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

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 59, NO. 5, MAY 2011

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New Bounds and Optimal Binary Signature Sets–Part II: Aperiodic 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 aperiodic total squared correlation (ATSC) of binary antipodal signature sets for any number of signatures K and any signature length L. We then present optimal designs that achieve the new bounds for several (𝐾, 𝐿) cases. As interesting -arguably- side results, we show that individual maximal merit factor sequences (for example Barker sequences) are single-user ATSC-optimal, while neither the familiar Gold nor the Kasami set designs are ATSC-optimal in general. The ATSC-optimal signature set designs provided in this work are in this sense better suited for asynchronous and/or multipath code-division multiplexing applications. Index Terms—Aperiodic correlation, aperiodic complementary sequences, Barker sequences, code-division multiple access (CDMA), cyclic correlation, Gold sequences, Karystinos-Pados bounds, Kasami sequences, maximal merit factor, periodic total squared correlation, total squared correlation, Welch bound.

I. I NTRODUCTION AND F ORMULATION

T

HERE has been renewed interest in recent years for optimized signature sets for the growing number of code-division multiplexing applications, for example multiuser ultra-wideband (UWB) systems, plain or multiple-input multiple-output (MIMO) code-division multiple access (CDMA), multiuser orthogonal-frequency-divisionmultiplexing (OFDM), etc. In general code-division multiplexing systems, each of the 𝐾 participating signals/users is assigned a unique identifying signature vector s𝑘 ∈ ℂ𝐿 , ∣∣s𝑘 ∣∣ = 1, 𝑘 = 1, 2, . . . , 𝐾. The 𝐾 signatures organized in the form of columns define the signature matrix (or signature set) △

S = [s1 s2 . . . s𝐾 ] ∈ ℂ𝐿×𝐾 .

(1)

For synchronous code-division multiplexing transmissions over well-behaved Nyquist channels, we are interested in using Paper approved by G. E. Corazza, the Editor for Spread Spectrum of the IEEE Communications Society. Manuscript received July 23, 2009; revised August 11, 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 International Conf. on Communications (ICC), Comm. Theory Symposium, Glasgow, Scotland, June 2007. 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.2011.020811.090405

a signature set that exhibits minimum total squared correlation (TSC) [1]-[6] △

TSC(S) =

𝐾 ∑ 𝐾 ∑  𝐻 2 s𝑖 s𝑗 

(2)

𝑖=1 𝑗=1

where 𝐻 denotes the Hermitian operator. We know that for overloaded (𝐾 > 𝐿) complex/real-valued signature sets 2 S ∈ ℂ𝐿×𝐾 or ℝ𝐿×𝐾 , TSC(S) ≥ 𝐾𝐿 [1]. Of course, TSC(S) ≥ 𝐾 if 𝐾 ≤ 𝐿. Overloaded (𝐾 > 𝐿) sets with 2 TSC equal to 𝐾𝐿 have been known as Welch-bound-equality (WBE) sets. Algorithms and studies for the design of complex or real-valued WBE signature sets can be found in [3][9]. Digital forms of communication require, however, finitealphabet 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 [10]-[12]. The sum capacity of minimum-TSC optimal binary sets and several other signature set designs under, potentially, a binary or quaternary alphabet was examined in [13], [14]. 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 propagation behavior, in addition to the total squared correlation between signatures we are also concerned about the periodic and aperiodic auto and cross-correlation values [15] among signatures. The periodic total squared correlation (PTSC) [1], [16], [17] is a metric that has been used effectively in the past to design signature sets with favorable multiple-access interference behavior. New lower bounds on the PTSC of binary antipodal signature sets are derived, together with optimal designs for many signature lengths and set sizes, in the companion (Part I) paper [18] of this present work. In the paper herein, we focus solely on designing binary signature sets with optimal aperiodic correlation properties. The new developments and results that we obtain are made possible to a great extent due to the problem formulation that we introduce below. Let △

. . 0 s𝑘 (1) s𝑘 (2) . . . s𝑘 (𝐿) 0 . . . 0] a𝑇𝑘∣𝑙 =[0 . 𝑙

𝐿−1−𝑙

1×(2𝐿−1)

∈ {0, ±1}

, 𝑙 = 0, 1, . . . , 2𝐿 − 2,

(3)

1 The case 𝐾 = 𝐿 ≡ 1 (mod 4) remains open. Ding, Golin, and Kl𝜙ve [11] showed that the TSC Karystinos-Pados bound [10] is tight for 𝐾 = 𝐿 = 5 or 13, but not for 𝐾 = 𝐿 = 9.

c 2011 IEEE 0090-6778/11$25.00 ⃝

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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 59, NO. 5, MAY 2011

denote the zero-padded-by-(L-1), cyclic-right-shifted-by-l po1×𝐿 sitions version of s𝑇𝑘 ∈ {±1} , 𝑙 = 0, 1, 2, . . . , 2𝐿 − 2, 𝑘 = 1, 2, . . . , 𝐾, where 𝑇 denotes the transpose operator and s𝑘 (𝑖) refers to the 𝑖th element of vector s𝑘 , 𝑖 = 1, 2, . . . , 𝐿, 𝑘 = 1, 2, . . . , 𝐾. We define the zero-padded cyclic extension matrix Szpc ∈ {0, ±1}(2𝐿−1)×𝐾(2𝐿−1) of the signature set 𝐿×𝐾 S ∈ {±1} as △

Szpc =[a1∣0 a2∣0 . . . a𝐾∣0 a1∣1 a2∣1 . . . a𝐾∣1 . . . a1∣2𝐿−2 a2∣2𝐿−2 . . . a𝐾∣2𝐿−2 ].

... (4)

The aperiodic correlations among the signatures in S become plain correlations among the columns of Szpc and the aperiodic total squared correlation (ATSC) of the signature set S is the total squared correlation (TSC) of the matrix Szpc , ATSC(S) = TSC(Szpc ).

while some are shown to be triple, ATSC, PTSC, and TSC, optimal. The ATSC optimal designs that we present serve as proof-by-construction of the tightness of the corresponding ATSC bounds. The rest of the paper is organized as follows. In Section II, we derive the new ATSC bounds for binary signature sets. In Section III, we construct ATSC-optimal binary signature sets. Section IV is devoted to discussion and examples and some concluding remarks are drawn in Section V.

(5)

Then,

II. N EW B OUNDS ON THE ATSC OF B INARY A NTIPODAL S IGNATURE S ETS Consider the zero-padded cyclic extension matrix Szpc ∈ (2𝐿−1)×𝐾(2𝐿−1) in (4) and denote its ith row by {0, ±1} △

d𝑇𝑖 =[a1∣0 (𝑖) . . . a𝐾∣0 (𝑖) ...

