Skip to main content
SpringerLink
Log in

On the reliability function for a noisy feedback Gaussian channel: Zero rate

  • Information Theory
  • Published:
Problems of Information Transmission Aims and scope Submit manuscript

Abstract

A discrete-time channel with independent additive Gaussian noise is used for information 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).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Shannon, C.E., The Zero Error Capacity of a Noisy Channel, IRE Trans. Inform. Theory, 1956, vol. 2, no. 3, pp. 8–19.

    Article  MathSciNet  Google Scholar 

  2. Dobrushin, R.L., Asymptotic Bound on the Error Probability for Message Transmission over a Memoryless Channel with Feedback, Probl. Kibern., 1962, vol. 8, pp. 161–168.

    Google Scholar 

  3. Horstein, M., Sequential Decoding Using Noiseless Feedback, IEEE Trans. Inform. Theory, 1963, vol. 9, no. 3, pp. 136–143.

    Article  MATH  Google Scholar 

  4. Berlekamp, E.R., Block Coding with Noiseless Feedback, PhD Thesis, Cambridge: MIT, 1964.

    Google Scholar 

  5. Schalkwijk, J.P.M. and Kailath, T., A Coding Scheme for Additive Noise Channels with Feedback. I: No Bandwidth Constraint, IEEE Trans. Inform. Theory, 1966, vol. 12, no. 2, pp. 172–182.

    Article  MathSciNet  MATH  Google Scholar 

  6. Pinsker, M.S., The Probability of Error in Block Transmission in a Memoryless Gaussian Channel with Feedback, Probl. Peredachi Inf., 1968, vol. 4, no. 4, pp. 3–19 [Probl. Inf. Trans. (Engl. Transl.), 1968, vol. 4, no. 4, pp. 1–14].

    MathSciNet  MATH  Google Scholar 

  7. Burnashev, M.V., Data Transmission over a Discrete Channel with Feedback. Random Transmission Time, Probl. Peredachi Inf., 1976, vol. 12, no. 4, pp. 10–30 [Probl. Inf. Trans. (Engl. Transl.), 1976, vol. 12, no. 4, pp. 250–265].

    MathSciNet  MATH  Google Scholar 

  8. Burnashev, M.V., On the Reliability Function of a Binary Symmetrical Channel with Feedback, Probl. Peredachi Inf., 1988, vol. 24, no. 1, pp. 3–10 [Probl. Inf. Trans. (Engl. Transl.), 1988, vol. 24, no. 1, pp. 1–7].

    MathSciNet  MATH  Google Scholar 

  9. Yamamoto, H. and Itoh, R., Asymptotic Performance of a Modified Schalkwijk-Barron Scheme for Channels with Noiseless Feedback, IEEE Trans. Inform. Theory, 1979, vol. 25, no. 6, pp. 729–733.

    Article  MATH  Google Scholar 

  10. Burnashev, M.V. and Yamamoto, H., On BSC, Noisy Feedback and Three Messages, in Proc. IEEE Int. Symp. on Information Theory, Toronto, Canada, 2008, pp. 886–889.

  11. Burnashev, M.V. and Yamamoto, H., On the Zero-Rate Error Exponent for a BSC with Noisy Feedback, Probl. Peredachi Inf., 2008, vol. 44, no. 3, pp. 33–49 [Probl. Inf. Trans. (Engl. Transl.), 2008, vol. 44, no. 3, pp. 198–213].

    MathSciNet  Google Scholar 

  12. Burnashev, M.V. and Yamamoto, H., Noisy Feedback Improves the BSC Reliability Function, in Proc. IEEE Int. Sympos. on Information Theory, Seoul, Korea, 2009, pp. 1501–1505.

  13. Burnashev, M.V. and Yamamoto, H., On the Reliability Function for a BSC with Noisy Feedback, Probl. Peredachi Inf., 2010, vol. 46, no. 2, pp. 3–23 [Probl. Inf. Trans. (Engl. Transl.), 2010, vol. 46, no. 2, pp. 103–121].

    MathSciNet  Google Scholar 

  14. Draper, S.C. and Sahai, A., Noisy Feedback Improves Communication Reliability, in Proc. IEEE Int. Symp. on Information Theory, Seattle, USA, 2006, pp. 69–73.

  15. Kim, Y.-H., Lapidoth, A., and Weissman, T., The Gaussian Channel with Noisy Feedback, in Proc. IEEE Int. Symp. on Information Theory, Nice, France, 2007, pp. 1416–1420.

  16. Xiang, Y. and Kim, Y.-H., On the AWGN Channel with Noisy Feedback and Peak Energy Constraint, in Proc. IEEE Int. Sympos. on Information Theory, Austin, Texas, USA, 2010, pp. 256–259.

  17. Shannon, C.E., Probability of Error for Optimal Codes in Gaussian Channel, Bell Syst. Tech. J., 1959, vol. 38, no. 3, pp. 611–656.

    MathSciNet  Google Scholar 

  18. Burnashev, M.V. and Yamamoto, H., On Decoding Error Exponent of Gaussian Channel with Noisy Feedback: Nonexponential Number of Messages, in Proc. IEEE Int. Sympos. on Information Theory, Boston, USA, 2012, pp. 2964–2968.

Download references

Author information

Authors and Affiliations

  1. Kharkevich Institute for Information Transmission Problems, Russian Academy of Sciences, Moscow, Russia

    M. V. Burnashev

  2. School of Frontier Sciences, The University of Tokyo, Tokyo, Japan

    H. Yamamoto

Authors
  1. M. V. Burnashev
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. H. Yamamoto
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. V. Burnashev.

Additional information

Original Russian Text © M.V. Burnashev, H. Yamamoto, 2012, published in Problemy Peredachi Informatsii, 2012, Vol. 48, No. 3, pp. 3–22.

Supported in part by the Russian Foundation for Basic Research, project no. 12-01-00905a.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Burnashev, M.V., Yamamoto, H. On the reliability function for a noisy feedback Gaussian channel: Zero rate. Probl Inf Transm 48, 199–216 (2012). https://doi.org/10.1134/S0032946012030015

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0032946012030015

Keywords

Search

Navigation