Near-Complementary Sequences of Various Lengths ... - IEEE Xplore

Near-complementary Sequences of Various Lengths and Low PMEPR for Multicarrier Communications Nam Yul Yu∗ and Guang Gong† ∗ Department

† Department

of Electrical Engineering, Lakehead University of Electrical and Computer Engineering, University of Waterloo

Abstract—New families of near-complementary sequences are presented for peak power control in multicarrier communications. A framework for near-complementary sequences is given by the explicit Boolean expression and the equivalent array structure. The framework transforms the seed pairs to the nearcomplementary sequences by the aid of Golay complementary sequences. As the examples, new families of near-complementary sequences with the peak-to-mean envelope power ratio (PMEPR) < 4 are presented, where the sequences provide various lengths by employing the seeds of shortened or extended Golay complementary pairs. The sequence families can find the potential applications for peak power control requiring codewords or sequences of various lengths as well as low PMEPR.

I. I NTRODUCTION Multicarrier communications have recently attracted much attention in wireless communications. The orthogonal frequency division multiple access (OFDMA) has been employed in several wireless communication standards [1][2]. Also, the multicarrier code-division multiple access (MC-CDMA) is of interest as a future wireless multiple access scheme. Their popularity is mainly due to the robustness to multipath fading channels and the efficient hardware implementation employing fast Fourier transform (FFT) techniques. However, multicarrier communications have the major drawback of the high peakto-average power ratio (PAPR) of transmitted signals, which may nullify all the potential benefits [3]. A constructive way for peak power control in multicarrier communications is to use Golay complementary sequences [4] for subcarriers such that the sequences provide low peak-tomean envelope power ratio (PMEPR) of at most 2 for transmitted signals [5], where the PAPR of the signals is bounded by the PMEPR [6]. In the brilliant theoretical study [7], Davis and Jedwab showed that the sequences can be constructed by the second order cosets of the first order Reed-Muller codes. Further theoretical researches on Golay complementary sequences have been accomplished in [3] and [8]. As an alternative to the sequences, numerous approaches have been made for a large number of near-complementary sequences, where the PMEPR is bounded by a finite value larger than 2. In [9], Parker and Tellambura presented a primitive framework for construction of near-complementary sequences. Based on the work, Schmidt [10] gave an explicit 0 The

authors were supported by the NSERC Discovery Grant of Canada.

Boolean expression for the framework of near-complementary sequences of length 2m . This paper presents a framework of near-complementary sequences of length n2m−k by the explicit Boolean expression and the equivalent array structure. The framework is a generalization of the results in [9] and [10], motivated by [8]. While it transforms the seed pairs of length n to the near-complementary sequences of length n2m−k by the aid of Golay complementary sequences of length 2m−k , the framework preserves the PMEPR bound of the seed pairs. Consequently, the framework provide in a constructive way a large number of near-complementary sequences of various lengths and limited PMEPR. As the examples, we present new families of q-ary nearcomplementary sequences of various lengths and PMEPR < 4 for even q. The near-complementary sequences of length (2k ± 1)2m−k , k ≤ m − 1, are constructed by the framework employing shortened and extended Golay complementary pairs as the seeds. Numerical results show that the new families produce a large number of distinct binary sequences for 5 ≤ m ≤ 7, equivalent to a large number of distinct codewords with valid code rates. Thus, the families can give the flexibility to coding solutions for peak power control in multicarrier communications by providing many codewords with variable lengths and low PMEPR. II. P RELIMINARIES The following notations will be used throughout this paper. j 2π q is a primitive − q is an even positive integer and √ ω=e q-th root of unity, where j = −1. − Zq is a ring of integer modulo q and Zm q is an mdimensional vector space where each component is an element in Zq . A. Golay complementary sequences Let a = (a0 , a1 , · · · , an−1 ) be a sequence over Zq of length n. Then, the aperiodic autocorrelation of a is defined by ρa (τ ) =

n−1−τ

ω ai −ai+τ ,

0 ≤ τ < n.

i=0

A q-ary sequence pair (a, b) of length n is called a Golay complementary pair [4] if ρa (τ ) + ρb (τ ) = 0 for all τ = 0

(1)

978-1-4244-4148-8/09/$25.00 ©2009 This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE "GLOBECOM" 2009 proceedings.

where a (or b) is called a Golay (complementary) sequence. For a q-ary sequence a = (a0 , a1 , · · · , an−1 ), its associated polynomial A(z) is defined by A(z) =

n−1

ω ai z i .

