Inverse power-law spectrum

Consequently the inverse power-law spectrum obtained in this way has a slope related to the fractal dimension by a + 1 = 5 — 2D. 

Thus, since the fractional-difference dynamics are linear, the system response is Gaussian, the same as the statistics for the white noise process on the right-hand side of Eq. (22). However, whereas the spectrum of fluctuations is flat, since it is white noise, the spectrum of the system response is inverse power law. From these analytic results we conclude that Xj is analogous to fractional Brownian motion. The analogy is complete if we set a = // 1/2 so that the... 

The Fourier spectrum (Figure 6b, presented as a log-log plot) displays an inverse power law behavior at high frequencies, i.e., the broad plateau in the region of 500 kHz to 20 MHz is followed by l//P-like behavior at still higher frequencies. A least-squares fit of the high-frequency coefficients on a log-log plot yielded (3 = 2.1 0.1. Mandelbrot has noted that fractal sets are characterized by Fourier spectra with inverse power law behavior. (74,75) He has shown that the exponent in the inverse power law, p, is related to the fractal dimension, Df, of the corresponding nominally one-dimensional data by... 

The other form of noise found in semiconductors is 1// power law noise, characterized by a spectrum in which the noise power depends approximately inversely upon frequency. The general expression for the noise current is... 

Figure 7.16 Dielectric relaxation spectrum of a 47.7 kDa Mw di-polyisoprene having nonparallel dipoles due to one inversion at the molecular center, dissolved in 700 Da polybutadiene at a concentration of 27 g/1, using original measurements by Watanabe, etal.Cil), with simple-exponential and power-law fits.

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