Phase Unwrapping and A Robust Chinese Remainder Theorem

7
L=3,Γ 1
=4,Γ 2
=5,Γ 3
=7, relative errors
3.5 x 10−3 maximal remainder error level τ
3
estimation error and its upper bound of n
2.5
2
1.5
1
0.5
estimation error of n divided by the mean of n
error upper bound divided by the mean of n
0
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Fig. 3.
Relative estimate error of ˆn in (23) and its upper bound in (24) using proposed robust CRT.
APPENDIX: PROOF OF THEOREM 1
From Condition (16) on x, it is not hard to see that the true solution n i in (8) falls in the range
0 ≤ n i < γ i for 1 ≤ i ≤ L. Thus, for 2 ≤ i ≤ L and any (¯n 1 , ¯n i ) ∈ S i , we have
∣
∣¯n iΓ i + ˜k i Γ i
M − ¯n 1Γ 1 − ˜k ∣
1 Γ ∣∣∣∣
1
M ∣ ≤ n i Γ i + ˜k i Γ i
M − n 1Γ 1 − ˜k 1 Γ 1
M ∣ . (27)
From (8),
Γσx = n i Γ i + ˜k i
M Γ i + ǫ i Γ i , 1 ≤ i ≤ L. (28)
Let µ i
∆ = ¯ni − n i for 1 ≤ i ≤ L. From (28), we replace ˜k i
M Γ i by Γσx − n i Γ i − ǫ i Γ i in both sides of (27)
and have
Thus, using (9) and (17) we have
|µ i Γ i − µ 1 Γ 1 − (ǫ i Γ i − ǫ 1 Γ 1 )| ≤ |ǫ i Γ i − ǫ 1 Γ 1 |.
|µ i Γ i − µ 1 Γ 1 | ≤ 2|ǫ i Γ i − ǫ 1 Γ 1 | ≤ 1 + 2τ
M (Γ 1 + Γ i ) < 1. (29)
Since µ i , Γ i , µ 1 , and Γ 1 are all integers, (29) implies
µ i Γ i = µ 1 Γ 1 , i = 2,3,...,L. (30)