Variations on a Theme by Schalkwijk and Kailath

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Gallager R. G. , NAKİBOĞLU B.

IEEE TRANSACTIONS ON INFORMATION THEORY, vol.56, no.1, pp.6-17, 2010 (Journal Indexed in SCI) identifier identifier

  • Publication Type: Article / Article
  • Volume: 56 Issue: 1
  • Publication Date: 2010
  • Doi Number: 10.1109/tit.2009.2034896
  • Page Numbers: pp.6-17
  • Keywords: Additive memoryless Gaussian noise channel, block codes, error probability, feedback, reliability, Schalkwijk-Kailath encoding scheme, ADDITIVE NOISE CHANNELS, CODING SCHEME, FEEDBACK, RELIABILITY, CAPACITY, LENGTH


Schalkwijk and Kailath (1966) developed a class of block codes for Gaussian channels with ideal feedback for which the probability of decoding error decreases as a second-order exponent in block length for rates below capacity. This well-known but surprising result is explained and simply derived here in terms of a result by Elias (1956) concerning the minimum mean-square distortion achievable in transmitting a single Gaussian random variable over multiple uses of the same Gaussian channel. A simple modification of the Schalkwijk-Kailath scheme is then shown to have an error probability that decreases with an exponential order which is linearly increasing with block length. In the infinite bandwidth limit, this scheme produces zero error probability using bounded expected energy at all rates below capacity. A lower bound on error probability for the finite bandwidth case is then derived in which the error probability decreases with an exponential order which is linearly increasing in block length at the same rate as the upper bound.