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The fractal energy measurement and the singularity energy spectrum analysis

机译：分形能量测量和奇异能谱分析

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摘要

The singularity exponent (SE) is the characteristic parameter of fractal and multifractal signals. Based on SE, the fractal dimension reflecting the global self-similar character, the instantaneous SE reflecting the local self-similar character, the multifractal spectrum (MFS) reflecting the distribution of SE, and the time-varying MFS reflecting pointwise multifractal spectrum were proposed. However, all the studies were based on the depiction of spatial or differentiability characters of fractal signals. Taking the SE as the independent dimension, this paper investigates the fractal energy measurement (FEM) and the singularity energy spectrum (SES) theory. Firstly, we study the energy measurement and the energy spectrum of a fractal signal in the singularity domain, propose the conception of FEM and SES of multifractal signals, and investigate the Hausdorff measure and the local direction angle of the fractal energy element. Then, we prove the compatibility between FEM and traditional energy, and point out that SES can be measured in the fractal space. Finally, we study the algorithm of SES under the condition of a continuous signal and a discrete signal, and give the approximation algorithm of the latter, and the estimations of FEM and SES of the Gaussian white noise, Fractal Brownian motion and the multifractal Brownian motion show the theoretical significance and application value of FEM and SES.
机译：奇异指数（SE）是分形和多重分形信号的特征参数。基于SE，提出了反映整体自相似特征的分形维数，反映局部自相似特征的瞬时SE，反映SE分布的多重分形谱（MFS）以及反映点状多重分形谱的时变MFS。 。但是，所有研究都是基于对分形信号的空间或微分特征的描述。本文以SE为独立维度，研究了分形能量测量（FEM）和奇异能谱（SES）理论。首先，我们研究了奇异域中分形信号的能量测量和能谱，提出了多分形信号的有限元和SES的概念，并研究了Hausdorff测度和分形能量元素的局部方向角。然后，我们证明了有限元法和传统能源之间的兼容性，并指出可以在分形空间中测量SES。最后，研究了连续信号和离散信号条件下的SES算法，并给出了后者的近似算法，以及高斯白噪声，分形布朗运动和多重分形布朗运动的FEM和SES估计。证明了有限元和SES的理论意义和应用价值。

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