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#### Chapter: Analog and Digital Transmission Section: CDMA, Direct Sequence CDMA, Codes. # Gold Sequences

Gold sequences have been proposed by Gold in 1967 and 1968. These are constructed by EXOR-ing two m-sequences of the same length with each other. Thus, for a Gold sequence of length m = 2l-1, one uses two LFSR, each of length 2l-1.

If the LSFRs are chosen appropriately, Gold sequences have better cross-correlation properties than maximum length LSFR sequences.

## Prefered sequences

Gold (and Kasami) showed that for certain well-chosen m-sequences, the cross correlation only takes on three possible values, namely -1, -t or t-2. Two such sequences are called preffered sequences. Here t depends solely on the length of the LFSR used. In fact, for a LFSR with l memory elements,

if l is odd, t = 2(l+1)/2 + 1, and

if l is even, t = 2(l+2)/2 + 1.

 lNumber of LFSR elements m = 2l-1Sequence length Number of m sequences Max. cross correlation of m-sequenceNormalized tcross-correlation of Gold sequence t/(2l-1) cross correlation of Gold-sequence, Normalized 3 7 2 0.71 5 0.71 4 15 2 0.60 9 0.60 5 31 6 0.35 9 0.29 6 63 6 0.36 17 0.27 7 127 18 0.32 17 0.13 8 255 16 0.37 33 0.13 10 1023 60 0.37 65 0.06 12 4095 144 0.34 129 0.03 ... ...
Enter a value for l, and m, t and the cross correlation.

Thus, a Gold sequence formally is an arbitrary phase of a sequence in the set G(u,v) defined by

G(u,v)= {u,v,u * v, u * Tv, u *T2 v, U * T(N-1) v}

Tk denotes the operator which shifts vectors cyclically to the left by k places, * is the exclusive OR operator and u, v are m-sequences of period generated by different primitive binary polynomials.

It is well known that the "partial crosscorrelation" values can be altered by changing the phases of the code sequences. In theory, then, it is possible to find optimal phases which minimize the interference in the desired data signal. However, for K users each employing a sequence of period N, there are a total of N K different sets of sequence phases possible. For a realistic system, e.g. , direct computation becomes intractable. Even when direct computation is performed, the reduction in interference of the optimal set of phases over the worst set of phases is 30%.    