  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.