(2)

i=0

In analysis of peak power control, it is convenient to assume that z in (2) lies on the unit√circle in a complex plane, i.e., z ∈ {ej2πt | 0 ≤ t < 1, j = −1}. Then, 2

|A(z)| = A(z)A∗ (z) = n +

n−1

ρa (τ )z −τ +

τ =1

n−1

ρ∗a (τ )z τ

τ =1

(3) where ‘*’ denotes a complex conjugate and |z| = 1. Assume A(z) and B(z) are the associated polynomials of a q-ary sequence pair a and b of length n, respectively. From (1), if the pair (a, b) forms a Golay complementary pair, then |A(z)|2 + |B(z)|2 = 2n.

(4)

For a brief overview to Golay complementary sequences, see [11]. B. Peak power control in multicarrier communications In multicarrier communications with n subcarriers and qPSK modulation, the transmitted signal for a q-ary sequence a = (a0 , · · · , an−1 ) can be modeled as the real part of n−1 1 ω ai ej2π(f0 +iΔf )t , t ∈ 0, sa (t) = Δf i=0 where Δf is the frequency separation between adjacent subcarriers and f0 is the carrier frequency. Then, the peak-to-mean envelope power ratio (PMEPR) of sa (t) is determined by 2 1 PMEPR(a) = · max A(ej2πt ) (5) n t∈[0,1) where A(ej2πt ) is the associated polynomial of a at z = ej2πt . From (4) and (5), if a is a Golay complementary sequence, then PMEPR(a) ≤ 2 since |A(z)|2 ≤ 2n. In general, if a is a q-ary sequence of length n and |A(z)|2 ≤ vn for any z with |z| = 1, then PMEPR(a) ≤ v. This implies that the study of associated polynomials and aperiodic autocorrelations plays a crucial role in PMEPR of sequences for multicarrier communications. C. Generalized Boolean functions Let x = (x0 , · · · , xm−1 ) be a vector in Zm 2 . A generalized Boolean function f (x) is defined by a mapping f : Zm 2 → Zq , which is represented by the sum of all possible products of xl ’s with coefficients in Zq , i.e., f (x) = f (x0 , · · · , xm−1 ) =

m 2 −1

i=0

ci

m−1 l=0

xil l

form of f . In (6), the order of the ith monomial with nonzero ci m−1 is given by l=0 il , and the highest order of the monomials with nonzero ci ’s is called the degree of the Boolean function f , denoted by deg(f ). Associated with a generalized Boolean function f , a sequence (or codeword) over Zq of length 2m can be , f1 , · · · , f2m −1 ), where fj = generated by f = (f0 m−1 l f (j0 , j1 , · · · , jm−1 ), j = l=0 jl 2 ∈ Z2m and jl ∈ Z2 . Hence, the associated sequence f of length 2m is obtained by the generalized Boolean function fj while j runs through 0 to 2m − 1 in the increasing order. In this paper, we generalize the concept of the associated sequences so as to associate a sequence of arbitrary length with a generalized Boolean function. Definition 1: Let m and n be positive integers, 2m−1 < n ≤ 2m . Let f be a generalized Boolean function of m variables in (6). Let X = {i0 , · · · , in−1 } ⊆ Z2m be a set of index ij , 0 ≤ j ≤ n − 1, where i0 < i1 < · · · < in−1 . Consider a sequence f over Zq of length n, i.e., f = (fi0 , fi1 , · · · , fin−1 ) where fij = f (ij0 , ij1 , · · · , ijm−1 ) where (i , i , · · · , ijm−1 ) is a binary representation of ij = m−1 j0 l j1 l=0 ijl 2 , ijl ∈ Z2 . In other words, f is generated by the generalized Boolean function fij while ij runs through the elements in X as the increasing order. Then we say that the sequence f of length n is associated with the generalized Boolean function f . Thus, the conventional associated sequence of length 2m is a special case of n = 2m and X = Z2m in Definition 1. D. Reed-Muller codes Let C be a linear code over Zq , where each codeword c of length n is a vector in Znq . For any vector a in Znq , the coset of C is defined by a + C = {a + c | c ∈ C} where a is called the coset representative. For q ≥ 2 and 0 ≤ r ≤ m, the generalized r-th order Reed-Muller code RMq (r, m) is defined by the set of sequences (or codewords) f of length 2m associated with f , where f is a generalized Boolean function of degree at most r [7][3]. The minimum Hamming distance of RMq (r, m) is 2m−r [3]. In [7], Davis and Jedwab showed that Golay complementary sequences of length 2m can be obtained by the second order cosets of the generalized first order Reed-Muller codes RM2h (1, m) for h ≥ 1. Paterson [3] then generalized the alphabet size to an even positive integer q. Theorem 1: [3] Let q be even and π be a permutation in {0, 1, · · · , m−1}. Let f : Zm 2 → Zq be a generalized Boolean function defined by

