The performance (bit-error rate vs. signal-to-noise ratio) of two different interleaving systems, block interleaving and the newer helical interleaving are compared. Both systems are studied with and without error forecasting. Without error forecasting, the two systems have identical performance. When error forecasting is used with shallow interleaving, helical interleaving gains, but less than 0.05 dB, over block interleaving. For higher interleaving depth, the systems have almost indistinguishable performance

Cheung, K.-M.

Swanson, L.

English

NASA Technical Reports Server

TDA Progress Report 42-98
N9 0" 12 79 "
August 15, 1989
A Performance Comparison Between Block
Interleaved and Helically Interleaved
Concatenated Coding Systems
K.-M. Cheung and L. Swanson
Communications Systems Research Section
This article compares the performance (bit-error rate vs. signal-to-noise ratio)
of two different interleaving systems, block interleaving and the newer helical in-
terleaving. Both systems are studied with and without error forecasting. Without
error forecasting, the two systems have identical performance. When error fore-
casting is used with shallow interleaving, helical interleaving gains, but less than
0.05 dB, over block interleaving. For higher interleaving depth, the systems have
almost indistinguishable performance.
I. Introduction
As NASA wants to receive more data from plane-
tary missions, and as technologies like data compression
make projects tighten their error-rate requirements, many
missions are turning to concatenated coding schemes with
Reed-Solomon (RS) codes as "outer" codes concatenated
with "inner" convolutional codes, which have been used in
the Deep Space Network for years (Fig. 1). RS decoders
for use beginning in the mid-1990s will be installed in the
network by February 1, 1992.
When concatenated coding is used, symbols in RS
words are generally "interleaved," so that the error cor-
recting ability of one RS word is not used up entirely by
one or two long error bursts from the Viterbi (convolu-
tional) decoder. Some missions planning to use RS codes
in the Deep Space Network will have "block" interleaving
depth five. This means that consecutive symbols in a RS
word are separated on the channel by exactly four other
RS symbols, one from each of the other four words in the
block (Fig. 2).
Recently, other interleaving schemes have been sug-
gested, such as E. Berlekamp's "helical interleaving" [1,
2]. In this article, the initial analysis of this scheme is
presented. In helical interleaving, words go into the inter-
leaver/deinterleaver in the staggered way shown in Fig 3.
95
https://ntrs.nasa.gov/search.jsp?R=19900003476 2020-03-19T23:47:56+00:00Z
(Notice that if a block-interleaved code block were pasted
onto a cylinder, the set of symbols consisting of the jth
symbol in each word lies on a circle around the cylinder;
if the helically interleaved words were pasted onto a cylin-
der, the set consisting of the jth symbol in each word lies
on a helix.) Helical interleaving is defined precisely in Sec-
tion II.
It is worth noticing that for a given interleaving depth,
the two schemes yield statistically identical code perfor-
mance. As far as any one codeword is concerned, it con-
sists of certain RS symbols; its decoding does not depend
on which other codewords the intervening symbols belong
to. So why this apparently more complicated interleaving
scheme? There are two ways in which we might expect
helical interleaving to be different from block interleaving.
The first is synchronization: if a synchronization marker
is placed before or at the beginning of each codeword in a
block-interleaved block, then several markers appear con-
secutively at the beginning of each code block; with helical
interleaving, the markers would be uniformly scattered.
This should allow faster (though possibly more compli-
cated) synchronization, but we do not address this issue
here. The second way in which helical interleaving can be
expected to perform differently from block interleaving is
in error forecasting. For several years, the DSN's planning
for RS decoders has taken into account the slight perfor-
mance gain available from the fact that RS codes can use
quality information in decoding; a symbol that is believed
questionable and flagged as an "erasure" costs only half
an error [3].
The error-forecasting scheme used in this article is a
simple but effective one: that is to create erasures in those
words that have too many symbol errors [4] by assuming
