12

# Gold codes obviously do not meet the meet the lower bound of

1/N derived by Welch.

# Kasami sequences use a similar method of to generate a smaller set of M = 2^{n/2 }binary sequences

of period N = 2^{n }-1 (for even n). Kasami sequences do so by taking a m-seqeunce (called c) and

forming a second binary sequence from it by taking every 2^{n/2 }+ 1 bit (called d). Then c is added

with a time shifted version of d using modulo two. The set which is created by taking all Kasami

sequences generated by different time shifts of d, as well as the original c and d sequences, form

the Kasami set of sequences. This set has is known to have 2^{N/2}(M) different sequences of length

2^{n}-1 (N). Thus, the _{ }= 2^{/2 }+ 1 , which satisfies the Welsh lower bound, making Kasami

sequences optimal for cross-correlation [17].

# 2.3 Analysis Method – Merit Factor

# A classical problem of digital sequence design is to determine those binary sequences

whose aperiodic autocorrelations are collectively small according to some suitable measure. This

is achieved by what is called the merit factor. It is used to determine whether coded signal is a

good or poor spreading signal. For example, let a real sequence of length N be represented by S

= _{0}, _{1}, … , _{−1 }. The aperiodic autocorrelation function of sequence S of length N is:

(5)

=0

_{ }_{ + }

;

+−1

_{ }_{ − }

;

−−1

0

≤ ≤ −1

=

− + 1 ≤ ≤ 0

=0