(6)

where ci ∈ Zq and il ∈ Z2 is obtained by a binary m−1 representation of i = l=0 il 2l . Note that the addition in (6) is computed modulo-q. (6) is also called the algebraic normal

f (x0 , · · · , xm−1 ) =

m−2 m−1 q xπ(i) xπ(i+1) + ui xi + e (7) 2 i=0 i=0

where ui , e ∈ Zq . Then, the sequence f of length 2m associated with f is a Golay complementary sequence.


Theorem 1 constructs a set of (m!/2) · q m+1 distinct Golay complementary sequences of length 2m from the (m!/2) second order cosets of RMq (1, m). Since all sequences (or codewords) are contained in RMq (2, m), the minimum Hamming distance is at least 2m−2 . III. A F RAMEWORK FOR N EAR - COMPLEMENTARY S EQUENCES This section introduces a framework for a large number of near-complementary sequences. The framework is given as the explicit Boolean expression and the equivalent array structure. The following notations will be used throughout this section. − k and m are nonnegative integers with 0 ≤ k ≤ m − 1. − g1 , g2 : Zk2 → Zq are the generalized Boolean functions of k variables. − g1 and g2 are the sequences of length n, associated with g1 and g2 , respectively. − G1 (z) and G2 (z) are the associated polynomials of g1 and g2 , respectively, where |z| = 1. A. Basic framework Generalizing the Rudin-Shapiro extension, Parker and Tellambura [9] presented a primitive framework for nearcomplementary sequences employing a seed pair, where the PMEPR bound of the seed sequences is preserved through the extension. Based on the work, Schmidt [10] gave a framework of a generalized Boolean function for near-complementary sequences of length 2m , which is represented by a coset of RMq (1, m). In [8], a framework of Golay complementary sequences of length n2m−k was presented by a 2t × n2m−k−t matrix, where 0 ≤ t ≤ m−k, employing seed pairs of length n and controlling pairs of length 2m−k . If n = 2k , the framework is then given by the algebraic normal form in Lemma 9 of [8]. In the following, we present a framework for nearcomplementary sequences of length n2m−k by integrating the previous works. Theorem 2: For 0 ≤ t ≤ m − k, let π be a permutation in a nonnegative integer set Ω = {0, 1, · · · , t−1, t+k, · · · , m−1}. Let f (t) be a generalized Boolean function defined by f (t) (x0 , · · · , xm−1 ) t−2

=

q q xπ(r) xπ(r+1) + xπ(t−1) xπ(t+k) 2 r=0 2

+

m−2 q xπ(r) xπ(r+1) + ur xr + e 2 r=t+k

(8)

r∈Ω

+ xπ(φ) · (−g1 (xt , · · · , xt+k−1 ) + g2 (xt , · · · , xt+k−1 )) + g1 (xt , · · · , xt+k−1 ) where ur , e ∈ Zq . In (8), φ = k if t = 0, and φ = 0 otherwise. (t) m−k For 2k−1 < n ≤ 2k , let a q-ary sequence m−1 f l of length n2 (t) be generated by f while x = l=0 xl 2 runs through the k−1 elements in Xt = {x | x ∈ Z2m , 0 ≤ l=0 xt+l 2l ≤ n − 1} in the increasing order. Then, the PMEPR of f (t) is given by PMEPR(f (t) ) ≤