ATSC(S) =(2𝐿 − 1)

𝐾 2𝐿−2 ∑ ∑ 

2  𝑇  a𝑘∣0 a𝑘∣𝑙 

𝑘=1 𝑙=0 𝐾 𝐾 ∑ ∑

+ (2𝐿 − 1)

2𝐿−2 ∑ 

𝑖=1 𝑗=1,𝑖∕=𝑗 𝑙=0

2  𝑇  a𝑖∣0 a𝑗∣𝑙  .

a1∣1 (𝑖) . . . a𝐾∣1 (𝑖)

...

a1∣2𝐿−2 (𝑖) . . . a𝐾∣2𝐿−2 (𝑖)], 𝑖 = 1, 2, . . . , 2𝐿 − 1. (7)

Then, by (5) and the “row-column equivalence” for the TSC metric of matrices [23], (6)

A detailed derivation of (6) is provided in the Appendix. The first term in (6) contains all aperiodic auto-correlation contributions and the second term contains all aperiodic crosscorrelation contributions. Minimizing ATSC addresses the problem of minimizing aperiodic auto- and cross-correlations together.2 Direct application of the Welch TSC bound [1] on Szpc for sets of complex/real-valued signatures of squared norm equal to L proves that ATSC(S) ≥ 𝐾 2 𝐿2 (2𝐿 − 1); a similar note over the complex-roots-of-unity alphabet in particular was made in [20]. The bound 𝐾 2 𝐿2 (2𝐿 − 1) on ATSC can be translated to a corresponding lower bound 𝐾 2 𝐿2 on the aperiodic mean-squared correlation (MSQC) defined as 1 2𝐿−1 ATSC(S) for which optimal binary signature sets were constructed by Schotten [21] for a limited } ( {and Elders-Boll range of values 𝐾 ∈ 2𝑛 , 22𝑛 , 𝐿 = 2(2𝑛 − 1)) , 𝑛 = 1, 3, 5, . . .. Hence, the constructions in [21] establish tightness of the ATSC bound 𝐾 2 𝐿2 (2𝐿 − 1) over the binary antipodal field for the specific (𝐾, 𝐿) set sizes. The main contributions in this paper are as follows: (i) We introduce a new formulation to tackle the aperiodic total squared correlation problem in the form of equations (3)-(6) above. (ii) We derive new tight or tightest-known bounds on the ATSC of binary signature sets for all possible values of 𝐾 (number of signatures) and 𝐿 (signature length). (iii) We prove a new theoretical hierarchical relation between the ATSC and PTSC metrics. (iv) We present new ATSC-optimal designs for several (𝐾, 𝐿) pairs. Some of the constructions utilize known sequences (such as maximal merit factor and aperiodic complementary sequences), thereby uniting this body of knowledge under the ATSC framework. Furthermore, some of our designs are shown to be double, ATSC and PTSC, optimal, 2 We note that the ATSC metric captures signature correlations between users over all relative delays in contrast to the total squared asynchronous correlation (TSAC) metric of Ulukus and Yates [19], which is a function of a particular delay profile.

ATSC(S) = 𝐾 2 𝐿2 (2𝐿 − 1) +

2𝐿−1 ∑

2𝐿−1 ∑

 𝑇 2 d𝑖 d𝑗  . (8)

𝑖=1 𝑗=1, 𝑗∕=𝑖

∑2𝐿−1 ∑2𝐿−1  𝑇 2   To obtain a bound on the term 𝑖=1 𝑗=1, 𝑗∕=𝑖 d𝑖 d𝑗 for all (𝐾, 𝐿) cases, we will make use of the following theorem. Theorem 1 (On the Properties of Zero-padded Cyclic Extension Matrices) 𝑇 𝐾(2𝐿−1) Let Szpc = [d1 d2 . . . d2𝐿−1 ] , d𝑖 ∈ {0, ±1} , 𝑖 = 1, 2, . . . , 2𝐿 − 1, be the zero-padded cyclic extension matrix of S = [s1 s2 . . . s𝐾 ] , s𝑘 ∈ {±1}𝐿 , 𝑘 = 1, 2, . . . , 𝐾. (i) Then, for 𝑖 < 𝑗, 𝑖, 𝑗 = 1, 2, . . . , 2𝐿 − 1, ⎧ 𝐾 ∑ 𝑇   a𝑘∣0 a𝑘∣𝑗−𝑖 , 𝑗 − 𝑖 = 1, 2, . . . , 𝐿 − 1, ⎨ 𝑇 𝑘=1 d𝑖 d 𝑗 = 𝐾 ∑   ⎩ a𝑇𝑘∣0 a𝑘∣2𝐿−1−(𝑗−𝑖) , 𝑗 − 𝑖 = 𝐿, . . . , 2𝐿 − 2, 𝑘=1

(9)

and 2𝐿−1 ∑

𝐿 ∑  𝑇 2  𝑇 2 d𝑖 d𝑗  = 2(2𝐿 − 1)  d1 d𝑗  .

2𝐿−1 ∑

𝑖=1 𝑗=1, 𝑗∕=𝑖

(10)

𝑗=2

Furthermore, tying back to the original signature form d𝑇1 d𝑗 =

𝐿−(𝑗−1) 𝐾 ∑ ∑ 𝑙=1

s𝑘 (𝑙)s𝑘 ([𝑙 +(𝑗 −1)]−), 𝑗 = 1, 2, . . . , 𝐿,

𝑘=1

(11) where [𝑥]− = 𝑥 − 𝐿 if 𝑥 > 𝐿 and [𝑥]− = 𝑥 if 𝑥 ≤ 𝐿. (ii) If 𝐾𝐿 ≡ 2 (mod 4) ⌋ ⌊   𝑇   𝑇 d1 d𝑗  + d1 d𝐿−(𝑗−2)  ≥ 2, 𝑗 = 2, 3, . . . , 𝐿 + 1 − 1. 2 (12) When, in addition, 𝐿 ≡ 1(mod 2)      𝑇    (13) d1 d⌊ 𝐿 +1⌋  + d𝑇1 d𝐿−⌊ 𝐿 −1⌋  ≥ 2, 2

2

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

while when 𝐿 ≡ 2(mod 4)     𝑇 d1 d 𝐿2 +1  ≥ 1.

(14)

□ From (8) and Theorem 1, Part (i), the ATSC of a binary signature set S equals ATSC(S) = 𝐾 2 𝐿2 (2𝐿 − 1) + 2(2𝐿 − 1)

𝐿 ∑  𝑇 2 d1 d𝑗  (15) 𝑗=2

and —interestingly— depends only on the sum of the aperiodic ∑ auto-correlations of the signatures for each shift 𝐾 (d𝑇1 d𝑗 = 𝑘=1 a𝑇𝑘∣0 a𝑘∣𝑗−1 , 𝑗 = 2, 3, . . . , 𝐿).3 With a revised goal (from (15)) of calculating lower bounds on the term ∑𝐿  𝑇 2 d1 d𝑗  for all (𝐾, 𝐿) cases, we write the following 𝑗=2 property which was presented as Lemma 1 in [25]: { 1 (mod 2), 𝐿 − 𝑗 ≡ 0 (mod 2), 𝑇 a𝑘∣0 a𝑘∣𝑗−1 ≡ 0 (mod 2), 𝐿 − 𝑗 ≡ 1 (mod 2), 𝑘 = 1, 2, . . . , 𝐾, 𝑗 = 2, 3, . . . , 𝐿.

(16)

From (16) and Theorem 1, Part (i), we lower }𝐿 { can calculate bounds on the cross-correlation set d𝑇1 d𝑗  𝑗=2 for the following cases. If 𝐾 ≡ 1 (mod 2), 𝐿 ≡ 0 (mod 4), then {  𝑇  1, 2 ≤ 𝑗 ≤ 𝐿, 𝑗 ≡ 0 (mod 2),  d1 d𝑗  ≥ (17) 0, otherwise. Similarly, if 𝐾 ≡ 1 (mod 2), 𝐿 ≡ 1 (mod 2), then {  𝑇  1, 3 ≤ 𝑗 ≤ 𝐿, 𝑗 ≡ 1 (mod 2),  d d𝑗  ≥ 1 0, otherwise.

