New Bounds and Optimal Binary Signature Sets ... - Semantic Scholar

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 59, NO. 4, APRIL 2011

1123

New Bounds and Optimal Binary Signature Sets–Part I: Periodic Total Squared Correlation Harish Ganapathy, Student Member, IEEE, Dimitris A. Pados, Member, IEEE, and George N. Karystinos, Member, IEEE

Abstract—We derive new bounds on the periodic (cyclic) total squared correlation (PTSC) of binary antipodal signature sets for any number of signatures K and any signature length L. Optimal designs that achieve the new bounds are then developed for several (𝐾, 𝐿) cases. As an example, it is seen that complete (𝐾 = 𝐿 + 2) Gold sets are PTSC optimal, but not, necessarily, Gold subsets of 𝐾 < 𝐿 + 2 signatures. In contrast, arguably against common expectation, the widely used Kasami sets are not PTSC optimal in general. The optimal sets provided herein are in this sense better suited for asynchronous and/or multipath code-division multiplexing applications. Index Terms—Binary sequences, code-division multiple access (CDMA), cyclic correlation, Gold sequences, Karystinos-Pados bounds, Kasami sequences, periodic correlation, signature design, total squared correlation, Welch bound.

I. I NTRODUCTION N code-division multiplexing applications, for example direct-sequence code-division multiple-access (DS-CDMA) cellular communication systems, each of the K participating signals/users is equipped with a unique identifying signature vector s𝑘 ∈ ℂ𝐿 , ∣∣s𝑘 ∣∣ = 1, 𝑘 = 1, 2, . . . , 𝐾. All signatures put together in the form of a matrix define what we call the signature matrix (or signature set)

I

△

S = [s1 s2 . . . s𝐾 ] ∈ ℂ𝐿×𝐾 .

(1)

In synchronous code-division multiplexing transmissions over well-behaved Nyquist channels, we are interested in using a signature set with the smallest possible total squared correlation (TSC) value [1]-[12] △

TSC(S) =

𝐾 ∑ 𝐾 ∑  𝐻 2 s𝑖 s𝑗 

(2)

𝑖=1 𝑗=1

Paper approved by G. E. Corazza, the Editor for Spread Spectrum of the IEEE Communications Society. Manuscript received July 23, 2009; revised August 1, 2010. This work was supported in part by the National Science Foundation under Grant CCF-0219903, and the U.S. Air Force Office of Scientific Research under Grant FA9550-04-1-0256. Material in this paper was presented at the IEEE Military Communications Conference (MILCOM), Atlantic City, NJ, October 2005. H. Ganapathy was with the Department of Electrical Engineering, State University of New York at Buffalo, Buffalo, NY 14260 USA. He is now with the Department of Electrical and Computer Engineering, The University of Texas, Austin, TX 78712 USA (e-mail: [email protected]). D. A. Pados is with the Department of Electrical Engineering, State University of New York at Buffalo, Buffalo, NY 14260 USA (e-mail: [email protected]). G. N. Karystinos is with the Department of Electronic and Computer Engineering, Technical University of Crete, Chania, 73100 Greece (e-mail: [email protected]). Digital Object Identifier 10.1109/TCOMM.2011.020411.090404

where H denotes the Hermitian operator. For complex/realvalued signature sets S ∈ ℂ𝐿×𝐾 or ℝ𝐿×𝐾 , if 𝐾 ≥ 𝐿, 2 TSC(S) ≥ 𝐾𝐿 [2]; of course, TSC(S) ≥ 𝐾 if 𝐾 < 𝐿. Over2 loaded (𝐾 ≥ 𝐿) sets with TSC equal to 𝐾𝐿 have been known as Welch-bound-equality (WBE) sets. Algorithms and studies for the design of complex or real WBE signature sets can be found in [3]-[12]. In digital transmission systems, however, it is necessary to have finite-alphabet signature sets. Recently, new bounds were derived on the TSC of binary antipodal signature sets together with optimal designs for almost all1 signature lengths and set sizes [13]-[15]. The sum capacity, total asymptotic efficiency, and maximum squared correlation of minimum-TSC optimal binary sets were evaluated in [16]. The sum capacity of several other signature set designs under potentially a binary or quaternary alphabet was examined in [17]. In this present paper, all developments that follow 𝐿×𝐾 . refer to binary antipodal signature sets in {±1} When asynchronous code-division multiplexing is attempted and/or the channel exhibits multipath behavior, apart from the total squared correlation between signatures we are also concerned about the individual periodic (cyclic) autocorrelation and the periodic (cyclic) cross-correlation values [18]. For notational simplicity, let s𝑇𝑘∣𝑙 (T is the transpose operator) denote the cyclic right-shifted version of s𝑇𝑘 ∈ 1×𝐿 {±1} , 𝑘 = 1, 2, . . . , 𝐾, by 𝑙 bit positions, 𝑙 = 0, 1, 2, . . . (hence, s𝑘∣0 = s𝑘∣𝐿 = . . . = s𝑘 ). First, we define the 𝐿×𝐾𝐿 cyclic extension matrix Sc ∈ {±1} of the signature set 𝐿×𝐾 S ∈ {±1} △ [ Sc = s1∣0 s2∣0 . . . s𝐾∣0 s1∣1 s2∣1 . .]. s𝐾∣1 . . . (3) . . . s1∣𝐿−1 s2∣𝐿−1 . . . s𝐾∣𝐿−1 . Then, we define the periodic total squared correlation (PTSC) of the signature set S as the TSC of Sc , △

PTSC(S) = TSC(Sc ),

(4)

and calculate PTSC(S) explicitly in (5). A detailed derivation of (5) is provided in the Appendix. The first two terms of the PTSC expressions in (5) (for L odd or L even) contain all periodic auto-correlation contributions. The third, triple summation term contains all periodic cross-correlation contributions. Minimizing PTSC2 1 The case 𝐾 = 𝐿 ≡ 1 (mod 4) remains open. Ding, Golin, and Kl𝜙ve [14] showed that the Karystinos-Pados TSC bound [13] is tight for 𝐾 = 𝐿 = 5 or 13, but not for 𝐾 = 𝐿 = 9. What happens when 𝐾 = 𝐿 = 17, 21, . . . is still unknown. 2 The PTSC metric has been referred to as mean periodic correlation in the past [19].

c 2011 IEEE 0090-6778/11$25.00 ⃝

1124

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 59, NO. 4, APRIL 2011

⎧  𝑇 2 2 ∑ ∑ 𝐿−1 ∑𝐿−1  𝑇  2   + 2𝐿 ∑𝐾 ∑𝐾  ⎨ 𝐾𝐿3 + 2𝐿 𝐾 𝑘=1[ 𝑙=1 s𝑘 s𝑘∣𝑙 𝑖=1 𝑗=1,𝑖 0 . Case 1: 𝐾 = 2, 𝐿 ∈ ℳG Set S to be a Golay set of size (2, 𝐿) [30], [31]. Then, 𝐾𝐿 ≡ 0 (mod 4), the Golay sets are PCS, therefore, PTSCoptimal. □ Another simple case of four-user (𝐾 = 4) PTSC-optimal sets can come from R. J. Turyn’s [31] “base sequences” b1 , b2 ∈ {±1}𝑀+1 , b3 , b4 ∈ {±1}𝑀 that have been found/tabulated for 𝑀 ≤ 32 [32], [33] and can be used to construct directly “aperiodic complementary quadruples” for all lengths in ∞ { } ∪ ′ △ 2 𝑚 𝑀 : 𝑀 ∈ ℳT where (16) ℳT =

Below, we develop two more involved design procedures with proofs of optimality included in the Appendix.

