NEW ORTHOGONAL BINARY USER CODES
FOR MULTIUSER SPREAD SPECTRUM COMMUNICATIONS
Radha Poluri and Ali N. Akansu
New Jersey Institute of Technology
Department of Electrical & Computer Engineering
University Heights, Newark, NJ 07102 USA
Emails: {rp74, Akansu}@NJIT.edu
ABSTRACT paper. A sample list of these new codes is also given in
the paper for further studies.
Walsh codes are perfectly orthogonal binary (antipodal)
block codes that found many popular applications over 2. MATHEMATICAL REMARKS AND SEARCH
many decades including synchronous multi-user
communications. It is well known that they perform Earlier studies in the literature on orthogonal binary
poorly for asynchronous multi-user communications. transforms suggest that linear phase, zero mean codes
Therefore, the Gold codes are the preferred user codes in outperform non-linear and non-zero mean codes in their
asynchronous CDMA communications. In this paper, BER performance [3-4]. Accordingly, sample space for
new sets of binary orthogonal user codes are introduced the design of the proposed new orthogonal codes is only
for asynchronous spread spectrum multi-user limited to zero mean and linear phase codes in this study.
communications. It is shown in this paper that the Therefore, for 8-bit (length 8) codes, binary sample space
proposed binary user code family outperforms the Walsh consists of 22 unique codes, for 16-bit codes there are
codes significantly and they match in performance with 326 candidate codes, and for 32-bit codes there are about
the popular, nearly orthogonal Gold codes closely for 38,000 codes.
asynchronous multi-user communications. We present
that there are a good number of such independent code Walsh codes perform poorly in AWGN channels for
sets of different sizes available in the binary space. They asynchronous communications when the circular shift of
might help us to increase service/multi-service the second code (following code in a stream of codes)
capabilities of future communications systems. matches with the first code or the complement of the first
code for any particular chip delay. Note that the decimal
1. INTRODUCTION values of all n-bit (size n) Walsh codes in a set are
multiples of 2(n/2) +1 or 2(n/2) -1. For example, 16-bit
This paper introduces complementary orthogonal sets of codes of the size 16 Walsh set are multiples of 255 or
binary space to the popular Walsh codes for spread- 257. In our orthogonal code design and search, such
spectrum service communications applications. Note that strict code conditions are avoided and the number of
Walsh family is a small subset of binary space with well- codes in the set that are multiples of 2(n/2) +1 or 2(n/2) -1 is
regulated functions in the bases of transforms [1-2]. In minimized. Moreover, we excluded the condition of
addition to the linear-phase property of the basis having only one code in the set for a given number of
functions, Walsh sets do not include any two sequences zero-crossings that was a requirement in the Walsh
in the set with the same number of zero-crossings. If one family.
relaxes the latter, complementary orthogonal sets in the
binary space with still linear phase are obtained. Orthogonal code sets with linear phase are iteratively
The computational power available today allowed us to selected from the binary sample space for the given
search for and obtain those new binary transform sets dimension. Our studies have shown that in any n-bit
reported in this paper. We compared their performance sample space (n-dimensional space), n-1 orthogonal code
with widely used codes like Gold codes and Walsh sets can be formed. DC code is added to the set to make
codes. Our comparisons include their time and frequency it a complete binary code set. Using this method, a
domain properties along with their BER performance in number of independent binary code sets can be generated
AWGN channels. It is shown that these new orthogonal from the sample space.
binary codes outperform Walsh codes significantly and
provide a comparable performance with Gold codes in all Note that the degrees of freedom is very limited when the
the measures and service scenarios considered in this dimension of the space is low, i.e. short codes or small
values of n. As an example, for the 8-bits and 16-bits
cases, there are not any other binary orthogonal sets 5 1915216974 4042322160
available that do not share some of their basis functions 6 1634031993 2779096485
with the Walsh family. These orthogonal binary code 7 1011666371 3284386755
sets can be formed by carefully including certain Walsh 8 116291424 2526451350
codes common in both sets. In contrast, for the cases of 9 1059591420 4278255360
32 bits or higher, a number of unique orthogonal binary 10 1455520917 2857740885
code sets can be generated. Table I displays the functions 11 1850886774 3425946675
of a typical new 16-bit code family in decimals for 12 1539580890 2573637990
convenience where the 0’s replaced by -1 value of the 13 429217383 4027576335
binary codes along with the Walsh set. Note that 8 of 14 902022060 2774181210
these functions are common with size 16 Walsh set. 15 1956395310 3275539260
16 1891164913 2523502185
Table I: A typical new 16-bit code family where 17 774992779 4294901760
common functions with Walsh set are bolded. 18 1792447657 2863289685
19 199813167 3435934515
20 304520119 2576967270
Function Index New Code Set Walsh Set 21 446421336 4042264335
22 477474360 2779077210
1 65535 65535 23 593345220 3284352060
2 383 43690 24 699821460 2526439785
3 3727 52428 25 939458579 4278190335
4 39321 39321 26 1202818530 2857719210
5 12979 61680 27 1286539981 3425907660
6 42405 42405 28 1292890290 2573624985
7 50115 50115 29 1366974090 4027518960
8 15683 38550 30 1567797573 2774162085
9 21717 65280 31 2064330529 3275504835
10 43605 43605 32 1760139030 2523490710
11 52275 52275
12 23333 39270
13 61455 61455
14 26393 42330 3. PERFORMANCE COMPARISONS
15 26857 49980
16 38505 38505 Typical 32-bit orthogonal Walsh and proposed codes
along with a 31-bit, nearly orthogonal, Gold codes are
Similarly, a typical set of size 32 binary sequences for displayed in Figure 1. Magnitude response functions of
the proposed family along with the Walsh set is these codes are shown in Figure 2. Note that the sample
displayed in Table II. Note that except the first function, sequence of the proposed orthogonal codes has more
which is DC, none of the basis functions of the new evenly spread frequency spectrum compared to sample
orthogonal binary set is a common function with the Walsh code of the same length.