that errors in adjacent words continue in the undecodable
word. Based on the above idea, there are two ways to flag
an erasure; that is, either flag a symbol of an undecod-
able word as an erasure if either one of the code symbols
next to it on the channel is detected as erroneous (single-
sided forecasting), or flag a symbol as an erasure only if
the code symbols on both sides of it on the channel are
detected as erroneous (double-sided forecasting). Simula-
tions show that in all cases the more aggressive single-sided
forecasting scheme performs better than the conservative
double-sided forecasting scheme. Thus we choose to use
the single-sided forecasting scheme in all our simulations
in this article. Since the Viterbi decoder, which decodes
the inner c0nvolutional code, creates errors in bursts, pre-
vious analysis [5] shows that in the case of block interleav-
ing, using this simple error-forecasting strategy allows a
gain of about 0.04 dB.
For block interleaved data, words go into the inter-
leaver as shown in Fig. 2. Each interleaving block is sep-
arated from others, and all possible error forecasting can
be done within the same block. Thus a finite buffer can
do all error forecasting. In the light of this finite buffer
feature, error forecasting with redecoding to each inter-
leaving block was selected, disregarding the order of de-
coding of codewords in each block. That is, when a code-
word in a block is decoded successfully, erasure informa-
tion is generated for both of its adjacent words in the block.
After the first round of decoding in a block, the previ-
ously undecoded codewords are then redecoded using the
additional erasure information. This process goes on until
no more undecodable words can be recovered. For heli-
cally interleaved data, words go into the interleaver in the
staggered way shown in Fig. 3. Unlike the block inter-
leaving scheme in which each codeword is "related" only
to words in the same block, each codeword in the helical
interleaving scheme is theoretically "related" to all words
before and after it, and this makes error forecasting with
redecoding impractical, if not impossible, since it would
require a buffer with infinite memory. Thus in helical
interleaving one does error forecasting without redecoding.
That is, when a codeword is successfully decoded, erasure
information is generated to those symbols (of other not yet
decoded codewords) that are adjacent to the previously er-
roneous symbols of the decoded codeword. Each codeword
is then decoded in order, first without using the erasure
information. If the codeword is decodable, the decoder
goes on to decode the next codeword. If not, the code-
word is decoded once again using the erasure information
generated from previously decoded words. Whether the
codeword is decodable or not (with erasure information),
the decoder goes on to decode the next codeword.
For deep interleaving, both block and helical inter-
leaving give nearly identical error forecasting capability
because both schemes are essentially "ideal interleaving"
schemes which allow nearly statistically independent RS
symbols. But for shallow helical interleaving (e.g., depth 2,
which is used by Galileo), a word with too many symbol
errors to decode is preceded by a part of one word and part
of another. On the one hand, this means that a Viterbi
decoder error burst is less likely to keep both different
codewords from decoding, and so the error forecasting ca-
pability should be enhanced. On the other hand, only
some of the symbols in a given codeword are preceded by
symbols in a codeword that is decoded before it. For the
exact schemes we compare in Section III, the trade-off be-
tween these effects depends on the interleaving depths and
the inner convolutional codes. For interleaving depth 2,
helical interleaving gives a slight performance gain over
block interleaving with both the (7,1/2) and (15,1/4) inner
96
convolutionalcodes.Forinterleavingdepth4or8,helical
interleavingisslightlybetterorslightlyworsethanblock
interleaving,dependingonwhetherthe (15,1/4)codeor
the(7,1/2)codeisusedastheinnercode.
II. Helical Interleaving
In this section, mathematical definitions of the inter-
leaving schemes are included for completeness, but the
reader is advised to consider Figs. 2 and 3 to illustrate
these definitions.
Definition 1. A block code with n code symbols
per codeword is block interleaved to depth d if the code
symbols sent on the channel are packed into codewords
in such a way that the code symbol that follows (on the
