Variations on a Theme by Schalkwijk and Kailath
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 he...
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| Format: | Article |
| Language: | en_US |
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Institute of Electrical and Electronics Engineers
2011
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| Online Access: | http://hdl.handle.net/1721.1/61702 https://orcid.org/0000-0003-1406-6484 |
| _version_ | 1826215171776839680 |
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| author | Gallager, Robert G. Nakiboglu, Baris |
| author2 | Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science |
| author_facet | Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science Gallager, Robert G. Nakiboglu, Baris |
| author_sort | Gallager, Robert G. |
| collection | MIT |
| description | 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. |
| first_indexed | 2024-09-23T16:17:44Z |
| format | Article |
| id | mit-1721.1/61702 |
| institution | Massachusetts Institute of Technology |
| language | en_US |
| last_indexed | 2024-09-23T16:17:44Z |
| publishDate | 2011 |
| publisher | Institute of Electrical and Electronics Engineers |
| record_format | dspace |
| spelling | mit-1721.1/617022022-09-29T19:26:55Z Variations on a Theme by Schalkwijk and Kailath Gallager, Robert G. Nakiboglu, Baris Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science Massachusetts Institute of Technology. Research Laboratory of Electronics Gallager, Robert G. Nakiboglu, Baris Gallager, Robert G. 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. 2011-03-16T17:10:42Z 2011-03-16T17:10:42Z 2009-12 2009-08 Article http://purl.org/eprint/type/JournalArticle 0018-9448 INSPEC Accession Number: 11024797 http://hdl.handle.net/1721.1/61702 Gallager, R.G., and B. Nakiboglu. “Variations on a Theme by Schalkwijk and Kailath.” Information Theory, IEEE Transactions on 56.1 (2010): 6-17. https://orcid.org/0000-0003-1406-6484 en_US http://dx.doi.org/10.1109/TIT.2009.2034896 IEEE Transactions on Information Theory Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. application/pdf Institute of Electrical and Electronics Engineers Baris Nakiboglu |
| spellingShingle | Gallager, Robert G. Nakiboglu, Baris Variations on a Theme by Schalkwijk and Kailath |
| title | Variations on a Theme by Schalkwijk and Kailath |
| title_full | Variations on a Theme by Schalkwijk and Kailath |
| title_fullStr | Variations on a Theme by Schalkwijk and Kailath |
| title_full_unstemmed | Variations on a Theme by Schalkwijk and Kailath |
| title_short | Variations on a Theme by Schalkwijk and Kailath |
| title_sort | variations on a theme by schalkwijk and kailath |
| url | http://hdl.handle.net/1721.1/61702 https://orcid.org/0000-0003-1406-6484 |
| work_keys_str_mv | AT gallagerrobertg variationsonathemebyschalkwijkandkailath AT nakiboglubaris variationsonathemebyschalkwijkandkailath |