|G1 (z)|2 + |G2 (z)|2 . n

In (8), note that only the terms with valid subscripts and arguments exist. Also, the generalized Boolean functions g1 and g2 generate the seed pair g1 and g2 of length n, respectively, as ) runs through (0, · · · , 0) to (n0 , · · · , nk−1 ), (xt , · · · , xt+k−1 k−1 where n − 1 = l=0 nl 2l , nl ∈ Z2 . We have to mention that the framework for t = 0 is equivalent to the one in [9] for near-complementary sequences of length n2m−k . Before proving Theorem 2, we describe the array structure equivalent to the framework. B. Equivalent array structure Given the notations in Theorem 2, Table I displays the arrangement of f (t) in a 2m−k × n array, where each subscript m−1 of f is x = l=0 xl 2l ∈ Xt . Apparently, reading the array as the increasing order of x is equivalent to choosing a 2t × n sub-matrix from top to bottom and reading each column-wise. In the array, each row vector si , 0 ≤ i ≤ 2m−k − 1, is obtained when the row index (xm−1 , · · · , xt+k , xt−1 , · · · , x0 ) is given as a constant vector. From (8), g1 + ci · 1, if xπ(φ) = 0, (10) si = if xπ(φ) = 1 g2 + ci · 1, where 1 = (1, 1, · · · , 1) of length n and ci ∈ Zq . The successive increase of the row index allows its conversion (xm−1 , · · · , xt+k , xt−1 , · · · , x0 ) → (xm−k−1 , · · · , x0 ) by changing xr to xr−k for r ∈ Ω ≥ t+k. Then, the permutation π in Ω induces its equivalence π in {0, · · · , m−k−1}, where π (j), if π (j) ≤ t − 1 π(r) = if π (j) ≥ t π (j) + k, where j = r if r ≤ t − 1 and j = r − k if r ≥ t + k. From the equivalence and the index conversion, it is easy to show that π (0) can replace π(φ) in (10). Moreover, we see from (8) that ci is the ith element of a Golay complementary sequence c of length 2m−k associated with a generalized Boolean function of (m−k) variables, which is represented by (7) with the new row index (xm−k−1 , · · · , x0 ) and the permutation π . Thus, (10) implies that the array structure is equivalent to the arrangement of the seed pair g1 and g2 in each row, controlled by xπ (0) and c. Choosing a 2t × n sub-matrix from top to bottom and reading each one column-wise then produce f (t) . We are now ready to prove Theorem 2. Proof of Theorem 2: Applying the coordinate variation of (x0 , · · · , xk−1 ) → (xt , · · · , xt+k−1 ) to the framework in Corollary 15 of [10], we obtain f (t) in (8). Controlled by xπ (0) and c in the array, f (t) is ultimately equivalent to the sequence generated by the matrix M of Lemma 5 in [8], where the seeds are replaced by g1 and g2 . From the proof of the lemma, it is immediate that the PMEPR of f (t) is given by t

t

2m−k · (|G1 (z 2 )|2 + |G2 (z 2 )|2 ) n2m−k 2 |G1 (z)| + |G2 (z)|2 . = n

PMEPR(f (t) ) ≤

(9)

2


TABLE I T HE ARRAY STRUCTURE OF f (t)

xm−1

···

xt+k

xt−1

···

x0

0 0 .. . 0 0 0 .. . 0 .. . 1 1 .. . 1

··· ···

0 0 .. . 0 1 1 .. . 1 .. . 1 1 .. . 1

0 0 .. . 1 0 0 .. . 1 .. . 0 0 .. . 1

··· ···

0 1 .. . 1 0 1 .. . 1 .. . 0 1 .. . 1

··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ···

··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ···

xt+k−1 .. . xt

0 .. . 0

0 .. . 1

···

f0 f1 .. .

f 2t f2t +1 .. .

f2t −1 f2t+k

f2t+1 −1 f2t+k +2t

··· ···

f2t+k +1 .. .

f2t+k +2t +1 .. .

f2t+k +2t −1 .. . f2m −2t+k f2m −2t+k +1 .. .

f2t+k +2t+1 −1 .. . f2m −2t+k +2t f2m −2t+k +2t +1 .. .

