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Faculty of Engineering and Information Sciences
2005
Performance comparison of sequences designed
from the Hall Difference Set and Orthogonal Gold
Sequences of Length 32
Jennifer Seberry
University of Wollongong, [email protected]
Beata J. Wysocki
University of Wollongong, [email protected]
Tadeusz A. Wysocki
University of Nebraska-Lincoln, [email protected]
Publication Details
Seberry, J. R., Wysocki, B. J. & Wysocki, T. A. (2005). Performance comparison of sequences designed from the Hall Difference Set
and Orthogonal Gold Sequences of Length 32. In M. Blaum, R. Carrasco & M. Darnell (Eds.), International Symposium on
Communication Theory and Applications (pp. 104-107). United Kingdom: HW Communications Ltd.
Research Online is the open access institutional repository for the University of Wollongong. For further information contact the UOW Library:
[email protected]
Performance comparison of sequences designed from the Hall Difference
Set and Orthogonal Gold Sequences of Length 32
Abstract
In the paper we perform a performance comparison between two sets of orthogonal spreading sequences i.e.
sequences based on the Hadamard matrix of order 32 constructed using the Hall difference set and
orthogonal Gold sequences of length 32. Both considered sets of sequences are characterised by low peaks in
the aperiodic cross-correlation functions and have also good aperiodic auto-correlation properties.
Disciplines
Physical Sciences and Mathematics
Publication Details
Seberry, J. R., Wysocki, B. J. & Wysocki, T. A. (2005). Performance comparison of sequences designed from
the Hall Difference Set and Orthogonal Gold Sequences of Length 32. In M. Blaum, R. Carrasco & M.
Darnell (Eds.), International Symposium on Communication Theory and Applications (pp. 104-107). United
Kingdom: HW Communications Ltd.
This conference paper is available at Research Online: http://ro.uow.edu.au/infopapers/2849
Performance Comparison of Sequences designed from the
Hall Difference Set and Orthogonal Gold Sequences
of Length 32
Jennifer Seberry, Beata J W ysocki, Tadeusz A W ysocki
Faculty o f Informatics, University o f W ollongong
N SW 2522, Australia
Wvsocki@ uow.edu.au
A bstract: In the paper w e perform a performance
comparison between tw o sets o f orthogonal spreading
sequences i.e. sequences based on the Hadamard
matrix o f order 32 constructed using the Hall
difference set and orthogonal Gold sequences o f length
32. Both considered sets o f sequences are characterised
by low peaks in the aperiodic cross-correlation
functions and have also good aperiodic auto-correlation
properties.
It is w ell known, e.g. [1], that if the sequences have
good aperiodic cross-correlation properties, the
transmission performance can be improved for those
C DM A system s w here different propagation delays
exist. In [2], W ysocki and W ysocki have shown that
spreading sequences derived from different Hequivalent matrices [3] o f Sylvester’s construction
have different aperiodic correlation properties, and that
by choosing the appropriate H-equivalent matrix, one
can significantly reduce the peaks in aperiodic cross­
correlation functions for the w hole set o f sequences.
The low est value o f peaks in the aperiodic cross­
correlation functions for the sequences derived from a
Hadamard matrix H-equivalent to the SylvesterHadamard matrix o f order /V = 32 published in [2] is
0.4063. This result is much lower than 0.9688 for
sequences derived from the Sylvester-Hadamard matrix
o f order jV = 32 in its w ell-know n canonical form. On
the other hand, the value o f 0.4063 is still much greater
than the Levenshtein bound [4] o f 0.1410 for the set o f
32 sequences o f order 32. O f course, the bound is
derived for sets o f bipolar sequences without imposed
condition o f orthogonality for their perfect alignment.
In [5], w e proposed a set o f orthogonal spreading
sequences o f order 32 derived from the Hall difference
set (31,15,7) [6], usually referred to as H32.o3 [7]. We
have found through computer search that the low est
value o f peaks in the aperiodic cross-correlation
functions for the sequences derived from a Hadamard
0 .3750.
It
H-equivalent to the matrix H jmb is
is
still
significantly
In the paper, w e present a performance comparison in
terms o f BER for direct sequences C DM A (DS
CDM A) system s utilizing sequences derived from the
matrix W 32.o3 and the best o f the orthogonal Gold
sequences sets o f length 32.
