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## Self-similarity of Noise

The scale invariancy of the signal can be explained by the simple scaling rule of Fourier transforms.
 (4.9) (4.10) (4.11)

From equations 4.10 and 4.11 we can conclude:
 (4.12)

which means that our auto-correlation function is scale independent, or in the other words the auto-correlation function is a fractal. We should note that most observed signals are random signals. Mandelbrot[28] suggests that these signals should be treated as nonstationary random signals to get around the infinite invariance problems. Thus, the autocorrelation and spectrum of noise would be time-dependent. The problem with infinite invariance is that a true self-similar signal has an infinite amount of energy in its high spectral region. In the case of the signal the integral:
 (4.13)

is finite when , and infinite when . This border is where we make the distinction between random and deterministic signals. Notice that the equation 4.12 holds for any , and that fact should be interpreted as the statistical behavior of the signal. There is no one-to-one relationship between a power spectrum and a signal. Many different signals in time may have the same power spectrum. If we were dealing with a deterministic signal and not a random process, one way we could explain equation 4.12 is that the auto-correlation function has to be a DC function. However, the fluctuations of the observed phenomena which exhibit a power spectrum are far more erratic than unit functions. (For rigorous mathematical treatment of noise see Keshner[20], Flandrin[12], Wornell[51].)

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