frame length J while the other parameters have been fixed at I = 6, I2 = 1, N = A4 = 1, and E = 3. It is seen that the efficiency decreases with the number of frames. This is expected because the receiver has a greater probability of locking on data words when there are more data words. Fig. 5
0995 8
4
3
Fig. 5. Efficiency versus number of A2 codewords 1 andE=3. N=I.M=
T,,=T,+(a+i3(J+B-1))
0
2
Number of A2 codewords
-i
1
I
I
2
3
Number of A I codewords Fig. 6. Efficiency versus number of Al codewords N=l,M= 1 andE=3.
4
I1
I, with J = 48, Iz = 2,
536
H.N. KundaeWComputer Communications 21 (1998) 530-537
\
0.96
0
12
3
4
5
6
7
Recovery verify number
8
9
10
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
Error rate (-E)
N
verify number N with J = 48, I = 3,
Fig. 9. Efficiency versus error rate P, = lo-” with J = 48, I = 3, I2 = 2. N=l andM= I.
shows how the efficiency varies with the number of A2 codewords (Z2) while the other parameters have remained unchanged. It is seen that the system has a critical value of two A2 codewords below which the efficiency drops drastically, and above which it remains almost constant. Variations in the other parameters also produced the same result. In Fig. 6, it is seen that the efficiency decreases with I,, the number of Al codewords while the other parameters have been fixed at I2 = 2, N = 1, M = 1 and E = 3. This decrease of efficiency with I, was found to be consistent even when the other parameters were varied. The explanation is that the important detection during synchronization recovery is the change from the Al sequence of codewords to the A2 sequence. Increasing the number of Al codewords in the framing pattern, therefore, only lowers the performance because the Al to A2 detection is more decisive than the mere detection of Al codewords. Also, only a few cheeks on the A2 codewords are necessary to verify correct synchronization after the Al to A2 change. Therefore, increasing the number of A2 codewords beyond a certain value does not improve the performance very much. Following the results of Figs. 5 and 6, I has been fixed at three codewords with I2 = 2 codewords in the next results. In Fig. 7, it is seen that the efficiency decreases
with the recovery verify number, whereas in Fig. 8 it increases with the loss verify number. The other parameters have been fixed at E = 3 and either N = 1 or M = 1. Finally, Fig. 9 shows the characteristic decrease of the efficiency to very low values at very high error rates, and the increase towards unity at low error rates. The results of Figs. 4, 7-9 show the same trend as those obtained for systems employing single-code markers [6,4]. Also, the synchronization efficiency which was used to obtain the optimum codewords for single-code systems in Ref. [4], has been decisive in establishing the minimum codeword combination of one Al and two A2 codewords in this analysis. In the analysis of Luan et al. [7], the minimum framing pattern of six codewords with I2 = 3 has been proposed. On the other hand, in a prototype receiver developed for SONETsystems, the framing pattern of three codewords with f2 = 2 was found satisfactory [3].
Fig. 7. Efficiency versus recovery I>=Z,M= 1 andE=3.
1.00
I
1
5. Conclusion The analysis of a communication system employing frames based on the SONET scheme has been performed. It has been found that the performance of the receiver decreases with the number of frames employed, although not too rapidly. It has also been found that the performance of the system decreases with the recovery verify number but increases with the loss verify number. The important result, however, was the possibility to determine a minimum length for the framing pattern and its composition of codewords. This was found to be the combination of one Al codeword and two A2 codewords.
Acknowledgements 0
12
3
4
5
6
Loss verify number Fig. 8. Efficiency N=I andE=3.
7
8
9
IO
M
versus loss verify number M with J = 48, I = 3, 12 = 2,
The author thanks the International Centre for Theoretical Physics, Italy, where this work was carried out, and the Swedish Research Agency, SAREC for providing the financial support.
H.N. KundaeWComputer Communications 21 (1998) 530-537
References
[l] A. Nilsson, M. Perry, M. Sutton, Frame synchronization failure: detection and recovery, IEEE Trans. Commun. 39 (1991) 613-618. [2] R. Ballart, Y.-C. Ching, SONET: It’s now the standard optical network, IEEE Commun. Mag. (March 1989) 8- 15. [3] D.T. Kong, 2.488 Gb/s SONET multiplexer/demultiplexer with frame detection capability, IEEE JSAC 9 (1991) 726-731.
537
[4] H.N. Kundaeli, The effect of synchronization codes on system design parameters, Int. J. Electron. 80 (1996) 693-701. [5] R.A. Howard, Dynamic Probabilistic Systems, vol. I: Markov Models, Wiley, New York, 1971. [6] H.N. Kundaeli, Design parameters for a code-synchronized transmission system, Int. J. Electron. 78 (I 995) 37-53. [7] 2. Luan, J.F. Hayes, M.K. Mehmet Ali, Frame synchronization performance of SONET signals, Computer Networks ISDN Systems 25 (1992) 183-190.