IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 45, NO. 6, JUNE 1997
625
Erasure Generation and Interleaving for Meteor-Burst Communications
with Fixed-Rate and Variable-Rate Coding
Carl W. Baum and Clint S. Wilkins
Abstract— Erasure-generation techniques are investigated in
conjunction with symbol interleaving as methods to improve the
performance of meteor-burst communication systems. Erasure
generation via Viterbi’s ratio-threshold test (RTT) and a Bayesian
scheme are both considered. It is found that a system using fixedrate coding, interleaving, and erasure generation can significantly
outperform more complex variable-rate coding schemes which do
not use erasures.
Index Terms— Meteor-burst communications, Reed–Solomon
codes, variable-rate codes.
I. INTRODUCTION
M
UCH attention has been devoted to the performance
of meteor-burst systems using error-control coding
[1]–[9]. Published work includes analysis of fixed- and
variable-rate Reed–Solomon (RS) coding [3], [4] and
automatic-repeat-request schemes [5], [6].
This paper investigates the use of erasure generation and
symbol interleaving on the performance of meteor-burst systems that employ RS coding. Two erasure-generation techniques are considered. The first technique involves comparisons of envelope detector outputs and is known as Viterbi’s
ratio-threshold test (RTT) [10]–[12]. The second technique
is based on Bayesian decision theory and is an extension
of a method used for mitigating partial-band interference in
frequency-hop spread-spectrum communication systems [13].
II. SYSTEM
AND
CHANNEL MODELS
The communications protocol considered here is the same
as that described in [3], in which detection of a meteor trail
requires that the transmitting terminal signal monitor the probe
signal power. Once the probe is detected, a single packet of
fixed information content is sent over the trail.
We let denote the number of RS code words in a particular
packet. All code words are singly extended and have length ,
which is also the alphabet size and a power of two. The system
.
employs -ary orthogonal modulation, where
The meteor-burst channel is characterized by a time-varying
signal amplitude and additive white Gaussian noise with two. The analysis of this paper
sided power spectral density
is limited to the performance of underdense trails, which occur
Paper approved by S. B. Wicker, the Editor for Coding Theory and
Techniques of the IEEE Communications Society. Manuscript received June
12, 1995; revised August 9, 1996. This paper was presented in part at the
1995 IEEE Military Communications Conference, San Diego, CA, November
5–8, 1995.
The authors are with the Department of Electrical and Computer Engineering, Clemson University, Clemson, SC 29634-0915 USA.
Publisher Item Identifier S 0090-6778(97)04170-6.
far more frequently than other trails. For such trails, the signalis modeled as
,
to-noise ratio
is an exponential decay constant in seconds and
where
is the initial signal-to-noise ratio for a particular trail.
is approximately equal
For a signaling rate of bits/s,
during the transmission
to the constant
.
of the th symbol in the packet, where
This approximation introduces negligible error in analysis if
.
parallel envelope
The demodulator consists of a bank of
detectors followed by a decision device that, for each symbol,
either declares that a particular symbol was sent or declares
an erasure. The decision to erase or not is made independently
for each channel symbol. For those symbols not erased, the
envelope
symbol that corresponds to the largest of the
detector outputs is selected.
We analyze two different erasure-generation schemes in this
paper. The first scheme, the RTT, can be described as follows.
Erase the th received symbol in the th code word if
2nd largest
where
is a fixed number between 0 and 1 and
is
the envelope detector output that corresponds to code symbol
(for the th symbol position in the th code word). The
second scheme will be referred to as the Bayesian scheme and
is described as follows. Erase the th received symbol in the
th code word if
where
is a threshold,
is the zeroth-order modified
and
satisfy the relationship
Bessel function, and
(assuming the
symbol is the th in the packet). The Bayesian erasure
rule is obtained by utilizing decision–theoretic minimization
techniques [13] on a risk function consisting of a linear combination of error and erasure probabilities. This minimization
is motivated by the fact that a tight upper bound on the
probability of not decoding correctly is a monotonic function
of this linear combination [13].
Two types of packet configurations are considered in this
paper. The first configuration assumes that fixed-rate coding is
employed and utilizes block interleaving. The interleaving is
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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 45, NO. 6, JUNE 1997
limited to individual packets and is the standard “read in by
rows, read out by columns” method for block interleaving.
The second configuration assumes the use of variable-rate
coding instead of interleaving. An algorithm that determines
the optimal packet configuration for maximizing the probability of packet success with variable-rate coding and errorsonly decoding has been developed previously [3], [4]. The
algorithm uses an iterative procedure that requires knowledge
of code-word success probabilities.
An issue arises when using erasures with variable-rate
for the RTT
coding—how should the erasure thresholds (
for the Bayesian scheme) be set? We consider two
and
methods. In the first method, the threshold is held constant
throughout the duration of the packet. The variable-rate codeselection algorithm is run once for each candidate threshold,
and the particular combination of threshold and packet configuration, that maximizes the probability of packet success, is
chosen. We call this technique the prior-erasure method.
In the second method, the thresholds are the same for all
symbols in a code word, but different from code word to code
word in a packet. Because the number of candidate thresholds
is much larger than in the prior-erasure case, we simply use the
packet configuration that is produced by the errors-only codeselection algorithm. From the resulting packet configuration,
thresholds are chosen that maximize the probability of packet
success. We refer to this technique as the posterior-erasure
method.
