Limited live-time measurements of frequency spectra

Frequency-difference-of-arrival (FDOA) can be used to monitor and track an emitter's location by observing Doppler frequency shifts of the carrier during short pulsed transmissions. This specific application needs to consider two frequency-stability levels: the short τ_on-average frequency measurements and the sampling time interval between measurements, denoted by τ_s. We show the advantage of using “dynamic” ThêoH for τ_on frequency measurements. Between τ_on transmissions while powered off, the emitter's reference-oscillator frequency will change due to power-start thermal variations, vibration, stress, and other frequency disturbances. If we regard long-term frequency instability as an uncertainty on an oscillator's expected or designated frequency, then the dominant error of frequency prediction is likely to be the error due to frequency drift and/or random walk FM, indicating the non-stationary behavior or disturbances. A two-sample variance is devised, called “Psi-variance,” that has desirable statistical properties similar to those of the Allan variance. From this, we compute the power spectral density of frequency fluctuations, S_y(f), from which phase noise, L(f), can be derived.