The Galileo spacecraft employs concatenated coding schemes with Reed-Solomon interleaving depth 2. The bit error rate (BER) performance of Galileo's concatenated codes, assuming different interleaving depths (including infinite interleaving depth) are compared. It is observed that Galileo's depth 2 interleaving, when used with the experimental (15, 1/4) code, requires about 0.4 to 0.5 dB additional signal-to-noise ratio to achieve the same BER performance as the concatenated code with ideal interleaving. When used with the standard (7, 1/2) code, depth 2 interleaving requires about 0.2 dB more signal-to-noise ratio than ideal interleaving

Cheung, K.-M.

Dolinar, S. J., Jr.

English

NASA Technical Reports Server

TDA Progress Report 42-95 July-September 1988 
Performance of Galileo’s Concatenated Codes With 
Nonideal Interleaving 
1. Background 
K.-M. Cheung and S.  J. Dolinar, Jr. 
Communications Systems Research Section 
The Galileo spacecraft employs concatenated coding schemes with Reed-Solomon 
interleaving depth 2. This article compares the bit eiror rate (BER) perfomance of 
Galileo’s concatenated codes, assuming different interleaving depths (including infinite 
interleaving depth). I t  is observed that Galileo 3 depth 2 interleaving, when used with the 
experimental (15, 1/41 code, requires about 0.4 dB to 0.5 dB additional signal-to-noise 
ratio to achieve the same BER perfomance as the concatenated code with ideal inter- 
leaving. When used with the standard (7, 1/21 code, depth 2 interleaving requires about 
0.2 dB more signal-to-noise ratio than ideal interleaving. 
The Galileo spacecraft employs a communication system 
which uses either a (7 ,  1/2) convolutional code or a (15, 1/4) 
convolutional code as the inner code, and a (255 ,  223) Reed- 
Solomon code as the outer code. By using soft, maximum- 
likelihood decoding on the received symbols, the convolu- 
tional codes perform well at low signal-to-noise ratios. How- 
ever, maximum-likelihood decoding of convolutional codes 
creates bursty errors. An interleaver is placed between the 
convolutional code and the Reed-Solomon code to  randomize 
the bursty errors before they are fed to  the Reed-Solomon 
decoder. 
Concerns were initially expressed last summer1t2 and re- 
cently repeated3 [ 11 about the adequacy of Galileo’s interleav- 
ing depth for the constraint length 15 code, even when it was 
~ 
‘S. D o h a ,  “Alternative Code Considerations for Galileo,” JPL Inter- 
office Memorandum 33 1-87.2-308 (Appendix), (internal document), 
Jet Propulsion Laboratory, Pasadena, California, July 1, 1987. 
S. Lushbough, “Galileo Convolutional Coder Meeting with Project, 
3 1 July 1987,” JPL Interoffice Memorandum 3 13/5-181-SL:jlc, 
(internal document), Jet Propulsion Laboratory, Pasadena, California, 
August 3, 1987. 
L. Swanson, “Interleaving Depths for Reed-Solomon Decoders,” 
JPL Interoffice Memorandum 33 1-88.2-042, (internal document), 
Jet Propulsion Laboratory, Pasadena, California, July 13, 1988. 
https://ntrs.nasa.gov/search.jsp?R=19890010972 2020-03-20T03:52:37+00:00Z
first mistakenly assumed that Galileo’s interleaver was the same 
as Voyager’s. Depth 4 interleaving was selected for Voyager to  
sufficiently randomize the error bursts created by the (7, 1/2) 
convolutional decoder. Performance degradation for the (7,1/2) 
code with depth4 interleaving is insignificant (less thanO.l dB) 
relative to ideal interleaving at bit error rates between and 
However, the error bursts from the (15, 1/4) decoder are 
about twice as long (on the average) as the bursts from the 
(7, 1/2) decoder, and thus the longer-constraint-length code 
would seem to require about double the interleaving depth. In- 
stead, Galileo’s actual interleaving depth is only half of Voya- 
ger’s, and this can potentially cause significant concatenated 
system performance degradation for both of Galileo’s codes 
relative to theoretical predictions based on ideal interleaving. 
Previous studies of the effects of interleaving depth on con- 
catenated system performance included some test data for the 
(7, 1/2) code4 but no in-depth analyses that would allow 
extrapolation to the case of the (15, 1/4) code. Direct simula- 
tion tests of concatenated system performance using the 
(15, 1/4) code were completely unfeasible because of the huge 
amount of data that would have to  be collected to  verify bit 
error rates (BERs) in the to  range, and because of 
the slowness of the software Viterbi decoder simulation (about 
30 hours of CPU time on a Sun-3/260 computer per 100,000 
decoded bits for the (15, 1/4) code). 
Recently the completion of C. R. Lahmeyer’s 1-kbit/sec 
(currently constrained to  run at about 0.1 kbit/sec-still a 
hundredfold increase in speed relative to last summer’s soft- 
ware simulation) hardware Viterbi decoder5 has allowed us to 
