Modeling the relationship between Higuchi’s fractal dimension and Fourier spectra of physiological signals

Abstract

The exact mathematical relationship between FFT spectrum and fractal dimension (FD) of an experimentally recorded signal is not known. In this work, we tried to calculate signal FD directly from its Fourier amplitudes. First, dependence of Higuchi’s FD of mathematical sinusoids on their individual frequencies was modeled with a two-parameter exponential function. Next, FD of a finite sum of sinusoids was found to be a weighted average of their FDs, weighting factors being their Fourier amplitudes raised to a fractal degree. Exponent dependence on frequency was modeled with exponential, power and logarithmic functions. A set of 280 EEG signals and Weierstrass functions were analyzed. Cross-validation was done within EEG signals and between them and Weierstrass functions. Exponential dependence of fractal exponents on frequency was found to be the most accurate. In this work, signal FD was for the first time expressed as a fractal weighted average of FD values of its Fourier components, also allowing researchers to perform direct estimation of signal fractal dimension from its FFT spectrum.

Appendix

We shall derive an exact expression linking the frequency of a mathematical sinusoid with its fractal dimension, according to the procedure described in [7]. In order to apply trigonometric summation rules, instead of the absolute difference of signal samples, as in (1), we shall observe the sum of squares

$$ L_{m}^{2} (k) = \frac{1}{k}\left( {\sum\limits_{i = 1}^{{\left[ {\frac{N - m}{k}} \right]}} {\left( {x(m + ik) - x(m + (i - 1)k)} \right)^{2} } } \right)\frac{N - 1}{{\left[ {\frac{N - m}{k}} \right]k}}. $$

(18)

Consequently, FD should be calculated according to a slightly modified formula:

$$ {\text{FD}} = \frac{1}{2}({\text{slope}}(\ln (1/k),\ln (L^{2} (k)) + 1) = k = \, 1, \ldots ,k_{\max } , $$

where

$$ L^{2} (k) = \frac{1}{k}\sum\limits_{m = 1}^{k} {L_{m}^{2} (k)} . $$

(19)

For a given sinusoid $x(i) = a\sin\, (\omega i\Updelta t + \varphi )$, where i = 1, …, N; a and φ representing its amplitude and initial phase and Δt = 1/f _samp, formula (18) becomes

$$ L_{m}^{2} (k) = \frac{1}{k}\left( {\sum\limits_{i = 1}^{{\left[ {\frac{N - m}{k}} \right]}} {\left( {a\sin \left( {(m + ik)\omega \Updelta t + \varphi } \right) - a\sin \left( {(m + (i - 1)k)\omega \Updelta t + \varphi } \right)} \right)^{2} } } \right)\frac{N - 1}{{\left[ {\frac{N - m}{k}} \right]k}}. $$

After transforming the difference of sinuses into product, and application of summation rule of squared cosines over the index i, we can write

$$ L_{m}^{2} (k) = A\left( {B + C\cos \left( {2((m - \frac{k}{2})\omega \Updelta t + \phi ) + \left[ {\frac{N - m}{k}} \right]k\omega \Updelta t} \right) - \cos^{2} \left( {(m - \frac{k}{2})\omega \Updelta t + \phi } \right)} \right) $$

where

$$ A = \frac{{4(N - 1)a^{2} }}{{\left[ {\frac{N - m}{k}} \right]k^{2} }}\sin^{2} \left( {\frac{k}{2}\omega \Updelta t} \right);\quad B = \frac{{\left[ {\frac{N - m}{k}} \right] + 1}}{2};\quad C = \frac{{\sin \left( {\left( {\left[ {\frac{N - m}{k}} \right] + 1} \right)k\omega \Updelta t} \right)}}{2\sin (k\omega \Updelta t)}. $$

As m appears within the square brackets as well, cosine summation rule over index m cannot be performed directly. Respecting that for most signals N ≫ m, therefore$\left[ {\frac{N - m}{k}} \right] \approx \left[ \frac{N}{k} \right]$, after some elementary operations equation (19) could be written as

$$ L^{2} (k) = \frac{1}{k}\sum\limits_{m = 1}^{k} {\left( {AB + AC\cos\left( {m2\omega \Updelta t + \left( {\left[ \frac{N}{k} \right] - 1} \right)k\omega \Updelta t + \varphi } \right) - A\cos^{2}\left( {m\omega \Updelta t - \frac{k}{2}\omega \Updelta t + \varphi } \right)} \right)}. $$

Finally, according to the summation rules,

$$ L^{2} (k) = \frac{AB}{k} + \frac{AC}{k}L_{1} - \frac{A}{k}L_{2} $$

(20)

where

$$ L_{1} = \frac{{\cos\left( {\left[ \frac{N}{k} \right]k\omega \Updelta t + \varphi } \right)\sin \left( {(k + 1)\omega \Updelta t} \right)}}{{\sin \left( {\omega \Updelta t} \right)}} - \cos\left( {\left( {\left[ \frac{N}{k} \right] - 1} \right)k\omega \Updelta t + \varphi } \right); $$

$$ L_{2} = \left( {\frac{k + 1}{2} + \frac{{\sin \left( {(k + 1)\omega \Updelta t} \right)\cos (2\varphi )}}{2\sin (\omega \Updelta t)} - \cos^{2} \left( { - \frac{k}{2}\omega \Updelta t + \varphi } \right)} \right). $$

Expression (20) was verified by calculating FD of sinusoids for $f = \frac{\omega }{2\pi } = 2, 4,\ldots, 30\,{\hbox{Hz}},$ for f _samp = 256 Hz, and $f = \frac{\omega }{2\pi }$  = 1, 2,…, 14 Hz, for f _samp = 128 Hz and the results presented as symbols in Fig. 1.