Abstract

The main peak in the autocorrelation function of bar codes is here utilized as the signal of accurate alignment. Attention is therefore directed to the problem of how to increase the contrast of the main autocorrelation peak above the remaining part of the correlation signal and to the problem concerned with the upper limit that the contrast may attain. To meet different requirements of accurate alignment, a series of bar codes is designed. They are classified into several categories according to their performance characteristics. The internal relations between these categories are discussed in greater detail.

© 1988 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. M. I. Skolnik, Radar Handbook, (McGraw-Hill, New York, 1978), Sec. 20.5, pp. 20-18–20-24.
  2. R. H. Barker, “Group Synchronizing of Binary Digital Systems,” in Communication Theory, W. Jackson, Ed. (Academic, New York, 1953), pp. 273–287.
  3. M. Hall, “A Survey of Different Sets,” Proc. Am. Math. Soc. 7, 975 (1956).
    [CrossRef]
  4. J. E. Storer, R. Turyn, “Optimum Finite Code Groups,” Proc. IRE 46, 1649 (1958).
  5. R. Turyn, J. E. Storer, “On Binary Sequences,” Proc. Am. Math. Soc. 12, 394 (1961).
    [CrossRef]
  6. R. Turyn, “On Barker Codes of Even Length,” Proc. IEEE 51, 1256 (1963).
    [CrossRef]
  7. Y. Li, “Autocorrelation Function of a Bar Code System,” Mod. Opt. 34, 1571 (1987).
    [CrossRef]

1987 (1)

Y. Li, “Autocorrelation Function of a Bar Code System,” Mod. Opt. 34, 1571 (1987).
[CrossRef]

1963 (1)

R. Turyn, “On Barker Codes of Even Length,” Proc. IEEE 51, 1256 (1963).
[CrossRef]

1961 (1)

R. Turyn, J. E. Storer, “On Binary Sequences,” Proc. Am. Math. Soc. 12, 394 (1961).
[CrossRef]

1958 (1)

J. E. Storer, R. Turyn, “Optimum Finite Code Groups,” Proc. IRE 46, 1649 (1958).

1956 (1)

M. Hall, “A Survey of Different Sets,” Proc. Am. Math. Soc. 7, 975 (1956).
[CrossRef]

Barker, R. H.

R. H. Barker, “Group Synchronizing of Binary Digital Systems,” in Communication Theory, W. Jackson, Ed. (Academic, New York, 1953), pp. 273–287.

Hall, M.

M. Hall, “A Survey of Different Sets,” Proc. Am. Math. Soc. 7, 975 (1956).
[CrossRef]

Li, Y.

Y. Li, “Autocorrelation Function of a Bar Code System,” Mod. Opt. 34, 1571 (1987).
[CrossRef]

Skolnik, M. I.

M. I. Skolnik, Radar Handbook, (McGraw-Hill, New York, 1978), Sec. 20.5, pp. 20-18–20-24.

Storer, J. E.

R. Turyn, J. E. Storer, “On Binary Sequences,” Proc. Am. Math. Soc. 12, 394 (1961).
[CrossRef]

J. E. Storer, R. Turyn, “Optimum Finite Code Groups,” Proc. IRE 46, 1649 (1958).

Turyn, R.

R. Turyn, “On Barker Codes of Even Length,” Proc. IEEE 51, 1256 (1963).
[CrossRef]

R. Turyn, J. E. Storer, “On Binary Sequences,” Proc. Am. Math. Soc. 12, 394 (1961).
[CrossRef]

J. E. Storer, R. Turyn, “Optimum Finite Code Groups,” Proc. IRE 46, 1649 (1958).

Mod. Opt. (1)

Y. Li, “Autocorrelation Function of a Bar Code System,” Mod. Opt. 34, 1571 (1987).
[CrossRef]

Proc. Am. Math. Soc. (2)

R. Turyn, J. E. Storer, “On Binary Sequences,” Proc. Am. Math. Soc. 12, 394 (1961).
[CrossRef]

M. Hall, “A Survey of Different Sets,” Proc. Am. Math. Soc. 7, 975 (1956).
[CrossRef]

Proc. IEEE (1)

R. Turyn, “On Barker Codes of Even Length,” Proc. IEEE 51, 1256 (1963).
[CrossRef]

Proc. IRE (1)

J. E. Storer, R. Turyn, “Optimum Finite Code Groups,” Proc. IRE 46, 1649 (1958).

Other (2)

