Preferred Pairs m-sequences generation for Gold Codes

Preferred Pair m-Sequences for Gold Codes:

Spreading sequences in a spread spectrum systems can be generated with help of diversified codes like m-sequences,Gold Codes,Kasami Codes, Walsh Codes etc.

Compared to m-sequences (maximum length PN Sequences), Gold codes have worst auto-correlation properties but they have better cross-correlation properties.

The sequences associated with Gold Codes are produced by binary addition (modulo-2) of cyclic shifted versions of two m-sequence of length $N = 2^{n} - 1$ . If two m-sequences of cyclic shift $t_{1}$ and $t_{2}$ are used in Gold code generation, then the generated gold code will be unique for each combination of $t_{1}$ and $t_{2}$ . This means that a plenty of gold codes are generated with just two m-sequences which implies larger number of users can be accommodated in a given spread spectrum system.

When the two m-sequences are picked randomly for Gold code generation, then the cross-correlation property of the generated Gold code might not be as good as expected. Gold codes are generated using “Preferred” pairs of sequences that will guarantee good cross-correlation (as well as auto-correlation) properties of the generated Gold code.A method for selecting the preferred pairs for Gold Code generation was given by Gold [1] and is detailed here.

Generating Preferred Pairs of m-sequences:

1) Take a m-sequence $(d)$ for given length $N$ $(N = 2^{n} - 1)$ ,where $n$ is the number of registers in the LFSR).

2) Decimate the m-sequence by a decimation factor of q. This is our second sequence. $d^{'} = d [q]$

3) If the value of q is chosen according to the following three conditions, then the two m-sequences $d$ and $d^{'}$ will be preferred pairs.

A) $n$ is odd or $m o d (n, 4) = 2$

B) $q$ is odd and either $q = 2^{k} + 1$ or $q = 2^{2 k} - 2 k + 1$ for an integer $k$ .

C) The greatest common divisor of $n$ and $k$ satisfies the following conditions:
$g c d (n, k) = 1$ , when $n$ is odd
$g c d (n, k) = 2$ , when $m o d (n, 4) = 2$

Example:

Lets consider generation a preferred pairs of m-sequence of length N=63 (n=6). Since mod(n=6,4)=2, the first condition is satisfied. Taking $q = 5$ and $k = 2$ satisfies condition 2 and 3. So we will use a decimation factor of $q = 5$ in our simulation for preferred pairs generation.

Cross-Correlation :

The cross-correlation property of the preferred pairs for Gold code generation are three valued and the values are,

[- \frac{1}{N} t (n), - \frac{1}{N}, \frac{1}{N} (t (n) - 2)]

where t(n) is given by

t (n) = {\begin{cases} 2^{(n + 1) / 2} + 1 & if n is odd \\ 2^{(n + 2) / 2} + 1 & if n is even \end{cases}

For our example, the theoretical cross-correlation values for the preferred pairs of m-sequences are $[- 0.2698, - 0.0159, 0.2381]$ .

Matlab Code:

Results:

The theoretical and simulated cross-correlation values for the generated preferred pairs (with N=63 and q=5) matches perfectly (which can be seen from data cursors on the second plot).

Go here for : Hardware implementation of Gold code generator

Cross Correlation of Preferred Pairs m sequences

Reference:

[1]Gold, R. (1967), Optimal binary sequences for spread spectrum multiplexing (Corresp.), IEEE Transactions on Information Theory, 13 (4), pp. 619–621.

Preferred Pair m-Sequences for Gold Codes:

Generating Preferred Pairs of m-sequences:

Example:

Cross-Correlation :

Matlab Code:

File 1: preferredPairs.m

File2: genPNSequence.m

File3 : genDecimatedPNSequence.m

Reference:

See also: