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Modern Geomatics Technologies and Applications

PRN or the C/A code generator contains two shift registers known as Gold polynomial 1 (G 1) and Gold polynomial 2 (G 2),
which are known to Gold code [14]. A Gold code, also known as Gold sequence, is a type of binary sequence, used
in telecommunication (CDMA) and GNSS satellites. Gold codes are named after Robert Gold. Gold codes have bounded,
small cross-correlations within a set, which is useful when multiple devices are broadcasting in the same frequency range. A set
n
n
of Gold code sequences consists of 2 + 1 sequences each one with a period of 2 − 1. A set of Gold codes can be generated with
n
the following steps. Pick two maximum length sequences of the same length 2 − 1 such that their absolute cross-correlation is
less than or equal to 2 (n+2)/2 , where n is the size of the linear feedback shift register used to generate the maximum length sequence
(Gold '67). The set of the 2 − 1 exclusive-OR of the two sequences in their various phases together with the two maximum
n
n
length sequences form a set of 2 + 1 Gold code sequences. The highest absolute cross-correlation in this set of codes is 2 (n+2)/2 +
1 for even n and 2 (n+1)/2 + 1 for odd n [14]. The exclusive OR of two different Gold codes from the same set is another Gold code
in some phase. Within a set of Gold codes about half of the codes are balanced – the number of ones and zeroes differs by only
one. The sequence would have 512 ones and 511 zeroes, and these would appear to be distributed at random.

3. The Gold codes and their properties
The Gold codes are selected as spreading sequences for the GPS signals because of their characteristics. The most
important characteristics of the C/A codes are their correlation properties [15]. A general PRN code can be seen as an infinite
sequence of independent, identically distributed, random binary variables taking one of two values (0 or 1) with equal probability.
The shift register can be described as Equation 2:

y(n) = h y(n − k), (2)

where m is the number of adders, and h can take either of two values, 1 or 0, where 0 would indicate an absence of the
corresponding adder in the shift register. In polynomial form, the shift register can be expressed as Equation 3:

G(X) = X (3)

The essential qualities of the PRN sequences are defined by the correlation measures. Two infinite random sequences
should be uncorrelated. The closer a given PRN sequence comes in a truly random sequence, the better the PRN sequence is.
The correlation between two sequences x(k) and z(k) is described by the following Equation 4:

1
R(n) = y(k) z(n + k) (4)
N

Moreover, a random sequence would have very specific autocorrelation properties, i.e. a random code would correlate
with its own replica at only one point. The autocorrelation function can be found as a result of the multiplication of the sequence
and its own shifted version as Equation 5:

1
R(n) = y(k) y(n + k) (5)
N

where N is the sequence period. The PRN codes are generated by a deterministic system and have finite length. In general,
PRN codes are almost orthogonal to each other. In other words, they are almost uncorrelated. The orthogonal codes also have an
autocorrelation function, with a single correlation peak. Gold codes are not orthogonal, but they guarantee uniformly low cross-
correlation properties with other Gold codes. This property is very important for GPS signals, because all satellites transmit on
the same frequency. Gold codes are short, with a maximum length of 1023 chips. The GPS satellite uses a 10.23 MHz onboard
satellite clock to generate signals on all frequencies. The L1 C/A code is of 1 ms duration. The chip rate is therefore 1023
chips/ms.
Furthermore, the Gold Codes are generated using a pair of shift registers (gold polynomials) with feedback. A shift
register is just a list of bits that has an input end and an output end. The GPS registers are length 10. We’ll start them out with
all 1’s, calculate the feedback, and then shift all the numbers over to the right. Gold codes are used the GPS C/A ranging codes
are Gold codes of period 1,023 [11]. Every 1023rd period, the shift registers are reset with all ones, making the code start over.
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