ISSN 0032-9460, Problems of Information Transmission, 2012, Vol. 48, No. 3, pp. 199–216.
Pleiades Publishing, Inc., 2012.
Original Russian Text
M.V. Burnashev, H. Yamamoto, 2012, published in Problemy Peredachi Informatsii, 2012, Vol. 48, No. 3, pp. 3–22.
On the Reliability Function for a Noisy
Feedback Gaussian Channel: Zero Rate
M. V. Burnashev
and H. Yamamoto
Kharkevich Institute for Information Transmission Problems,
Russian Academy of Sciences, Moscow
School of Frontier Sciences, The University of Tokyo, Japan
Received April 9, 2012; in ﬁnal form, June 25, 2012
Abstract—A discrete-time channel with independent additive Gaussian noise is used for in-
formation transmission. There is also a feedback channel with independent additive Gaussian
noise, and the transmitter observes all outputs of the forward channel without delay via this
feedback channel. Transmission of a nonexponential number of messages is considered (i.e., the
transmission rate is zero), and the achievable decoding error exponent for such a combination
of channels is investigated. It is shown that for any ﬁnite noise in the feedback channel the
achievable error exponent is better than the similar error exponent for a no-feedback channel.
The transmission/decoding method developed in the paper strengthens the method earlier used
by the authors for a BSC. In particular, for small feedback noise, it provides a gain of 23.6%
(instead of 14.3% obtained earlier for a BSC).
1. INTRODUCTION AND MAIN RESULTS
We consider a discrete-time channel with independent additive Gaussian noise; i.e., if x =
) is an input codeword, then a received block y =(y
where ξ =(ξ
) are independent N (0, 1) Gaussian random variables (i.e., E ξ
= 1). There is also a noisy feedback channel which allows the transmitter to observe (without
delay) all outputs of the forward channel
where η =(η
) are independent (and independent of ξ) N (0, 1) Gaussian random variables
(i.e., E η
=1). Thequantityσ>0, characterizing the feedback channel noise
intensity, is given. No coding is used in the feedback channel (i.e., the receiver simply retransmits
all received outputs to the transmitter). In other words, the feedback channel is “passive” (see the
We assume that an input block x satisﬁes the constraint
≤ nA, (3)
Supported in part by the Russian Foundation for Basic Research, project no. 12-01-00905a.