An Algorithm to Estimate the Power Spectral Density From Allan Deviation

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 time domain. However, stochastic processes constituting the noise are more thoroughly described in the frequency domain by the power spectral density function (PSD). Both are second-moment measures of the time series, but 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, a rather unrealistic case. This note presents 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 as 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 computation. Coupling with well-established algorithms relying on the PSD for power-law noise generation, the ensuing method encompasses the capability for generating multicolored noise in end-to-end simulations, as demonstrated hereby for NASA's deep space atomic clock. We also report on the limitations of the algorithm and analytical expressions of the continuous version of the algorithm.