f2m −2t+k +2t −1

f2m −2t+k +2t+1 −1

Using the 2m−k × n array with the row index (xm−k−1 , · · · , x0 ), we establish a general procedure to transform the seed pair g1 and g2 of length n to nearcomplementary sequences f (t) of length n2m−k . Procedure: Input: a seed pair g1 and g2 of length n. Output: a near-complementary sequence f (t) of length n2m−k , where 0 ≤ t ≤ m − k. 1. Arrange the row index (xm−k−1 , · · · , x0 ) as the increasm−k−1 ing order of i = l=0 xl 2l in a 2m−k × n array, and generate a permutation π in {0, · · · , m − k − 1}. 2. Transform the seeds by (10) using a Golay sequence c = (c0 , · · · , c2m−k −1 ) with the permutation π , where π (0) replaces π(φ) in (10). Place the transformed seed si in the ith row of the array for 0 ≤ i ≤ 2m−k − 1. 3. Choose a 2t × n sub-matrix from the top of the array, and read it column-wise from left to right. Repeat for all the sub-matrices from top to bottom. Remark 1: The above procedure does not need the restriction on k, i.e., 2k−1 < n ≤ 2k , which is considered for the Boolean expression in Theorem 2. Therefore, we ignore the restriction in applying the procedure for near-complementary sequences in next section. In summary, the near-complementary sequences of length n2m−k can be obtained by the transformation of the seed pairs of length n by the Golay complementary sequences of length 2m−k . From the equivalence to Theorem 2, the PMEPR of the sequences is bounded by the PMEPR of the seeds in (9). IV. N EAR C OMPLEMENTARY S EQUENCES OF VARIOUS L ENGTHS AND PMEPR < 4 This section applies the procedure in Section III for new families of near-complementary sequences.

··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ···

nk−1 .. . n0 f2t n−2t f2t n−2t +1 .. . f2t n−1

f2t+k +2t n−2t

f2t+k +2t n−2t +1 .. . f2t+k +2t n−1 .. . f2m −(2k −n+1)2t

f2m −(2k −n+1)2t +1 .. . f2m −(2k −n)2t −1

s0 s1 .. . s2t −1 s 2t s2t +1 .. . s2t+1 −1 .. . s2m−k −2t

s2m−k −2t +1 .. . s2m−k −1

A. Shortened Golay complementary pairs Definition 2: Let k ≥ 2 be a positive integer and n = 2k −1. Let g1 and g2 be the generalized Boolean functions given by k−2 k−1 q xπ1 (r) xπ1 (r+1) + ur xr + e, 2 r=0 r=0 q g2 (x0 , · · · , xk−1 ) = g1 (x0 , · · · , xk−1 ) + xπ1 (0) 2 (11)

g1 (x0 , · · · , xk−1 ) =

where ur , e ∈ Zq and π1 is a permutation in {0, · · · , k − 1}. k−1 While x = l=0 xl 2l runs through 0 to 2k − 2, the sequences g1 and g2 associated with g1 and g2 form a shortened Golay complementary pair of length n = 2k − 1. 1 While x runs through all elements in Z2k , the sequences g 2 associated with (11) form a Golay complementary pair and g of length 2k [3]. In fact, g1 and g2 are equivalently obtained 2 , respectively. We now 1 and g by eliminating the last bits of g discuss the PMEPR of the shortened Golay pair. Lemma 1: Let G1 (z) and G2 (z) be the associated polynomials of a shortened Golay complementary pair g1 and g2 of 2 2 2 (z)| length n = 2k − 1, respectively. Then, |G1 (z)| +|G < 4. n 1 = (a0 , · · · , an ) = (g1 , an ) and g 2 = Proof: Let g (b0 , · · · , bn ) = (g2 , bn ) be a Golay complementary pair of length 2k . Then, ρg1 (τ ) = ρg 1 (τ ) − ω an−τ −an and ρg2 (τ ) = ρg 2 (τ ) − ω bn−τ −bn . From (3), |G1 (z)|2 + |G2 (z)|2 = 2n +

n−1

(ρg 1 (τ ) + ρg 2 (τ ) − ω an−τ −an − ω bn−τ −bn )z −τ

τ =1

+

n−1 τ =1

(ρ∗g 1 (τ ) + ρ∗g 2 (τ ) − ω −an−τ +an − ω −bn−τ +bn )z τ


where ρg 1 (τ ) + ρg 2 (τ ) = ρ∗g 1 (τ ) + ρ∗g 2 (τ ) = 0. Thus,

let g1 = ( g1 , an−1 ) and g2 = ( g2 , bn−1 ). Then the pair (g1 , g2 ) is defined as an extended Golay complementary pair of length n.