1. Introduction
matrix W m
Levenshtein bound but is a significant improvement
compared to the best result obtained from H-equivalent
Sylvester-Hadamard matrices. The value o f 0.3750 can
be also achieved for som e sets o f orthogonal Gold
sequences [ 8 ] or their H-equivalents.
higher than
the
The paper is organised as follow s. In section 2 , w e
introduce the Hall difference set construction and apply
it to produce a H 32-03 Hadamard matrix in its canonical
form and show the method to search for the matrix
W 32.03. In section 3, w e describe the method used to
generate orthogonal Gold sequences. Section 4 presents
simulation results for the DS C DM A system s utilizing
both sets o f sequences, and the paper is concluded in
section 5.
2. Sequences derived from
Hadamard matrix constructed
using the Hall difference set
Let a be a primitive root o f 31 ( a = 2 or 3 or 5) [9]. To
construct the matrix H32 .o3 w e first create a set:
^ = ^ , a 6^ '3 , e 6j,+5}
; = 0,1,23,4
(1)
which in the considered case is a set o f 15 integers:
A = {a l , . . . , a l5 )
(2)
when taken modulo 31.
Then w e create a circulant matrix B o f order 31, with
first row elem ents b \ d e f i n e d as:
b n A 1’
[ - 1,
11
otherw ise
(3)
The matrix H 32.03 is created from the matrix B by
adding the first row and the first column containing all
‘ is’, i.e.:
'l
H 3 2 -0 3 =
1
1
• -
I
d,
>
h ,l
*1,2
■•
*1.31
1
*1.3]
*1,1
' '
*1,30
1
*1,2
*1,3
'
'
(4)
The matrix H 32-o3 is therefore equal to [7]:
-
++++++++++++++++++++++++++++++++
+ - + +++ - + - + — + - - + ++
++--+-++
+ + - + + + + - + - + — + - - + + + ----------+ + - - + - +
+ + + - + + + + - + - + — + - - + + + ---------- + + - - + + - + +- ++ ++ - +
- + — + - - + ++• ++- - +
+ + - + + - + + + + - + - + — + - - ++ +
++ - + - + - ++ - + + ++ - + - + — + - - + + +
++ + - - + - + + - + ++ + - + - + — + - - ++ +
++
+ + - - + - + + - ++
++ - + - + — + - - + + + + + + - - + - + + - + + + + - + - + — + - - + + + --------+ - + + - - + - + + - + + + + - + - + — + - - + + + ------+ - - + + - - + - + + - + + + + - + - + — + - - +++ —
+ — + + — + - + + - +++ + - + - + — + - - + + + - +
++--+-++-++++-+-+ — +--++*+
+ + - - + - + + - + + + + - + - + — + - - ++ +
++
+ + - - + - + + - + + + + - + - + — + - - ++
H 3 2 -0 3 = + + +
++ - - +- + + - + + + + - + - + — +- - +
++ + +
++- - +- ++- ++++- +- + — + -+- + + +
++--+-++-++++-+-+ — ++ - - + ++
++--+-++-++++-+-+— +
++ - - ++ +
+ + - - + - + + - +•(•+ + -+ - + —
+ - +- - +++
+ + - - + -+ + - + + + + - + - + - +--+--+++
++- - + - ++- ++++- +- ++— +- - +++
++--+-++-++++-+-+
++ — + - - + + +
+ + - - + - ++ - + + + + - + + - + — +- - +++
+ + - - + - ++ - ++++- +
++ - + — + - - + ++
+ + - - + - + + - +++ + + - + - + — + - - + ++
++--+-++-++++
+ + - + - + — + - - ++ + --------- + + - - + - + + - + + +
+ ++ - + - +— +- - + ++
++ - - + - ++ - + +
+ + ++ - + - + — + - - + + +
++--+-++-+
+++++- + - + — + - - + + +
++- - + - + + The m odification is achieved by taking another
orthogonal N x N matrix Dm and the new set o f
sequences is based on a matrix Wm given by:
(5)
O f course, the matrix W w is also orthogonal [2].
