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Proper calculation of a modulation half cycle from the Allan deviation

The Allan deviation is commonly used to determine the long term stability and precision of oscillators or otherwise continuous measurements. When periodic, fixed frequency modulation is present, it has been previously stated that the resulting peak in the estimate coincides with the half cycle of the modulation period. However, this peak is located at ~74.2% of the actual half cycle. In this work, we provide a generalized derivation of the modulation half cycle time constant from the Allan deviation estimate. Furthermore, the peak locations for the Hadamard, Modified Allan, and time deviations are shown to be approximately 84.2, 61.0, and 84.2%, respectively of the modulation half cycle time constants.