09-05-2013, 03:01 PM
Method for generating sets of orthogonal sequences
ABSTRACT
A new, systematic method of generating orthogonal sets of
sequences with good correlation properties is described. An
orthogonal set is defined as a collection of n sequences, of length
n chps, that are mutually orthogonal. Although there are many
possible orthogonal sets of a specified length, few have been
identified with a structured method of generation such as Walsh
codes and orthogonal Gold codes.
Introduction:
Orthogonal sequences are utilised in many applications,
in particular CDMA spread spectrum systems to improve
the bandwidth eficiency. The most common orthogonal
sequences, and those employed in or proposed for today’s communications
systems, are Walsh codes [ 11 and more recently orthogonal
Gold codes [2]. The new algorithm proposed is related to that
used to generate orthogonal Gold codes but produces large numbers
of different orthogonal sets with favourable cross-correlation
values between sets of the same size. The procedure generates (n -
1) distinct, orthogonal sets of n sequences with length n. Sequences
are represented by the notation (x,} = (xo, xl, x2 ... x + ~ a)n d {xk)
denotes a set of sequences: {x,O}, {xr’}, {x;} ... {x;-l}. The
sequences contain elements of the alphabet { 1, -1). Equivalent
definitions can be used by mapping 1 -+ 0 and -1 + 1 and replacing
multiplication operations between elements with modulo-2
addition.
Method of construction:
The orthogonal sequences are developed
from a set of sequences created using the Gold sequence construct.
Gold sequences are constructed from a preferred pair of maximal
length sequences, by the element-by-element multiplication of one
sequence with every time shift of the second sequence. Orthogonal
Gold sequences can then be constructed from this family of Gold
sequences by appending an additional ‘1’ to the end of each
sequence. Although, for optimum periodic cross-correlation the
two m-sequences should be preferred pairs [3], this construct can
be applied to any pair of m-sequences, of the same length, to produce
orthogonal sets of sequences. Using the new method, orthogonal
sequences are developed from a family of sequences
generated using the Gold construct, i.e. sequences generated by
the multiplication of one m-sequence with all shifts of a second msequence.