A self-synchronizing coding scheme for NASA's TDRSS satellite system is a concatenation of a (2,1,7) inner convolutional code with a (255,223) Reed-Solomon outer code. Both symbol and word synchronization are achieved without requiring that any additional symbols be transmitted. An important parameter which determines the performance of the word sync procedure is the ratio of the decoding failure probability to the undetected error probability. Ideally, the former should be as small as possible compared to the latter when the error correcting capability of the code is exceeded. A computer simulation of a (255,223) Reed-Solomon code as carried out. Results for decoding failure probability and for undetected error probability are tabulated and compared

Costello, D. J., Jr.

Lin, S.

English

NASA Technical Reports Server

PERFORMANCE ANALYSIS OF THE WORD
SYNCHRONIZATION PROPERTIES OF THE
OUTER CODE IN A TDRSS DECODER
Technical Report
to
NASA
Goddard Space Flight Center
Greenbelt, Maryland
Grant Number 5-234
February, 1984
Principal Investigators
Daniel J. Costello, Jr.
Department of Electrical Engineering
University of Notre Dame*
Notre Dame, IN 46556
Shu Lin
Department of Electrical Engineering
University of Hawaii at Manoa
Honolulu, HI 96822
On leave from the Illinois Institute of Technology, .Chicago, IL 60616
https://ntrs.nasa.gov/search.jsp?R=19840016853 2020-03-20T22:22:47+00:00Z
PERFORMANCE ANALYSIS OF THE WORD SYNCHRONIZATION PROPERTIES
OF THE OUTER CODE IN A TDRSS DECODER
Daniel J, Costello, Jr.
Dept. of Elec. Eng.
University of Notre Dame1
Notre Dame, IN 46556
Huijie Deng
Dept. of Elec. Eng.
111. Inst. of Tech.
Chicago, IL 60616
Farid F. El-Wailly
TRW Defense & Space Systems2
One Space Park
Redondo Beach, CA 90278
In [1], a self-synchronizing coding system for NASA's TDRSS satellite
system was described. The coding system used is a concatenation of a (2,1,7)
inner convolutional code with a (255,223) Reed-Solomon outer code. The scheme
described in [1] achieves both symbol and word synchronization without
requiring that any additional symbols be transmitted. In this report we
discuss the performance of the word synchronization properties of this
scheme.
The outer code used is a (255,223) Reed-Solomon code over GF(q), where
q=2 . It has a minimum distance d=33, and therefore can be used to correct
t=16 or fewer errors. Suppose that the code word v is transmitted. At the
receiver, misframing is either due to a synchronization loss of X, symbols of
\r, as shown in Figure 1, or to a synchronization gain of X- symbols from the
preceding code vector, as shown in Figure 2, where _r is the received word.
255-1 a 255-1 a
Figure 1
I0n leave from the 111. Inst. of Tech.
2Formerly with the-Ill. Inst. of Tech.
a 255-2, a 255-H
Figure 2
It was shown in [1] that the RS outer code can be used for simultaneously
recovering from a sync loss (or gain) of £ U<16) symbols and any combination
of 16-& symbol errors due to noise.
At the receiver, three outcomes are possible: (1) decoding is correct
and word synchronization is acquired; (2) the decoder fails, indicating a lack
of synchronization; or (3) decoding is completed but is incorrect, resulting
in a false declaration of synchronization. This corresponds to an undetected
error. If outcome (2) occurs, the received word is shifted 32 positions and
decoded again. This procedure is repeated until correct word synchronization
