Problems of Information Transmission, Vol. 37, No. 4, 2001, pp. 307–324. Translated from Problemy Peredachi Informatsii, No. 4, 2001, pp. 36–55.
Original Russian Text Copyright
2001 by Burnashev.
On Upper Bounds for the Decoding Error Probability
of Convolutional Codes
M. V. Burnashev
Received February 5, 2001; in ﬁnal form, August 27, 2001
Abstract—A new approach to upper bounding the decoding error probability of convolutional
codes is proposed. Its idea is, instead of evaluating the individual contribution of each funda-
mental path, to compare it with contribution of another (lighter) fundamental path. This allows
us to (1) take into account the dependence between diﬀerent fundamental paths based on the
code tree structure; (2) represent the decoding error probability through the contribution of
the ﬁrst fundamental paths and a correction factor; (3) get much more accurate estimates.
Transmission of binary information sequence over the binary symmetric channel (BSC) with
crossover probability 0 <p<1/2 is considered. A noncatastrophical time-invariant convolutional
encoder and Viterbi decoder are used [1, 2]. In the case of a “tie,” the survived path is chosen
equiprobably among all candidates.
Two most important characteristics of such communication system are the ﬁrst-error event
, and bit-error probability, P
. Recall that P
is the probability that the ﬁrst edge
in the trellis diagram is decoded incorrectly. In the “union bounds” most commonly used, for any
of the above-mentioned characteristics [1,2], the individual contribution of each fundamental path
(see the deﬁnition below) is evaluated (in fact, upper bounded) by comparing it with the all-zero
path. These union bounds give rather accurate estimates for small p (usually, for p<0.01). For
larger p, union bounds become very inaccurate and, for p ≥ p
, they stop working (“blow up”).
The critical value, p
, depends on a code. Its typical values for many codes are in the range
For the ﬁrst-error event probability P
, there are some improvements of union bounds [3–5] for p
not too small; however, they are not much eﬃcient.
Here, we present another approach to evaluation of probabilities P
. Its idea is, instead of
evaluating the individual contribution of each fundamental path, to compare it with the individual
contribution of another (lighter) fundamental path. As a result, this allows us to get much more
accurate estimates than those given in [3–5].
For simplicity, we consider convolutional codes of rate R =1/b only. Memory m is the number of
encoder memory cells. States in the trellis diagram are denoted by S
with M =2
,... be a binary information sequence at the encoder input, x
,... be a sequence of
binary code blocks at the encoder output (i.e., BSC input), and y
,... be a sequence of binary
blocks at the BSC output (blocks x and y have length b).
According to the Viterbi algorithm, at the BSC output, at each time moment n, based on received
we ﬁnd a survived path in the trellis diagram for each state S
Recall that a survived path for state S
at moment n is a path that connects the initial state S
at moment n and has the minimal possible Hamming distance from the received blocks
Supported in part by the INTAS, Grant no. 00-738
2001 MAIK “Nauka/Interperiodica”