channel) the jth symbol of the ruth codeword, m < d, is
the jth code symbol of the (m + 1)th codeword, and the
code symbol that follows the jth code symbol of the dth
codeword, j < n, is the (j + 1)th code symbol of the first
codeword, and the code symbol following the nth code
symbol of the dth codeword is the first symbol of the first
codeword in the next block.
Definition 2. A block code with n code symbols
per codeword is helically interleaved to depth d, where
dr = n- 1 and r is any integer, if code symbols are packed
into codewords in such a way that the code symbol that
follows (on the channel) the jth code symbol of the mth
word, m < d, is the (j - r)th code symbol of the (m+ 1)th
codeword; if j - r < 0, then this is the (j - r + n)th code
symbol of the mth codeword of the "previous" group. The
jth code symbol of the dth codeword is followed by the
(j + (d - 1)r)th code symbol of the first codeword of this
group if j < r and by the (j - r)th code symbol of the first
codeword of the next group if j > r.
Helical interleaving can be defined for codeword
length n and interleaving depth d as long as n and d are
relatively prime. While this more complicated definition
for interleaving depths 4 and 8 was needed in the simula-
tions, technical definitions will not be written here. The
interested reader can find them in [1].
Helical interleaving was introduced in 1982 [1, 2] by
E. Berlekamp. It has since been successfully implemented
in several hardware decoders. Figure 3 depicts a helical
interleaver for a code of length 4 interleaved to depth 3. As
discussed in Section I, use of a helical interleaving scheme
in general does not enhance the gain performance of a
concatenated coding system.
For many applications, it is better than block inter-
leaving because of synchronization and interleaving delay.
Helical interleaving facilitates synchronization in the fol-
lowing way: in block interleaving, the jth symbols of two
adjacent codewords in a block are adjacent to each other
in the channel; in helical interleaving, the jth symbols of
two adjacent codewords are separated from each other in
the channel by n - 1 symbols in a block-interleaved sys-
tem. If a synchronization marker is placed before or at the
beginning of each codeword in a block-interleaved block,
several markers appear consecutively at the beginning of
each code block in the channel symbol streams, and the re-
ceiver can acquire only synchronization modulo nd. With
helical interleaving, the markers are uniformly scattered,
and this allows synchronization modulo n. The difference
in interleaving delay is that the end-to-end delay of a block
interleaver is 2nd, exclusive of the channel delay, whereas
the delay of a helical interleaver is only nd.
III. Simulation
The performance of the concatenated coding systems
described in this section were obtained by simulation. The
inner codes used were the standard (7,1/2) convolutional
code, which was simulated with a software Viterbi decoder,
and Galileo's experimental (15,1/4) code, which was de-
coded using the Communications Systems Research Sec-
tion's long constraint length "Little Viterbi Decoder" built
for Advanced Systems. The outer codes used were the
standard (255,223) RS code, which has been adopted for
use by a number of present and future missions, and the
proposed (1023,959) RS code for deep-space missions of
the future.
Figures 4 through 11 give the SNR performance com-
parisons of various concatenated schemes using block inter-
leaving and helical interleaving. Figures 4 and 5 show the
performance curves of the concatenated coding schemes
using the (7,1/2) convolutional code as inner code, and the
(255,223) RS code as outer code, without and with error
forecasting respectively. Similarly, Figs. 6 and 7 show the
performance curves of the concatenated coding schemes
with the (7,1/2) convolutional code and the (1023,959)
RS code. Figures 8 and 9 show the performance curves of
the concatenated schemes using the (15,1/4) convolutional
code and the (255,223) RS code. And finally, Figs. 10 and
11 show the performance curves of the concatenated cod-
ing schemes using the (15,1/4) convolutional code and the
(1023,959) RS code.
97
IV. Conclusion
It is observed from Figs. 4 through 11 that, as pointed
out in Section I, when no error-forecasting strategy is used,
both helical interleaving and block interleaving perform
identically for all interleaving depths. When error forecast-
ing is used, helical interleaving gains less than 0.05 dB at
shallow interleaving. For deeper interleaving (e.g., depth 4