is acquired.
Let PCU), PfU), and PUU) be the probabilities corresponding to the
three outcomes described above, respectively, where X., 0<&<255, denotes the
number of symbols gained (or lost). Obviously we have
PCU) + PfU) + PUU) = 1. (1)
First, for simplicity, assume that the channel error rate e=0. Then we
have
PfU) =0 for 0 < 4 < 16 (2.1)
PfU)
PCU)
PCU)
pf(256-Z)
1
0
for 17 <_ I <_ 128
for 129 £ a £ 255
for 0 <_ SL £ 16
for 17 < i < 128
(2.2)
(2.3)
(3.1)
(3.2)
PUU)
Pc(256-£)
0
1 - PfU)
Pu(256-*)
for 129 £ A £ 255
for 0 £ * _< 16
for 17 £ A £ 128
for 129 < * < 255
(3.3)
(4.1)
(4.2)
(4.3)
Pf(£) and Pu(£), based on (2) and (4), are given in table 1 and table 2,
respectively.
Now consider the effect of channel errors due to noise. Suppose that any
symbol which is transmitted has a probability (1-e) of being received
correctly and a probability e/(q-l) of being transformed into each of the q-1
other symbols. Furthermore, assume that successive symbols incur errors
independently. Thus, the probability that the received word differs from the
transmitted word in exactly i positions is given by the expression
(5)*} (q-l)i( — )i(l-e)n-i = (*\ ei(i-e)n-i
Then the probabilities Pf(&), PC(A), and Pu(&) are given by
Pf(0)
PfU)
16
255-X.
h £
i=17
255
m=0
2'55-i\
}
 ei(i-e)255-A-i
i '
for 1 < A < 16 (6.1)
(25
i=17 \ x
255\
ei(l-e)255-i
7
i=0
-17
m=0
for 4=0
-
q
(6.2)
9SS-2,\
)
 £i(l-e)255-i-i1 7 for 17 < A < 128 (6.3)
= Pf(256-*) for 129 <_ 4 £ 255 (6.4)
PCU) - 1-PfU) for 0 £ * _ < 1 6 (7.1)
PCU) = 0 for 17 £ A £ 128 (7.2)
PCU) = Pc(256-4) for 129 £ * £ 255 (7.3)
Pu(4) " 0 for 0 £ 4 <_ 16 (8.1)
Pu(4) - l-Pf(4) for 17 £ * £ 128 (8.2)
PUU) s Pu<256-4) for 129 £ A £ 255 (8.3)
If we let e=0, (6)-(8) become (2)-(4). Evaluation of Pf(£) based on
(6.1)-(6.3) is given in Table 3.
In the following we evaluate the average number of decoding trials, and
the probability of false declaration of sync. Let P(m) be the probability of
performing exactly m decoding trials before word sync is acquired. Then the
average number of decoding trials, E(m), is given by
00
E(m) = I mp(m) (9)
m=l
where we can show (see Appendix A) that
1 16
P(l) = . - (pc(0) + 2 I PCU)}, (10.1)
256 £=1
! 32 16
P(m) - {I PfU)Pc(32-*) + I PC(JO + tl-Pc(16)]Pc(16)l,
256 4-17 4-1
for m = 2,3,4,5,6, (10.2)
P(7) - P(2) + [1-PC(16)]PC(16), (10.3)
256
P(8) » P(2) + {[1-PC(16)]2 PC(16)-PC(16)}. (10.4)
256
1 32
P(m) » {I [PfU)]K+1[l-Pc(32-i)]K Pc(32-4)
256 1=17
16
+ I [1-PC(4)JK Pc(4) + [1-PC(16)]K+1 Pc(16)},
*=1
for 8K+1 <_ m £ 8(K+1), K- 1,2,3, (10.5)
Evaluation of E(m) based on (6) , (7 ) , (9) , and (10) is given in table 4 and
shown in Figure 3.
Let Pu(m) be the probability of a false declaration of synchronization on
the mtn decoding trial, and let R denote the total probability of false
declaration of sync. Then we have
oo
R- I Pu(m), (11)
m=l
where we can show (see Appendix B) that
Pu(l) « 0, (12.1)
1 20
pu(m) m { I pu(4)}, for m - 2,3,...,8, (12.2)
256 4-17
1 20
pu(m) « { I .[Pf(4)]K[l-Pc(32-4)]K Pu(4)},
256 i=17
for 8K + 1 £ m £ 8(K+1), where K = 1,2 (12.3)
Evaluation of R based on (6), (8), (11), and (12) is given in table 5 and
shown in Figure 4.
From Figure 3 we see that E(m) increases very quickly when e >10~3. When
the channel becomes too noisy (e > 5 x 10~3), it is possible that word sync
may never be acquired. In this case the best strategy to achieve word sync
