GPS C/A Code Signal Structure - Fundamentals of Global Positioning System Receivers: A Software Approach - page 87

TABLE 5.5 Parity Encoding Equations

D 30

d 1 = D 1 ⊕

D 30

d 2 = D 2 ⊕

D 30 d 3 = D 3 ⊕

.
d 24 = D 24 ⊕ D 30
D 25 = D 29 ⊕ d 1 ⊕ d 2 ⊕ d 3 ⊕ d 5 ⊕ d 6 ⊕ d 10 ⊕ d 11 ⊕ d 12 ⊕ d 13 ⊕ d 14 ⊕ d 17 ⊕ d 18 ⊕ d 20
⊕ d 23
D 26 = D 30 ⊕ d 2 ⊕ d 3 ⊕ d 4 ⊕ d 6 ⊕ d 7 ⊕ d 11 ⊕ d 12 ⊕ d 13 ⊕ d 14 ⊕ d 15 ⊕ d 18 ⊕ d 19 ⊕ d 21
⊕ d 24
D 27 = D 29 ⊕ d 1 ⊕ d 3 ⊕ d 4 ⊕ d 5 ⊕ d 7 ⊕ d 8 ⊕ d 12 ⊕ d 13 ⊕ d 14 ⊕ d 15 ⊕ d 16 ⊕ d 19 ⊕ d 20
⊕ d 22

D 28 = D 30 ⊕

d 2 ⊕ d 4 ⊕ d 5 ⊕ d 6 ⊕ d 8 ⊕ d 9 ⊕ d 13 ⊕ d 14 ⊕ d 15 ⊕ d 16 ⊕ d 17 ⊕ d 20 ⊕ d 21

d 23
D 29 = D 30 ⊕

⊕

d 1 ⊕ d 3 ⊕ d 5 ⊕ d 6 ⊕ d 7 ⊕ d 9 ⊕ d 10 ⊕ d 14 ⊕ d 15 ⊕ d 16 ⊕ d 17 ⊕ d 18 ⊕ d 21 ⊕

d 22 ⊕ d 24

D 30 = D 29 ⊕ d 3 ⊕ d 5 ⊕ d 6 ⊕ d 8 ⊕ d 9 ⊕ d 10 ⊕ d 11 ⊕ d 13 ⊕ d 15 ⊕ d 19 ⊕ d 22 ⊕ d 23 ⊕ d 24

sign change. If D 30 = 1, then D i = 0 changes to d i = 1and D i = 1 changes to

d i 1 to 24). This operation changes the signs of the source bits.

These values of d i are used to check the parity relation given through D 25
to D 30 .
In a receiver the polarity of the navigation data bits is usually arbitrarily
assigned. The operations listed in Table 5.5 can automatically correct the polarity.
If D 30 =

= 0 (for i =

0, the polarity of the next 24 data bits does not change. If the D 30 = 1,

the polarity of the next 24 data will change. This operation takes care of the
polarity of the bit pattern.
The equations listed in Table 5.5 can be calculated from a matrix operation.
This matrix is often referred as the parity matrix and defined as ( 7 )

123456789 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4

111011000111110011010010
011101100011111001101001
101110110001111100110100
010111011000111110011010
101011101100011111001101
001011011110101000100111

H =

( 5 . 3 )
This matrix matches the last six equations in Table 5.5. If a certain d i is present a
1 will be placed in the matrix. If a certain d i does not exist, a zero will be placed
in the matrix. Note that each row in H is simply a cyclic shift of the previous row
except for the last row. In order to use the parity matrix, the following property
must be noted. The similarity between modulo-2 and multiplication of + 1and

− 1 must be found first. The results in Table 5.2 are listed in Table 5.6 again

for comparison.