M. I. Skolnik, Radar Handbook, (McGraw-Hill, New York, 1978), Sec. 20.5, pp. 20-18–20-24.

R. H. Barker, “Group Synchronizing of Binary Digital Systems,” in Communication Theory, W. Jackson, Ed. (Academic, New York, 1953), pp. 273–287.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (12)

Fig. 1
Fig. 1

Simplest structures of light valves for accurate alignment [(a) and (b)]. The autocorrelation functions of single slits (a) and bar codes (BCs) (b). The output signals of the photocell and their A–D conversion. Δ1 and Δ2 in (c) and (d) are the errors in locating the reference pulse of alignment on the time axis due to the change of the direction of relative displacement (c) and due to the decrease in illumination level (d), respectively. It is seen that Δ1 and Δ2 for the BCs are much smaller than those for the single slit.

Fig. 2
Fig. 2

The BCs of a more complicated structure.

Fig. 3
Fig. 3

Explanation of the implications of Eqs. (9) and (17).

Fig. 4
Fig. 4

Estimation of the secondary maximum c m , the models used for calculating its lower bound c1 (a) and its upper bound c2 (b). The actual distribution of the autocorrelation function U (c).

Fig. 5
Fig. 5

Values of the secondary maximum c m when the constructional data of the BCs are known.

Fig. 6
Fig. 6

Light valves using two identical BCs with the distinguishing feature in light utilization (a) and in the rise (or fall) speed of the main peak (b), respectively.

Fig. 7
Fig. 7

Light valves using two identical BCs with the distinguishing feature in light utilization (a) and in the rise (or fall) speed of the main peak (b), respectively.

Fig. 8
Fig. 8

Light valves using two identical BCs with the distinguishing feature in expanding the range of detecting the relative displacement (a) and in a higher rise (or fall) speed of the main peak (b), respectively.

Fig. 9
Fig. 9

Light valves using two identical BCs with the distinguishing feature in expanding the range of detecting the relative displacement (a) and in an all-around moderate improvement on the performance characteristics (b), respectively.

Fig. 10
Fig. 10

Change of the state of a light valve from switching on (a) to off (b) in a short distance of ξ = w when all the light modules in a BCs have been separated by dark bars. Explanation of the method used for improving the rise (or fall) speed of the main peak.

Fig. 11
Fig. 11

Series of binary sequences with the initial elements [1,1,0,1, …] and their autocorrelation functions. An analytical approach to the optimization of the performance parameters β and SNR of a light valve.

Fig. 12
Fig. 12

Series of binary sequences with the initial elements [1,0,1,0, …], and their autocorrelation functions. An analytical approach to the optimization of the performance parameters β and SNR of a light valve.

Tables (1)

Tables Icon

Table I Comparison of the Performance Characteristics of the BCs Listed in Eqs. (31)

Equations (57)

Equations on this page are rendered with MathJax. Learn more.

