The Fourier Frequency Integral Interval and its Impact on Conversion from Power Spectral Density to Allan Variance Characterizing Precise Oscillators

Precise oscillators such as quartz oscillators and atomic clocks are the heart-beat sources for power, banking, communication, satellite navigation, aerospace systems and so on. Two domains are applied to characterize the frequency instability of the precise oscillators, the Fourier frequency domain and the averaging time domain. The conversion exists from the Fourier frequency domain to the averaging time domain by an integral with the integral interval from the low cut-off frequency f_l to the high cut-off frequency f_h. It is pointed out firstly in this paper that the definitions of f_h in history are ambiguous and there are two choices already in use: the measurement bandwidth in Fourier frequency domain and the one in the averaging time domain. Both choices are usually different and will give different conversion results. As for f_l, it determines the achievable maximal averaging time τ_max for the converted Allan variance within the endurable conversion loss. With the defined converted loss L≤10%, τ_max is 0.2081/ fl.