Wiener Numbers
المؤلف:
Papoulis, A.
المصدر:
"The Wiener Numbers." The Fourier Integral and Its Applications. New York: McGraw-Hill
الجزء والصفحة:
...
4-5-2021
2764
Wiener Numbers
A sequence of uncorrelated numbers 
developed by Wiener (1926-1927). The numbers are constructed by beginning with
![<span style=]() {1,-1}, " src="https://mathworld.wolfram.com/images/equations/WienerNumbers/NumberedEquation1.gif" style="height:15px; width:46px" /> |
(1)
|
then forming the outer product with ![<span style=]()
{1,-1}" src="https://mathworld.wolfram.com/images/equations/WienerNumbers/Inline2.gif" style="height:15px; width:42px" /> to obtain
![<span style=]() {{{1,1},{1,-1}},{{-1,1},{-1,-1}}}. " src="https://mathworld.wolfram.com/images/equations/WienerNumbers/NumberedEquation2.gif" style="height:15px; width:226px" /> |
(2)
|
This row is repeated twice, and its outer product is then taken to give
![<span style=]() {{{1,1,1},{1,1,-1},{1,-1,1},{1,-1,-1}},
{{-1,1,1},{-1,1,-1},{-1,-1,1},{-1,-1,-1}}}. " src="https://mathworld.wolfram.com/images/equations/WienerNumbers/NumberedEquation3.gif" style="height:35px; width:314px" /> |
(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, 
, 
, ....
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.
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