IEEE COMMUNICATIONS LETTERS, VOL. 16, NO. 4, APRIL 2012 425

A Note on “A Novel ZCZ Code Based on m-Sequences and itsApplications in CDMA Systems”

Sujit Jos, Member, IEEE, and Jinesh P. Nair, Member, IEEE

Abstract—This short communication shows that zero correla-tion zone sequences could be generated by using any odd-lengthbinary sequence with ideal two-level autocorrelation functionin place of the m-sequence using the exact same procedureproposed in the original paper. This removes the restrictionon the spreading factor of the m-ZCZ sequences which couldassume only the form 2n for some integer n. The spreading factorcan now be any N for which an ideal two-level autocorrelationsequence of period N − 1 exists.

Index Terms—ZCZ sequences, spreading codes, DS-CDMA.

I. INTRODUCTION

IN the letter [1], m-ZCZ codes are proposed for MAI-freereception over multi-path fading channels in DS-CDMA

systems. The despreading codes are configured differentlyfrom the transmitted codes as is the case with the approachof sequence design using mismatched filtering [2], [3].

In [1], the periodic autocorrelation(ACF) and periodic cross-correlation function(CCF) of m-ZCZ codes are defined by [1,eq. (1)]. The property of the m-sequence used in designing them-ZCZ codes is the ideal two-level autocorrelation functionof the m-sequence [4]. A binary sequence x0, x1, ...., xN−1 ofodd period N is said to have ideal two-level autocorrelationfunction if the periodic autocorrelation function R(k) of thesequence satisfies

R(k) =

{N if k = 0−1 if 0 < k < N

}, (1)

where, R(k) at delay k is defined as

R(k) =

N−1∑i=0

xix(i+k)modN . (2)

Therefore, the m-sequence used to generate the m-ZCZcodes could be replaced with any odd-length binary sequencewith ideal two-level autocorrelation function. Many well de-fined constructions of binary sequences with ideal two-level

Manuscript received December 8, 2011. The associate editor coordinatingthe review of this letter and approving it for publication was G. Karagiannidis.

S. Jos and J. P. Nair are with Samsung Advanced Institute of Tech-nology India Labs, Samsung Bangalore, India (e-mail: {sujit.jos, ji-nesh.p}@samsung.com).

Digital Object Identifier 10.1109/LCOMM.2012.020712.112492

autocorrelation function has been reported in the literature.Some examples of such binary sequences include Legendresequences [5], Halls sextic residue sequences [6], GMWsequences [7] and multiple trace-term sequences [4].

The binary sequences with ideal two-level autocorrelationfunction are known for odd periods of the following threeforms [8]:

1) P = 2n − 1, where n is a positive integer.2) P = 4m − 1, P being a prime and m being a positive

integer.3) P = p(p+ 2), P being a product of primes p and p+2.

The spreading factor of the resulting ZCZ codes obtainedfrom an ideal two-level autocorrelation sequence is nowL=P+1, where P can assume any of the above forms. Therestriction in [1] that the spreading factor be of the form 2n ishence removed. This gives enhanced flexibility in the choiceof spreading factor/processing gain of the spread spectrumsystem. The size of the family is retained as J = �P/wmin�as in [1].

REFERENCES

[1] G. Ye, J. Li, A. Huang. and H.-H. Chen, “A novel ZCZ code based onm-sequences and its applications in CDMA systems,” IEEE Commun.Lett., vol. 11, no. 6, pp. 465–467, June 2007.

[2] H. Rohling and W. Plagge, “Mismatched-filter design for periodicalbinary phased signals,” IEEE Trans. Aerosp. Electron. Syst., vol. 25, no.6, pp. 890-897, Nov. 1989.

[3] Q. K Trinh, P. Fan, D. Peng, and M. Darnell, “Construction of optimalmismatched periodic sequence sets with zero correlation Zone,” IEEESignal Proc. Lett., vol. 15, pp. 341–344, 2008.

[4] S. W Golomb and G. Gong, Signal Design with Good Correlation forWireless Communications, Cryptography and Radar. Cambridge Univer-sity Press, 2004.

[5] J.-S. No, H.-K. Lee, H. Chung, H.-Y. Song, and K. Yang, “Tracerepresentation of Legendre sequences of Mersenne prime period,” IEEETrans. Inf. Theory, vol. 42, no. 6, pp. 2254–2255, Nov. 1996.

[6] J.-H. Kim and H.-Y. Song, “On the linear complexity of Hall’s sexticresidue sequences,” IEEE Trans. Inf. Theory, vol. 47, no. 5, pp. 2094–2096, July 2001.

[7] R. Scholtz and L. Welch, “GMW sequences (corresp.),” IEEE Trans. Inf.Theory, vol. 30, no. 3, pp. 548–553, May 1984.

[8] J. H. Kim, “On the Hadamard sequences,” Ph.D. dissertation, YonseiUniv., Dep. Electron. Eng., 2002.

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