Case 4: 𝐾 = 2 or 𝐾 ≡ 0 (mod 4), 𝐿 ≡ 0 (mod 2), 𝐾 ≥ 𝐿2 Calculate 𝑟 = 𝐿 − 𝐾. Obtain a Hadamard matrix H𝐾 and create the initial template matrix Θ𝐿×𝐾 = [H𝐾 h1 h2 . . . h𝑟 ]𝑇 . Define the diagonal correction matrix

(18)

6

⎧ ⎡⎡ ⎤𝑇 ⎡ ⎤𝑇 ⎤⎫ ⎨ ⎬ ⎦ ⎣1 − 1 1 − 1 . . . 1 − 1⎦ ⎦ . C=diag Vec ⎣⎣1 1 . . . 1  

  ⎩ ⎭ 𝐾

𝑟

(19)

Calculate the PTSC-optimal set S by S𝑇 = Θ𝑇 C. Proof of optimality of S is included in the Appendix.

(20) □

A small but arguably worth mentioning side result from Case 3a is the design of a PTSC-optimal set of size 𝑇 (𝐾 = 3, 𝐿 = 4). The length-four signature [−1 1 1 1] is single-user PTSC-optimal (cf. (13)). A three-user, length-four PTSC-optimal set is produced below. Case 5: 𝐾 = {1, 3}, 𝐿 = 4 ′ Obtain the signature set S𝐿×(𝐾−1) from Case 3a and form the set ] [ ′ 𝑇 (21) S𝐿×𝐾 = S [−1 1 1 1] . By (13), the set is PTSC-optimal.

□

In some of our PTSC-optimal set designs that follow we utilize specific well-known types of individual seℳT = {(2𝑝 + 1) : 𝑝 ∈ {1, 2, 3, . . . , 32} ∪ ℳG } quences, namely Cyclic Hadamard, Barker [34], Lempel∪ {(2𝑝 + 1)(2𝑞 + 1) : 𝑝, 𝑞 ∈ {1, 2, 3, . . . , 32} ∪ ℳG , 𝑝 ∕= 𝑞} Cohn-Eastman [35], and Ding-Helleseth-Martinsen [36] se∪ {3𝑝 − 1 : 𝑝 ∈ {1, 2, 3, . . . , 24}}. (17) quences. Cyclic Hadamard sequences g ∈ {±1}𝑀 of length 𝑀 Case 2: 𝐾 = 4, 𝐿 ∈ ℳT 6 In our work, we identify and use such matrices to suitably modify the Set S to be an “aperiodic complementary quadruple” set of correlation structure of the initial template matrix in order to achieve the size (4, 𝐿) [32], [33]. Then, 𝐾𝐿 ≡ 0 (mod 4), the aperiodic PTSC bounds. ′

𝑚=0

1126

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 59, NO. 4, APRIL 2011

𝑀 ≡ 3 (mod 4) have ideal two-level periodic auto-correlation —hence, Cyclic Hadamard sequences are single-user PTSCoptimal,— 𝑇 g𝑀 g𝑀 ∣𝑚 =

{

𝑀, −1,

𝑚 = 0, 𝑀 ≡ 3 (mod 4), 𝑚 = 1, 2, . . . , 𝑀 − 1, (22)

and have been used in the past to construct Cyclic Hadamard matrices [37]-[39]. In particular, if △

ℳCH = {𝑀 : (i) 𝑀 prime congruent to 3 (mod 4) or (ii) 𝑀 congruent to 3 (mod 4) and product of 𝑛

“twin primes” 𝑝 and 𝑝 + 2 or (iii) 𝑀 = 2 − 1, = 2, 3, . . . } , (23) then there exists at least one systematic method [37]-[39] (reproduced in the Appendix) to construct a Cyclic Hadamard sequence g𝑀 when 𝑀 ∈ ℳCH . We also utilize Barker sequences (which are single-user PTSC-optimal with the same correlation spectrum as in (22)) of length five and length 𝑀 thirteen, specifically, g𝑀 ∈ {±1} , 𝑀 ∈ ℳB with △

ℳB = {5, 13}.

(24)

All known Barker sequences are tabulated in [40]. Finally, we 𝑀 use Lempel-Cohn-Eastman (LCE) sequences g𝑀 ∈ {±1} of lengths 𝑀 ∈ ℳLCE where △

ℳLCE = {𝑀 : 𝑀 ≡ 2 (mod 4), 𝑀 = 𝑝𝑛 − 1 where 𝑝 is an odd prime and 𝑛 = 1, 2, . . .} (25) and Ding-Helleseth-Martinsen (DHM) sequences g𝑀 𝑀 {±1} of lengths 𝑀 ∈ ℳDHM where △

ℳDHM = {𝑀 : 𝑀 = 2𝑝, 𝑝 ≡ 5 (mod 8)}.

⌈√ ⌉ ⌈ ⌉ Case 7a: 𝐾 ≡ 1 (mod 8) and 8 84𝐿 + 1 ≤ 𝐾 < 4 𝐿8 , 𝐿 ∈ ℳLCE ∪ ℳDHM ⌈ ⌉ 7b: 𝐾 ≡ 1 (mod 4) and 𝐾 ≥ 4 𝐿8 + 1, 𝐿 ∈ ℳLCE ∪ ℳDHM ′ For Case 7a, obtain the signature set S𝐿×(𝐾−1) from Case 3b. ′ For Case 7b, obtain the signature set S𝐿×(𝐾−1) from Case 4. Then, form the PTSC-optimal set (cf. (13)) ] [ ′ S𝐿×𝐾 = S g𝐿∣𝑙1 (29) where shift 𝑙1 ∈ {0, 1, . . . , 𝐿 − 1} is selected such that the first 𝐾 2 or 𝐾 elements of vector g𝐿∣𝑙1 , for Case 7a and 7b, respectively, contain an unequal number of ones and minus ones. □ ⌈√ ⌉ Case 8: 𝐾 ≡ 1 (mod 8), 𝐾 ≥ 8 84𝐿 + 1, 𝐿 ∈ ℳCH ∪ ℳB ′ Obtain the signature set S𝐿×(𝐾−1) from Case 3b and form the PTSC-optimal set (cf. (13)) ] [ ′ S𝐿×𝐾 = S g𝐿∣𝑙1 (30) where shift 𝑙1 ∈ {0, 1, . . . , 𝐿 − 1} is selected such that the first 𝐾 2 elements of vector g𝐿∣𝑙1 contain an unequal number of ones and minus ones. □ Case 9: 𝐾 = 𝐿, 𝐿 ∈ ℳCH ∪ ℳB Set 𝑁 = 𝐾 + 1, obtain a Hadamard matrix H𝑁 and remove ′ its first row and column. Call the resulting size-K matrix H𝑁 . “Correct” with the diagonal matrix C = diag {g𝐿 }

∈ to obtain the set

′

S𝑇 = H𝑁 C.

