Unbiased estimation of an initial phase

This paper explores the problem of the unbiased estimation of the initial phase of a tone burst. Two types of initial phase estimators are investigated. The first one is based on the modification of a well-known statistically-optimal approach of estimation the phase of a continuous sinusoidal signal in the presence of a noise. The modified version permits nonlinear compensation of a priori unknown bias which is a function of a tone frequency, length of the observation interval, and of an unknown initial phase itself. The second approach employs a notch filter at the burst's ‘steady-state’ frequency. The transient of such a filter resulting from the onset of a burst can be used to deduce the signal's initial phase while the steady-state response is eliminated. Analysis shows that an estimation is unbiased and, therefore, it might be reasonably applied in high SNR environments where an estimate must be made within an extremely short length of an observation interval (less than one period).