When Can Interleaving Be Avoided for Reed-Solomon Coded Binary Markov Channels?

Abstract

The performance of Reed-Solomon codes over the binary additive Markov noise channel (BAMNC) is analyzed. A recursive expression for the probability of m error symbols in a block of n symbols is derived using the generating series approach, thus facilitating the exact calculation of the probability of codeword error under bounded distance decoding. An approximation to this probability is obtained, and it is shown to be tight when the noise correlation is not very large. In this case, interleaving the channel at the symbol level can be avoided. Furthermore, a wide range of channel conditions, under which channel interleaving at the bit level can be avoided, is identified.

Full text

Page 1

When Can Interleaving Be Avoided for
Reed-Solomon Coded Binary Markov Channels ?
Haider Al-Lawati, Fady Alajaji and Cecilio Pimentel
Department of Mathematics and Statistics
Queen’s University, Kingston, ON K7L 3N6, Canada
Email:
?haider,fady,pimentel?@mast.queensu.ca
Abstract—The performance of Reed-Solomon codes over the
binary additive Markov noise channel (BAMNC) is analyzed.
A recursive expression for the probability of
in a block of
approach, thus facilitating the exact calculation of the proba-
bility of codeword error under bounded distance decoding. An
approximation to this probability is obtained, and it is shown
to be tight when the noise correlation is not very large. In this
case, interleaving the channel at the symbol level can be avoided.
Furthermore, a wide range of channel conditions, under which
channel interleaving at the bit level can be avoided, is identified.
? error symbols
? symbols is derived using the generating series
I. INTRODUCTION
Burst-error correcting codes are of prime theoretical and
practical interest due to the bursty nature of real-world wireless
digital communication channels. An important class of non-
binary burst-error correcting codes used widely in data trans-
mission and storage systems is the family of Reed-Solomon
(RS) codes (e.g., [1], [2]). Conventional communication sys-
tems employing these codes are designed for memoryless
channels, which is not an accurate model for wireless fading
channels. As a consequence, interleaving is used to render
the channel memoryless; this introduces additional delay and
complexity to the system. Furthermore, such interleaved sys-
tem fails to exploit the benefits of the statistical memory of
the channel noise. When non-binary codes are sent over a
stationary binary additive noise channel with memory, two
interleaving strategies are worth considering: interleaving the
code (or channel) bits which reduces the channel to the
memoryless binary symmetric channel (BSC) (under perfect or
infinite interleaving depth) and interleaving the code symbols.
The performance of non-interleaved RS codes on corre-
lated fading channels is analyzed in [3]–[6] using a two
step procedure. First, a channel model is introduced for the
generation of the bit or symbol error process, and then a