(26)

Both LCE and DHM sequences are single-user PTSC-optimal with periodic auto-correlation { 𝑀, 𝑚 = 0, 𝑇 g𝑀 g𝑀∣𝑚 = (27) ±2, 𝑚 = 1, 2, . . . , 𝑀 − 1, where 𝑀 ∈ ℳLCE ∪ ℳDHM . Construction procedures for LCE and DHM sequences can be found in [41]. We continue with the presentation of the design cases. ⌈√ ⌉ Case 6: 𝐾 ≡ 2 (mod 8) and 𝐾 ≥ 8 84𝐿 + 2, 𝐿 ∈ ℳCH ∪ ℳB

By (13), the set is PTSC-optimal.

(31) (32) □

Case 10: 𝐾 ∈ {𝐿 − 4, 𝐿 − 2} , 𝐿 ∈ ℳCH ∪ ℳB Design directly the PTSC-optimal set (cf. (13)) ⎤ ⎡⎡ ⎤𝑇 ⎥ ⎢ S𝐿×𝐾 = ⎣⎣1 1 . . . 1 − 1⎦ g𝐿 g𝐿∣1 . . . g𝐿∣𝐾−2 ⎦ . (33) 𝐿−1

□ Next, we proceed with the presentation of our optimal designs for overloaded systems.

′

Obtain the signature set S𝐿×(𝐾−2) from Case 3b and form the set ] [ ′ (28) S𝐿×𝐾 = S g𝐿∣𝑙1 g𝐿∣𝑙2 where shifts 𝑙1 , 𝑙2 ∈ {0, 1, . . . , 𝐿 − 1}, 𝑙1 ∕= 𝑙2 , are selected such that the first 𝐾 2 elements of vectors g𝐿∣𝑙1 and g𝐿∣𝑙2 contain an unequal number of ones and minus ones. By (13), the set is PTSC-optimal. □

B. Overloaded Systems (𝐾 > 𝐿) Case 1a: 𝐾 ≡ 0 (mod 2), 𝐿 ∈ ℳG 1b: 𝐾 ≡ 0 (mod 4), 𝐿 ∕∈ ℳG Calculate 𝑁 = 4⌊ 𝐾 4 ⌋, obtain a Hadamard matrix H𝑁 and keep only the first 𝐿 columns, h1 , h2 , . . . , h𝐿 . Design the 𝐿× 𝑁 set 𝑇 (34) S𝐿×𝑁 = [h1 h2 . . . h𝐿 ] . For Case 1a, if 𝐾 ≡ 0 (mod 4), then S in (34) is an 𝐿 × 𝐾 PTSC-optimal signature set. If 𝐾 ≡ 2 (mod 4), ′ design a PTSC-optimal set S𝐿×(𝐾−2) as above and obtain

GANAPATHY et al.: NEW BOUNDS AND OPTIMAL BINARY SIGNATURE SETS–PART I: PERIODIC TOTAL SQUARED CORRELATION ′′

S𝐿×2 from Underloaded Case 1 or Case 3 (if 𝐿 ∈ ℳG ′′ or 𝐿 = 2𝑛 , 𝑛 = 1, 2, . . . , respectively). Append S𝐿×2 to ′ S to form the 𝐿 × 𝐾 PTSC-optimal signature set ] [ 𝐿×(𝐾−2) ′ ′′ S S . For Case 1b, S in (34) is directly an 𝐿 × 𝐾 PTSC-optimal signature set.7 Proof of PTSC optimality is given in the Appendix. □ Case 2: 𝐾 ≡ 2 (mod 4), 𝐿 ∈ ℳCH ∪ ℳB Set 𝑁 = 𝐾 − 2, obtain a Hadamard matrix H𝑁 and keep only ′ the first 𝐿 columns. Call the resulting matrix H𝑁 . Then, ] [ ′𝑇 (35) S𝐿×𝐾 = H𝑁 g𝐿 g𝐿∣1 is PTSC-optimal. The proof is given in the Appendix.

□

Case 3: 𝐾 ≡ 1 (mod 4), 𝐿 ∈ ℳLCE ∪ ℳDHM ∪ ℳCH ∪ ℳB Set 𝑁 = 𝐾 − 1, obtain a Hadamard matrix H𝑁 and keep ′ only the first 𝐿 columns. Call the resulting matrix H𝑁 . Then, ] [ ′𝑇 (36) S𝐿×𝐾 = H𝑁 g𝐿 □

is PTSC-optimal.

Case 4: 𝐾 ≡ 3 (mod 4), 𝐿 ∈ ℳLCE ∪ ℳDHM ∪ ℳCH ∪ ℳB Set 𝑁 = 𝐾 + 1, obtain a Hadamard matrix H𝑁 and remove the first row and 𝑁 − 𝐿 columns. Call the resulting matrix ′ H𝑁 . “Correct” with the diagonal matrix C = diag {g𝐿 } ′

to {1, 2, . . . , 256} (at present, it does not appear of much practical interest to consider code-division applications outside this parameter range), we can calculate that Underloaded Cases 1 through 10 and Overloaded Cases 1 through 4 together represent 36.23% of all possible combination pairs 2 (𝐾, 𝐿) ∈ {1, 2, . . . , 256} . Certainly, tightness of the bounds and optimal PTSC designs under the remaining cases is an important open research problem. Direct comparison of our PTSC-optimal designs with the TSC bounds and optimal sets in [13]–[15], shows that Underloaded Case 3 when 𝐿 ≡ 0 (mod 𝐾), Underloaded Cases 7, 11, and all Overloaded cases except Case 2 are doubly, both PTSC and TSC, optimal. Furthermore, we can now establish that the familiar Gold sets [44], which have been widely used for their periodic correlation properties [18], are PTSC-optimal when full-sized (complete). To that respect, we recall [18], [44] that complete Gold sets are defined for 𝐾 = 𝐿 + 2, 𝐿 = 2𝑛 − 1, 𝑛 ≥ 3 and 𝑛 ∕≡ 0 (mod 4), and for every 𝑖 ∕= 𝑗, 𝑖, 𝑗 = 1, . . . , 𝐾, the periodic signature cross𝑛+2 𝑛+2 correlations have value −1, − 2⌊ 2 ⌋ − 1, or 2⌊ 2 ⌋ − 1. Consider such a Gold set G𝐿×𝐾 = [g1 g2 . . . g𝐾 ] where g1 , g2 are the two preferred m-sequences [45] with g𝑗𝑇 g𝑗∣𝑙 = −1, 𝑙 = 1, . . . , 𝐿 − 1 and 𝑗 = 1, 2.

(39)

The signatures g𝑗 , 𝑗 = 3, . . . , 𝐿 + 2, are constructed by the operation [18]

(37)

g𝑗 = g1 ⊙ g2∣𝑗−2 , 𝑗 = 3, . . . , 𝐿 + 2,

(38)

where ⊙ represents the Hadamard product (i.e. element-wise multiplication). We denote the cyclic extension matrix of G𝐿×𝐾 by G𝑐 and the rows of the extension matrix G𝑐 by d𝑇𝑘,𝑔 . Then, by (7),

to obtain the PTSC-optimal set S𝑇 = H𝑁 C.

