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.