(18)

Consider now the case 𝐾 ≡ 2 (mod 4), 𝐿 ≡ 1 (mod 2). From  𝑇 (16) and Theorem 1, Part (i), we can show that d d𝑗  ≡ 0 (mod 2), 𝑗 = 2, 3, . . . , 𝐿. Furthermore, 1 in 1, Part (ii), we ⌊see that  to  satisfy Theorem ) ( 𝑇order ⌋ d1 d𝑗  , d𝑇1 d𝐿−(𝑗−2)  ∕= (0, 0) , 𝑗 = 2, 3, . . . , 𝐿 + 1 . 2 Hence, either     (𝑖) d𝑇1 d𝑗  ≥ 0 and d𝑇1 d𝐿−(𝑗−2)  ≥ 2 or    𝑇  (𝑖𝑖) d1 d𝑗  ≥ 2 and d𝑇1 d𝐿−(𝑗−2)  ≥ 0, ⌊ ⌋ 𝐿 𝑗 = 2, 3, . . . , +1 . 2

(19)

Similarly, for the case 𝐾 ≡ 1 (mod 2), 𝐿 ≡ 2 (mod 4), from (16) and Theorem 1, Part (i), we can show that  𝑇  d d𝑗  ≡ 1 (mod 2), 𝑗 = 2, 4, 6, . . . , 𝐿. (20) 1

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on the aperiodic total squared correlation of binary signature sets. Although general (𝐾 ≥ 1, 𝐿) user signature set design forms the primary focus of this work, we would like to highlight two findings that can be used to tighten the bounds in (22) for single-user (𝐾 = 1, 𝐿) systems in particular. First, consider S ∈ {±1}𝐿×1 where 𝐿 ∈ {8, 12} ∪ {𝐿 : 𝐿 ≡ 0 (mod 4), 64 ≤ 𝐿 ≤ 4(165)2}. We can show [18] that for these lengths the vectors d1 and d 𝐿 +1 obtained from  2   ∑     𝐾 Szpc satisfy d𝑇1 d 𝐿 +1  =  𝑘=1 a𝑇𝑘∣0 a𝑘∣ 𝐿  ≥ 2. Thus, for 2 2 these particular (𝐾 = 1, 𝐿) cases, by applying (17) to the remaining correlations ∣d𝑇1 d𝑗 ∣, 𝑗 = 2, 3, . . . , 𝐿2 , the ATSC lower bound can be tightened to ATSC(S) ≥ 𝐿2 (2𝐿 − 1) + (2𝐿 − 1)(𝐿 + 8). Next, we revisit the literature on the design of individual binary sequences s ∈ {±1}𝐿 that have maximum “merit factor” (MF) [26], [27], which following our notation and definition in (3) is directly given by △

MF(s) =

𝐿2 )2 . ∑𝐿−1 ( 2 𝑙=1 a𝑇∣0 a∣𝑙

(23)

From (9) and (15), maximization of MF in (23) is equivalent to minimization of ATSC. Over the last decade, many maximal merit factor sequences have been obtained numerically through exhaustive search for lengths up to what can be handled computationally; [28], for example, includes a comprehensive list of maximal merit factor sequences for 𝐿 ∈ {1, 2, 3, . . . , 60}. For any 𝐿 ∈ {9, 15, 16, 17, 19, 20, 21, 22, . . ., 60}, in particular, the ATSC of the maximal-MF (hence, minimum-ATSC) sequences exceeds our given bounds. Therefore, merit factor literature provides tighter ATSC bounds for those specific single-user (𝐾 = 1, 𝐿 ∈ {9, 15, 16, 17, 19, 20, 21, 22, . . ., 60}) cases; we produce and list the tight bound values explicitly in Table 1 for completeness purposes and ease in reference. Our new aperiodic bounds in (22) show immediately that the Welch aperiodic bound for complex/real-valued sets 𝐾 2 𝐿2 (2𝐿−1) is loose for binary signature sets of sizes (𝐾, 𝐿) that do not satisfy 𝐾𝐿 ≡ 0 (mod 4) and 𝐾 ≡ 0 (mod 2). In the following section, we develop new binary signature set designs that meet the ATSC bounds with equality. The designs are based primarily on Hadamard matrix transformations, an approach also followed in [10], [18], and [22]. III. D ESIGN OF M INIMUM ATSC B INARY A NTIPODAL S IGNATURE S ETS

Applying (17)-(21) to (15) we obtain (22) given at the top of the following page. Expression (22) defines our new bounds

In view of the formulation that was introduced in Section I, minimum ATSC design of a signature set S is equivalent to minimum TSC design of the corresponding zero-padded cyclic extension matrix Szpc . Below, we provide ATSC optimal signature set designs under several (𝐾, 𝐿) cases for both underloaded (𝐾 ≤ 𝐿) and overloaded (𝐾 > 𝐿) systems. In our presentation, we will make occasional use of what are known as “aperiodic complementary sequences” (ACS). 𝐿×𝐾 ACS sets are sets of sequences4 [s1 s2 . . . s𝐾 ] ∈ {±1}

3 This dependence solely on (aperiodic herein) auto-correlations has been highlighted in the past by Stalder and Cahn [24] and Sarwate and Pursley [15].

4 In general, ACS sets are defined over the entire ℂ𝐿×𝐾 space. We are concerned only with binary ACS sets herein.

For 𝑗 = 3, 5, 7, . . . , 𝐿 − 1, using similar reasoning as in (19), we conclude that either     (𝑖) d𝑇 d𝑗  ≥ 0 and d𝑇 d𝐿−(𝑗−2)  ≥ 2 1

1

or

    (𝑖𝑖) d𝑇1 d𝑗  ≥ 2 and d𝑇1 d𝐿−(𝑗−2)  ≥ 0.

(21)

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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 59, NO. 5, MAY 2011

⎧ 2 2 𝐾 𝐿 (2𝐿 − 1),     ⎨ 𝐾 2 𝐿2 (2𝐿 − 1) + (2𝐿 − 1)𝐿, 𝐾 2 𝐿2 (2𝐿 − 1) + (2𝐿 − 1)(4𝐿 − 4), ATSC(S) ≥    𝐾 2 𝐿2 (2𝐿 − 1) + (2𝐿 − 1)(3𝐿 − 4),  ⎩ 2 2 𝐾 𝐿 (2𝐿 − 1) + (2𝐿 − 1)(𝐿 − 1), TABLE I T IGHT LOWER BOUNDS ON ATSC FOR SINGLE - USER (𝐾 = 1) SYSTEMS 𝐿 9 15 16 17 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Lower bound 1785 7395 9424 11649 15503 17628 20213 24166 28035 30456 34153 39066 42559 48620 55005 60062 66795 72576 79105 86162 94599 103660 112493 121350 132363 143464 153657 163178 175695 189660 201229 216398 230547 245480 259281 277794 293607 312708 330645 349462 367003 390720 409625 432170 455247 480284

𝑘=1

a𝑇𝑘∣0 a𝑘∣𝑙

{ =

𝐾𝐿, 𝑙 = 0, 0, otherwise.

