IEICE TRANS. FUNDAMENTALS, VOL.E89–A, NO.9 SEPTEMBER 2006 2341 LETTER Special Section on Sequence Design and its Application in Communications Searching for the Best Biphase and Quadriphase Quasi-Barker Sequences * Ka Ming HO † , Nonmember and Wai Ho MOW †a) , Member SUMMARY Barker sequences have been used in many existing com- munications and ranging systems. Unfortunately, the longest known biphase and quadriphase Barker sequences are of lengths 13 and 15, re- spectively. In this paper, we introduce the so-called quasi-Barker sequences which achieve the minimum peak sidelobe level one within a certain win- dow centered at the mainlobe. As our key results, all the best biphase and quadriphase quasi-Barker sequences of lengths up to 36 and 21, respec- tively, were obtained by an efficient computer search. These sequences may provide better multipath resistance and tracking accuracy in ranging applications. key words: Barker sequences, aperiodic autocorrelation, peak sidelobe, window size, quasi-synchronous applications 1. Introduction Sequences with low aperiodic autocorrelation sidelobe lev- els are useful for spread spectrum communication and radar applications [1], [2]. Sequences with the aperiodic autocor- relation sidelobe levels at most one (i.e., the Barker condi- tion) are called the Barker sequences [3]. Unfortunately, the longest known biphase and quadriphase Barker sequences are of lengths 13 and 15, respectively [3]. To find longer biphase or quadriphase spreading sequences for applications which require a sufficient processing gain, the Barker condi- tion must be relaxed. To meet this practical need, search re- sults on the minimum peak sidelobe (MPS) sequences have been reported in the literature [4], [5]. In in-door ranging applications, the multipath delay spread is relatively small and hence the correlation function at the receiver side is mainly distorted by multipath echos at small relative de- lay (c.f. [6]). In addition, ranging for a (moving) object can be achieved in two modes, acquisition and tracking. In the tracking mode, quasi-synchronism (i.e., synchronization to within a certain number of chips) achieved during the ac- quisition mode may be exploited to improve the ranging ac- curacy. While the MPS sequences are probably ideal for the acquisition mode, sequences which can better exploit the quasi-synchronism available should be used in the tracking Manuscript received December 14, 2005. Manuscript revised March 31, 2006. Final manuscript received May 8, 2006. † The authors are with the Department of Electrical & Elec- tronic Engineering, Hong Kong University of Science and Tech- nology, Clear Water Bay, Hong Kong. ∗ This work was supported by the National Natural Science Foundation of China (NSFC)/Hong Kong Research Grants Coun- cil (RGC) Joint Research Scheme (Project No. N HKUST617/02/ No. 60218001). a) E-mail: w.mow@ieee.org DOI: 10.1093/ietfec/e89–a.9.2341 mode. For sequences to be used in the aforementioned ap- plication scenarios, the autocorrelation sidelobe levels close to the mainlobe are more important than those far away. Motivated by this observation, we introduce the so-called quasi-Barker sequences, which achieve the minimum peak aperiodic autocorrelation sidelobe level one within a certain window centered at the mainlobe (i.e., the quasi-Barker con- dition). If this window is larger than the multipath spread plus the uncertain timing offset in the tracking mode, the quasi-Barker sequences can provide a better ranging accu- racy than the MPS sequences, at those lengths or processing gains for which Barker sequences do not exist. The quasi- Barker sequences may be considered as a special class of the so-called low autocorrelation zone sequences [7]. By definition, quasi-Barker sequences exist for all lengths, but some can achieve a larger window than the others for the same length. The best quasi-Barker sequences of a given length are those that achieve the largest window and can thus tolerate the largest extent of delay spread and uncer- tain timing offset. In this work, as a first approach to find such sequences, an efficient exhaustive computer search is implemented to discover the best biphase and quadriphase quasi-Barker sequences, together with their attainable max- imum window sizes. 2. Definitions The peak sidelobe level (PSL) of a sequence a = (a 0 , a 1 ,... a L-1 ) with an M-phase alphabet A M =  exp(i2πk/M))  M-1 k=0 (where i = √ -1) and length L is de- fined as the peak magnitude of the aperiodic autocorrelation function C a (t) over all non-zero shifts t, i.e., PS L(a) = max 1≤t≤L-1 |C a (t)|, (1) where C a (t) = L-t-1  n=0 a n a ∗ n+t . (2) An M-phase sequence a can be equivalently represented by its M-ary index sequence m = (m 0 , m 1 ,..., m L-1 ) such that a n = exp(i2πm n /M), for n = 0, 1,..., L - 1, and C a (t) = L-t-1  n=0 e i2π(m n -m n+t )/M . (3) Besides, if the sequence satisfies the Barker condition Copyright c  2006 The Institute of Electronics, Information and Communication Engineers