would be to reduce the number of positions shifted between decoding trials to
16 from 32.
From the discussion above we see that an important parameter which
determines the performance of the word sync procedure is the ratio of the
decoding failure probability Pf(&) to the undetected error probability Pu(&).
Ideally, Pu(£) should be as small as possible compared to Pf(&) when the
error—correcting-capability of the code is exceeded (& > 16). A computer
simulation of a (255,223) Reed-Solomon code was carried out, and results for
Pf(i) and Pu(&) are given in tables 6 and 7, respectively. Comparing them
with the formula results in tables 1-3, we see that they are very close.
Reference
[1] S. Lin, D.J. Costello, W. Miller, and J. Morakis, "Self-Synchronizing
Outer Codes for the TDRSS Decoder," Proceedings IEEE Global
Telecommunications Conference, pp. 30.7.1-30.7.4, San Diego, CA, November
1983.
APPENDIX A
From (7.1) -(7.3) we see that correct decoding and word sync acquisition
is possible only when £,. the amount of sync loss (or gain) satisfies the
following inequalities:
0 £ £ £ 16 (Al.l)
or
240 £ £ £ 255. (A1.2)
Note that it is possible that the decoder cannot achieve correct decoding due
to noise even though £ satisfies (Al.l) or (A1.2), and that decoding is
certainly incorrect if £ is not within the region specified by (Al.l) -(A1.2).
In either case the received word must be shifted and decoded a number of times
in order to acquire word sync, with each shift equal to 32 symbol positions.
Let P(m) denote the probability of exactly m decoding trials. Then we have
I 16 255
P(l) = • { I Pc(£) + I Pc(256-£)}
256 £=0 £=240
1 16
= (Pc(0) + 2 I Pc(£)}. (A2)
256 £=1
1 32 48
P(2) = ' { I Pf(£)Pc(32-£) + I Pf(£)Pc(£-32)
256 £=17 £=33
+ [1-PC(16)]PC(256-240)}. (A3)
1 6,4 80
P(3) - { I Pf(£)Pf(£-32)Pc(64-£) + I Pf(£)Pf(£-32)Pc(£-64)
256 £=49 £=65
+ Pf(48)[l-Pc(48-32)]Pc(256-240)}. (A4)
From Tables 1 and 3 we see that Pf(£) is very close to 1 for £ ^  20. So (3)
and (4) can be reduced to
! 32 16
P(2)»P(3)» { I Pf(£)Pc(32-£) + I Pc(£) + [1-PC(16)]PC(16)}. (A5)
256 £=17 £=1
10 the Same w
*y ve can obtain
P(m) •» p(2)
 f
^
}> for m « 4,5,6,
P(7) . p(2) + 1 , (A6)
^ ~fl-Pca6)]Pc( l6).
(A7)
P(8) .
256 -^cU6;j^ Pc(l6)-Pc(I6)h
+
 1 < » < 8 ( K + 1 ) , w h e r e K a s
»*».*,. . . ,
 we have
1 32
16
* I [1-PCU)]Kc
'
 J]
ti-Pc(l6)]K+l U ^ * '-'X J .
(A9)
. APPENDIX B
Let Pu(m) denote the probability of a false declaration of word sync on
the mth decoding trial. From (8.l)-(8.3) and tables 1-3 we see that Pu(&) a 0
for 0 £ i < 16, 240 < 4 £ 255, and 20 £ 1 <_ 236. Using this fact we have
I 16 255
Pu(i) = —- { I PU(£) + I pu(ft)> - 0 CB1)
256 4=0 4=240
i 32 48 i 20
Pu(2) * { I ?u(ft) + I Pu(ft)} - (I Pu(ft)}. (B2)
256 A-17 ^33 256 &=17
I 64 80
256 i=49 ft-65
, 20
» { I pu(ft)>. (B3)
256 4=17
Similarly, we can obtain
1 20
Pu(m) • { £ -PuCft)}, for m - 4,5,6,7,8 (B4)
256 ft-17
! 20
256 4=17
for 8K. + 1 < m < 8(K+1) , where K « 1,2,3, ...
10
Table 1. Probability of decoding failure Pf(i) with e =* 0.
I
<
—
16
17
18
19
20
21
PfU)
0
.93562896
.99776057
.99994449
.99999893
>. 99999900
Table 2. Probability of undetected error Pu(£) with e « 0.
A
1 16
17
18
19
20
2, 21
Pu(^ )
0
.643710 x
.223943 x
.551120 x
.107302 x
£.175672 x
10-1
10-2
lo-^
10-5
10-7
11
Table 3 (1) Probability of Decoding Failure Pf(£)
\^^  e
*^ v^.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
IS
19
20