formula for the probability of codeword error (PCE) under
bounded distance decoding is derived for the proposed model.
In [3], the channel is modeled via the Gilbert-Elliott channel
[7] whose parameters are calculated using a simple threshold
model. An approximation to the PCE is derived under the
assumption that the channel state does not change during
each symbol transmission. In [4], level crossing statistics are
This work was supported in part by NSERC of Canada and CNPq of Brazil.
Cecilio Pimentel is on leave from the Department of Electronics and Systems,
Federal University of Pernambuco, 50711-970, Recife, PE, Brazil (Email:
cecilio@ufpe.br).
applied to characterize the fading arrival process and the fading
durations, and the PCE is expressed in terms of the probability
distribution of the fading durations. In [5], the bit error
process resulting from the hard-decision demodulation of bi-
nary frequency-shift keying modulated signals over correlated
Rician fading channels is modeled via a Fritchman channel.
Furthermore, an analytical method based on the generating
series approach for calculating the PCE of RS codes over finite
state channels is presented. In [6] an
is proposed to characterize the correlation of symbol errors.
Imperfect (finite-length) symbol interleaving is also considered
in [4], [5], [8].
The objective of this paper is to identify the range of
channel parameters for which perfect interleaving of RS codes
(at either the bit or symbol level) does not yield improved
performance. To make the analytical derivation simple, we
consider the binary additive (first-order) Markov noise channel
(BAMNC), but this study can also be conducted for higher-
order Markov models which can accurately model correlated
fading channels [9]. Using the approach of [5], we first derive
a recursive expression for the probability of
in a block of
RS codes are sent over the BAMNC. Then we derive an
approximation to the PCE under the assumption that the noise
within a symbol is Markovian but is independent from symbol
to symbol (i.e., the PCE under perfect symbol interleaving).
We show that the PCE under perfect symbol interleaving is
superior to that under perfect bit interleaving. We compare
the approximated PCE with the exact PCE numerically for
four different RS codes to find channel conditions – described
in terms of the channel bit error rate (CBER) and noise
correlation – for which the approximation is accurate. For
such conditions, symbol interleaving can be discarded or
avoided. Finally, we compare the exact PCE for the BAMNC
with the exact PCE for the BSC (the BAMNC under perfect
bit interleaving) and determine a wider range of channel
parameters under which bit interleaving can be avoided.
?-state Markov chain
? error symbols
? symbols and determine the exact PCE when
II. SYSTEM DESCRIPTION
A. Channel Model
We consider a BAMNC whose output symbol
?
? at time
? is described by
?
?
??
?
??
?,
?????????? where
? denotes addition modulo-2,
symbol and
?
?
?????? is the
?th input
?
?
?????? is the
?th noise symbol. We assume