1127

□ In the following section, we discuss these design findings and present some examples.

2

3

PTSC(G) = TSC(G𝑐 ) = 𝐾 𝐿 + 2𝐿

𝐿−1 2 +1

∑   d𝑇1,𝑔 d𝑗,𝑔 2 . 𝑗=2

IV. D ISCUSSION AND E XAMPLES In asynchronous code-division multiplexing communication  applications all possible periodic correlations s𝑇𝑖 s𝑗∣𝑙  , 𝑖, 𝑗 = 1, 2, . . . , 𝐾, 𝑙 = 0, 1, . . . , 𝐿 − 1, may appear in the sufficient statistic of the maximum likelihood multiuser detector [42]. This is also the case in synchronous codedivision multiplexing communications over multipath propagation channels when the number of chip-level resolvable paths is greater than or equal to ⌈ 𝐿+2 2 ⌉. The PTSC metric captures precisely the contribution of all periodic correlations and PTSC optimization aims at minimizing their negative effect in individual signal recovery by means of optimal maximum-likelihood multiuser detection or otherwise (signature matched-filtering, decorrelation, minimum-mean-squareerror filtering or else [43]). The PTSC-optimal design cases presented in the previous section constitute proof-by-construction of the tightness of the corresponding PTSC bounds developed in Section II. To acquire a quantitative feeling of the extend/coverage of the presented designs, if we restrict the domain of 𝐾, 𝐿 7 Hence, interestingly, overloaded direct Hadamard designs as described herein are optimal under both the PTSC and TSC metric as shown in [13].

(40)

(41) From (8) and (40) using the notation [𝑥]+ = min{𝑥, 𝐿 − 𝐿 𝑥, 2𝐿 − 𝑥}, we calculate the row correlations of G𝑐 in (42), where the last equality follows from (39). Substituting (42) in (41), we obtain PTSC(G) = 𝐾 2 𝐿3 + 𝐿(𝐿 − 1), which is equal to the bound in (13). Hence, complete (𝐾 = 𝐿 + 2) Gold signature sets are PTSC-optimal. In contrast, maybe against common belief among codedivision multiplexing practitioners, the bounds and designs of Sections II and III show that Gold subsets (choice of 𝐾 < 𝐿 + 2 sequences) are not PTSC-optimal in general. Fig. 1 shows as an example a (16, 31) Gold subset which has PTSC = 8213760 together with our optimal (16, 31) design (under Underloaded Case 3b) with minimum PTSC = 𝐾 2 𝐿3 = 7626496. Other signature sets well known for their periodic correlation properties are the small and largeset Kasami designs [18], [46]. We recall that small-set Kasami designs are defined for lengths 𝐿 = 2𝑛 − 1, 𝑛 ≡ 0 (mod 2), 𝑛 and have size 𝐾 ≤ 2 2 . The attained periodic cross-correlation 𝑛 𝑛 values are −1, − 2 2 − 1, 2 2 − 1, but their frequency of occurrence is not known in closed form as a function of 𝑛. Fig. 2 presents a (2, 15) small-set Kasami design with calculated PTSC = 15300 together with our optimal (2, 15)

1128

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 59, NO. 4, APRIL 2011

d𝑇 1, 𝑔 d𝑗,

𝑔

=

2 ∑ 𝑘=1

g𝑘𝑇 g𝑘∣𝑗−1 +

𝐿+2 ∑ 𝑘=2

g𝑘𝑇 g𝑘∣𝑗−1 = −2 +

𝐿+2 𝐿 ∑∑ 𝑘=2 𝑖=1

+ + 𝑔1 (𝑖)𝑔2 ([𝑖 + (𝑘 − 2)]+ 𝐿 )𝑔1 ([𝑖 + (𝑗 − 1)]𝐿 )𝑔2 ([𝑖 + (𝑘 − 2) + (𝑗 − 1)]𝐿 )

)( ) ( g2𝑇 g2∣𝑗−1 = −2 + g1𝑇 g1∣𝑗−1

(42)

= −1

G31×16

⎡− + − − − − + + + + + − − + − −⎤

⎡ + + + + + + + + + + + + + + + +⎤

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

+ − − + + − − + + − − + + − −⎥ ⎢+ − − + + − − + + − − + + − − +⎥ ⎢+ + + + − − − − + + + + − − − −⎥ ⎢+ − + − − + − + + − + − − + − +⎥ ⎢+ + − − − − + + + + − − − − + +⎥ ⎢+ − − + − + + − + − − + − + + −⎥ ⎢+ + + + + + + + + + + + + + + + +⎥ ⎢− − + − + − + − + − + − + − +⎥ ⎢+ + + − − + + − − + + − − + + − −⎥ ⎢− + + − − + + − − + + − − + + −⎥ ⎢+ + + + − − − − + + + + − − − −⎥ ⎢− + − + + − + − − + − + + − + −⎥ ⎢+ + − − − + + + + − − − − + +⎥ ⎢ ++−+− − − + − + + − + − − +⎥ = ⎢− + + + + + + + + + + + + + + + +⎥ ⎢+ − + − + − + + − + − + − + −⎥ ⎢− − + + − − + − + − − + + − − + +⎥ ⎢− + + − − + + − + + − − + + −⎥ ⎢+ + + + − − − − − + + + − − − −⎥ ⎢+ − + − − + − + + + − + − − + − +⎥ ⎢− − + + + + − − − − + + + − −⎥ ⎢− + + − + − − + − + + + − + − − +⎥ ⎢+ + + + + + + + − − − − − − −⎥ ⎢+ − + − + − + − − + − + − + − +⎥ ⎢+ + − − + + − − − − + + − − + +⎥ ⎢+ − − + + − − + − + + − − − + −⎦ ⎣+ + + + − − − − − − − − + + +++

−++−− −+++− −−+++ −+−++ −++−+ ++−−+ −−+++ +++−− −++−+ −+++− +−−−− +−+−− +−++− −−−−− −−−−− −+−−− +−−++ −−−+− −+−−+ −−+−− −−−+− −+−−+ ++−++ ++−−+ ++−−− ++−−− +−−−− −−−++ +−++− −−−−−

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ =⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

−−+++++ −−−++++ −−−−+++ +−−−−++ ++−−−−+ −−−++++ −+++−−− −+−−−++ ++−+++− +++−+++ +−−−+−− −+−−−+− −−+−−−+ +++−+++ −+++−++ −−+++−+ +++−−−+ −−−−+++ −−−−−++ +−−−−−+ −+−−−−− −−+−−−− −++−+++ +−++−++ ++−++−+ −++−++− −−++−++ +++−−+− −−−−++− +++++−−

−−+− +−−+ ++−− +++− ++++ −−−− −+++ ++−− −−−+ −−−− −+++ −−++ −−−+ −+++ +−++ ++−+ −−−+ −+++ +−++ ++−+ +++− −+++ ++−− +++− ++++ ++++ −+++ −+−− ++−+ +−−+

+−+−+−+−+−+−+−+−

Sopt 31×16

+−+−−+−+−+−++−+− ++−−−−++−−++++−−

(b)

(a) opt

Fig. 1. (a) G31×16 Gold subset with PTSC = 8213760. (b) Optimal signature set S31×16 designed under Underloaded Case 3b with PTSC = (16)2 (31)3 = 7626496.