do some long decoding runs not previously feasible for the 
(15, 1/4) code. With the advent of this hardware decoder, the 
following research tools have been developed: 
(1) Long decoding runs (several megabits) for the (15, 1/4) 
convolutional code were performed on the hardware 
decoder, and the error bursts are stored in data files 
conforming to the data compression format described 
in recent These data files can be used to  
~ ~ 
4F. H. J. Taylor, “Project Galileo OrbiterlDeep Space Network Com- 
munications Design Document,” Project Document 625-257, Jet Pro- 
pulsion Laboratory, Pasadena, California, pp. 5-12, January 15, 1981. 
’C. Lahmeyer, “1 Kilobit per Second Viterbi Decoder,” JPL Inter- 
office Memorandum 33 1-88.3-042, (internal document), Jet Propul- 
sion Laboratory, Pasadena, California, August 19, 1988. 
6C. Lahmeyer and K. Cheung, “Long Decoding Runs for the (15, 1/4) 
Convolutional Code,” JPL Interoffice Memorandum 33 1-88.3-040, 
(internal document), Jet Propulsion Laboratory, Pasadena, California, 
July 27, 1988. 
7K. Cheung, “More Long Decoding Runs of the Convolutional Codes 
(including the (15, 1/4) convolutional code and the (7, 1/2) convolu- 
tional code),” JPL Interoffice Memorandum 331-88.2-048, (internal 
document), Jet Propulsion Laboratory, Pasadena, California, August 1, 
1988. 
replicate the long, time-consuming runs exactly and are 
useful to anyone who wants to analyze the burst 
statistics of the Viterbi decoder. 
Similar long decoding runs were performed for the 
(7, 1/2) convolutional code using the software simu- 
lation, and the error bursts from those runs are also 
saved in the compressed format. 
Simulation software was developed which reads the 
compressed burst data obtained from the long decod- 
ing runs and simulates the operation of the entire 
concatenated coding system with different interleav- 
ing depths. 
II. Performance Results 
Simulated BERs of concatenated coding systems with vari- 
ous interleaving depths are given in Figs. 1 and 2. Figure 1 
shows the performance of the concatenated system using the 
(15, 1/4) convolutional code as the inner code, and Fig. 2 
shows the performance of the concatenated system using the 
(7, 1/2) convolutional code. The BERs of the concatenated 
coding systems with finite interleaving depths are compared to 
the BERs with ideal interleaving (infinite interleaving depth). 
In both figures, the concatenated code BER is shown as a 
function of two bit-energy-to-spectral-noise-density ratios, 
(1) for the convolutional code alone and 
(2) for the overall concatenated system. 
The difference between the two Eb/No scales is the overhead 
of 0.58 dB accounting for the redundancy of the outer (255, 
223) Reed-Solomon code. 
The data points plotted in Figs. 1 and 2 were obtained from 
a series of long decoding runs varying in length from 5 million 
to 40 million decoded bits. Smooth curves were fitted through 
the data points corresponding to BERs greater than 
The l o  statistical uncertainty in the data points at BERs lower 
than is more than 100% for the cases of finite interleav- 
ing depth. The corresponding uncertainty in the data points 
at BERs higher than ranges from about 5% to about 
100% of the simulated BER. The la uncertainty in the data 
points for ideal interleaving is between 10% and 30% of the 
BER. 
An example illustrates the difficulty of obtaining accurate 
simulated concatenated system performance at low BERs. 
The rightmost data point in Fig. 1 (interleaving depth 2, con- 
volutional code Eb/NO = 0.7 dB) required about 120 hours 
(5 days) of running time on the hardware decoder to decode 
40 Mbits. The same decoding run would have consumed 
149 
1.4 years of CPU time on the software decoder. Furthermore, 
the entire 40-Mbit run produced only three observed code- 
word errors, and the l o  statistical uncertainty in the simulated 
BER is around 100%. 
111. Performance Comparison 
Tables 1 and 2 show the minimum convolutional code 
Eb/No to achieve Galileo’s concatenated and unconcatenated 
system performance requirements assuming ideal (infinite- 
depth) interleaving and depth 2 interleaving, respectively. 
Table 3 shows the performance degradation caused by depth 2 
interleaving relative to ideal interleaving for Galileo’s two 
alternative convolutional codes. Galileo’s experimental (1 5 ,  
1/4) code requires about 0.4 dB to 0.5 dB additional signal-to- 
noise ratio to overcome the insufficiencies of depth 2 inter- 
leaving and achieve concatenated code BERs between and 
Galileo’s standard (7, 1/2) code is hurt less by depth 2 
interleaving, but still suffers about 0.2 dB degradation. The 
relative performance advantage of the (15, 1/4) code over the 
(7, 1/2) code is reduced by about 0.2 dB to 0.3 dB from the 
amount predicted in earlier studies (e.g., [ 11 ) based on ideal 
interleaving. With depth 2 interleaving, concatenated system 