In [2], it has been shown that the correlation properties
o f the sequences defined by
can be significantly
different to those o f the original sequences.
A sim ple class o f orthogonal matrices o f any order are
diagonal matrices with their elements d ,j fulfilling the
condition:
k-l={
for I* m
for l - m
with k being any non-zero real constant. That way, the
matrix \ ) N satisfies the condition:
»
a
where I# is the N x N
=
d
» d ; = * !i »
0
fo r
l*m
expO’0/)
fo r
l =m
(8)
l,m = I , . . . , N
The parameters $ > / = 1, ...» A/-, are phase coefficients,
which are real numbers with values from [0, 2 7 1 ). From
the implementation point o f view , the best class o f
sequences is the one o f binary sequences, i.e. when $ =
0 or n.
* u .
W„=H„D„
=
T o find the best possible m odifying diagonal matrix D*
w e can do an exhaustive search o f all possible bipolar
sequences o f length N, and choose the one, which leads
to the best performance o f the modified set o f
sequences.
However,
this
approach
is
very
computationally intensive, and even for a modest
+
values o f N, e.g. N = 20, it is rather impractical. Hence,
other search m ethods, like a random search, must be
considered, e.g. M onte Carlo algorithm.
In the considered case, w e have performed 10,000
random drawings o f binary sequences o f length 32, and
used them as diagonals o f the m odifying matrix D 32.
The matrix W 32.03< H-equivalent to H 32_03, giving the
low est maximum peaks in any pair o f aperiodic cross­
correlation functions is obtained when the diagonal o f
the matrix D32 is
[- + - + - + + + + --- + - - + + + - + + + + + + + + + + + + +]
The matrix W 32.03 is then:
- + - +- ++++
+-- +++
f ++ + + + + + + +
+-+- +- - + + -- +- ++
■-++- - +-++
- + ++ - + + - + +- + - +
+- +
+
-+
+ + + - - + - - + ++
++
++- - +++ +++ + + - ++ + + - + - + + ++
++- - +
-+ ++ - - ++ + - +
+ - - ++ - +++
++- + -+ + - - + + +
- - +++++ - +
-+ +
++
+++
++ - ' + + - - + ++
++- - + - + - + - - + + +
++-
++
+
++- +-+ - +- +
+- + - - +++
--+ +
+
+ +- - + - + + +
+ - - + + + ----+
++
+
+- - + ++-- - + - + ++
+ - + + - + + +++ - +
+ - - ++ + + - + - + + - + - + + - + + ++
+ ------ + - - + + +
- + + - + - - + +++ ' ++ - + - + - +
+ - - ++
w 32-03 =
+
+ - + ++ + - + - + - + +- + - +
+-- +
- + - ++ + - - ++ +- - + ++ - + - +
+-+
+
+ - - ++ +++ + + - + - +
+-- ++- +
+ + •-+ + + + - + - - + + + + - + - +
+
+++ + - + - +
+ + + - + -----+ ++ + - + ++
+ + + + - + + ++ - + - + - - - + + +- + + + + + + - + - + + + - + + + +- + - + - - + ------ + + - + + - + + + +- + + - +
++- +
+ +- - +
+ - + + - + ++ + - +
- + + ++ - - + - + - - +++
+ - - + - + + - + ++ + + ++ - + - +
++ + - - + - + + - + + + +
- + + +++
++ +
+ - + + - - + - + +- + + +
-+
+-- +- +
+
+- ++
+
-+
++- +- ++
+
++-- +- ++- +
- + - +- - + - + ++ + +++ ++ ++
++-- +- ++-
(?)
identity matrix.
T o preserve the normalization o f the sequences, the
elem ents o f D m being in general complex numbers,
must be o f the form:
The plot o f the peaks in the aperiodic cross-correlation
functions between any two pairs o f the sequences
derived from the matrix W 32.03 is given in Figure I the solid line. The set o f sequences derived from the
+
matrix W 32.o3 is characterized by
aperiodic correlation characteristics:
the
follow ing
C max = 0 .3 7 5 0
second element, i.e. the sequence (0,6) [8]. Hance, w e
have
G O n (a , b) = {(0, a), (0, a © b) , ... (0, a © r 306 )}( 10)
Rcc = 0.9682
R ac = 0 .9 8 4 4
where Cmax denotes the maximum peak value in the
magnitude o f aperiodic cross-correlation functions
between any pair o f the sequences in the set, RCc is the
mean square aperiodic cross-correlation for the set o f
sequences [1], and RAC is the mean square aperiodic
auto-correlation for the set o f sequences [1].
