247.

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		1.5.12— Statistical Properties and the Power Spectra

		A sequence of values measured in time can be broken down into a series of periodic oscillations of different frequencies. A plot of the amount present in each frequency is called the power spectrum.

		The power spectra of fractal objects in space or fractal processes in time reflect the self-similarity, scaling relationship, and statistical properties of fractals.

		Low frequencies in the power spectrum correspond to coarse resolution. High frequencies in the power spectrum correspond to fine resolution. Thus the power spectrum is a measure of the amount present in structures of different sizes.

		Self-similarity means that there is a relationship between the power at high frequencies (fine resolution) and the power at low frequencies (coarse resolution). This relationship has a power law scaling that the energy at a given frequency is proportional to 1/fa. This form is called 1/f (''one-over-f') noise, even when the exponent a is not equal to 1.

		Depending on the value of a, the total power at all the frequencies in the power spectrum and the average power do not exist. When , then the total energy increases as the lowest frequency used to measure it decreases. That is, the longer the interval of data that is analyzed, the larger the total power in the power spectrum. When a £ 1, then the total power increases as the highest frequency used to measure it increases. In this case, the ever shorter intervals of data contain ever larger amounts of power.

		An example of l/f noise in time is the electrical signal generated by the contraction of the heart. An example of l/f noise in space is the spatial distribution of a radioactive tracer in the liver.

 

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