Fractional scaling digital filters and the generation of standardized noise and synthetic data series

Generation of standardized noise signals that provide mathematically correct noise with no errors and no loss of data, and can generate the noise of specific environments based on the transfer function of that environment are discussed. Various embodiments can generate synthetic data sets based on natural data sets that have similar scaling behavior. Fractional scaling digital filters, containing the fractional scaling characteristics of one or more of the eleven fundamental forms of basic building block transfer functions which incorporate the scaling exponent, can be encoded on FPGA devices or DSP chips for use in digital signal processing. Fractional Scaling Digital Filters allow fractional calculus, and thus fractional filtering (e.g., fractional scaling, fractional phase shifting, fractional integration, or fractional differentiation), to be performed on any signal, represent exact filtering solutions rather than approximations, and demonstrably are extremely accurate, highly efficient, and exhibit a higher level of performance than traditional DSP filters.