The bipolar spreading sequences.are obtained from the
sequences G 0 32(a,6) through a mapping in which ‘0 ’
corresponds to ‘ 1’ and ‘ 1’ corresponds to ‘-1 ’.
According to [9], there are 6 cyclically different msequences o f period 31. These are:
(an) = (0000100101100111110001101110101)
(a3n) = (0001010110100001100100111110111)
(aSn) = (0011011111010001001010110000111)
(a7n) = (0111110010011000010110101000111)
(« n n ) = (0 0 1 0 1 111 1 0 1 1 0 0 1 1 1 0 0 0 0 1 1 0 1 0 1 0 0 1 0 )
(flisn) = (0 1 1 1 0 1 1 0 0 0 1 1 1 1 1 0 0 1 1 0 1 0 0 1 0 0 0 0 1 0 1 )
By choosing these sequences, (a 3n) and ( a nn) as the
seed sequences in construction o f orthogonal Gold
sequences w e obtain a set o f orthogonal sequences o f
order 32 characterised by the follow ing correlation
parameters
C max = 0 .3 7 5 0
Rcc = 0 .9 6 7 5
R ac = 1 .0 0 7 8
which are almost the same as for the sequences derived
from the matrix W 32.03 .
F igure 1: Peaks in the magnitude of: aperiodic cross­
correlation fu nction s betw een an y tw o p a irs o f the
sequ en ces d eriv ed fro m m atrix W32-03 - so lid line,
a p eriodic auto-correlation functions - d o tted line
Synchronisation amenability o f the derived sequences
can be assessed b y exam ining the maximum off-peak
values in the magnitudes o f aperiodic autocorrelation
functions for the whole sequence set. From Figure 1 the dotted line, it is visible that the sequences derived
from the matrix W 32.03 have a very distinctive peak for
the perfect alignm ent and that there are no other
significant peaks for any non-zero shift.
3.
Orthogonal Gold sequences
Relative Shift between Sequences [chips]
The Gold sequences or Gold codes can be constructed
from a preferred pair o f m-sequences [9]. For a pair o f
preferred sequences a = {a n} and b = {6n} o f period N
= 2 5 - 1 = 3 1 the set
G (a , b) = {a, b, a © b, a ® Tb, . . . , a ® F 3
0
(9)
is a set o f Gold sequences o f order 31, where f b ; k=*
1 ,...,3 0 , denotes a cyclical shift o f a sequence b by k
p o sitio n s to the right.
The
orthogonal
Gold
sequences
G O (a,b)
are
constructed from G (a,b) b y adding a zero on the first
position o f all elem ents o f G (a,b), and disregarding the
Figure 2: Peaks in the magnitude of: a p eriodic cross­
correlation functions betw een an y tw o p a irs o f the
orthogonal G o ld sequences derived fro m m-sequences,
(a3n) a n d (a n ,,)- s o lid line, aperiodic auto-correlation
fu n c tio n s- d o tted line.
It should be noted here, that not all the preferred pairs
o f m -sequences lead to the same values o f those
correlation parameters. In fact, the value Cmax —0.3750
is the low est value one can achieve for the orthogonal
Gold codes. For the comparison with sequences
derived from the W 32. 03 matrix, in Fig.2, we present the
plots o f maximum magnitudes o f autocorrelation cross­
correlation functions and auto-correlation functions for
the constructed set o f orthogonal Gold sequences.
5.