Walsh set for 32-bit codes.
Cross-correlation sequences between a typical pair of
There are several other 32 dimensional unique binary codes (2-user case) for the three families under
sets available in the binary space. consideration are displayed in Figure 3. It is observed
that Gold and proposed codes have similar cross-
correlation (inter-code correlation) while sample Walsh
Table II: A typical new 32-bit code family along with pair has worse correlation properties.
the Walsh set.
For 32 bit codes, comparisons of sums and variances of
Function Index New Code Set Walsh Set the maximum, and sum of aperiodic cross correlations
between all the pairs of codes for Walsh, Gold, and the
1 4294967295 4294967295 proposed codes (three distinct orthogonal codes; Ortho1,
2 5973503 2863311530 Ortho2, Ortho3) are given in Table III. It is observed
3 629325403 3435973836 from the table that the cross-correlation properties of the
4 1193068317 2576980377 proposed codes and Gold codes are comparable while the
cross-correlations of Walsh code pairs are inferior to
others. These inter-code and intra-code correlation
properties dictate the performance of a service
communications system. Therefore, choosing the best
possible user codes with minimum intra-code and inter-
code correlation properties will improve the system
performance.
We considered an asynchronous communications
scenario with 2 users in the system. The goal here is to
investigate the BER performance of the communications
system with AWGN noise assumption and employing
different user code families. This helps us to understand
better the variations of the inter-code and inter-code
correlations of the codes whenever the noise is at
presence. The randomness of channel noise will perturb Fig. 1: Time domain representations of typical 32-length
the noise-free correlation properties of the user codes codes for Walsh and proposed (Ortho) families along
presented in Table III. Communications performance is with a 31-length Gold code.
computed by taking the average of BER performances
over all the possible pairs of codes. Figure 4 displays
BER performances of 16-bit Walsh and proposed code
families. It is clearly seen from this figure that the latter
significantly outperforms the first. Similarly, Figure 5
displays the BER curves for the case of 32-bits length
orthogonal and 31-bits Gold codes. It is observed from
these BER curves that the performance of the proposed
orthogonal code (Ortho3; set3) outperforms Walsh codes
significantly and it closely matches with that of Gold
code. Similarly, the other sets of the proposed orthogonal
binary code family, namely Ortho1 (set1) and Ortho2
(set2) perform comparable to the popular Gold codes.
4. CONCLUSIONS Fig. 2: Magnitude response functions of Walsh, Gold,
and proposed orthogonal (ortho) binary codes plotted in
Figure 1.
The growing demand for orthogonal, fixed power
(binary/antipodal) user codes require additional codes to
be available for service spread spectrum communications
applications. We presented in this paper that the Walsh
codes utilize only a small portion of the orthogonal
binary space due to their restrictions that are not
necessarily important for service communications. We
proposed a design methodology and derived a number of
orthogonal code sets that outperform Walsh codes and
closely match with Gold codes for asynchronous CDMA
applications. Service capabilities of existing
communications systems might be improved by
simultaneously using these independent code sets for
different users in software radios. Formulating the
problem with more elegance and extending the code
design for any length using mathematical analysis tools
are currently under study. Fig. 3: Cross-correlation (inter-code correlation)
sequences between typical pairs of codes.
Fig. 4: BER performance of length 16 Walsh and
proposed codes for asynchronous communications
channel in 2-users scenario.
Fig. 5: BER performance of various codes for
asynchronous communications channel in
2-users scenario.
TABLE III: Cross - Correlation (inter-code correlations) Comparisons of 31-bit Gold code and 32-bit Orthogonal Walsh
and proposed codes.
Walsh Gold Ortho1 Ortho2 Ortho3
Max Total Max Total Max Total Max Total Max Total
Sum: 103 935 138 1343 140 1336 133 1225 144 1361
Variance: .041 2.21 .0018 .1844 .0076 .4086 .0132 .8240 .0054 .2435
REFERENCES
[1] Walsh, J., L., “A closed set of Normal Orthogonal
Functions”, America. Journal of Mathematics, Vol. 55,
pp. 5-24, 1923.
[2] J.J. Sylvester, “Thoughts on Inverse Orthogonal [5] A.N. Akansu, P. Duhamel, X. Lin, and M.
Matrices, Simultaneous Sign Successions, and deCourville, “Orthogonal Transmultiplexers in
Tessellated Pavements in Two or More Contours,” Phil. Communications: A Review,” IEEE Trans. Signal
Mag.34, pp. 461-475, 1867. Processing, Vol. 46, No 4, pp. 979-995, April 1998.
[3] V. Dasilva and E.S. Sousa, “Performance of [6] A.N. Akansu, M.V. Tazebay and R.A. Haddad, “A
orthogonal CDMA codes for Quasi – Synchronous New Look at Digital Orthogonal Transmultiplexers for
Communication Systems”, Universal Personal CDMA Communications,” IEEE Trans. Signal
Communications, Vol. 2, 12-15 pp. 995-999, 1993. Processing, Vol. 45, No. 1, pp. 263-267, Jan. 1997.
[4] D. Gerakoulis and S. Ghassemzadeh, “Extended [7] A.N. Akansu and R.A. Haddad, Multiresolution
Orthogonal Code Designs With Applications in CDMA”, Signal Decomposition: Transforms, Subbands, Wavelets.
Spread Spectrum Techniques and Applications, 2000 2nd Edition, Academic Press, 2001.
IEEE Sixth International Symposium on, Volume: 2,
Vol. 2, pp. 657-661, Sept. 2000.
