The following documents are referred to in this document:                [1] C. E. Shannon, “The zero-error capacity of a noisy channel,” IEEE Trans. Inf. Theory, vol. IT-2, pp. 8-19, September 1956.        [2] J. P. M. Schalkwijk and T. Kailath, “A coding scheme for additive noise channels with feedback part I: No bandwidth constraint,” IEEE Trans. Inf. Theory, vol. IT-12, pp. 172-182, April 1966.        [3] J. P. M. Schalkwijk, “A coding scheme for additive noise channels with feedback part II: Band-limited signals,” IEEE Trans. Inf. Theory, vol. IT-12, pp. 183-189, April 1966.        [4] M. Horstein, “Sequential transmission using noiseless feedback,” IEEE Trans. Info. Theory, vol. IT-9, pp. 136-143, July 1963.        [5] O. Shayevitz and M. Feder, “Optimal feedback communication via posterior matching,” IEEE Trans. Inf. Theory, vol. 57, no. 3, pp. 1186-1222, March 2011.        [6] Y.-H. Kim, A. Lapidoth, and T. Weissman, “On the reliability of Gaussian channels with noisy feedback,” in Proc. 41st Allerton Conf. Communication, Control Computing, September 2006, pp. 364-371.        [7] Z. Chance and D. J. Love, “Concatenated coding for the AWGN channel with noisy feedback,” IEEE Trans. Inf. Theory, vol. 57, pp. 6633-6649, October 2011.        [8] A. Sato and H. Yamamoto, “Error exponents of discrete memoryless channels and awgn channels with noisy feedback,” in ISITA, 2010, pp. 452-457.        [9] M. V. Burnashev and H. Yamamoto, “On reliability function of BSC with noisy feedback,” Problems of Information Transmission, vol. 46, pp. 2-23, 2010.        [10] M. V. Burnashev and H. Yamamoto, “Noisy feedback improves the gaussian channel reliability function,” in ISIT, 2014, pp. 2554-2558.        [11] M. Tomlinson, “New automatic equalizer employing modulo arithmetic,” Electronics Letters, vol. 7, no. 5, pp. 138-139, 1971.        [12] H. Harashima and H. Miyakawa, “Matched-transmission technique for channels with intersymbol interference,” IEEE Transactions on Communications, vol. 20, no. 4, pp.        774-780, 1972.        [13] Y. Kochman and R. Zamir, “Joint Wyner-Ziv/Dirty-Paper Coding by Analog Modulo-Lattice Modulation,” IEEE Trans. Inf. Theory, vol. 55, pp. 4878-4889, 2009.        
Feedback cannot improve the capacity of point-to-point memoryless channels [1]. Nevertheless, noiseless feedback can significantly simplify the transmission schemes and improve the error probability performance, see e.g. [2]-[5]. These elegant schemes fail however in the presence of arbitrarily small feedback noise, rendering them grossly impractical. This fact has been initially observed in [3] for the Additive White Gaussian Noise (AWGN) channel, and further strengthened in [6]. A handful of works have tackled the problem of noisy feedback as means for improving error performance, see e.g. [7]-[10]. However, these works attain their superior error performance at the cost of a significant increase in complexity w.r.t. their noiseless feedback counterparts. There appears to be no simple scheme (in the spirit of [3]-[5]) that is robust to feedback noise known hitherto.
USPTO Backgrounds
USPTO Backgrounds