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.

[1]  LJubisa Stankovic,et al.  Instantaneous frequency estimation using the Wigner distribution with varying and data-driven window length , 1998, IEEE Trans. Signal Process..

[2]  Gary H. Glover,et al.  Phase unwrapping of MR phase images using Poisson equation , 1995, IEEE Trans. Image Process..

[3]  Benjamin Friedlander,et al.  The discrete polynomial-phase transform , 1995, IEEE Trans. Signal Process..

[4]  Asad Zaman,et al.  Statistical Foundations for Econometric Techniques , 1996 .

[5]  Dennis C. Ghiglia,et al.  Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software , 1998 .

[6]  Dorothy E. Denning,et al.  Cryptography and Data Security , 1982 .

[7]  José Tribolet,et al.  A new phase unwrapping algorithm , 1977 .

[8]  Reinaldo Castro Souza,et al.  A bootstrap simulation study in ARMA (p, q) structures , 1996 .

[9]  D. Wehner.  High Resolution Radar , 1987 .

[10]  Satoru Goto,et al.  On-line spectral estimation of nonstationary time series based on AR model parameter estimation and order selection with a forgetting factor , 1995, IEEE Trans. Signal Process..

[11]  Messaoud Benidir,et al.  Polynomial phase signal analysis based on the polynomial derivatives decompositions , 1999, IEEE Trans. Signal Process..

[12]  Ahmed S. Abutaleb,et al.  A genetic algorithm for the maximum likelihood estimation of the parameters of sinusoids in a noisy environment , 1997 .

[13]  Edward J. Powers,et al.  Time-varying spectral estimation using AR models with variable forgetting factors , 1991, IEEE Trans. Signal Process..

[14]  W. Leveque.  Fundamentals of number theory , 1977 .

[15]  Boualem Boashash,et al.  The bootstrap and its application in signal processing , 1998, IEEE Signal Process. Mag..

[16]  Boualem Boashash,et al.  Estimating and interpreting the instantaneous frequency of a signal. II. A/lgorithms and applications , 1992, Proc. IEEE.

[17]  K Itoh,et al.  Analysis of the phase unwrapping algorithm. , 1982, Applied optics.

[18]  Dimitris N. Politis,et al.  Computer-intensive methods in statistical analysis , 1998, IEEE Signal Process. Mag..

[19]  D R Haynor,et al.  Resampling estimates of precision in emission tomography. , 1989, IEEE transactions on medical imaging.

[20]  H. E. Rose.  A course in number theory , 1988 .

[21]  Robert Tibshirani,et al.  An Introduction to the Bootstrap , 1994 .

[22]  Benjamin Friedlander,et al.  Asymptotic statistical analysis of the high-order ambiguity function for parameter estimation of polynomial-phase signals , 1996, IEEE Trans. Inf. Theory.

[23]  Steven A. Tretter,et al.  Estimating the frequency of a noisy sinusoid by linear regression , 1985, IEEE Trans. Inf. Theory.

[24]  Benjamin Friedlander,et al.  Estimation and statistical analysis of exponential polynomial signals , 1998, IEEE Trans. Signal Process..

[25]  C. Werner,et al.  Satellite radar interferometry: Two-dimensional phase unwrapping , 1988 .

[26]  Boaz Porat,et al.  Estimation and classification of polynomial-phase signals , 1991, IEEE Trans. Inf. Theory.

[27]  Tze Fen Li.  Multipath time delay estimation using regression stepwise procedure , 1998, IEEE Trans. Signal Process..

[28]  Kenneth Steiglitz,et al.  Phase unwrapping by factorization , 1982 .