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Golay in [18] defined the merit factor as the ratio of main lobe energy to side lobes energy of

autocorrelation function of sequence S. The merit factor can be mathematically is defined as,

(6)

  •  

    =

2

−1 ≠0 0  2

2

the denominator term represents the energy in the side lobes. The merit factor must be as large

as possible for good sequences [19]. The larger the merit factor of a binary sequence that is

used to transmit information by modulating a carrier signal, the more uniformly the signal energy

is distributed over the frequency range; this is particularly important in spread-spectrum

communication.

When the merit factor is applied to m-sequence, Gold, and Kasami sequences found in

the previous section, [20] concluded that, through simulation, when the sequence length becomes

large, the merit factors of the of the Gold sequence and Kasami sequence converge to a value of

one. In fact, there have been studies that show the asymptotic merit factor of any maximal

length shift register sequence is three and the asymptotic merit factor of a twin-prime, Legendre,

and Jacobi (modified or not) sequences is six for the optimal shift [21]. So, from the standpoint

of the merit factor, the Gold and Kasami sequences seem less appealing than m-sequence or

other methods.

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