Phase Unwrapping and A Robust Chinese Remainder Theorem

3
and
Theorem 1: Assume Γ i and Γ j defined in (10) are co-prime for 1 ≤ i ≠ j ≤ L. If
x < 1
σΓ Γ 1Γ 2 · · · Γ L (16)
M > (1 + 2τ)(Γ 1 + Γ L ), (17)
then, set S defined above contains only element n 1 , i.e., S = {n 1 }, and (n 1 , ¯n i ) ∈ S i implies ¯n i = n i
for 1 ≤ i ≤ L, where n i , 1 ≤ i ≤ L, are the true solution in (8).
□
Its proof is in Appendix. Note that the case when τ = 0 has been considered in [4] but there is an
error in the result (Theorem 1 in [4]) obtained in [4] where the set S i may not necessarily contain only
one pair (n 1 ,n i ) and also the proof in [4] has errors. From the above result, we can see that, when
Conditions (16) and (17) hold, the folding integers n i in (8) can be uniquely solved and the above results
in fact provide an algorithm for the solution of n i from the erroneous remainders ˜k i . When n i in (8) are
correctly solved, the unknown parameter x can be estimated as
(
ˆx = 1 L∑
n i + ˜k
)
i
λ i (18)
σL M
and the estimate error can be upper bounded by
i=1
|x − ˆx| ≤ 1 + 2τ
2Mσ
1
L
L∑
λ i . (19)
The above estimate error of x is due to the precision errors ǫ i and the remainder errors k i − ˜k i .
i=1
III. A ROBUST CHINESE REMAINDER THEOREM
We now go back to the CRT problem (4)-(6) where the precision errors ǫ i = 0 is because the CRT
concerns only integers and there is no fractional errors. From (4) and (5), we can see that, for each i
with 1 ≤ i ≤ L, modulo M i = MΓ i and remainder r i have a common factor Γ i and thus integer n also
has factor Γ i . Since a remainder is known in a priori to have a factor Γ i , its erroneous version ˜r i has a
factor Γ i too, i.e., ˜r i = ˜k i Γ i . Thus,
Assume
Then, for 1 ≤ i ≤ L,
˜r i − r i = ǫ i Γ i . (20)
|ǫ i | ≤ τ, i.e., |˜k i − k i | ≤ τ. (21)
n = n i M i + ˜k i Γ i + ǫ i Γ i . (22)