Number theory and bootstrapping for phase unwrapping

The problem of the phase unwrapping and the estimation of time-varying frequency is considered. The phase is first modeled as a polynomial in time. Using Lagrange interpolation polynomial approximation, for the modulo operation, where the modulus is a prime, a unique estimate for the phase is obtained. This estimate, however, is sensitive to noise. Using the method of bootstrapping, one is able to obtain good estimate even at SNR values as low as 10 dB. The method is applied to several examples, and compared to the minimum mean square polynomial fit for the phase. It is shown that the proposed approach has superior performance.