⎡− −⎤

Kss 15×2

− − + − − + + − + − + + + +

⎢ ⎢ ⎢ ⎢ ⎢ =⎢ ⎢ ⎢ ⎢ ⎣

(a)

+ + + + + + − + + + − + − −

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

⎡− +⎤

Sopt 15×2

++ ++ +− −+ ++ +− −− −+ +− −+ +− −− −− −−

⎢ ⎢ ⎢ ⎢ ⎢ =⎢ ⎢ ⎢ ⎢ ⎣

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(b)

Fig. 2. (a) Kss 15×2 small-set Kasami with PTSC = 15300. (b) Optimal opt signature set S15×2 designed under Underloaded Case 5 with PTSC = (2)2 (15)3 + 4(15)(14) = 14340.

design (under Underloaded Case 5) with minimum PTSC = 𝐾 2 𝐿3 + 4𝐿(𝐿 − 1) = 14340. We directly conclude that, in general, small-set Kasami designs are not PTSC-optimal. Large-set Kasami designs [18], [46] are defined for 𝐿 = 𝑛 2𝑛 − 1 and even 𝑛. If 𝑛 ≡ 2 (mod 4), 𝐾 ≤ 2 2 (2𝑛 + 1); 𝑛 if 𝑛 ≡ 0 (mod 4), 𝐾 ≤ 2 2 (2𝑛 + 1) − 1. Fig. 3(a) shows a (67, 15) large-set Kasami design that has PTSC = 15157785. Fig. 3(b) shows our PTSC-optimal set Sopt 15×67 designed under Overloaded Case 4 with minimum PTSC value 15150585. Hence, in general, large-set Kasami are not optimal either. Arguably, in future communication systems overloaded code-division will be of primary interest. The overloaded signature set results presented in this paper constitute an early contribution toward improving our understanding and tools for this problem [47]-[50]. We conclude this section with an example of an overloaded (42 user signatures of length

31) PTSC-optimal design Sopt 31×42 given in Fig. 4. The set is designed by our Overloaded Case 2 procedure and has minimum PTSC value 52555044. V. C ONCLUSIONS We derived new bounds on the periodic (cyclic) total squared correlation (PTSC) of binary signature sets for any signature length 𝐿 and set size 𝐾 and provided optimal constructions for a variety of 𝐾, 𝐿 values that establish the tightness of the corresponding bounds. The constructions include underloaded (𝐾 ≤ 𝐿) and overloaded (𝐾 > 𝐿) design cases and cover, as an example, 36.23% of all possible combinations of 𝐾, 𝐿 in {1, 2, . . . , 256}. Side results of the presented research include proof of the PTSC optimality of full-sized Gold sets. In contrast, Gold subsets were readily seen to lack optimality. PTSC-optimal constructions described herein for small and large-set Kasamicompatible (𝐾, 𝐿) pairs, establish that the Kasami sets are not PTSC-optimal in general. In view of these findings, the developed PTSC-optimal sets take precedence whenever the periodic correlation properties of signatures is of concern in code-division multiplexing applications. This is particularly true for the new overloaded PTSC-optimal sets as candidates for future overloaded code-division communication applications.

GANAPATHY et al.: NEW BOUNDS AND OPTIMAL BINARY SIGNATURE SETS–PART I: PERIODIC TOTAL SQUARED CORRELATION

1129

⎡− + + + + − − + − − + − − − + + − + + + + − − + − − + − − − + + + − − − − + + − + + − + + + − − + − − − − + + − + + − + + + − − − + +⎤ −−+++++−+−−+−−−+++−−−−−+−++−+++−−−+++++−+−−+−−−+++−−−−−+−++−+++−−++

Kls15×67

+ − + + + − + − + − + + − − − + − + − − − + − + − + − − + + + + − + − − − + − + − + − − + + + − + − + + + − + − + − + + − − − + − +⎥ ⎢− − − + − − + + − + − + − − + + + − − + − − + + − + − + − − + + − + + − + + − − + − + − + + − − − + + − + + − − + − + − + + − − − + +⎥ ⎢+ + − − + − − + + − + + + − − + − + + + − + + − − + − − − + + − + − − − + − − + + − + + + − − + − + + + − + + − − + − − − + + − + + −⎥ ⎢+ − − − − + + − + + − + + + − − − + + + + − − + − − + − − − + + − + + + + − − + − − + − − − + + + − − − − + + − + + − + + + − − + − −⎥ ⎢− − − + + + + + − + − − + − − − + − − + + + + + − + − − + − − − + + + − − − − − + − + + − + + + − + + − − − − − + − + + − + + + − + − −⎥ ⎢ −+− − − + − + − + − − + + + − + − + + + − + − + − + + − − − + − + − − − + − + − + − − + + + − + − + + + − + − + − + + − − − + − +⎥ = ⎢+ + − − + − + + − + − + − − + + − + + − + + − − + − + − + + − − − + + − + + − − + − + − + + − − + − − + − − + + − + − + − − + + + − −⎥ ⎢− + + + − + + − − + − − − + + − − + + + − + + − − + − − − + + − + − − − + − − + + − + + + − − + + − − − + − − + + − + + + − − + − − +⎥ ⎢− + + + − + − − + − − + − − − + + + − − − − + + − + + − + + + − − − + + + + − − + − − + − − − + + + − − − − + + − + + − + + + − − + − −⎥ ⎢+ + − − − − + − + + − + + + − − − + + + + + − + − − + − − − + − − + + + + + − + − − + − − − + + + − − − − − + − + + − + + + − − + +⎦ ⎣− + − + + + − −+−+−++−−−−+−+++−+−+−++−−−+−+−−−+−+−+−−++++−+−−−+−+−+−−+++−+− +−−+−−++−+−+−−++−++−++−−+−+−++−−+−−+−−++−+−+−−++−++−++−−+−+−++−−+−− −+++−++−−+−−−++−+−−−+−−++−+++−−++−−−+−−++−+++−−+−+++−++−−+−−−++−++−

