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Design of zero reference codes by means of a global optimization method

José Saez-Landete, José Alonso, and Eusebio Bernabeu

Open Access Open Access

Abstract

The grating measurement systems can be used for displacement and angle measurements. They require of zero reference codes to obtain zero reference signals and absolute measures. The zero reference signals are obtained from the autocorrelation of two identical zero reference codes. The design of codes which generate optimum signals is rather complex, especially for larges codes. In this paper we present a global optimization method, a DIRECT algorithm for the design of zero reference codes. This method proves to be a powerful tool for solving this inverse problem.

©2005 Optical Society of America

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Figures (3)
Fig. 1.
Fig. 1. Height of the second maximum of the autocorrelation signal respect to the number of slits in the ZRC. The ZRC has 50 elements. The continuous graph is the reached with the optimization and the dotted one is a lower bound calculated theoretically.

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Fig. 2.
Fig. 2. Six of the 136 autocorrelation signals found by the algorithm with n=50 and s=25. The value of the second maximum is S1 =11.

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Fig. 3.
Fig. 3. Six of the 136 ZRC’s found by the algorithm with n=50 and s=25.

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Equations (17)

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c = [ c 0 , c 1 , c 2 , … , c n ]
t ( x ) = ∑ j = 0 n c j rect ( x − j b )
rect ( x ) = { 1 if ∣ x ∣ < 1 2 , 0 if ∣ x ∣ ≥ 1 2 .
S ( τ ) = ∫ − ∞ + ∞ t ( x ) t ( x − τ ) dx = ∑ j = − n n a j · Λ ( τ − j b )
Λ ( x ) = { 1 − ∣ x ∣ if ∣ x ∣ ≤ 1 0 if ∣ x ∣ > 1
a k = a − k = ∑ j = 0 n − k c j c j + k , k = 0 , 1 , … , n .
S ( kb ) = S k = a k , k = 0 , 1 , … , n .
K = σ S 0 ,
σ ≥ [ ( 2 n + 1 ) − ( 2 n + 1 ) 2 − 4 n 1 ( n 1 − 1 ) ] 2 .
min x f ( x )
L x ≤ x ≤ U x
LB ≤ A · X ′ ≤ UB
x i ∈ I integer .
f ( c ) = max { S 1 , … , S n } , S k = ∑ j = 0 n − k c j c j − k
[ 0 , … , 0 ] ≤ c ≤ [ 1 , … , 1 ]
n 1 L ≤ [ 1 , … , 1 ] · [ c 1 ⋮ c n ] ≤ n 1 U
c ∈ ℤ n + 1
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