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Wiener Numbers


A sequence of uncorrelated numbers alpha_n developed by Wiener (1926-1927). The numbers are constructed by beginning with

 {1,-1},
(1)

then forming the outer product with {1,-1} to obtain

 {{{1,1},{1,-1}},{{-1,1},{-1,-1}}}.
(2)

This row is repeated twice, and its outer product is then taken to give

 {{{1,1,1},{1,1,-1},{1,-1,1},{1,-1,-1}}, 
 {{-1,1,1},{-1,1,-1},{-1,-1,1},{-1,-1,-1}}}.
(3)

This is then repeated four times. The procedure is repeated, and the result repeated eight times, and so on. The sequences from each stage are then concatenated to form the sequence 1, -1, 1, 1, 1, -1, -1, 1, -1, -1, 1, 1, 1, -1, -1, 1, -1, -1, ....


See also

Uncorrelated Numbers

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References

Papoulis, A. "The Wiener Numbers." The Fourier Integral and Its Applications. New York: McGraw-Hill, pp. 258-259, 1962.Wiener, N. "The Spectrum of an Array and Its Applications to the Study of the Translation Properties of a Simple Class of Arithmetical Functions." J. Math. Phys. 6, 1926-1927.

Referenced on Wolfram|Alpha

Wiener Numbers

Cite this as:

Weisstein, Eric W. "Wiener Numbers." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/WienerNumbers.html

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