  Complex architectures for wireless communications, digital electronics and
space-based navigation interlink several oscillator-based devices such as
clocks, transponders and synthesizers. Estimators characterizing their
stability are critical for addressing the impact of random fluctuations (noise)
on the overall system performance. Manufacturers typically specify this as an
Allan/Hadamard Variance (AVAR/HVAR) profile in the _integration_ time domain,
yet, stochastic processes governing the noise take place in the _Fourier_
frequency domain in the shape of a Power Spectral Density (PSD) function. Both
are second-moment measures of the time series, however, it is only possible to
translate unambiguously from the PSD to the AVAR/HVAR, not vice versa, except
in the case of a single noise type, which is severely limiting in real-life
applications. This note elaborates an analytical method to generate an
approximated PSD expressed as a set of power-laws defined in specific intervals
in the frequency domain, starting from an AVAR/HVAR expressed a set of
power-laws in the time domain. The proposed algorithm is straightforward to
implement, applicable to all noise types (and combinations thereof) and can be
self-validated by reconstructing the corresponding AVAR/HVAR by direct
calculus. We also report on its limitations of and analytical expressions of
the continuous version of this algorithm. Coupling with well-established
algorithms relying on the PSD for power-law noise generation, the ensuing
method encompasses the capability for generating multi-colored noise in
end-to-end simulations, as demonstrated hereby for NASA's Deep Space Atomic
Clock.