Page 2

that the input and noise processes are independent of each
other. Furthermore, we assume that the noise process
is a stationary (first-order) Markov with transition probability
matrix given by
??
?
?
?
?
??
?
???
??
??
?
?? ?????????? ??????
???????????? ??????
?
where
?
??
?
????
?
????
?
??
???,
?????????. Here
?? ????
?
? ?? is the CBER, and
?
?
?????
?
????
?
??
?
??
process. We assume that
??
?
?????????? is the correlation coefficient of the noise
??????? and that
????
?, ensuring that the noise process is irreducible (it is also
general in the sense of being equivalent to all stationary binary
Markov processes with nonnegative correlation coefficient).
When
identically distributed and the resulting channel reduces to the
memoryless BSC channel with crossover probability
state of the Markov channel at time
given by
whose
Hence,
???, the noise process becomes independent and
?. The
? is denoted by
?
? and
?
?
??
?. For
???????, let
???? be a
??? matrix
??th entry is given by
????
?
????
?
????
?
??
???.
?????
?
?????????????
???????????
?
?
?????
?
???????
??? ??????
?
?
Note that
binary noise sequence of length
?????
?????
?. Let
?
?
???
?
??
?
??????
?
? be a
?, then
?? ??
?
??
?
??
?
?
?
?
?
???
???
?
?
?
1
where the superscript
???
?indicates the transpose of a matrix,
?
column matrix with all entries being 1. This channel model is
a special case of the Gilbert-Elliott channel, realized by setting
the cross-over probabilities of the “good” and “bad” channel
components [7] equal to zero and one, respectively.
????????
?is the stationary distribution, and 1 is a
B. Reed-Solomon Codes
An
binary linear block code whose codewords are of length
????? RS code over the Galois field GF??
?
? is a non-
?
symbols, the rate of the code is
correct up to
decoding). Each symbol in GF??
to a binary
sent over the BAMNC by transmitting the equivalent binary
representation for each codeword. A transmitted symbol is
received correctly if the noise corrupting it is a sequence
of zeros of length
received incorrectly.
??
?
?? symbols. Each codeword contains
? information
????? and the code can
???
???
?
? symbols (under bounded distance
?
? can be mapped one-to-one
?-tuple. As a result, the non-binary codewords are
?. Otherwise, the transmitted symbol is
III. PERFORMANCE ANALYSIS
A. Exact Probability of Codeword Error
For a
ing, the probability of correct decoding
?-error correcting code using bounded distance decod-
?
?and the probability
of codeword error PCE are given by
?
?
?
?
?
???
??????
andPCE
????
?
?
respectively, where
errors occur in a block of
nates
?
?????? is the probability that
? symbol
? symbols. Given indetermi-
? and
?, define the formal power series
???????
?
?
???
?
???
???????
?
?
?. For a
?
?-ary code transmitted
over a binary finite-state channel,
?????? is given by [5]
??????
?
?
?
?I
???????
?
????
?
?
????
?
??
?
??1
(1)
where I is the
derived as the coefficient of
For the BAMNC, it can be shown by induction (see [10])
that for any integer
??? identity matrix. Thus
?????? can be
?
?
?
?in
?????? above [5].
?
?
?
?
?
?
?
??????????
?
??????
?
?
?????????
?
??
?
? ????
?
??
?
?
It can also be shown by induction that for any integer
?
????
?
?
?
??????????????
?
?
????????????? ???????????
???
?
?
?
Since
recursive expression for
the denominator polynomial, which is the determinant of the
matrix I
?????? in (1) is a ratio of two polynomials, a
?????? is obtained by examining
???????
?
????
?
?
????
?
??. Specifically,
?????????? ???????????
?
????????
????????????????
?
?????
?
????????????
????????????????
???
??
?
????????????????????
???
?
???
?
?????? ?????? ???????????
???
?
????????? ??
(2)
for
???, with initial conditions given by
????????
if
????? or
???
? ??? ????
? ??? ??? ???????????????????
???
? ??? ????? ???????? ???????????
???
?
If
???, then we have binary codes, and for this special case
?????? reduces to
?????????? ????????????????? ???
??? ??????????????? ??? ????????? ???
This is a simpler expression than the one derived in [10] for
the same binary system as it contains one less term.
B. Approximate Probability of Codeword Error
We herein assume that the binary noise process is only
Markovian within each symbol (of length
is independent between symbols. Obviously, this assumption
(which can be realized via perfect symbol interleaving) does
not hold when RS codes are directly sent over the BAMNC.
Thus, the corresponding
? bits), and it
?????? for this new system, denoted