III. ANALYSIS
OF
Fig. 1. The effects of interleaving and erasure generation on performance of
systems with fixed-rate coding ( = 0:5):
have
where “*” denotes discrete convolution and
represents the
vector
Let
denote the probability that the th code word does
not decode successfully. It follows that
SYSTEM PERFORMANCE
The th code word in a packet is not decoded successfully if
The packet is successfully decoded only if each code word
within the packet is successfully decoded. Thus, the probability
is given by
of packet error
where
denotes the number of information symbols in the
th code word and
is equal to 2 if the th symbol in
the code word is in error, 1 if it is erased, and 0 if it is
denote the probability mass function for
correct. Let
, i.e,
. It follows that
if
if
; and
if
;
is the probability of symbol error and
is the
where
probability of erasure. For the RTT, closed-form expressions
and
have been determined previously [13]. For the
for
Bayesian technique, closed-form expressions have not been
determined and, therefore, simulation is used to determine
these probabilities.
denote the probability that the sum of the
Now, let
number of erasures and twice the number of errors is equal to
for the th code word in the packet. Observe that
For the meteor-burst channel, the
pendent random variables. Thus, for
s are mutually indewe
IV. NUMERICAL RESULTS
The following performance results are for a system using
) RS codes, 700 information bits (140 information
(
symbols) per packet, and a signaling rate of 16 000 bit/s.
Our results are given for decay constant values
and
. These values are chosen because they are
representative of slow- and fast-trail decay rates, respectively.
Fig. 1 presents performance of erasure generation and interleaving combinations for fixed-rate coding with a (32,
(slow
14) RS code under the assumption that
decay). The (32, 14) code is chosen because it gives the
best performance for all the schemes shown in the figure
. The figure shows that both interleaving and
when
erasure generation significantly improve performance, and a
combination of Bayesian erasure generation and interleaving
gives the best performance.
The results are plotted assuming that a single threshold is
used for all code words in the packet, and the best single
threshold is used. This threshold for the RTT ranges from
BAUM AND WILKINS: ERASURE GENERATION AND INTERLEAVING FOR METEOR-BURST COMMUNICATIONS
Fig. 2. The effects of interleaving and erasure generation on performance of
systems with fixed-rate coding ( = 0:1):
at
dB to
at
dB; the
to
Bayesian threshold correspondingly ranges from
Thus, the improved performance of the Bayesian
scheme comes at a cost of increased sensitivity to parameters.
In Fig. 2, a similar performance comparison is made for
(fast decay). The (optimal) code in this case is a (32,
28) RS code. This figure shows that interleaving again provides
significant performance gains. Erasure generation, however, is
completely ineffective—the optimal erasure thresholds result
and
).
in no symbols being erased (
Fig. 3 presents the performance of variable-rate coding with
The best and worst fixederasure generation for
rate coding results from Fig. 1 are also included. The figure
shows that a combination of erasure generation and variablerate coding outperforms the best fixed-rate system, but the
performance gains are not particularly large. Furthermore,
the best fixed-rate system performs better than the variablerate/errors-only system. There is little difference between
prior- and posterior-erasure performance.
A packet configurations for the systems in this figure vary
and as a function of the code-selection and
as a function of
dB, the errorserasure techniques. For example, at
only decoding and the RTT posterior-erasure schemes use 12
code words, whereas the RTT prior-erasure scheme uses 16
code words, and the Bayesian prior-erasure scheme uses 18.
dB, all but the Bayesian-prior erasure scheme use
At
13 code words; the latter scheme uses 16.
There is also a difference in erasure thresholds. For the
RTT posterior-erasure scheme, the thresholds (which vary for
each code word in the packet) vary from roughly 0.9 at the
beginning of the packet to about 0.8 at the end; the thresholds
For the RTT
do not change significantly as a function of
is nearly optimal
prior-erasure scheme, the fixed
over all cases shown in the figure. However, once again, the
Bayesian scheme is more sensitive to threshold, ranging from
at
dB to
at
dB.
. In this
Fig. 4 compares the various schemes for
case, we see that the variable-rate coding schemes signifi-
627
Fig. 3. The effects of interleaving and erasure generation on performance of
systems with variable-rate coding ( = 0:5):
Fig. 4. The effects of interleaving and erasure generation on performance of
systems with variable-rate coding ( = 0:1):
cantly outperform the fixed-rate schemes, and the Bayesian
prior-erasure scheme gives the best performance. The packet
configurations in all cases use seven or eight code words, and
much less redundancy is used (so as to finish using the channel
before the fast decay makes communication impossible).
A single recommendation for a “best” meteor-burst
system is difficult to provide. Although variable-rate coding
performs extremely well when the channel degrades rapidly,
determination of the decay rate is required to use the codeselection algorithm. Determination of a reasonably accurate
value of may be difficult in practice, especially since the
exponential decay model is itself only an approximation.
The tradeoff between performance and sensitivity and the
RTT and Bayesian methods is also difficult to quantify.
Regardless, we can safely conclude that erasure generation
and interleaving significantly benefit fixed-rate systems,
whereas erasure generation significantly benefits variablerate systems.
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