(22)

In our constructions, H𝑁 refers to a Hadamard matrix5 of size 𝑁 with columns h𝑖 ∈ {±1}𝑁 , 𝑖 = 1, 2, . . . , 𝑁 , vec[A] ∈ {±1}𝑃 𝑄×1 denotes the standard column-by-column vectorization operation on matrix A ∈ {±1}𝑃 ×𝑄, and diag{b} is the square diagonal matrix with the elements of vector b on its diagonal. Proofs of optimality of the suggested designs — necessary for some of the more complicated constructions— are included in the Appendix. A. Underloaded Systems (𝐾 ≤ 𝐿) We begin the presentation of our developments with a simple case/class of two-user (𝐾 = 2) ATSC-optimal sets. Define the set } △ { (25) ℳG = 2𝑎 10𝑏 26𝑐 : 𝑎, 𝑏, 𝑐 ≥ 0, 𝑎 + 𝑏 + 𝑐 > 0 . Case 1: 𝐾 = 2, 𝐿 ∈ ℳG Consider a Golay [32], [33] set S of size (2, 𝐿). S is ATSC-optimal. □ We now recall the notion of “aperiodic complementary quadruples” or “base sequences” originally introduced by R. J. Turyn [33]. Base sequences are sets of four binary sequences that can be used to construct directly a four-sequence ACS set [34], [35]. They are available for all lengths in ∞ { } ′ △ ∪ 2 𝑚 𝑀 : 𝑀 ∈ ℳT where (26) ℳT = ′

𝑚=0

ℳT = {(2𝑝 + 1) : 𝑝 ∈ {1, 2, 3, . . . , 32} ∪ ℳG } ∪ {(2𝑝 + 1)(2𝑞 + 1) : 𝑝, 𝑞 ∈ {1, 2, 3, . . . , 32} ∪ ℳG , 𝑝 ∕= 𝑞} ∪ {3𝑝 − 1 : 𝑝 ∈ {1, 2, 3, . . . , 24}} . (27)

that -in our notation- have the property 𝐾 ∑

𝐾𝐿 ≡ 0 (mod 4), 𝐾 ≡ 0 (mod 2), 𝐾𝐿 ≡ 0 (mod 4), 𝐾 ≡ 1 (mod 2), 𝐾 ≡ 2 (mod 4), 𝐿 ≡ 1 (mod 2), 𝐾 ≡ 1 (mod 2), 𝐿 ≡ 2 (mod 4), 𝐾 ≡ 1 (mod 2), 𝐿 ≡ 1 (mod 2).

(24)

A necessary condition for the existence of an ACS set of size (𝐾, 𝐿) is that 𝐾𝐿 ≡ 0 (mod 4) and 𝐾 ≡ 0 (mod 2) [25], [30]. Applying (24) to (15) and comparing to the corresponding bound in (22), we conclude that ACS sets are ATSC-optimal. This note was first made in [29] which states that ACS sets achieve Welch’s aperiodic bound 𝐾 2 𝐿2 (2𝐿−1). Numerous binary ACS set constructions that are either direct or recursive in nature are provided in [25], [30]-[35]. However, to guarantee no signature repetition for underloaded systems as required in code-division multiplexing applications, we make use of only selected constructions in addition to sets that we design independently. The following ACS property (Theorem 4 in [25]) shows that concatenating ACS sets results to an ACS set. Property 1: If S1 and S2 are ACS sets, then S = [S1 S2 ] is an ACS set. □

Case 2: 𝐾 = 4, 𝐿 ∈ ℳT Consider an aperiodic complementary quadruple [34], [35] set S of size (4, 𝐿). S is ATSC-optimal. □ Below, we develop two more involved design procedures with proofs of optimality included in the Appendix. Case 3a: 𝐾 = 2 or 𝐾 ≡ 0 (mod 4), 𝐿 = 𝑃 𝑄, 𝑄×𝑃 ACS , 𝐾≥𝑃 3b: 𝐾 ≡ 0 (mod 4), 𝐿 = 𝑃 𝑄 + 𝑟, 𝑄 × 𝑃 ACS , 𝐾 1≤𝑟≤ 𝐾 2, 2 ≥ 𝑃

Obtain an ACS set S of size 𝑄 × 𝑃 and select 𝑁 ≥ 𝑃 with 𝑁 = 2 or 𝑁 ≡ 4). Construct { 0 (mod [ 𝑇 ]} ′ (1𝑄 ⊗ H𝑃 [×𝑁 ) and vec S (a) S = diag ]) ] [ { [ 𝑇 ]} ( ′ ′′ 1𝑄 ⊗ H𝑃 ×𝑁 H𝑃 ×𝑁 diag vec S , (b) S = ′ − H𝑟×𝑁 H𝑟×𝑁 5 We recall that a Hadamard matrix H 𝑁 is an 𝑁 × 𝑁 matrix with elements +1 or −1 and orthogonal columns, H𝑇 𝑁 H𝑁 = 𝑁 I𝑁 , where I𝑁 is the size𝑁 identity matrix. A necessary condition for a Hadamard matrix to exist is that 𝑁 ≡ 0 (mod 4), except for the trivial cases of 𝑁 = 1 or 𝑁 = 2.

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

TABLE II M AXIMAL -MF SEQUENCES IN run-length NOTATION ( E . G ., 2 1 1 = [1 1 − 1 1]𝑇 ) 𝐿 1 2 3 4 5 6 7 10 11 13 14 18

′

where g𝐿 is the sequence obtained by negating alternate elements of the sequence g𝐿 . Then, S is ATSC-optimal. □

Maximal MF sequence 1 11 21 211 311 1113 1123 22114 112133 5221111 2221115 441112221

B. Overloaded Systems (𝐾 > 𝐿) Case 1: 𝐾 ≡ 0 (mod 4) Consider a Hadamard matrix H𝐾 and its 𝐿 × 𝐾 submatrix H𝐿×𝐾 . Then, S = H𝐿×𝐾 is ACS, hence ATSC-optimal. □

′

1 ≤ 𝑟 ≤ 𝑁 , where H𝑖×𝑁 and H𝑖×𝑁 are 𝑖 × 𝑁 matrices that consist of the first 𝑖 rows of Hadamard matrices ′ H𝑁 and H𝑁 , respectively, 𝑖 = 1, 2, . . . , 𝑁 , 1𝑄 is the 𝑄 × 1 all-one vector, and ⊗ denotes the Kronecker tensor ′ ′′ product. We can show that the sets S and S are ACS, hence ATSC-optimal. Proof of optimality is given in the Appendix.□ Case 4: 𝐾 = 2 or 𝐾 ≡ 0 (mod 4), 𝐿 ≡ 0 (mod 2), 𝐾 ≥

𝐿 2

Obtain a Hadamard matrix H𝐾 and calculate the ATSCoptimal set ⎡

⎤ H𝐾 ⎧⎡ ⎤𝑇 ⎫ ⎢ ⎥ ⎬ ⎨ ⎥ S=⎢ ⎣ diag ⎣1 − 1 1 − 1 . . . 1 − 1⎦ H(𝐿−𝐾)×𝐾 ⎦

  ⎭ ⎩ 𝐿−𝐾

(28)

of size 𝐿 × 𝐾. Proof of optimality of S is included in the Appendix. □ Some of our designs that follow make use of maximal-MF individual sequences (available in [28] and reproduced herein in Table 2 for ease in reference) of length 𝐿 ∈ ℳMF where6 △

ℳMF = {1, 2, 3, 4, 5, 6, 7, 10, 11, 13, 14, 18} .

1415

(29)

Case 5a: (𝐾, 𝐿) = (3, 10) Case 5b: 𝐾 = 5, 𝐿 ∈ ℳMF Case 5c: (𝐾, 𝐿) = (3, 4) Case 5d: 𝐾 = 9, 𝐿 ∈ {11, 13} Case 5e: (𝐾, 𝐿) ∈ {(9, 10), (17, 18)} For Cases 5a, 5b, 5c, 5d, and 5e, design the signature set ′ S𝐿×(𝐾−1) from Underloaded Case 1, 2, 3a, 3b, and 4, respectively, and draw a maximal-MF sequence g𝐿 of length 𝐿. Then, form the ATSC-optimal set ] [ ′ (30) S𝐿×𝐾 = S g𝐿 .

