Problems of Information Transmission, Vol. 38, No. 1, 2002, pp. 55–64. Translated from Problemy Peredachi Informatsii, No. 1, 2002, pp. 64–74.
Original Russian Text Copyright
2002 by St˚ahl, Johannesson.
A Note on Systematic Tailbiting Encoders
P. St˚ahl and R. Johannesson
Received May 22, 2001; in ﬁnal form, October 19, 2001
Abstract—Tailbiting codes encoded by convolutional encoders are studied. An explanation
is given for the fact that, at low signal-to-noise ratios, a systematic feedback encoder results
in fewer decoding bit errors than a nonsystematic feedforward encoder for the same tailbiting
code. The analysis is based on a recently introduced code property, namely, the weight density
of distance-d codewords. For a given distance-d weight density, the decoding bit error prob-
ability depends on an encoder property, viz., the number of taps in the tap-minimal encoder
pseudoinverse. Among all convolutional encoders that encode a given tailbiting code, the sys-
tematic one has the tap-minimal encoder pseudoinverse with fewest taps and, hence, gives the
smallest bit error probability.
Consider communication using convolutional codes. It is well known that errors in the output
from a Viterbi decoder are grouped into error bursts. At high signal-to-noise ratios, the burst
error probability depends primarily on the free distance of the convolutional code, which is a code
property. Hence, it is the same if the convolutional code is encoded by a nonsystematic feedforward
encoder or by a systematic feedback encoder. The bit-error probability, however, depends on the
particular encoder that is used. It is well known that, at high signal-to-noise ratios, the systematic
feedback encoder results in more bit errors than a nonsystematic feedforward encoder. At rates
close to the channel capacity, the typical error bursts are very long and, thus, for low signal-to-noise
ratios, codewords with weights essentially larger than the free distance have a strong inﬂuence on
the performance of the communication system. It is well known that, at low signal-to-noise ratios,
a systematic feedback encoder results in fewer bit errors than a nonsystematic feedforward encoder.
An explanation is given in [1, 2].
Here, we will consider block codes obtained from terminating the convolutional codes using the
tailbiting technique. As in the case of the burst error probability of convolutional codes, the block
error probability of tailbiting codes, which is a code property, is the same whether the tailbiting code
is encoded by a nonsystematic feedforward encoder or its systematic feedback equivalent. At high
signal-to-noise ratios, for short and moderate tailbiting lengths (relative to the encoder memory), a
systematic feedback encoder results, in general, in fewer bit errors than a nonsystematic feedforward
encoder but, for long tailbiting lengths, a nonsystematic feedforward encoder has the best bit error
performance [3, 4]. In this paper, we give an explanation for the observation that, at low signal-
to-noise ratios, a systematic feedback encoder results in fewer bit errors than a nonsystematic
feedforward encoder. For our analysis, we use the same technique as in [1, 2] for unterminated
convolutional codes. A short summary of those results is as follows:
The analysis is based on a code property, viz., the weight density of distance-d detours, p
deﬁned as the fraction of 1s in the detours of weight d in a binary convolutional code. For large
values of d, the code parameter p
tends towards an asymptotic value p
, which is essentially
Supported in part by the Foundation for Strategic Research—Personal Computing and Communication,
Grant no. PCC-9706-09.
2002 MAIK “Nauka/Interperiodica”