|G1 (z)|2 + |G2 (z)|2 = 2n − 2 ≤ 2n + 2

n−1 τ =1 n−1

Re (ω an−τ −an + ω bn−τ −bn )z −τ

Equivalently, g1 and g2 are obtained by appending an−1 and bn−1 to a Golay complementary pair of length 2k , respectively.

a −a ω τ n + ω bτ −bn .

Lemma 2: Let G1 (z) and G2 (z) be the associated polynomials of an extended Golay complementary pair g1 = g2 , bn−1 ) of length n = 2k + 1, respec( g1 , an−1 ) and g2 = ( tively, defined in Definition 3. If bn−1 ∈ {an−1 , an−1 + 2q } 2 2 2 (z)| for an−1 ∈ Zq , then |G1 (z)| +|G < 4. n

τ =1

From (11), q bτ = g2 (τ0 , · · · , τk−1 ) = aτ + τπ1 (0) , 2 q bn = g2 (1, · · · , 1) = an + 2 k−1 l where 1 ≤ τ = l=0 τl 2 ≤ n − 1, τl ∈ Z2 . Finally, |G1 (z)|2 + |G2 (z)|2 ≤ 2n + 2 = 2n + 2

n−1

aτ ω

(12)

q − ω aτ + 2 τπ1 (0)

τ =1 n−1

q 1 − ω 2 τπ1 (0)

τ =1

= 2n + 2 · N (τπ1 (0) = 1) · 2 = 4n − 2 < 4n where N (τπ1 (0) = 1) = n−1 2 is the number of occurrences of 2 τπ1 (0) = 1 while τ runs through 1 to n − 1. Applying the shortened Golay complementary pairs as the seeds, the transformation procedure in Section III leads us to the construction of a new family of near-complementary sequences of PMEPR < 4. Construction 1: For a positive integer m ≥ 4, let 3 ≤ k ≤ m − 1 and 0 ≤ t ≤ m − k. Let g1 and g2 be the shortened Golay complementary pair of length n = 2k −1 in Definition 2. Employing the pair as the seeds, we construct a new family of near-complementary sequences f (t) of length n2m−k = 2m − 2m−k by the procedure described in Section III. Obviously, the PMEPR of f (t) is less than 4 from Theorem 2 and Lemma 1. Remark 2: In Definition 2, if k = 2, a set of all shortened Golay pairs of length 3 is equivalent to a set of the sequence pairs of length 3 extended from Golay complementary pairs of length 2, where the extension will be discussed in Construction 2. The case of k = 2 is therefore included in the construction from the extended Golay pairs of k = 1. Note that we may vary t, the seed pair (g1 , g2 ), and the Golay complementary sequence c in the procedure to obtain a variety of sequences of length 2m − 2m−k . Applying g1 and g2 in (11) to the framework in (8) could give the equivalent Boolean expression [12] for the near-complementary sequences in Construction 1. B. Extended Golay complementary pairs Definition 3: Let k ≥ 1 be a positive integer and n = 2k +1. 2 = (b0 , · · · , bn−2 ) be a Golay 1 = (a0 , · · · , an−2 ) and g Let g complementary pair of length 2k associated with the generalized Boolean functions in (11). For arbitrary an−1 , bn−1 ∈ Zq ,

Proof: From the extension, ρg1 (τ ) = ρg 1 (τ ) + ω an−τ −1 −an−1 and ρg2 (τ ) = ρg 2 (τ ) + ω bn−τ −1 −bn−1 . Similar to the proof of Lemma 1, |G1 (z)|2 + |G2 (z)|2 ≤ 2n + 2

n−2

a −a ω τ n−1 + ω bτ −bn−1 .