Page 3

by
We obtain
?
?
?????, will approximate the actual
?????? given in (2).
?
?
??????
?
?
?
?
???1 symbol error?
?
???No symbol error?
???
?
?
?
?
?
?
?
?
??
?
????
???
?
????
?
???
???
?
?
?
??
?
????
? ??
?
????
?
???
???
?
?????
?
(3)
In this case, the probability of correct decoding for a
error correcting RS code is
probability of codeword error for this code (under bounded
distance decoding) is simply P ?
the channel is a BSC (i.e., when
is exact as the channel becomes memoryless.
?-
?
?
?
?
?
?
???
?
?
?????, while the
?
?
????
?
?. Note that when
???), this approximation
C. Symbol Interleaving is Always Better than Bit Interleaving
For
?????, let
????
?
?
?? ???????? ???????????
???
?
?
?
?
???????????????????
???
?
???
?
????
?
?
???????
?
?
?
?
?????
?
?
???
?
We notice that if
???? and
???,
?????? ??? and
?
or monotone increasing with
is larger than, equal to, or less than 1/2, respectively. A similar
behavior is observed for
????????. Also,
???? is monotone decreasing, constant
? depending on whether
?????
?
???? depending on how
????????
???????????
???compares with
???. Therefore, if
???? and
?
exists a unique value of
Specifically,
??,
?????
?
????????????????????
???and there
?, say
?
?, such that
???
?
?????
?
?.
?
?is given by
?
?
?? ??? ??????, where
?
?
?? ???
?
???????? ???????????
???
?
?
?
??? ???????????????????
???
??????
?
????????? ???????????
???
???? ?????
?
?
?
For example, if
?????? and
?????, then
?
?
?????? for
?
less than or equal to
?? and
?
?
????? for
???. Now letting
? be an integer
??
?
? yields
?
?
???
?
?
?
?
?????
?
?
???
?
?
?
?
?????
(4)
The right-handside of (4) is the probability of correct decoding
for a
the BSC, while the left-hand side is
larger than
?-error correcting RS code over GF ??
?
? when sent over
?
?
?. Furthermore, if
? is
??
?
?, then
????????? for
???????. Then
???
?
?
?
?
?????
?
?
?
?
?????
?
?
?????
?
?
?
?
????
(5)
where the right-hand side of (5) is the probability of codeword
error for the BSC. Thus we have shown that the code’s perfor-
mance under perfect symbol interleaving is always better when
compared with its performance under perfect bit interleaving.
This result can also be shown for a larger class of
Markov channels [9] as well as the Gilbert-Elliott channel [7].
?-order
IV. RESULTS
We consider four RS codes given in Table I. First, we vali-
date our analytical derivation of
the PCE calculated using
via simulations (implemented using the Berlekamp-Massey
decoding algorithm). The results, shown in Fig. 1 for code
and
simulations and the numerical calculations (a similar behavior
is observed for the other codes). Thus (2) provides an effective
tool for determining PCE without the need for simulations
which can be complex and long for low PCE values.
?????? in (2) by comparing
?????? with the PCE obtained
?
?
?????????, indicate a complete agreement between the
A. When Can Symbol Interleaving Be Avoided ?
Equipped with (2) and (3), we determine the regions of
? and
absolute relative error less than or equal to
codes of Table I. In Table II, the
approximation is accurate are given in the form
for values of
Thus for these values of
avoided.
? for which our approximation is accurate (within an
???) for the four
????? values for which our
?????
???
? chosen so that PCE is between
????and
????.
?????, symbol interleaving can be
B. When Can Bit Interleaving Be Avoided ?
We evaluate the RS codes of Table I on the BAMNC
using (2) to systematically identify the
the codes performance without interleaving (with
superior to that with perfect bit interleaving (with
results, with
given, are summarized in Tables III-VI (the dash symbols in
the tables indicate that perfect bit interleaving yields better
performance for the specified
conditions, not only can one forgo additional delay and com-
plexity by avoiding bit interleaving, but improved performance
can also be achieved as illustrated in Fig. 2.
????? values for which
???) is
???). The
? shown in the form
?
???
????
??? for
?
? value). Thus for such channel
V. CONCLUSIONS
The performance of non-interleaved RS codes over a simple
binary channel with memory, the BAMNC, was analyzed and
evaluated. It was shown that for any given RS code using
standard encoding/decoding procedures (that do not exploit
the channel memory), there is a (sometimes wide) range of
channel conditions for which the code is well matched to the
channel in such way that the code provides the best perfor-
mance when no (symbol or bit) interleaving is employed. The
design of an RS decoding technique that exploits the channel
memory can lead to further performance improvements and is
an interesting topic for future work.
REFERENCES
[1] J. Roberts, A. Ryley, D. Jones, and D. Burke, “Analysis of error-
correction constraints in a optical disk,” Applied Optics, vol. 35,
pp. 3915-3924, July 1996.
[2] H. Chang, C. Shung, and C. Lee, “A Reed-Solomon product-code (RS-
PC) decoder chip for DVD applications,” IEEE Journal of Solid-State
Circuits, vol. 36, pp. 229-238, Feb. 2001.
[3] H. Labiod “Performance of Reed Solomon error-correcting codes on
fading channels,” Proc. IEEE Int. Conf. Personal Wireless Commun.,
1999, pp. 259-263.

