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The sequency numbers of the 8 rows of a version of the Hadamard-Walsh matrix of order 8.
4

%I #28 Jun 29 2015 06:16:54

%S 0,7,3,4,1,6,2,5

%N The sequency numbers of the 8 rows of a version of the Hadamard-Walsh matrix of order 8.

%C The Hadamard (Hadamard-Walsh) matrix is widely used in telecommunications and signal analysis. It has 3 well-known forms which vary according to the sequency ordering of its rows: "natural" ordering, "dyadic" or Payley ordering, and sequency ordering. In a mathematical context the sequency is the number of zero crossings or transitions in a matrix row (although in a physical signal context, it is half the number of zero crossings per time period). The matrix row sequencies are a permutation of the set [0,1,2,...n-1], where n is the order of the matrix. For spectral analysis of signals the sequency-ordered form is needed. Unlike the dyadic ordering (given by A153141), the natural ordering requires a separate list for each matrix order. This sequence is the natural sequency ordering for an order 8 matrix.

%H N. J. A. Sloane, <a href="http://neilsloane.com/hadamard">A Library of Hadamard Matrices</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HadamardMatrix.html">Hadamard Matrix</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Walsh_matrix">Walsh matrix</a>

%F Recursion: H(2)=[1 1; 1 -1]; H(n) = H(n-1)*H(2), where * is Kronecker matrix product.

%e This is a fixed length sequence of only 8 values, as given.

%Y Cf. A240909 "natural order" sequencies for Hadamard-Walsh matrix, order 16.

%Y Cf. A240910 "natural order" sequencies for Hadamard-Walsh matrix, order 32.

%Y Cf. A153141 "dyadic order" sequencies for Hadamard-Walsh matrix, all orders.

%Y Cf. A000975(n) is sequency of last row of H(n). - _William P. Orrick_, Jun 28 2015

%K nonn,fini,full

%O 1,2

%A _Ross Drewe_, Apr 14 2014

%E Definition of H(n) corrected by _William P. Orrick_, Jun 28 2015