τ =0

If bn−1 ∈ {an−1 , an−1 + 2q }, then ω bn−1 = (−1)δ ω an−1 where δ = 0 if bn−1 = an−1 , and δ = 1 otherwise. From (12), |G1 (z)|2 + |G2 (z)|2 ≤ 2n + 2

n−2

q 1 + (−1)δ ω 2 τπ1 (0)

τ =0

= 2n + 2 · N (τπ1 (0) = δ) · 2 = 4n − 2 < 4n where N (τπ1 (0) = δ) = n−1 2 is the number of occurrences of 2 τπ1 (0) = δ ∈ {0, 1} for 0 ≤ τ ≤ n − 2. Construction 2: For a positive integer m ≥ 2, let 1 ≤ g1 , an−1 ) and k ≤ m − 1 and 0 ≤ t ≤ m − k. Let g1 = ( g2 , bn−1 ) be the extended Golay complementary pair g2 = ( of length n = 2k + 1 in Definition 3, where an−1 ∈ Zq and bn−1 ∈ {an−1 , an−1 + 2q }. Employing the pair as the seeds, we construct a new family of near-complementary sequences f (t) of length n2m−k = 2m + 2m−k by the procedure in Section III. Varying t from 0 to m − k, we can generate a variety of sequences f (t) from different seed pairs (g1 , g2 ) and different Golay complementary sequences c in the procedure. Obviously, the PMEPR of f (t) is less than 4. Remark 3: In [9], the authors pointed out that the full usefulness of the seed technique would be apparent only if a method could be found to construct seeds. (as opposed to computational search.) They left the method as an open problem, which has been partly solved in [10] by the construction example of near-complementary sequences of length 2m . In Constructions 1 and 2, we could also solve the problem by employing the shortened and the extended Golay pairs as the seeds, producing many near-complementary sequences of various lengths. Furthermore, we could construct a large number of distinct near-complementary sequences of length 2m using the seed technique in a constructive way [12]. C. Numerical results and discussions (s)

be the maximum number of the nearLet Nk complementary sequences of length 2m − 2m−k from Con-


TABLE II T HE CODE RATES AND THE MAXIMUM PMEPR S FOR THE BINARY NEAR - COMPLEMENTARY SEQUENCES FROM MODIFIED G OLAY PAIRS m

k

5

1 2 3 4 1 2 3 4 5 2 3 4 5 6

6

7

Construction 1 (Shortened Golay pair) length Rs Pmax 28 30

0.399 0.386

3.695 3.333

56 60 62

0.253 0.236 0.240

3.711 3.310 2.845

112 120 124 126

0.156 0.143 0.141 0.147

3.714 3.333 2.842 2.571

Construction 2 (Extended Golay pair) length Re Pmax 48 0.311 3.333 40 0.340 3.600 36 0.366 3.176 34 0.400 2.941 96 0.193 3.333 80 0.211 3.594 72 0.225 3.223 68 0.238 2.917 66 0.256 2.685 160 0.128 3.600 144 0.135 3.223 136 0.141 2.941 132 0.148 2.682 130 0.158 2.492

struction 1. Then, it is straightforward that (s)

Nk

= k! · (m − k)! · (m − k + 1) · q m+1 = k! · (m − k + 1)! · q m+1 ,

3≤k ≤m−1

(13)

for all possible (g1 , g2 ) pairs, c’s, and t’s in the procedure. Our (s) numerical experiments demonstrated that all Nk sequences are distinct for k’s and m’s in Table II for binary cases (q = 2), and for all k’s in m = 5 for quaternary cases (q = 4). (e) In the mean time, let Nk be the maximum number of the near-complementary sequences of length 2m + 2m−k from Construction 2. Similar to (13), (e)

Nk

= 2 · k! · (m − k + 1)! · q m+2 ,

1 ≤ k ≤ m − 1. (e)

Numerical experiments also confirmed that total Nk distinct sequences exist for k’s and m’s in Table II for q = 2 and for all k’s in m = 5 for q = 4. (s) Table II shows code rates Rs = log2 Nk /(2m − 2m−k ) (e) and Re = log2 Nk /(2m + 2m−k ) for 5 ≤ m ≤ 7, considering our binary near-complementary sequences (q = 2) as codewords. Also, Table II presents Pmax or maximum PMEPR measured over all codewords for each length, which verifies that the near-complementary sequences of various lengths achieve the PMEPR < 4. In the table, the code rates of the near-complementary sequences of short lengths appear to be acceptable for the application to peak power control for a small number of subcarriers in multicarrier communications. Remark 4: If t is fixed, we can determine the minimum Hamming distances of the codewords f (t) . From [12], all codewords for a fixed t in Construction 1 lie in punctured RMq (2, m) where 2m−k elements are removed from the same positions in each codeword of RMq (2, m). Thus, the minimum Hamming distance is at least 2m−2 −2m−k . On the other hand, Construction 2 obtains each codeword for a fixed t by inserting 2m−k elements to the same positions in each codeword of a subset of RMq (2, m). The minimum Hamming distance is