I21
10-7
.882116 x ID'87
.553494 x 10-81
.326877 x 10-75
.180889 x 10~69
.933214 x 10~64
.446193 x 10~58
.196344 x 10~52
.788615 x ID'47
.286227 x 10~41
.927185 x 10-36
.263875 x ID'30
.646339 x 10~25
.132472 x 10~19
.218107 x 10~14
.270443 x 10-9
.22449 x 10-4
.93563045
.99776062
.99994489
.99999893
X99999990
10~6
.881927 x 10-71
.553383 x 10-66
.326811 x 10~61
.180853 x 10-56
.93303 x 10-52
.446106 x 10~47
.196305 x 10~42
.788464 x 10-38
.286173 x 10-33
.927012 x 10-29
.263827 x 10~24
.646225 x 10-20
.13245 x 10~15
.218073 x 1Q-11
.270405 x 10-7
.224467 x 10-3
.93564381
.99776109
.99994490
.99999893
X99999990
10-5
.88004 x 10-55
.552273 x 10-51
.326159 x 10~47
.180494 x 10~43
.93119 x 10~40
.445233 x lO'36
.195925 x 10-32
.786951 x 10-29
.285631 x 10-25
.925286 x 10-22
.263347 x 10-18
.645084 x 10-15
.13226 x 10-11
.217729 x 10~8
.270031 x 10~5
.224242 x 10-2
.93577725
.99776576
.99994502
.99999893
X99999990
12
Table 3(2) Probability of Decoding Failure Pf(£)
10-3 5 x 10~3 10-2
1 .861389 x 10-39
2 .541298 x 10~36
3 .319709 x 10-33
4 .176944 x 10-30
5 .912989 x 10~28
6 .436594 x 10~25
7 .192157 x 10~22
8 .771928 x 10~20
9 .280268 x IQ"17
10 .908203 x 10-15
11 .258591 x 10-12
12 .63378 x 10-1°
13 .130009 x 10~7
14 .214324 x ID'5
15 .26632 x 10-3
16 .22201 x 10-1
17 .93709658
18 .99781195
19' .99994615
20 .99999895
>21 >.99999990
.695220 x 10~23
.442769 x 10~21
.261821 x 10-19
.145067 x 10-17
.74947 x 10-16
.358935 x 10-14
.158256 x 10-12
.637136 x 10-U
.231915 x 10-9
.753963 x 10-8
.215575 x 10~6
.531314 x ID"5
.109847 x 10-3
.183229 x 10~2
.23218 x 10-1
.20120591
.94889303
.99822460
.99995632
.99999915
>.99999990
.408272 x 10~12
.550947 x 10-U
.657607 x 10-1°
.732838 x 10-9
.761881 x 10~8
.734996 x 10~7
.653655 x ID"6
.531743 x 10~5
.392017 x 10~4
.2589691 x 10-3
.15116574 x 10~2
.7659994 x 10~2
.32929854 x lO'1
.116471360
.325018357
.675814520
.979742897
.999300290
.999982800
.999999665
>.999999995
.806700 x 10~8
.582823 x 10~7
.354656 x 10~6
.199566 x 10-5
.104689 x 10~4
.510228 x 10~4
.229693 x 10-3
.948368 x 10-3
.356140 x 10~2
.120448 x 10-1
.362575 x 10-1
.957902 x 10-1
.21845707
.42203251
.67780659
.89572830
.99366426
.99978266
.99999466
.99999990
>.99999999
13
Table 4. Average Number of Decoding Trials E(m)
e
0
10-7
10~6
10-5
io-4
10-3
5xlO-3
10-2
E(m)
4.4635
4.4635
4.4637
4.4648
4.4722
4.6804
5.9659
9.2061
Table 5. Probability of False Declaration of Synchronization R
e
0
10-7
io-6
10-5
10-*
10-3
5xlO"3
10-2
R
1.8229 x
1.8228 x
1.8224 x
1.8186 x
1.7813 x
1.4517 x
6.9107 x
3.9156 x
10-3
10-3
10-3
10-3
10-3
10-3
io-*
lo-^
14
Table 6. Probability of Decoding Failure
(computer simulation results)
S\^ e
J^ v.
15
16
17
18
19
>20
0
0
0
.93360
.99822
.99991
1
5 x 10~3
.29056
.66047
.98024
.99941
1
1
10-2
.64711
.88419
.99423
.99949
1
1
Table 7. Probability of Undetected Error Pu(&)
(computer simulation results)
\v e
*^ v
17
18
19
>20
0
.664 x 10'1
.178 x ID'2
.900 x 10~4
0
5 x 10-3
.1976 x 10-1
.5900 x 10-3
0
0
10-2
.5770 x ID"2
.5100 x 10-3
0
0
E(m)
10
5
15
ff'8 • 7io~8 io"
i . J . l , I . I . _ . L
io"6 io~5 10 4 io~3 io~2
Figure 3. The Average Number of Decoding Trials
R10-21
10-3
10.-4
I \ I I
~
10 10 ' 10 10 " 10~3 10
Figure 4. The Probability of False Declaration of Synchronization


Performance analysis of the word synchronization properties of the outer code in a TDRSS decoder

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