Filtering and specification of self-affinity

  • Benoit B. Mandelbrot


Chapter N1 argued that FƒB noises with the same exponent B can take any of many very different forms. This brief chapter takes the next step and faces the challenge of going beyond the spectrum and discriminating between those various possibilities. Two very different questions come to mind.

Is it legitimate to view the measurements performed on a finite sample as estimating an underlying Wiener-Khinchin spectrum? The key issue is whether or not the prefactor F is independent of the sample size T.

It is useful to evaluate and compare the Fourier spectra of suitably chosen non-linearly filtered functions G[X(t)], exemplified by x q . Often, the filtered function is also a ƒB noise, at least for low frequencies. If so, the B exponents that correspond to different filters G constitutes a (finite or infinite) “signature” of the process. This chapter focusses on three forms of 1/ƒ noise: Gaussian, “dustborne,” and multifractal, and shows that each has its own very distinctive signature.


Spectral Density Fourier Spectrum Hermite Polynomial Nonlinear Transformation Exponential Time 
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Copyright information

© Benoit B. Mandelbrot 1999

Authors and Affiliations

  • Benoit B. Mandelbrot
    • 1
    • 2
  1. 1.Mathematics DepartmentYale UniversityNew HavenUSA
  2. 2.IBM T.J. Watson Research CenterYorktown HeightsUSA

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