(a) ⎡+ + + + + + − + + + − − + + + − − − − − − + − − + − − + − + + + − − − − + − + + + + + − − − + + + − + − + − − + + − − − − − + + − − +⎤ +++−−++−++−++−+−+−−+++−+−−+−++−−+++−−−−+−+−+++++−−−−−−+−+−+−−−+−−++

opt

S15×67

+ + − + − − + + + + + − + + − + − − + + + + + − + − + − − + + + − + − − − + − + − − − + − − − − − + − + − + − − + − + − + − − + + − +⎥ ⎢− − − − + − − + − − − − + − + − − + + + − + + + − − − − + + + − + + + − − + + − + + + + + + + − − + − − − − + + + − − + + + − + − −⎥ ⎢− − − + − + − + − − + + + + − − − − − + + + − − + + − − + − − + + + − − − − − − − + + − + + + − + − − + + + + + + + − − + − − + − + +⎥ ⎢− − − + − + + − + + + − − − + + + − + + − − − + + − − − − + − − − − + − + − + − − − − − + − − + + + + + + + + + + − + − − + − + − + +⎥ ⎢+ − + − − + + − − − + + + − + + + + − − − + + + − + − + − + + − − − + − + − + + − − − + − + − + + + + + + − − + + − − − + + − − − − − −⎥ ⎢ −− − − + − + + + − − + + + + − + − − + + − − − − + − + − − + + − + + − + + + − − − + − − − − + − + + − − − + − + + + − + + + − + +⎥ = ⎢− − − − + − − − + − + + + + + − + + − + + − − + − − − + − + + − − − − − + + + − − − + + − + + − + − − + − − + − + − + + + − + + + − +⎥ ⎢+ + − − + + + − − + − + − + + + − − + + + + − − + − + + − − − − + + + − + + − + − − + − − − − + + + + − + − + − − − − − − − + + − + + −⎥ ⎢+ + − + − + − − + + − + + + − + + + − + − − − + + + − − + + + − + − + − + + − − + − − + − − + − − − + + + − − + − + − − + − − − + + −⎥ ⎢− − + − + − + − + − + − + + − − + − − − − − − − + − + − + − + − + + − + + − − + + − − + + − + + − + + + − + + + − + − − − − + + + − +⎦ ⎣+ + − − − − + ++−+++−++++−−+−++++−+−−−++−−+−−++−−−++−−++−−−+−++++−−−−−++−− −−−+−−+−−−−++−−+−+−−++++−−+−++−+−−+−++++++−−−−++++−−++++−−−−+−+−+−+ +++++++−+−+++−−+−++−−−−−−−−−+++++−+−−−++−−+−++−−−−−+−+++−−−+++++−−−

(b)

Fig. 3. (a) Kls15×67 large-set Kasami with PTSC = 15157785. (b) Optimal signature set Sopt 15×67 designed under Overloaded Case 4 with PTSC = (672 )(153 ) + (15)(14) = 15150585.

1≤𝑟≤

A PPENDIX A D ERIVATION OF PTSC EXPRESSION IN (5) From (4), PTSC(S) =

𝐾 ∑ 𝐾 𝐿−1 2 ∑ ∑ 𝐿−1 ∑   s𝑇𝑖∣𝑙1 s𝑗∣𝑙2  .

(43)

𝑖=1 𝑗=1 𝑙1 =0 𝑙2 =0

We split the quadruple summation in (43) into periodic autoand cross-correlations, 𝐾 𝐿−1 2 ∑ ∑ 𝐿−1 ∑   PTSC(S) = s𝑇𝑖∣𝑙1 s𝑖∣𝑙2  𝑖=1 𝑙1 =0 𝑙2 =0

+

𝐾 ∑

𝐾 ∑

(44)

𝐿−1 ∑ 𝐿−1 ∑

𝑖=1 𝑗=1, 𝑖∕=𝑗 𝑙1 =0 𝑙2 =0

2  𝑇  s𝑖∣𝑙1 s𝑗∣𝑙2  .

Since for 𝑙1 ≤ 𝑙2 , 𝑙1 , 𝑙2 = 0, 1, 2, . . . , 𝐿 − 1, 𝑖, 𝑗 = 1, 2, . . . , 𝐾, s𝑇𝑖∣𝑙1 s𝑗∣𝑙2

= =

s𝑇𝑖∣0 s𝑗∣𝑙2 −𝑙1 s𝑇𝑗∣0 s𝑖∣𝐿−(𝑙2 −𝑙1 )

(45)

and s𝑇𝑖 s𝑖 = 𝐿, 𝑖 = 1, 2, . . . , 𝐾, we can simplify (44) to PTSC(S) = 𝐿

𝐾 𝐿−1 𝐾 2 ∑ ∑  ∑  s𝑇 𝑖 s𝑖∣𝑙  + 𝐿 𝑖=1 𝑙=0

= 𝐾𝐿3 + 𝐿

𝐾 ∑

𝐿−1 ∑

𝑖=1 𝑗=1, 𝑖∕=𝑗 𝑙=0

𝐾 𝐿−1 𝐾 2 ∑ ∑  ∑  s𝑇 𝑖 s𝑖∣𝑙  + 2𝐿 𝑖=1 𝑙=1

2  𝑇  s𝑖 s𝑗∣𝑙 

𝐾 ∑

(46)

𝐿−1 ∑

𝑖=1 𝑗=1, 𝑖 𝐿) Case 1a: 𝐾 ≡ 0 (mod 2), 𝐿 ∈ ℳG 1b: 𝐾 ≡ 0 (mod 4), 𝐿 ∕∈ ℳG Recalling that the concatenation of two PCS sets is a PCS set [27], to establish optimality of the designs for both Cases 1a and 1b it suffices to prove that S in (34) is PTSC-optimal. Consider the cyclic  extension matrix Sc of S; all crosscorrelations d𝑇1 d𝑗  of Sc , 𝑗 = 2, 3, . . . , 𝐿, are zero by (9). We conclude that S is PTSC-optimal with PTSC(S) = 𝑁 2 𝐿3 . Case 2: 𝐾 ≡ 2 (mod 4), 𝐿 ∈ ℳCH ∪ ℳB ′𝑇 Partition set S into two sets S1 = H𝑁 and [ the signature ] S2 = g𝐿∣𝑙1 g𝐿∣𝑙2 . Consider the cyclic extension matrices S ) , and S2,c of ( c ,𝑇 S1,c ( S, S)1 , and S2 , respectively. Then, d1 d𝑗 S1,c = 0 and d𝑇1 d𝑗 S2,c = −2, 𝑗 = 2, . . . , 𝐿−1 2 + 1. (   𝑇 )  Hence,  d1 d𝑗 Sc  = 2, 𝑗 = 2, . . . , 𝐿−1 2 + 1. We conclude that S is PTSC-optimal with PTSC(S) = 𝐾 2 𝐿3 + 4𝐿(𝐿 − 1). A PPENDIX D C ONSTRUCTION OF C YCLIC H ADAMARD SEQUENCES g𝑀 , 𝑀 ∈ ℳCH Binary antipodal sequences g𝑀 = [ g𝑀 (0) g𝑀 (1) . . . 𝑇 g𝑀 (𝑀 − 1)] , 𝑀 ∈ ℳCH in (23), with ideal 2-level autocorrelation can be constructed as follows. (i) If 𝑀 is a prime congruent to 3 (mod 4) [51], ⎧ −1, if 𝑖 = 0,   ⎨ +1, if 𝑖 ∈ {1, 2, . . . , 𝑀 − 1} is a quadratic g𝑀 (𝑖) = residue mod 𝑀,   ⎩ −1, otherwise. (48) (ii) If 𝑀 is a prime congruent to 3 (mod 4) and product of twin primes 𝑝 and 𝑝 + 2 [52], define i//p to be +1 if 𝑖 is a