or depth 8), helical interleaving is slightly worse or slightly
better than block interleaving, depending on whether the
(7,1/2) convolutional code or the (15,1/4) convolutional
code is used as inner code. The difference in performance
between helical and block interleaving for higher depths
might be attributed to the fact that the (15,1/4) code pro-
duces longer bursts than the (7,1/2) code.
On the basis of these results, it is recommended that
helical interleaving be considered for possible use only in
particular situations, and not for general use. As usual,
the real moral seems to be that one should allow greater
interleaving depth, whatever interleaving scheme is used.
References
[1] Cyclotomics Corporation, Interleaved Coding for Bursty Channels, Final Re-
port on Phase I, Small Business Innovation Research, NSF Grant No. ECS-
8260180, Berkeley, California, April 12, 1983.
[2] E. R. Berlekamp and P. Tong, Improved Interleavers for Digital Communica-
tions, United States Patent Number 4559625 (granted December 17, 1985).
[3] R. E. Blahut, Theory and Practice of Error Control Codes, Reading, Mas-
sachusetts: Addison Wesley, 1983.
[4] R. J. McEliece and L. Swanson, "On the Decoder Error Probability for Reed-
Solomon Codes," IEEE Trans. Inform. Theory, IT-32, pp. 701-703, September
1986.
[5] G. H. Pitt, III, and L. Swanson, "Erasure Information for a Reed-Solomon
Decoder," TDA Progress Report 42-83, Jet Propulsion Laboratory, Pasadena,
California, pp. 39-44, November 15, 1985.
98
I REED-SOLOMON IENCODER I
_ r CONVOLUTIONAL
- I ENCODER
REED-SOLOMON
DECODER
NOISY CHANNEL I
DV="J I
Fig. 1. Concatenated coding system.
A0
A1
A2
INTERLEAVING
---,lll_
DEPTH
B0 CO DO E0
B1 C1 D1 E1
B2 C2 D2 E2
: : : : :
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SYMBOLS IN BLOCK INTERLEAVER
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1
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Fig. 2. Example of block interleaving.
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_e_INTERLEAVING___
DEPTH
B'3 C'2
A1 B0 C'3
A2 B1 CO l
/
£3
A3 B2 C1 " -r
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A"I B"0 1
SYMBOLS IN INTERLEAVER
n=4
d=3
• • • A0 iB31c2A180c31A21_1coA3s21Cl
SYMBOLS IN CHANNEL
Fig. 3. Example of helical interleaving.
A'0 • • •
9g
10-3
10_4
n-
O
r_
LU
F--
10-5
10-6
] I I I I
=4
1=2
BLOCK INTERLEAVING
-- HELICAL INTERLEAVING
I I I I J
2.0 2.1 2.2 2.3 2.4 2.5
SIGNAL-TO-NOISE RATIO
Fig. 4. (7,1/2) convolutional code and (255,223)
Reed-Solomon code; no error forecasting.
2.6
10-4
10-5
uJ
n-
n-
O
r_
n-
LU
i--
10 -6
10-7
10-8
I I I I I
BLOCK INTERLEAVING
m HELICAL INTERLEAVING
I I f I I
2.0 2.1 2.2 2.3 2.4 2.5 2.6
SIGNAL-TO-NOISE RATIO
Fig. 5. (7,1/2) convolutional code and (255,223)
Reed-Solomon code with error forecasting.
lO0
10-2
10-3
,<
_r"
rr"
0
mm 10-4
UJ
I'--
10-5
I
\
BLOCK INTERLEAVING
_ HELICAL INTERLEAVING
10 -6 I I [ I
1.7 1,8 1.9 2.0 2.1 2.2
SIGNAL-TO-NOISE RATIO
Fig. 6. (7,1/2) convolutional code and (1023,959)
Reed-Solomon code; no error forecasting.
2.3
10-2
10"3
n"
n,"
0
m 10_4
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rn
10-5
I I I 1 I
1=4
'_ 1=2
,k
1=8
\ \ \
\ \
\ \
\
\ \
\ \
BLOCK INTERLEAVING
_ HELICAL INTERLEAVING
10-6 I I I I I
1.7 1.8 1.9 2.0 2.1 2.2
SIGNAL-TO-NOISE RATIO
2.3
Fig. 7. (7.1/2) convolutional code and (1023.959)
Reed-Solomon code with error forecasting.
101
10-2
10-3
n,-
0
_. 10-4
uJ
I.-
10 -5
BLOCK INTERLEAVING
m HELICAL INTERLEAVING
10-6 t I I I
0.4 0.6 0.8 1.0 1.2 1.4
SIGNAL-TO-NOISE RATIO
Fig. 8. (15,1/4) convolutional code and (255,223)
Reed-Solomon code; no error forecasting.
10-2
10-3
n.-
n-
O
_: 10-4
LU
I'-
10-5
I I
BLOCK INTERLEAVING
m HELICAL INTERLEAVING
I I
1=8
\
\
\
\
\
\
\
\
\
\
[
0.6
lO_ i \ i i
0.4 0.8 1.0 1.2
SIGNAL-TO-NOISE RATIO
Fig. 9. (15,1/4) convotutionat code (255,223)
Reed-Solomon code with error forecasting.
1.4
102
10-2
10-3
LU
Q_
O
CE
UJ
1--
10"4
10-5
I I I
BLOCK INTERLEAVING
m HELICAL INTERLEAVING
10.6 I I I
0.2 0.4 0.6 0.8
SIGNAL-TO-NOISE RATIO
\
\
Fig. 10. (15,1/4) convolutional code (1023,959)
Reed-Solomon code; no error forecasting.
1.0
I
10"2
10-3 I = E
10-5
O
¢z;
iJJ
10_4
10-6
0.2
I I
1=2
1=4 \
\
\
\
\
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BLOCK INTERLEAVING
D_ HELICAL INTERLEAVING
[ I l
0.4 0.6 0.8
SIGNAL-TO-NOISE RATIO
Fig. 11. (15,1/4) convolutional code (1023,959)
Reed-Solomon code with error forecasting.
1.0
103


A performance comparison between block interleaved and helically interleaved concatenated coding systems

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