Optimization in Multi-Frequency Interferometry Ranging: Theory and   Experiment
  Multi-frequency interferometry (MFI) is well known as an accurate phase-based measurement scheme. The paper reveals the inherent relationship of the unambiguous measurement range (UMR), the outlier probability, the MSE performance with the frequency pattern in MFI system, and then provides the corresponding criterion for choosing the frequency pattern. We point out that the theoretical rigorous UMR of MFI deduced in the literature is usually optimistic for practical application and derive a more practical expression . It is found that the least-square (LS) estimator of MFI has a distinguished "double threshold effect". Distinct difference is observed for the MSE in moderate and high signal-to-noise ratio (SNR) region (denoted by MMSE and HMSE respectively) and the second threshold effect occurs during the rapid transition from MMSE to HMSE with increasing SNR. The closed-form expressions for the MMSE, HMSE and Cramer-Rao bound (CRB) are further derived, with HMSE coinciding with CRB. Since the HMSE is insensitive to frequency pattern, we focus on MMSE minimization by proper frequency optimization. We show that a prime-based frequency interval can be exploited for the purpose of both outlier suppression and UMR extension and design a special optimal rearrangement for any set of frequency interval, in the sense of MMSE minimization. An extremely simple frequency design method is finally developed. Simulation and field experiment verified that the proposed scheme considerably outperforms the existing method in UMR as well as MSE performance, especially in the transition from MMSE to HMSE, for Gaussian and non-Gaussian channel. 
Li Wei, Wangdong Qi
cs.IT math.IT