Case 2: 𝐾 ≡ 1 (mod 4), 𝐿 ∈ ℳMF Consider a Hadamard matrix H𝐾−1[ and its 𝐿 × (𝐾 ] − 1) submatrix H𝐿×(𝐾−1) . Then, S = H𝐿×(𝐾−1) g𝐿 where g𝐿 is a maximal-MF sequence of length 𝐿 is an 𝐿 × 𝐾 ATSC-optimal set. □ Case 3: 𝐾 ≡ 2 (mod 4), 𝐾 > 𝐿 + 1, 𝐿 ∈ ℳG ∪ ℳMF Consider a Hadamard matrix H𝐾−2 and its 𝐿 × (𝐾 − 2) submatrix H𝐿×(𝐾−2) . [ ] ′

(i) If 𝐿 ∈ ℳMF , then define S = H𝐿×(𝐾−2) g𝐿 g𝐿 where ′

g𝐿 is a sequence obtained by negating alternate elements of the maximal-MF sequence g𝐿 . ] [ ′ (ii) If 𝐿 ∈ ℳG , then define S = H𝐿×(𝐾−2) S𝐿×2 where ′

S𝐿×2 is an ACS set obtained from Underloaded Case 1. We can show that S is an ATSC-optimal 𝐿×𝐾 signature set.□ Case 4: 𝐾 ≡ 3 (mod 4), 𝐿 ∈ ℳMF Consider a normalized7 Hadamard matrix H𝐾+1 and its 𝐿 × 𝐾 submatrix H𝐿×𝐾 created by removing the 1𝐾 all-one column and 𝐾 − 𝐿 + 1 rows. Define S = diag {g𝐿 } H𝐿×𝐾 where g𝐿 is a maximal-MF sequence of length 𝐿. It can be shown that S is ATSC-optimal. □ We conclude our ATSC-optimal set design section with an important new theorem that explores the relationship between the aperiodic (ATSC) and periodic total squared correlation (PTSC) metrics [1], [16]-[18], [20]. We prove the theorem in the Appendix. Theorem 2 (On the Relationship Between ATSC and PTSC) For set sizes (𝐾, 𝐿) ∕∈ {(𝐾, 𝐿) : 𝐾𝐿 ≡ 0 (mod 4), 𝐾 ≡ 1 (mod 2)}, if there exists a binary signature set S with ATSC(S) that achieves the lower bounds in (22), then S is also PTSC-optimal. □ In the following section, we discuss our design findings and present some examples.

□ Case 6a: 𝐾 ∈ {2, 6}, 𝐿 ∈ {3, 5, 7} Case 6b: 𝐾 = 10, 𝐿 ∈ {11, 13} ′ For Cases 6a and 6b design the signature set S𝐿×(𝐾−2) from Case 2 and 3b, respectively, and take a maximal-MF sequence g𝐿 of length 𝐿. Then, form the ATSC-optimal set ] [ ′ ′ (31) S𝐿×𝐾 = S g𝐿 g𝐿

We begin this section with a comment on the relevance of ATSC as a signature set design metric for codedivision multiplexing applications. Consider a scenario of code-division multiplexed transmissions over an asynchronous and/or multipath channel with a RAKE signature-matched filter receiver [38]. Upon calculation of the RAKE-processed

6 We recall that maximal-MF sequences for the lengths 𝐿 ∈ {1, 2, 3, 4, 5, 7, 11, 13} are called Barker sequences [36]. As discussed in Section II, (9), (15), and (23) prove that every maximal-MF sequence is individually (𝐾 = 1) ATSC optimal.

7 We recall that a Hadamard matrix H is called “normalized” if it contains 𝑁 an all-one column. Every Hadamard matrix can be converted to an equivalent normalized form with respect to any column h𝑗 , 𝑗 = 1, 2, . . . , 𝑁 , by multiplying by −1 all rows 𝑖 ∈ {1, ⋅ ⋅ ⋅ , 𝑁 } for which ℎ𝑖𝑗 = −1.

IV. D ISCUSSION AND E XAMPLES

1416

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 59, NO. 5, MAY 2011

⎡−−++−−+−−+++++−+++−−−+−+−++−+−−−−⎤

G31×33

Fig. 1.

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

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

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

G31×33 Gold set with ATSC = 63860595.

received signal across all 𝐾 users, we can see that when the number of multipaths exceeds ⌊ 𝐿+2 to 2 ⌋ it is of interest  minimize all possible aperiodic correlations a𝑇𝑖 a𝑗∣𝑙  , 𝑖, 𝑗 = 1, 2, . . . , 𝐾, 𝑙 = 0, 1, . . . , 𝐿 − 1, to mitigate interference. The same aperiodic correlations appear in the sufficient statistic of the maximum likelihood multiuser detector, the output of the minimum-mean-squared-error filter receiver, and the zeroforcing decorrelator [39], [40]. The ATSC-optimal design cases presented in the previous section constitute proof-by-construction of the tightness of the corresponding ATSC bounds developed in Section II and offer maximal suppression of the total squared aperiodic correlation. Under the 𝐾𝐿 ≡ 0 (mod 4) design case, we relied heavily on ACS literature. Advancements were made under the 𝐾𝐿 ≡ 2 (mod 4) and 𝐾𝐿 ≡ 1 (mod 2) design cases. To acquire a quantitative feeling for the coverage of the presented designs, if we restrict the domain of 𝐾, 𝐿 to {1, 2, . . . , 256} (at present, it may not appear of practical interest to consider code-division applications outside this parameter range), we can calculate that Underloaded Cases 1 through 6 and Overloaded Cases 1 through 4 together represent 30.59% of 2 all possible combination pairs (𝐾, 𝐿) ∈ {1, 2, . . . , 256} . In overloaded systems, when 𝐿 ∈ ℳMF , optimal signature sets are available for all values of 𝐾 > 𝐿. Certainly, tightness of the bounds and optimal ATSC designs under the remaining cases is an important open research problem. Two results can be highlighted in this regard: The non-existence of Barker sequences for 60 < 𝐿 < 1022 [37] and the non-existence of (2, 𝐿) ACS sets if all prime divisors of 𝐿 are not congruent to 1 (mod 4) [31]. Furthermore, since each signature set constructed in Section III has ATSC equal to the corresponding lower bound, by Theorem 2 all designs8 are jointly ATSC and PTSC-optimal. In addition, direct comparison of our ATSC-optimal designs with the TSC bounds and optimal sets in [10], [11], [12] shows that Underloaded Cases 1 and 2 when 𝐾 is a power of 2, Case 3(a), and Overloaded Cases 1 through 4 are triple ATSC, PTSC, and TSC-optimal, which is an interesting testament to the robustness of the specific designs. 8 Applicable

to Underloaded Case 5 and Overloaded Cases 3 and 4, the Barker sequence g4 is double, ATSC and PTSC optimal.

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

S31×33

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

Fig. 2.

Signature set S31×33 with ATSC = 63846443.