quadratic residue mod p and −1 otherwise. Calculate ⎧ +1, if 𝑖 = 0 or a multiple of   ⎨ 𝑝 + 2, g𝑀 (𝑖) = −1, if 𝑖 is a multiple of 𝑝,   ⎩ 𝑖//𝑝 𝑖// (𝑝 + 2) , otherwise. (49) (iii) If 𝑀 = 2𝑛 − 1, 𝑛 = 1, 2, . . ., let g𝑀 be an msequence [45] of length 𝑀 . ACKNOWLEDGMENT The authors wish to express their gratitude to the three anonymous reviewers whose comments and keen insight helped improve significantly the quality of this manuscript. In particular, many thanks are extended to Reviewer 1 for the suggested simplification of the derivation of (13) and calculation of (42). R EFERENCES [1] M. Rupf and J. L. Massey, “Optimum sequence multisets for synchronous code-division-multiple-access channels," IEEE Trans. Inf. Theory, vol. 40, pp. 1261-1266, July 1994. [2] R. L. Welch, “Lower bounds on the maximum cross correlation of signals," IEEE Trans. Inf. Theory, vol. IT-20, pp. 397-399, May 1974. [3] P. Vishwanath, V. Anantharaman, and D. N. C. Tse, “Optimal sequences, power control, and user capacity of synchronous CDMA systems with linear MMSE multiuser receivers," IEEE Trans. Inf. Theory, vol. 45, pp. 1968-1983, Sep. 1999. [4] S. Ulukus and R. D. Yates, “Iterative construction of optimum signatures sequences sets in synchronous CDMA systems," IEEE Trans. Inf. Theory, vol. 47, pp. 1989-1998, July 2001. [5] C. Rose, “CDMA codeword optimization: interference avoidance and convergence via class warfare," IEEE Trans. Inf. Theory, vol. 47, pp. 2368-2382, Sep. 2001. [6] C. Rose, S. Ulukus, and R. D. Yates, “Wireless systems and interference avoidance," IEEE Trans. Wireless Commun., vol. 1, pp. 415-428, Mar. 2002. [7] P. Viswanath and V. Anantharam, “Optimal sequences for CDMA under colored noise: a Schur-Saddle function property," IEEE Trans. Inf. Theory, vol. 48, pp. 1295-1318, June 2002. [8] P. Cotae, “Spreading sequence design for multiple cell synchronous DS-CDMA systems under total weighted squared correlation criterion," EURASIP J. Wireless Commun. Netw., vol. 2004, no. 1, pp. 4-11, Aug. 2004.

GANAPATHY et al.: NEW BOUNDS AND OPTIMAL BINARY SIGNATURE SETS–PART I: PERIODIC TOTAL SQUARED CORRELATION

[9] O. Popescu and C. Rose, “Sum capacity and TSC bounds in collaborative multibase wireless systems," IEEE Trans. Inf. Theory, vol. 50, pp. 2433-2440, Oct. 2004. [10] J. A. Tropp, I. S. Dhillon, and R. W. Heath Jr., “Finite-step algorithms for constructing optimal CDMA signature sequences," IEEE Trans. Inf. Theory, vol. 50, pp. 2916-2921, Nov. 2004. [11] G. S. Rajappan and M. L. Honig, “Signature sequence adaptation for DS-CDMA with multipath," IEEE J. Sel. Areas Commun., vol. 20, pp. 384-395, Feb. 2002. [12] P. Xia, S. Zhou, and G. B. Giannakis, “Achieving the Welch bound with difference sets," IEEE Trans. Inf. Theory, vol. 51, pp. 1900-1907, May 2005. [13] G. N. Karystinos and D. A. Pados, “New bounds on the total squared correlation and optimum design of DS-CDMA binary signature sets," IEEE Trans. Commun., vol. 51, pp. 48-51, Jan. 2003. [14] C. Ding, M. Golin, and T. Kl𝜙ve, “Meeting the Welch and KarystinosPados bounds on DS-CDMA binary signature sets," Des., Codes Cryptogr., vol. 30, pp. 73-84, Aug. 2003. [15] V. P. Ipatov, “On the Karystinos-Pados bounds and optimal binary DSCDMA signature ensembles," IEEE Commun. Lett., vol. 8, pp. 81-83, Feb. 2004. [16] G. N. Karystinos and D. A. Pados, “The maximum squared correlation, sum capacity, and total asymptotic efficiency of minimum total-squaredcorrelation binary signature sets," IEEE Trans. Inf. Theory, vol. 51, pp. 348-355, Jan. 2005. [17] F. Vanhaverbeke and M. Moeneclaey, “Sum capacity of equal-power users in overloaded channels," IEEE Trans. Inf. Theory, vol. 53, pp. 228-233, Feb. 2005. [18] D. V. Sarwate and M. B. Pursley, “Crosscorrelation properties of pseudorandom and related sequences," Proc. IEEE, vol. 68, pp. 593619, May 1980. ˘ [19] M. Stular and S. Toma˘zi˘c, “Mean periodic correlation of sequences in CDMA," in Proc. IEEE Region 10 Conf. Electric. Electron. Tech., vol. 1, Aug. 2001, pp. 287-290. [20] D. Y. Peng and P. Z. Fan, “Generalised Sarwate bounds on the periodic autocorrelations and cross-correlations of binary sequences," Electron. Lett., vol. 38, pp. 1521-1523, Nov. 2002. [21] B. J. Wysocki and T. A. Wysocki, “Modified Walsh-Hadamard sequences for DS CDMA wireless systems," Intern. J. Adaptive Control Signal Process., vol. 16, pp. 589-602, Sep. 2002. [22] J. L. Massey and T. Mittelholzer, “Welch’s bound and sequence sets for code-division multiple-access systems," Sequences II, Methods in Communication, Security, and Computer Sciences, R. Capocelli, A. De Santis, and U. Vaccaro, editors. Springer-Verlag, 1993. [23] L. Bömer and M. Antweiler, “Binary and biphase sequences and arrays with low periodic autocorrelation sidelobes," in Proc. IEEE Intern. Conf. Acoust,. Speech, Signal Process., Apr. 1990, vol. 3, pp. 1663-1666. [24] J. E. Stalder and C. R. Cahn, “Bounds for correlation peaks of periodic digital sequences," Proc. IEEE, vol. 52, pp. 1262-1263, Oct. 1964. [25] D. V. Sarwate and M. B. Pursley, “Performance evaluation for phasecoded spread-spectrum multiple-access communication–part II: code sequence analysis," IEEE Trans. Commun., vol. 25, pp. 795-799, Aug. 1977. [26] B. Schmidt, “Cyclotomic integers and finite geometry," J. Amer. Math. Soc., vol. 12, pp. 929-952, 1999. [27] L. Bömer and M. Antweiler, “Periodic complementary binary sequences," IEEE Trans. Inf. Theory, vol. 36, pp. 1487-1494, Nov. 1990. [28] W. H. Mow, “Optimal sequence sets meeting Welch’s lower bound," in Proc. IEEE Intern. Symp. Inform. Theory, Sep. 1995, p. 90. [29] K. Feng, J.-S. Shiue, and Q. Xiang, “On aperiodic and periodic complementary binary sequences," IEEE Trans. Inf. Theory, vol. 45, pp. 296-303, Jan. 1999. [30] M. Golay, “Complementary series," IEEE Trans. Inf. Theory, vol. IT-7, pp. 82-87, Apr. 1961. [31] R. J. Turyn, “Hadamard matrices, Baumert-Hall units, four symbol sequences, pulse compression and surface wave encodings," J. Combin. Theory, Series A, vol. 16, pp. 313-333, 1974. [32] D. Ž. Dokovi´c, “Note on periodic complementary sets of binary sequences," Des., Codes and Cryptogr., vol. 13, pp. 251-256, Mar. 1998. [33] D. Ž. Dokovi´c, “Aperiodic complementary quadruples of binary sequences," J. Combin. Math. and Combin. Comput., vol. 27, pp. 3-31, 1998. [34] R. H. Barker, “Group synchronizing of binary digital sequences," in Commun. Theory, pp. 273-287, W. Jackson editor. Butterworths, 1953. [35] A. Lempel, M. Cohn, and W. L. Eastman, “A class of balanced binary sequences with optimal autocorrelation properties," IEEE Trans. Inf. Theory, vol. IT-23, pp. 38-42, Jan. 1977.

