Simply, the mathematics of sNoise answers the question of "What is inside the black box?" and is an effective tool in the study of systems that generate self-affine time series.
Formally, the fractional calculus of sNoise, first introduced in the dissertation of Dr. Smigelski, encompasses a new class of modified Laplace transfer functions incorporating a combination of a scaling exponent β and an altitude exponent
æ which mathematically define fractional control orders, fractional scaling, fractional phase shifting, fractional filtering, fractional integration, or fractional differentiation known to occur in a variety of systems such as those that generate stochastic time series.
In the scientific literature, noise observed in a time series is generally described as 1/f -noise (or 1 over f noise) where f is the simple frequency term. The term 1/f -noise to describe the scaling behavior of noise observed in stochastic time series is limited in function. As a simple frequency term, 1/f -noise can only describe scaling of magnitude at each frequency but cannot describe any phase shifting which naturally occurs in all waveforms generated as output of a simple system or stochastic self-affine time series generated as output of a complex system.
The term sNoise is shorthand for writing equations using 1/s-noise (or 1 over s noise) in which the Laplace term, denoted as s, represents a complex frequency (j ω) that fully describes how a system, across all frequencies, will scale (in magnitude) and/or shift (in phase) any input that passes into the system. Thus, the term 1/s-noise, or sNoise, is a more descriptive reference and more accurate mathematically allowing the degree of scaling and shifting behavior across all frequencies from input to output by a system to be expressed by one or more fractional differential equations, or transfer functions, also referred to as the Frequency Response Model (FRM) of the system. sNoise also represents the patented algorithms encoding the fractional calculus-based mathematics of Fractional Scaling Digital Signal Processing (FSDSP), Fractional Scaling Digital Filters (FSDF), and Fractional Order Control Systems (FOCS).
Magnitude with Scaling Exponent
Exhibiting both Fractional Integration and Fractional Differentiation
Phase Map of Scaling Exponent of Fractional Integration or Fractional Differentiation
A fundamental concept in Control Theory, a Black Box represents an unknown system or process that changes or "filters" an input signal into that system to yield an output signal. However, exactly how these changes occur or what is inside the system or process is obscured, "in the dark", or closed-off from direct observation and thus the system is a black box where one can only observe inputs into the system or outputs from the system but not the system itself, the Black Box, at least not directly.
The sNRL Black Box Logo represents the fractional calculus mathematics of sNoise which reveal what is inside the Black Box, illuminating the underlying system dynamics. The sNRL logo consists of two time series, a Gaussian white noise and a Brownian motion. The Gaussian white noise (in blue in the logo) as input into an integration system contains equal power, on average, at all frequencies and exhibits a scaling exponent of β = 0. Upon integration, the Gaussian white noise input becomes a Brownian motion (in green in the logo) as the output of that system which exhibits a single scaling exponent of β = 2 over all frequencies in the power spectrum in which there is more power in the lower frequencies. There is a 90 degree phase shift upon integration, hence, the two time series appear at a 90 degree right angle to each other on the cube face. With sNRL's patented algorithms, the Black Box becomes transparent revealing not only the dymnamic processes within the system, but also how the system or inputs to the system may be altered to achieve a specific output signal.