[ a ] = [ a ( 0 ) , a ( 1 ) , a ( 2 ) , , a ( n ) ] ,
a ( 0 ) = a ( n ) = 1.
[ a ( i ) ] 2 = a ( i ) ,
a ( i ) + a ( j ) + a ( k ) + = 0 when a ( i ) = a ( j ) = a ( k ) = = 0 ,
a ( i ) a ( j ) a ( k ) = { 1 when a ( i ) = a ( j ) = a ( k ) = = 1 , 0 otherwise ,
i = 0 n a ( i ) = σ .
t ( x ) = i = 0 n a ( i ) rect ( x - i ) ,
U = K - + t ( x ) t ( x - ξ ) d x             ( V ) ,
U = k = - n n c ( k ) Λ ( ξ - k )             ( V ) .
c ( k ) = c ( - k ) = i = 0 n a ( i ) a ( i + k ) ,
[ c ( 0 ) c ( 1 ) c ( 2 ) . . . . . . . . . c ( n - 1 ) c ( n ) ] = [ a ( 0 ) a ( 1 ) , a ( 2 ) , , a ( n - 1 ) , a ( n ) 0 , a ( 0 ) , a ( 1 ) , , a ( n - 1 ) , a ( n - 1 ) 0 , 0 , a ( 0 ) , , a ( n - 3 ) , a ( n - 2 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 , 0 , 0 , , a ( 0 ) , a ( 1 ) 0. 0 , 0 , , 0 , a ( 0 ) ]     [ a ( 0 ) a ( 1 ) a ( 2 ) . . . . . . . . . a ( n - 1 ) a ( n ) ] .
[ c ] = [ c ( 0 ) , c ( 1 ) , c ( 2 ) , , c ( n ) ] .
c ( 0 ) = i = 0 n a 2 ( i ) = i = 0 n a ( i ) = σ ,
c ( n ) = a ( 0 ) a ( n ) = 1.
[ a ] [ a ( n ) , a ( n - 1 ) , , a ( 0 ) ] ,
i = - n n c ( k ) = k = - n n i = 0 n a ( i ) a ( i + k ) = i = - n n [ i = 0 n a ( i + k ) ] a ( i ) = σ i = 0 n a ( i ) = σ 2 .
S = i = 1 n c ( k ) = σ ( σ - 1 ) / 2 ,
β = [ c ( 0 ) - c m ] / c ( 0 ) = [ σ - c m ] / σ ,
c m = max [ c ( 1 ) , c ( 2 ) , , c ( n ) ] .
S 1 = c 1 ( 2 n - c 1 + 1 ) / 2.
c 1 2 - ( 2 n + 1 ) c 1 + σ ( σ - 1 ) = 0.
c 1 = [ ( 2 n + 1 ) - ( 2 n + 1 ) 2 - 4 σ ( σ - 1 ) ] / 2.
S 2 = c 2 ( n + 1 ) / 2.
c 2 = σ ( σ - 1 ) / ( n + 1 ) .
c 1 c m < c 2 ,
[ ( 2 n + 1 ) - ( 2 n + 1 ) 2 - 4 σ ( σ - 1 ) ] / 2 c m < σ ( σ - 1 ) / ( n + 1 ) .
α = σ / ( n + 1 ) ,
S L = [ c ( 0 ) - c ( 1 ) ] / w = [ σ - c ( 1 ) ] / w ,
SNR = 10 log 10 ( P s / P n )             ( dB ) .
P s = c 2 ( 0 ) = σ 2             ( V 2 ) ,
P n = 1 n k = 1 n c 2 ( k )             ( V 2 )
C ( n - 1 , σ - 2 ) = ( n - 1 ) ! ( σ - 2 ) ! ( n - σ + 1 ) ! .
[ 1 , 1 , 1 , 0 , 0 , 0 , 0 , 1 ] ,
[ 1 , 1 , 0 , 1 , 0 , 0 , 0 , 1 ] ,
[ 1 , 0 , 1 , 1 , 0 , 0 , 0 , 1 ] ,
[ 1 , 1 , 0 , 0 , 1 , 0 , 0 , 1 ] ,
[ 1 , 0 , 1 , 0 , 1 , 0 , 0 , 1 ] ,
[ 1 , 0 , 0 , 1 , 1 , 0 , 0 , 1 ] ,
[ 1 , 1 , 0 , 0 , 0 , 1 , 0 , 1 ] ,
[ 1 , 0 , 1 , 0 , 0 , 1 , 0 , 1 ] ,
[ 1 , 0 , 0 , 1 , 0 , 1 , 0 , 1 ] ,
[ 1 , 0 , 0 , 0 , 1 , 1 , 0 , 1 ] ,
[ 1 , 1 , 0 , 0 , 0 , 0 , 1 , 1 ] ,
[ 1 , 0 , 1 , 0 , 0 , 0 , 1 , 1 ] ,
[ 1 , 0 , 0 , 1 , 0 , 0 , 1 , 1 ] ,
[ 1 , 0 , 0 , 0 , 1 , 0 , 1 , 1 , ] ,
[ 1 , 0 , 0 , 0 , 0 , 1 , 1 , 1 ] .
c ( 1 ) = i = 0 n a ( i ) a ( i + 1 ) = 0.
( S L ) max = σ / w .
a ( i ) = 0             or             a ( i + 1 ) = 0.
c m = 1 ,
c 2 ( k ) = c ( k ) ,
k = 1 n c 2 ( k ) = k = 1 n c ( k ) = σ ( σ - 1 ) / 2.
i = 0 i j n j = 0 n a ( i ) a ( j ) a ( i + k ) a ( j + k ) = 0.
a ( i ) = 0 ,             a ( j ) = 0 ,             a ( i + k ) = 0 ,             a ( j + k ) = 0.
β = 1 - 1 / σ ,
SNR = 10 log 10 [ 2 n σ / ( σ - 1 ) ]             ( dB ) .

Metrics