Finally, we are now in a position to establish that the familiar Gold and Kasami sets [41], [42] that have been widely used for their aperiodic correlation properties [15] are not ATSCoptimal in general.9 Fig. 1 provides an example of a G31×33 Gold set with ATSC = 63860595; Fig. 2 presents a competing signature set S31×33 with lower ATSC = 63846443. The set of Fig. 2 was designed following the method of Overloaded Case 2 with maximal-MF sequence g𝐿 = g31 . Note that, while the set of Fig. 2 is superior in ATSC to the Gold set in Fig. 1, ATSC optimality cannot be claimed based on the material developed in this paper (Overloaded Case 2) because 𝐿 = 31 is not in ℳMF . Next, in Fig. 3 we show a numerically generated (𝐾 = 2, 𝐿 = 15) signature set with lower (but not necessarily minimum) ATSC = 33292 than a Kss 15×2 small-set Kasami design with ATSC = 34220. We follow on with Fig. 4 which provides a large-set Kasami design Kls15×8 with ATSC = 441264 and a competing optimal signature set opt S15×8 designed under Underloaded Case 3b with minimum ATSC = 417600. It can also be seen that the complete (4, 15) and (67, 15) small/large Kasami sets are ATSC-inferior to competing sets from Underloaded Case 2, herein, and Overloaded Case 4 from [18], respectively. These examples establish that Gold and Kasami designs are not ATSC-optimal in general. 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 [43]-[47]. We conclude this section with an example of an overloaded ATSC-optimal design Sopt 14×25 given in Fig. 5. The set is designed under our Overloaded Case 2 procedure and has minimum ATSC value 3308526. V. C ONCLUSIONS We derived new bounds on the aperiodic total squared correlation (ATSC) of binary antipodal signature sets for any signature length 𝐿 and set size 𝐾. We provided optimal designs for a wide range of (𝐾, 𝐿) pairs that establish the 9 It is shown in [18] that complete (𝐾 = 𝐿 + 2) Gold sets (but not Gold subsets of 𝐾 < 𝐿 + 2 sequences) are PTSC-optimal, while Kasami sets are not PTSC-optimal in general.

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

⎡− −⎤

Kss 15×2

− − + − − + + − + − + + + +

⎢ ⎢ ⎢ ⎢ ⎢ =⎢ ⎢ ⎢ ⎢ ⎣

+ + + + + + − + + + − + − −

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

S15×2

⎡− +⎤

⎡ + + + + + + + + + + + + + + + + + + + + + + + + +⎤

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

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

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

⎢ ⎢ ⎢ ⎢ ⎢ =⎢ ⎢ ⎢ ⎢ ⎣

(a)

+−−+−−−+++−++−−+−−−+++−++

Sopt 14×25

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

opt

(b)

Fig. 3. (a) Kss 15×2 small-set Kasami with ATSC = 34220. (b) Numerically generated S15×2 signature set with ATSC = 33292.

⎡+ − − + + − − + ⎤

⎡+ + + + + + + + ⎤

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

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

++−+−−+−

Kls15×8

−−+−+−+− +−++−−++

opt

+ (2𝐿 − 1)

tightness of the corresponding lower bounds. The constructions include underloaded (𝐾 ≤ 𝐿) and overloaded (𝐾 > 𝐿) design cases and cover, as an example, 30.59% of all possible combinations of 𝐾, 𝐿 in {1, 2, . . . , 256}. Side results of the presented research include establishing that maximal-meritfactor sequences (and, hence, Barker sequences) are individually ATSC-optimal and that neither Gold nor small nor large-set Kasami sequences are ATSC-optimal in general. In view of these findings, the developed ATSC-optimal sets take precedence in consideration for code-division multiplexing applications. A PPENDIX A D ERIVATION OF (6) From (5), (32)

𝑖=1 𝑗=1 𝑙1 =0 𝑙2 =0

We split the quadruple summation in (32) into aperiodic auto and cross-correlations, 𝐾 2𝐿−2 2 ∑ ∑ 2𝐿−2 ∑   a𝑇𝑖∣𝑙1 a𝑖∣𝑙2 

+

𝐾 ∑

𝐾 ∑

2𝐿−2 ∑ 2𝐿−2 ∑ 

𝑖=1 𝑗=1, 𝑗∕=𝑖 𝑙1 =0 𝑙2 =0

2  𝑇  a𝑖∣𝑙1 a𝑗∣𝑙2  .

2𝐿−2 ∑ 

2  𝑇  a𝑖∣0 a𝑗∣𝑙  . (35)

A PPENDIX B P ROOF OF T HEOREM 1

Fig. 4. (a) large-set Kasami with ATSC = 441264. (b) Optimal signature set designed under Underloaded Case 3b with ATSC = (8)2 (15)2 (2(15) − 1) = 417600.

𝑖=1 𝑙1 =0 𝑙2 =0

𝐾 ∑

𝑖=1 𝑗=1, 𝑗∕=𝑖 𝑙=0

(b)

𝐾 ∑ 𝐾 2𝐿−2 2 ∑ ∑ 2𝐿−2 ∑   a𝑇𝑖∣𝑙1 a𝑗∣𝑙2  .

𝐾 2𝐿−2 2 ∑ ∑   a𝑇𝑖∣0 a𝑖∣𝑙  𝑖=1 𝑙=0 𝐾 ∑

Kls 15×8 Sopt 15×8

ATSC(S) =

we can simplify (33) to

+−+−−+−+ ++−−−−++

(a)

ATSC(S) =

Fig. 5. Optimal signature set S14×25 designed under Overloaded Case 2 with minimum ATSC = (25)2 (14)2 (2(14)−1)+(2(14)−1)(3(14)−4) = 3308526.

ATSC(S) =(2𝐿 − 1)

+−+−+−+−

S15×8

1417

(33)

Since a𝑇𝑖∣𝑙1 a𝑗∣𝑙2 = a𝑇𝑖∣0 a𝑗∣𝑙2 −𝑙1 , 𝑙1 ≤ 𝑙2 , 𝑙1 , 𝑙2 = 0, 1, 2, . . . , 2𝐿 − 2, 𝑖, 𝑗 = 1, 2, . . . , 𝐾, (34)

Expression (9) follows directly from the definitions of the extension matrix Szpc in (4) and d𝑖 , 𝑖 = 1, 2, . . . , 2𝐿 − 1, in (7). By (9), d𝑇𝑖 d𝑗 = d𝑇𝑖 d𝑖+2𝐿−1−(𝑗−𝑖) if 𝑗 − 𝑖 ≥ 𝐿, 𝑖 < 𝑗, 𝑖, 𝑗 = 1, 2, . . . , 2𝐿 − 1, and d𝑇𝑖 d𝑗 = d𝑇𝑘 d𝑛 if 𝑗 − 𝑖 = 𝑛 − 𝑘, 𝑖, 𝑗, 𝑘, 𝑛 = 1, 2, . . . , 𝐿, 𝑖 ≤ 𝑗, 𝑘 ≤ 𝑛. Then, 2𝐿−1 ∑

2𝐿−1 ∑

𝐿 ∑  𝑇 2  𝑇 2 d𝑖 d𝑗  = 2(2𝐿 − 1)  d1 d𝑗  .

𝑖=1 𝑗=1, 𝑖∕=𝑗

(36)

𝑗=2

Expression (11) follows again by inspection of the formulation in (3), (4), and (7). To prove the result in Part (ii), we make use of the triangular inequality in conjunction with (9) to obtain

𝐾  𝐾      ∑  ∑   𝑇   𝑇      𝑇 𝑇 a𝑘∣0 a𝑘∣𝑗−1  +  a𝑘∣0 a𝑘∣𝐿−𝑗+1  d1 d𝑗  + d1 d𝐿−(𝑗−2)  =      𝑘=1 𝑘=1 𝐾  ⌊ ⌋ ∑ ( ) 𝐿   ≥ +1 . a𝑇𝑘∣0 a𝑘∣𝑗−1 + a𝑇𝑘∣0 a𝑘∣𝐿−𝑗+1  , 𝑗 = 2, 3, . . . ,   2 𝑘=1 (37)

∑ ( )   𝐾  𝑘=1 a𝑇𝑘∣0 a𝑘∣𝑗−1 + a𝑇𝑘∣0 a𝑘∣𝐿−𝑗+1  is the absolute sum of the periodic auto-correlations of the signatures for shift 𝑗 − 1 for which [18] provides the following lower bound if 𝐾𝐿 ≡ 2 (mod 4)   𝐾 ∑    𝑇 𝑇 a𝑘∣0 a𝑘∣𝑗−1 + a𝑘∣0 a𝑘∣𝐿−𝑗+1  ≥ 2,    𝑘=1 ⌊ ⌋ 𝐿 𝑗 = 2, 3, . . . , + 1 − 1. (38) 2 ⌊ ⌋ When 𝐿 ≡ 1(mod 2) and 𝑗 = 𝐿2 + 1 , from [18]      𝑇    (39) d1 d⌊ 𝐿 +1⌋  + d𝑇1 d𝐿−⌊ 𝐿 −1⌋  ≥ 2. 2

2

Finally, when 𝐿 ≡ 2(mod 4) and 𝑗 = 𝐿2 + 1, from [18]    𝑇   𝑇  d d𝑗  + d d𝐿−(𝑗−2)  = 2 d𝑇 d 𝐿  ≥ 2. (40) 1 1 1 2 +1 The result follows.