Page 4

[4] J. Lai and N. Mandayam, “Performance of Reed-Solomon codes for
hybrid-ARQ over Rayleigh fading channels under imperfect interleav-
ing,” IEEE Trans. Commun., vol. 48, pp. 1650-1659, Oct. 2000.
[5] C. Pimentel and I. F. Blake, “Concatenated coding performance for
FSK modulation on time-correlated Rician fading channels,” IEEE
Trans. Commun., vol. 46, pp. 1610-1618, Dec. 1998.
[6] T. Berman and J. Freedman “Non-interleaved Reed-Solomon coding
over a bursty channel,” Proc. IEEE Military Commun. Conf., 1992, pp.
580-583.
[7] M. Mushkin and I. Bar-David, “Capacity and coding for the Gilbert-
Elliott channel,” IEEE Trans. Inform. Theory, vol. 35, pp. 1277-1290,
Nov. 1989.
[8] C. Pimentel and I. F. Blake, “Non-interleaved Reed-Solomon coding
performance on finite state channels,” Proc. IEEE Int. Conf. Commun.,
1997, pp. 1493-1497.
[9] L. Zhong, F. Alajaji and G. Takahara, “A queue-based model for
wireless Rayleigh fading channels with memory,” in Proc. IEEE 62nd
Semiannual Veh. Technol. Conf., 2005, pp. 1362-1366.
[10] J. Yee and E. Weldon “Evaluation of the performance of error-
correcting codes on a Gilbert channel,” IEEE Trans. Commun., vol. 43,
pp. 2316-2323, Aug. 1995.
TABLE I
PARAMETERS OF THE CONSIDERED RS CODES.
code
????
?
?
255
255
127
127
221
129
111
65
17
63
8
31
0.867
0.506
0.874
0.511
?
?
?
?
?
?
TABLE II
????? INTERVALS FOR WHICH TO AVOID SYMBOL INTERLEAVING.
code
??
???
0.2
0.1
0.06
0.3
0.13
0.06
0.38
0.12
0.08
0.26
0.13
0.04
?
?
??
??
?
?
?
?
??
??
?
?
??
??
?
?
??
??
?
?
?
?
??
??
???
?
??
??
?
?
??
??
?
?
?
?
??
??
?
?
??
??
?
?
??
??
?
?
?
?
??
??
?
?
??
??
TABLE III
????? VALUES FOR WHICH TO AVOID BIT INTERLEAVING FOR CODE
?
?.
??
???
0
0
0
0.14
-
?
???
1
0.9
0.83
0.62
-
?
?
?
??
??
?
?
??
??
?
?
??
??
?
?
??
??
?
?
?
??
??
TABLE IV
??? ?? VALUES FOR WHICH TO AVOID BIT INTERLEAVING FOR CODE
?
?.
??
???
0
0
0
0
0.11
0.33
-
?
???
1
0.87
0.73
0.64
0.61
0.54
-
?
?
?
??
??
?
?
??
??
?
?
??
??
?
?
??
??
???
?
??
??
???
?
??
??
?
???
?
??
??
TABLE V
????? VALUES FOR WHICH TO AVOID BIT INTERLEAVING FOR CODE
?
?.
??
???
0
0
0
0
0.29
-
?
???
1
0.87
0.78
0.69
0.53
-
?
?
?
??
??
?
?
??
??
?
?
??
??
???
?
??
??
???
?
??
??
?
?
?
??
??
TABLE VI
????? VALUES FOR WHICH TO AVOID BIT INTERLEAVING FOR CODE
?
?.
??
???
0
0
0
0.05
-
?
???
1
0.82
0.73
0.58
-
?
?
?
??
??
?
?
??
??
?
?
??
??
?
?
??
??
?
???
?
??
??
1e-05
0.0001
0.001
0.01
0.1
1
0 0.02 0.04 0.06 0.08 0.1
PCE
p
RS (127,65)
epsilon = 0.1; calc.
epsilon = 0.1; sim.
epsilon = 0.9; calc.
epsilon = 0.9; sim.
Fig. 1.PCE for code
?
?: simulation (sim.) vs analytical (calc.) results.
1e-14
1e-12
1e-10
1e-08
1e-06
0.0001
0.01
1
0.001 0.01 0.1
PCE
p
RS (255,221)
epsilon = 0.0
epsilon = 0.3
epsilon = 0.9
epsilon = 0.99
Fig. 2. PCE for code
?
?: BAMNC vs BSC.