performance will be improved by only about 1.2 dB when the 
(15, 1/4) code is substituted for the (7, 1/2) code. The corre- 
sponding improvement for an unconcatenated system or for a 
concatenated system with ideal interleaving is between 1.4 dB 
and 1.5 dB. 
i 
, 
the constraint length of the inner convolutional code. For 
example, interleaving depth 8 appears sufficient to keep the 
degradation under 0.1 dB for the (15, 1/4) code, as does inter- 
leaving depth 4 for the (7, 1/2) code. 
As alternatives to simply increasing the interleaving depth 
of conventional block interleaving schemes, new techniques to  
combat bursty errors, such as convolutional interleaving, heli- 
cal interleaving, and burst forecasting, should also be investi- 
gated. These techniques are superior to conventional block 
interleaving schemes. Also, a new Reed-Solomon decoder 
which can correct both errors and erasures is being developed 
in the Communications Systems Research Section at JPL. It is 
expected that the performance of concatenated systems will 
be substantially improved by the use of error-forecasting 
techniques together with erasure-correcting Reed-Solomon 
decoders. We propose to investigate the possibility of develop- 
ing better interleaving schemes for future deep space missions 
and to analyze the performance of concatenated coding 
systems using these new interleaving schemes. 
Even though the hardware Viterbi decoder has allowed us 
to  simulate many million decoded bits at a time, the data is 
still insufficient to accurately simulate concatenated code 
performance at BERs less than about lom5.  The amount of 
data required for an accurate estimate increases in proportion 
to the interleaving depth, and so it is even more difficult to 
simulate directly the performance of deeply interleaved 
schemes than it was for Galileo’s interleaving depth 2. Hence, 
notwithstanding the recent advance in decoding speed, it is 
still important to develop theoretical models for the decoded 
output of the Viterbi decoder, from which concatenated code 
performance can be accurately estimated without directly 
simulating the entire concatenated system. The geometric 
burst model of [2] should be reexamined for applicability tc 
long-constraint-length codes, and new models need to be 
developed for estimating Reed-Solomon code performance 
based on the theoretical model for the Viterbi decoder output. 
IV. Conclusions and Recommendations 
Galileo is unfortunately stuck with depth 2 interleaving, 
and so the immediate consequence of our simulations is simply 
to quantify the amount of expected degradation for Galileo’s 
concatenated codes. However, future missions should select 
interleaving schemes that produce minimal degradation. The 
required interleaving depth increases roughly in proportion to  
I 
References 
[ l ]  S .  Dolinar, “A New Code for Galileo,” TDA ProgressReport 42-93, January-March 
1988, Jet Propulsion Laboratory, Pasadena, California, pp. 83-96, May 15, 1988. 
[2] R. L. Miller, L. J. Deutsch, and S. A. Butman, On the Error Statistics of Viterbi 
Decoding and the Performance of  Concatenated Codes, JPL Publication 8 1-9, Jet 
Propulsion Laboratory, Pasadena, California, September 1, 198 1. 
150 
Table 1. Minimum Eb/N, to achieve concatenated and uncon- 
catenated system performance requirements under ideal inter- 
leaving assumption 
~ 
(7 ,  1/2) code (15 ,1 /4 )  code Difference 
Unconcatenated 
BER = 5 x 10-3 2.02 dB 0.52 dB 1.50 dB 
Concatenated 
BER = 1.79 dB 0.33 dB 1.46 dB 
Concatenated 
BER = 1 0 - ~  1.70 dB 0.27 dB 1.43 dB 
Table 2. Minimum Eb/N, to achieve concatenated and uncon- 
catenated system performance requirements for interleaving 
depth 2 
(7 ,  1/2) code (15,  1/4) code Difference 
Unconcatenated 
BER = 5 x 10-3 2.02 dB 0.52 dB 1 .SO dB 
Concatenated 
BER = low6 2.01 dB 0.83 dB 1.18 dB 
Concatenated 
BER = 1 0 - ~  1.90 dB 0.69 dB 1.21 dB 
Table 3. Concatenated system performance degradation for inter- 
leaving depth 2 versus ideal interleaving 
(7 ,1 /2 )  code (15,  1/4) code Difference 
Concatenated 
BER = 0.22 dB 0.50 dB 0.28 dB 
Concatenated 
BER = 1 0 - ~  0.20 dB 0.42 dB 0.22 dB 
151 
CONCATENATED CODE EdNo, d B  
INTERLEAVING DEPTH I 
-2-l=2 
-4- I = 4 
-6- I = 6 
-8--1=8 
& I = -  
CONVOLUTIONAL CODE Eb/N0. d8 
Fig. 1. Concatenated code performance for Galileo’s experimental (15, 1/4) inner code with 
nonideal interleaving. 
CONCATENATED CODE EdNo, dB 
2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 
INTERLEAVING DEPTH I 
-2-l=2 
-4- I = 4 
+I=- 
w 
I- 
1.4 1.5 1.6 1.7 1 .a 1.9 2.0 2.1 2.2 2.3 
CONVOLUTIONAL CODE Eb/No, dB 
Fig. 2. Concatenated code performance for Galileo’s standard (7, 1/2) inner code with 
nonideal interleaving. 
152 


Performance of Galileo's concatenated codes with nonideal interleaving

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