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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 59, NO. 5, MAY 2011

A PPENDIX C P ROOF OF O PTIMALITY OF U NDERLOADED D ESIGNS (𝐾 ≤ 𝐿) Case 3a: 𝐾 = 2 or 𝐾 ≡ 0 (mod 4), 𝐿 = 𝑃 𝑄, 𝑄×𝑃 ACS , 𝐾 ≥𝑃 3b: 𝐾 ≡ 0 (mod 4), 𝐿 = 𝑃 𝑄 + 𝑟, 𝑄 × 𝑃 ACS , 𝐾 1≤𝑟≤ 𝐾 2, 2 ≥𝑃

(a) Consider the zero-padded cyclic extension matrix Θzpc of Θ = 1𝑄 ⊗ H𝑃 ×𝑁 ( . The) only non-zero ( ⌊row⌋) crosscorrelations of Θzpc are d𝑇1 d𝑗 Θzpc = 𝑁 𝑃 𝑄 − 𝑃𝑗 , 𝑗 ≡ 1 (mod { 𝑃[), 1]} < 𝑗 ≤ ′ 𝐿. Pre-multiplying ( ) Θ by diag vec S𝑇 = to create S , we have d𝑇1 d𝑗 S′ zpc ) ( 𝑁 d𝑇1 d⌈ 𝑗 ⌉ = 0, 𝑗 ≡ 1 (mod 𝑃 ), 1 < 𝑗 ≤ 𝐿, since 𝑃

Szpc

S is ATSC-optimal. All other correlations are not affected by ( 𝑇the )pre-multiplication and remain equal to zero. Hence, d1 d𝑗 S′ = 0, 𝑗 = 2, . . . , 𝐿, and we conclude that ′

zpc

′

TSC(Szpc ) = 𝑁 2 𝐾 2 𝐿2 (2𝐾𝐿 − 1). By (5) and (22), S is ATSC-optimal. (b) Consider the zero-padded cyclic extension extension matrices S1,zpc and S2,zpc of [ { [ ]} ] diag vec S𝑇 (1𝑄 ⊗ H𝑃 ×𝑁 ) S1 = H𝑟×𝑁 [ { [ ]} ( ) ] diag vec S𝑇 1𝑄 ⊗ H′𝑃 ×𝑁 and S2 = , (41) −H′𝑟×𝑁 ′′

respectively, and observe that S = [S1 S2 ]. By ((9), the) only non-zero row cross-correlations of S1,zpc are d𝑇1 d𝑗 S1,zpc for 𝑗 = 𝑃 (𝑄 + 𝑚) + 2𝑟, 𝑚 = 0, 1, . . . , 𝑄 − 1, and 𝑗 ≡ 1 (mod 𝑃 ). Similarly, the ( ) only non-zero row crosscorrelations of S2,zpc are d𝑇1 d𝑗 S2,zpc for 𝑗 = 𝑃 (𝑄+𝑚)+2𝑟, 𝑚 𝑗 )≡ 1 (mod 𝑃 ). But, in particular, ( 𝑇= 0,)1, . . . , 𝑄 − 1,(and d1 d𝑗 S2,zpc = − d𝑇1 d𝑗 S1,zpc , 𝑗 = 2, . . . , 𝐿. Hence,  (  ( ) ) (    𝑇 )   d1 d𝑗 Szpc  =  d𝑇1 d𝑗 S1,zpc + d𝑇1 d𝑗 S2,zpc  = 0, 𝑗 = 2, 3, . . . , 𝐿, and we conclude that S is ATSC-optimal with ATSC(S) = 4𝑁 2 (𝐾𝐿 + 𝑟)2 [2(𝐾𝐿 + 𝑟) − 1]. Case 4: 𝐾 = 2 or 𝐾 ≡ 0 (mod 4), 𝐿 ≡ 0 (mod 2), 𝐾 ≥ 𝐿2 Consider the zero-padded cyclic extension matrix Θzpc of [ ] H𝐾 Θ= . (42) H(𝐿−𝐾)×𝐾 By (9) applied to cross-correlations are 𝑗 =⎧⎡𝐾 + 1 and 𝑗 =

 Θ𝑇zpc ,  the only non-zero row  d1 d𝑗  = 𝐾(𝐿 − 𝐾) for Θzpc 𝐿 − 𝐾 + 1. Premultiplying Θ ⎤⎫by

⎬ ⎨ ⎦ diag ⎣1 1 1 1 . . . 1 1 1 − 1 1 − 1 . . . 1 − 1

 

 ⎭ ⎩  𝐾

𝐿−𝐾

to create S, the non-zero cross-correlations are brought to zero while the remaining  zero  cross-correlations of Θzpc are unaffected. Hence, d𝑇1 d𝑗 S = 0, 𝑗 = 2, 3, . . . , 𝐿 zpc and we conclude that S is ATSC-optimal with ATSC(S) = 𝐾 2 𝐿2 (2𝐿 − 1). Case 6a: 𝐾 ∈ {2, 6}, 𝐿 ∈ {3, 5, 7} 6b: 𝐾 = 10, 𝐿 ∈ {11, 13} ′ Partition the signature set S into two sets, S1 = S and S2 =

] ′ g𝐿 g𝐿 . Consider the zero-padded cyclic extension matrices , S1,zpc , and S2,zpc of S, S1 , and S2 , respectively. Then, Szpc  d𝑇 d𝑗  = 0 and d𝑇1 d𝑗 S2,zpc = 2, 𝑗 = 3, 5, 7, . . . , 𝐿, 1 S1,zpc