then at least 2m−k−1 , which occurs when a different pair of the 2m−k elements with the distance 2m−k−1 is inserted to the same codewords in RMq (2, m). Both the minimum distances have been numerically confirmed at t = 0 for the cases in Table II. In the cases, however, we suffer slight rate losses (s) due to the reduced numbers of codewords, i.e., Nk |t=0 = (e) k!(m − k)!q m+1 and Nk |t=0 = 2k!(m − k)!q m+2 . We are currently taking numerical experiments to find the minimum Hamming distances of f (t) including all t’s. V. C ONCLUSION This paper has presented a framework for nearcomplementary sequences of length n2m−k by the generalized Boolean function and the equivalent array structure. The framework transforms the seed pairs of length n to nearcomplementary sequences of length n2m−k by the aid of Golay complementary sequences of length 2m−k , preserving the PMEPR bound of the seeds. As the examples, we have constructed new families of near-complementary sequences of various lengths and the PMEPR < 4 by applying the shortened and the extended Golay complementary pairs to the framework as the seeds. Providing a large number of distinct near-complementary sequences of various lengths in a constructive way, the sequence families could give the flexibility to coding solutions for peak power control in multicarrier communications. R EFERENCES [1] IEEE802.16e-2005, IEEE Standard for Local and Metropolitan Area Networks, Part 16 - Air Interface for Fixed and Mobile Broadband Wireless Access Systems, Amendment 2: Physical and Medium Access Control Layers for Combined Fixed and Mobile Operation in Licensed Bands. [2] 3GPP TS 36.212, v. 8.1.0, Technical Specification Group Radio Access Network; Evolved Universal Terrestrial Radio Access (E-UTRA); Multiplexing and channel coding (Release 8). [3] K. G. Paterson, “Generalized Reed-Muller codes and power control in OFDM modulation,” IEEE Trans. Inform. Theory, vol. 46, no. 1, pp. 104120, Jan. 2000. [4] M. J. E. Golay, “Complementary series,” IRE Trans. Inform. Theory, vol. IT-7, pp. 82-87, 1961. [5] B. M. Popovic, “Synthesis of power efficient multitone signals with flat amplitude spectrum,” IEEE Trans. Commun., vol. 39, pp. 1032-1033, 1991. [6] S. Litsyn, Peak Power Control in Multicarrier Communications, Cambridge University Press, 2007. [7] J. A. Davis and J. Jedwab, “Peak-to-mean power control for OFDM, Golay complementary sequences, and Reed-Muller codes,” IEEE Trans. Inform. Theory, vol. 45, no. 7, pp. 2397-2417, Nov. 1999. [8] F. Fiedler, J. Jedwab, and M. G. Parker, “A framework for the construction of Golay sequences,” IEEE Trans. Inform. Theory, vol. 54, no. 7, pp. 3113-3129, July 2008. [9] M. G. Parker and C. Tellambura, “Golay-Davis-Jedwab complementary sequences and Rudin-Shapiro constructions,” manuscript. Available in “http://www.ii.uib.no/∼matthew/ConstaBent2.pdf”, 2001. [10] K. -U. Schmidt, “On cosets of the generalized first-order Reed-Muller code with low PMEPR,” IEEE Trans. Inform. Theory, vol. 52, no. 7, pp. 3220-3232, July 2006. [11] M. G. Parker, K. G. Paterson, and C. Tellambura, “Golay complementary sequences,” Wiley Encyclopedia of Telecommunications, Edited by J. G. Proakis, Wiley Interscience, 2002. [12] N. Y. Yu, and G. Gong, “Near-complementary sequences with low PMEPR for peak power control in multicarrier communications,” submitted to IEEE Trans. Inform. Theory, 2009. Also appeared at CACR 2009-24, CACR Technical Report, University of Waterloo.