,X1(J
Total num bor of packets = 160,000
Packet length = 1024 bits
No. of sim ultaneously active u sers = 8
Average B E R = 0.0021
Max. number of errors per packet = 92
0
10
20
30
40
50
60
70
80
90
1
Conclusions
In the paper w e presented the performance comparison
o f the orthogonal spreading sequences derived from the
Hadamard matrix o f order 32 (W 32.o3) constructed from
the H all difference set and orthogonal Gold sequences
derived from /n-sequences (o 3n) and (<2 n„) [9]. It has
been found that both set o f sequences are characterised
by almost identical aperiodic correlation parameters. In
fact, the values o f peaks in the aperiodic crosscorrelation functions and the maximum value o f the
off-peak autocorrelation functions are the sam e for
both sets o f sequences. Additionally, the error
performance o f the DS CDM A BPSK system s utilizing
sequences from both sets is also almost identical with
the average BER for 8 active users being the sam e, i.e.
0.0021, and the maximum number o f erroneous bits per
1024 packets being equal to .... And 94 for orthogonal
Gold sequences.
N um ber of e rrors per packet
References
Figure 4: H istogram o f the num ber o f errors in
received packets f o r a system utilizing D S CDMA with
B PSK a n d orthogonal G o ld codes derived from matrix
[1] I.Oppermann
and
B.S.Vucetic:
“Com plex
spreading sequences with a w ide range o f
correlation properties,” IEEE Trans. on Commun.,
vol. COM -45, p p.365-375, 1997.
W 32-03-
.. x104 _________________ ________
[2] B.J.W ysocki, T.W ysocki: “M odified
WalshHadamard Sequences for D S CDM A W ireless
System s,” Ini. J. Adapt. C on trol Signal Process.,
vol. 1 6 ,2 0 0 2 , pp.589-602.
Total n um bor o f packots = 160,000
Packet length = 1024 bits
No. o f sim ultane ou sly active u se rs = 8
Ave rage B E R = 0.0021
Max. num be r o f erro rs p or packet = 94
0
1
2
3
4
5
6
7
8
N um ber of errors per packet
[3] A.V.Geramita,
and
J.Seberry:
“Orthogonal
designs, quadratic forms and Hadamard matrices,”
Lecture N otes in Pure a n d A p p lied M athem atics,
vol.43, Marcel Dekker, N ew York and Basel,
1979.
9
10
Figure 4: H istogram o f the number o f errors in
received p a ck ets f o r a system u tilizing D S CDMA with
B PSK a n d orthogonal G o ld codes d eriv ed from msequences (a3n) a n d (ctun).
4.
Simulations
In Fig. 3 and Fig. 4, w e present the simulation results
for the 32 channel asynchronous DS CDM A BPSK
system utilizing sequences derived from the W 32.o3
matrix and the orthogonal Gold sequences derived
from w -sequences (a3n) and (flun)> respectively. In both
cases, w e had simulated the same number o f 8
randomly chosen simultaneous active users, and
transmitted the sam e number o f 1024-bit frames in
each o f the 32 p ossible channels. The results have been
then averaged across the 32 channels. The transmission
channel w as assumed to be an AW GN channel with an
Eh/No = 2 0 dB.
[4] V.I.Levenshtein: “A new lower bound on
aperiodic crosscorrelation o f binary codes,” 4,h
International Symp. On Comm unication Theory
an d Applications, ISCTA ’97, A m bleside, U.K., 1318 July 1997, pp. 147-149.
[5] M. Hall Jr. “A survey o f difference sets,” Proc.
Amer. Math. S oc.t 1 (1956), pp.975-986.
[6] J. Seberry, L. C. Tran, Y . Wang, B. J W ysocki, T.
A .W ysocki, Y. Zhao: “Orthogonal Spreading
Sequences Constructed U sing H all’s Difference
Set,” Proc. o f Sym poT IC ’04, Bratislava, Slovakia,
2 4-26 Oct. 200 4 , pp.82-85.
[7] J.Seberry: Library o f H adam ard M atrices,
http://www.uow.edu.au/~jennie/hadamard.html
[8] G.V.S.Raju and J.Charoensakwiroj: “Orthogonal
Codes performance in Multi-Code C DM A,” IEEE
Int. Conf. on System s, Man & C ybern etics, San
A ntonio, U SA , 5-8 O ct.2003,vol.2, pp.1928-1931.
[9] P.Fan and M.Darnell: “Sequence D esign f o r
Comm unications Applications,” John W iley &
Sons, N ew York, 1996.