[

′

since the correlation spectra of g𝐿 and g𝐿 are the same and Barker in value (multiplication of alternate elements of a sequence by −1 does not change the correlation spectrum [25]). Hence,    ( ) (   𝑇 )  ( 𝑇 )  d1 d𝑗 Szpc  =  d1 d𝑗 S1,zpc + d𝑇1 d𝑗 S2,zpc  { 0, 𝑗 = 2, 4, 6, . . . , 𝐿 − 1, = (43) 2, 𝑗 = 3, 5, 7, . . . , 𝐿, and ATSC(S) = 𝐾 2 𝐿2 (2𝐿 − 1) + (2𝐿 − 1)(4𝐿 − 4). We conclude that S is ATSC-optimal. A PPENDIX D P ROOF OF O PTIMALITY OF OVERLOADED D ESIGNS (𝐾 > 𝐿) Case 4: 𝐾 ≡ 3 (mod 4), 𝐿 ∈ ℳMF Consider the zero-padded cyclic extension matrix Θzpc of Θ = H𝐿×𝐾 . By (11) applied to Θzpc , (d𝑇1 d𝑗 )Θzpc = ∑𝐿−(𝑗−1) 𝑇 h𝑙 h𝑙+(𝑗−1) = −(𝐿 − 𝑗 + 1) where h𝑙 is the 𝑙th 𝑙=1 row of H𝐿×𝐾 . In addition, by the properties of Hadamard matrices, h𝑇𝑖 h𝑗 = −1 for 𝑖 ∕= 𝑗. Now, premultiply Θ by diag {g𝐿 } to obtain S = diag ( {g𝐿)} H𝐿×𝐾 . The ( row) crosscorrelations of Szpc become d𝑇1 d𝑗 Szpc = (−1) d𝑇1 d𝑗 [g𝐿 ] . zpc Since g𝐿 is ATSC-optimal, we can calculate ATSC(S) =𝐾 2 𝐿2 (2𝐿 − 1) ⎧ 𝐿 ≡ 0 (mod 4), ⎨ (2𝐿 − 1)𝐿, (2𝐿 − 1)(3𝐿 − 4), 𝐿 ≡ 2 (mod 4), + ⎩ (2𝐿 − 1)(𝐿 − 1), 𝐿 ≡ 1 (mod 2), (44) and conclude that S is ATSC-optimal. A PPENDIX E P ROOF OF T HEOREM 2 The approach of the proof is to assume first that we have an ATSC-optimal signature set S that achieves the lower bounds in (22) and hence a zero-padded cyclic extension matrix Szpc that achieves the row cross-correlation bounds in (17)-(21). Under we then seek to compute ∑ this hypothesis, ∑    𝐾 𝑇 𝑇 a a + a a the quantity  𝐾 𝑘=1 𝑘∣0 𝑘∣𝑗−1 𝑘=1 𝑘∣0 𝑘∣𝐿−𝑗+1 . This quantity represents the absolute sum of the periodic autocorrelations of the signatures for shift 𝑗 − 1. From [18], we know that if S is such that   𝐾 𝐾 ∑  ∑   a𝑇𝑘∣0 a𝑘∣𝑗−1 + a𝑇𝑘∣0 a𝑘∣𝐿−𝑗+1     𝑘=1 {𝑘=1 ⌊ ⌋ 𝐿 2, 𝐾𝐿 ≡ 2 (mod 4), = 𝑗 = 2, 3, . . . , + 1 , (45) 1, 𝐾𝐿 ≡ 1 (mod 2), 2 then S is PTSC-optimal. For the case 𝐾𝐿 ≡ 0 (mod 4), 𝐾 ≡ 0 (mod 2), the theorem is proved in the context of ACS and PCS sets in [30]. Now consider the case 𝐾𝐿 ≡ 1 (mod 2). If there exists a binary signature set design S which is ATSC-optimal with ATSC(S) that achieves the lower bounds in (22),

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

  then the row cross-correlations d𝑇1 d𝑗  , 𝑗 = 2, . . . , 𝐿, of the zero-padded cyclic extension matrix Szpc achieve the bounds in (18) with equality. In other words, d𝑇1 d𝑗 + ∑𝐾 ∑𝐾 d𝑇1 d𝐿−(𝑗−2) = a𝑇 a + 𝑘=1 a𝑇𝑘∣0 a𝑘∣𝐿−𝑗+1 = ⌊ 𝐿𝑘=1 ⌋𝑘∣0 𝑘∣𝑗−1 ±1, 𝑗 = 2, . . . , 2 + 1 , which implies that S is PTSCoptimal by (45). For 𝐾𝐿 ≡ 2 (mod 4), 𝐿 ≡ 1 (mod 2), the row crosscorrelations of Szpc take again the bound values in (19). In ∑𝐾 𝑇 other words, d𝑇1 d𝑗 + d𝑇1 d𝐿−(𝑗−2) = 𝑘=1 a𝑘∣0 a𝑘∣𝑗−1 + ⌊ ⌋ ∑𝐾 𝐿 𝑇 𝑘=1 a𝑘∣0 a𝑘∣𝐿−𝑗+1 = ±2, 𝑗 = 2, . . . , 2 + 1 , which implies that S is PTSC-optimal by (45). For 𝐾𝐿 ≡ 2 (mod 4), 𝐿 ≡ 2 (mod 4), and 𝑗 even in {2, 3, . . . , 𝐿}, from (20) we have that the row cross-correlations of Szpc take values ∣d𝑇1 d𝑗 ∣ = 1. Together with Theorem 1, Part (ii), we also have the additional property that d𝑇1 d𝑗 = d𝑇1 d𝐿−(𝑗−2) or equivalently ∑𝐾 ∑𝐾 𝑇 𝑇 𝑇 𝑘=1 a𝑘∣0 a𝑘∣𝑗−1 = 𝑘=1 a𝑘∣0 a𝑘∣𝐿−𝑗+1 . Hence, d1 d𝑗 + ∑ ∑ 𝐾 𝐾 𝑇 𝑇 d𝑇1 d𝐿−(𝑗−2) = 𝑘=1 a𝑘∣0 a𝑘∣𝑗−1 + 𝑘=1 a𝑘∣0 a𝑘∣𝐿−𝑗+1 = 𝐿 ±2 for even 𝑗 in {2, 3, . . . , 2 + 1}, which implies that S is PTSC-optimal by (45). For 𝐾𝐿 ≡ 2 (mod 4), 𝐿 ≡ 2 (mod 4), and 𝑗 odd in {2, 3, . . . , 𝐿}, the row crosscorrelations of Szpc take again the bound values in (21). This ∑𝐾 would imply that d𝑇1 d𝑗 + d𝑇1 d𝐿−(𝑗−2) = 𝑘=1 a𝑇𝑘∣0 a𝑘∣𝑗−1 + ∑𝐾 𝐿 𝑇 𝑘=1 a𝑘∣0 a𝑘∣𝐿−𝑗+1 = ±2 for odd 𝑗 in {2, . . . , 2 + 1} and S is PTSC-optimal by (45). R EFERENCES [1] R. L. Welch, “Lower bounds on the maximum cross correlation of signals," IEEE Trans. Inf. Theory, vol. IT-20, pp. 397-399, May 1974. [2] 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. [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, S. Ulukus, and R. D. Yates, “Wireless systems and interference avoidance," IEEE Trans. Wireless Commun., vol. 1, pp. 415-428, Mar. 2002. [6] 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. [7] 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. [8] 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. [9] 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. [10] 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. [11] 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. [12] V. P. Ipatov, “On the Karystinos-Pados bounds and optimal binary DSCDMA signature ensembles," IEEE Commun. Lett., vol. 8, pp. 81-83, Feb. 2004. [13] 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.

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[44] G. Romano, F. Palmieri, and P. K. Willett, “Soft iterative decoding for overloaded CDMA," in Proc. IEEE Intern. Conf. Acoust., Speech, Signal Proc., Mar. 2005, vol. 3, pp. 733-736. [45] 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, Sep. 2006. [46] 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. [47] H. Ganapathy, R. Grover, and D. A. Pados, “Scalable TSC-optimal overloading of binary signature sets," in Proc. IEEE GLOBECOM, Comm. Theory Symp., Nov. 2006, pp. CTH06-5 1-6. 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.

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 59, NO. 5, MAY 2011

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 S IGNAL P ROCESSING L ETTERS from 2001 to 2004 and the IEEE T RANSACTIONS ON N EURAL N ETWORKS from 2001 to 2005. He received a 2001 IEEE International Conference on Telecommunications best paper award, the 2003 IEEE T RANSACTIONS ON N EURAL N ETWORKS 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. 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 T RANSACTIONS ON N EURAL N ETWORKS 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.