From equations 4.10 and 4.11 we can conclude:

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].)