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The Normalized Second Moment of the Binary Lattice Determined by a Convolutional Code

Publication ,  Journal Article
Calderbank, AR; Fishburn, PC
Published in: IEEE Transactions on Information Theory
January 1, 1994

We calculate the per-dimension mean squared error μ(S) of the two-state convolutional code C with generator matrix [1,1 + D], for the symmetric binary source S = [0,1], and for the uniform source S = [0,1]. When S = [0,1], the quantity μ(S) is the second moment of the coset weight distribution, which gives the expected Hamming distance of a random binary sequence from the code. When S = [0,1], the quantity μ(S) is the second moment of the Voronoi region of the modulo 2 binary lattice determined by C. The key observation is that a convolutional code with 2Vstates gives 2Vapproximations to a given source sequence, and these approximations do not differ very much. It is possible to calculate the steady state distribution for the differences in these path metrics, and hence, the second moment. In this paper we shall only give details for the convolutional code [1,1 + D], but the method applies to arbitrary codes. We also define the covering radius of a convolutional code, and calculate this quantity for the code [1,1 + D]. © 1994 IEEE

Duke Scholars

Published In

IEEE Transactions on Information Theory

DOI

EISSN

1557-9654

ISSN

0018-9448

Publication Date

January 1, 1994

Volume

40

Issue

1

Start / End Page

166 / 174

Related Subject Headings

  • Networking & Telecommunications
  • 1005 Communications Technologies
  • 0906 Electrical and Electronic Engineering
  • 0801 Artificial Intelligence and Image Processing
 

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Calderbank, A. R., & Fishburn, P. C. (1994). The Normalized Second Moment of the Binary Lattice Determined by a Convolutional Code. IEEE Transactions on Information Theory, 40(1), 166–174. https://doi.org/10.1109/18.272475
Calderbank, A. R., and P. C. Fishburn. “The Normalized Second Moment of the Binary Lattice Determined by a Convolutional Code.” IEEE Transactions on Information Theory 40, no. 1 (January 1, 1994): 166–74. https://doi.org/10.1109/18.272475.
Calderbank AR, Fishburn PC. The Normalized Second Moment of the Binary Lattice Determined by a Convolutional Code. IEEE Transactions on Information Theory. 1994 Jan 1;40(1):166–74.
Calderbank, A. R., and P. C. Fishburn. “The Normalized Second Moment of the Binary Lattice Determined by a Convolutional Code.” IEEE Transactions on Information Theory, vol. 40, no. 1, Jan. 1994, pp. 166–74. Scopus, doi:10.1109/18.272475.
Calderbank AR, Fishburn PC. The Normalized Second Moment of the Binary Lattice Determined by a Convolutional Code. IEEE Transactions on Information Theory. 1994 Jan 1;40(1):166–174.

Published In

IEEE Transactions on Information Theory

DOI

EISSN

1557-9654

ISSN

0018-9448

Publication Date

January 1, 1994

Volume

40

Issue

1

Start / End Page

166 / 174

Related Subject Headings

  • Networking & Telecommunications
  • 1005 Communications Technologies
  • 0906 Electrical and Electronic Engineering
  • 0801 Artificial Intelligence and Image Processing