A240908 The sequency numbers of the 8 rows of a version of the Hadamard-Walsh matrix of order 8.

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

Offset

1,2

Comments

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.

Links

Table of n, a(n) for n=1..8.

N. J. A. Sloane, A Library of Hadamard Matrices

Eric Weisstein's World of Mathematics, Hadamard Matrix

Wikipedia, Walsh matrix

Formula

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

Example

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

Crossrefs

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

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

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

Cf. A000975(n) is sequency of last row of H(n). - William P. Orrick, Jun 28 2015

Sequence in context: A097517 A127559 A066747 * A117043 A013664 A154173

Adjacent sequences: A240905 A240906 A240907 * A240909 A240910 A240911

Keyword

nonn,fini,full

Author

Ross Drewe, Apr 14 2014

Extensions

Definition of H(n) corrected by William P. Orrick, Jun 28 2015

Status

approved