1131

[36] C. Ding, T. Helleseth, and H. Martinsen, “New families of binary sequences with optimal three-level autocorrelation," IEEE Trans. Inf. Theory, vol. 47, pp. 428-433, Jan. 2001. [37] H.-Y. Song and S. W. Golomb, “On the existence of cyclic Hadamard difference sets," IEEE Trans. Inf. Theory, vol. 4, pp. 1266-1268, July 1994. [38] S. W. Golomb and H.-Y. Song, “A conjecture on the existence of cyclic Hadamard difference sets," J. Statist. Planning and Inference, vol. 62, pp. 39-41, 1997. [39] L. D. Baumert, “Cyclic difference sets," in Lecture Notes in Mathematics, vol. 182. Springer-Verlag, 1971. [40] S. W. Golomb and R. A. Scholtz, “Generalized Barker sequences," IEEE Trans. Inf. Theory, vol. IT-13, pp. 619-621, Oct. 1967. [41] H. D. Luke, H. D. Schotten, and H. Hadinejad-Mahram, “Binary and quadriphase sequences with optimal autocorrelation properties: a survey” IEEE Trans. Inf. Theory, vol. 49, pp. 3271-3282, Dec. 2003. [42] S. Verdu, “Minimum probability of error for asynchronous Gaussian multiple-access channels," IEEE Trans. Inf. Theory, vol. IT-32, pp. 8596, Jan. 1986. [43] S. Verdu, Multiuser Detection. Cambridge University Press, 1998. [44] R. Gold, “Optimal binary sequences for spread spectrum multiplexing," IEEE Trans. Inf. Theory, vol. IT-13, pp. 619-621, Oct. 1967. [45] N. Zierler, “Linear recurring sequences," J. Soc. Ind. Appl. Math., vol. 7, pp. 31-48, Mar. 1959. [46] T. Kasami, “Weight distribution formula for some class of cyclic codes," Coordinated Science Laboratory, University of Illinois, Urbana, tech. rep. R-285 (AD632574), 1966. [47] S. P. Ponnaluri and T. Guess, “Signature sequence and training design for overloaded CDMA systems," IEEE Trans. Wireless Commun., vol. 6, pp. 1337-1345, Apr. 2007. [48] G. Romano, F. Palmieri, and P. K. Willett, “Soft iterative decoding for overloaded CDMA," in Proc. IEEE Intern. Conf. Acoust., Speech, Signal Process., Mar. 2005, vol. 3, pp. 733-736. [49] M. K. Varanasi, C. T. Mullis, and A. Kapur, “On the limitation of linear MMSE detection," IEEE Trans. Inf. Theory, vol. 52, pp. 4282-4286, Sept. 2006. [50] J. H. Cho, Q. Zhang, and L. Gao, “A comparison of frequency-division systems to code-division systems in overloaded channels," IEEE Trans. Commun., vol. 56, pp. 289-298, Feb. 2008. [51] S. W. Golomb, Shift Register Sequences. Holden-Day, 1967; Aegean Park Press, 1982 (revised edition). [52] R. G. Stanton and D. A. Sprott, “A family of difference sets," Canadian J. Math., vol. 10, pp. 73-77, 1958.

Harish Ganapathy (S’05) was born in Chennai, India, on March 8, 1983. He received his B.S. and M.S. degrees in electrical engineering from the State University of New York at Buffalo in May 2003 and May 2005, respectively. He is currently a Ph. D. candidate in electrical engineering at The University of Texas, Austin. His current research interests lie broadly in optimization as applied to wireless networks, including both physical layer and networking aspects. His industry experience includes internships at Qualcomm Inc. in the years 2006 and 2008, and at Freescale Semiconductor Inc. in 2007.

Dimitris A. Pados (M’95) was born in Athens, Greece, on October 22, 1966. He received the Diploma degree in computer science and engineering (five-year program) from the University of Patras, Greece, in 1989, and the Ph.D. degree in electrical engineering from the University of Virginia, Charlottesville, VA, in 1994. From 1994 to 1997, he held an Assistant Professor position in the Department of Electrical and Computer Engineering and the Center for Telecommunications Studies, University of Louisiana, Lafayette. Since August 1997, he has been with the Department of Electrical Engineering, State University of New York at Buffalo, where he is presently a Professor. He served the Department as Associate Chair in 2009-2010. Dr. Pados was elected three times University Faculty Senator (terms 2004-06, 2008-10, 2010-12) and served on the Faculty Senate Executive Committee in 2009-10.

1132

His research interests are in the general areas of communication systems and adaptive signal processing with an emphasis on wireless multipleaccess communications, spread-spectrum theory and applications, coding and sequences, cognitive channelization and networking. Dr. Pados is a member of the IEEE Communications, Information Theory, Signal Processing, and Computational Intelligence Societies. He served as an Associate Editor for the IEEE Signal Processing Letters from 2001 to 2004 and the IEEE Transactions on Neural Networks from 2001 to 2005. He received a 2001 IEEE International Conference on Telecommunications best paper award, the 2003 IEEE Transactions on Neural Networks Outstanding Paper Award, and the 2010 IEEE International Communications Conference Best Paper Award in Signal Processing for Communications for articles that he coauthored with students and colleagues. Professor Pados is a recipient of the 2009 SUNY-system-wide Chancellor’s Award for Excellence in Teaching.

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 59, NO. 4, APRIL 2011

George N. Karystinos (S’98-M’03) was born in Athens, Greece, on April 12, 1974. He received the Diploma degree in computer science and engineering (five-year program) from the University of Patras, Greece, in 1997 and the Ph.D. degree in electrical engineering from the State University of New York at Buffalo in 2003. In August 2003, he joined the Department of Electrical Engineering, Wright State University, Dayton, OH as an Assistant Professor. Since September 2005, he has been an Assistant Professor with the Department of Electronic and Computer Engineering, Technical University of Crete, Chania, Greece. His current research interests are in the general areas of communication theory and adaptive signal processing with an emphasis on wireless and cooperative communications systems, low-complexity sequence detection, optimization with low complexity and limited data, spreading code and signal waveform design, and sparse principal component analysis. Dr. Karystinos received a 2001 IEEE International Conference on Telecommunications best paper award and the 2003 IEEE Transactions on Neural Networks Outstanding Paper Award. He is a member of the IEEE Communications, Signal Processing, Information Theory, and Computational Intelligence Societies and a member of Eta Kappa Nu.