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Almost Perfect Sequences and Periodic Complementary Sequence Pairs over the 16-QAM Constellation∗ Fanxin ZENG†,††a) , Member, Xiaoping ZENG† , Nonmember, Zhenyu ZHANG†† , Member, and Guixin XUAN†† , Nonmember
SUMMARY Based on quadriphase perfect sequences and their cyclical shift versions, three families of almost perfect 16-QAM sequences are presented. When one of two time shifts chosen equals half a period of quadriphase sequence employed and another is zero, two of the proposed three sequence families possess the property that their out-of-phase autocorrelation function values vanish except one. At the same time, to the other time shifts, the nontrivial autocorrelation function values in three families are zero except two or four. In addition, two classes of periodic complementary sequence (PCS) pairs over the 16-QAM constellation, whose autocorrelation is similar to the one of conventional PCS pairs, are constructed as well. key words: 16-QAM sequence, almost perfect sequence, periodic complementary sequence pair, quadriphase perfect sequence, autocorrelation
1.
Introduction
Sequences with good autocorrelation play an important role in communications, such as in synchronization. The best autocorrelation property of such sequences are with all zero nontrivial autocorrelation values, i.e., so-called perfect sequences [1]. Unfortunately, only a binary perfect sequence with length 4 has been found up to now, which implies there do not exist the binary perfect sequences with other length. In order to satisfy the requirements of applications, the replacers of perfect sequences have been widely investigated. Thereinto, one of the replacers is called an almost perfect sequence. In fact, an almost perfect sequence is a sub-optimal binary sequence with only a nonzero out-of-phase autocorrelation value, which is investigated by Refs. [2]–[6]. For instance, in Ref. [7], an almost perfect binary sequence with length 16 is given as {vr (t)} = (1, 1, −1, −1, −1, 1, −1, −1, 1, −1, 1, 1, 1, −1, 1, 1),
(1)
Manuscript received June 6, 2011. Manuscript revised August 29, 2011. † The authors are with College of Communication Engineering, Chongqing University, Chongqing 400030, China. †† The authors are with the Chongqing Key Laboratory of Emergency Communication, Chongqing Communication Institute, Chongqing 400035, China. ∗ The work is supported by the National Natural Science Foundation of China (NSFC) under Grants 60872164, 61171089, and 61002034, the Ministry of Industry and Information Technology of China (No.Equipment[2010]307), the Natural Science Project of CQ (CSTC, 2009BA2063, 2009DA0001, 2009AB2147, and 2010BB2203), and the Open Research Foundation of Chongqing Key Laboratory of Signal and Information Processing under Grant CQSIP-2010-01. a) E-mail:
[email protected] DOI: 10.1587/transfun.E95.A.400
whose autocorrelation is Rvr ,vr (0 ≤ τ ≤ 15) = (16, 0, 0, 0, 0, 0, 0, 0, −12, 0, 0, 0, 0, 0, 0, 0).
(2)
Subsequently, such sequences are generalized to almost perfect polyphase sequences by L¨uke and the number of the nonzero nontrivial autocorrelation values is enlarged to more than one [7]. An example given by L¨uke is with four nonzero out-of-phase autocorrelation values [7]. However, all methods mentioned above do not produced almost perfect QAM sequences. A complementary sequence pair is also one of many replacers, which consist of two sub-sequences and were firstly investigated by Golay [8]. Quickly, a complementary sequence pair is generalized to the periodic case from the original aperiodic case [9], and to polyphase case from binary case [10]. For more messages on complementary sequences, the reader is recommended to refer to Refs. [1] and [11]. In present communications, the signals over the quadrature amplitude modulation (QAM) constellation have been widely used, such as in 3GPP standard [12], where 16QAM and 64-QAM are recommended as modulation symbols. In addition, Ref. [13] shows that the communication system using the QAM sequences has a higher transmission data rate (TDR) than the one making use of traditional sequences with the same sequence length. Reference [14] states that the application of the QAM sequences with zero correlation zone (ZCZ) can hold both no multiple access interference (MAI) and high TDR, and Ref. [15] suggests that the QAM Golay complementary sequences can reduce the peak-to-mean envelope power ratio (PMEPR) of signals in an orthogonal frequency division multiplexing (OFDM) system. Therefore, the research on the QAM sequences with various properties has been widely considered. In this letter, the authors will focus on the investigation of almost perfect 16-QAM sequences, and three families of such sequences, resulting from quadriphase perfect sequences, are presented. In addition, it is worth mentioning, to the best authors’s knowledge, that there does not exist any almost perfect QAM sequence apart from ours. But, when the enlarged QAM alphabet set “QAM+” is consid ered, that is, QAM+=QAM {0} (very different from the traditional QAM constellation), only a paper [16] discussing such sequences by m-sequences is found by the authors. On the other hand, the periodic complementary sequence (PCS) pairs over the 16-QAM constellation are constructed as well.
c 2012 The Institute of Electronics, Information and Communication Engineers Copyright
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401
2.
construct a new sequence, called an interleaved sequence, I I (t)} = {vr (k)}{v s (k)} or vr,s = vr v s , which is denoted by {vr,s as follows.
Preliminaries
In this section, for convenience of the reader, we will review the related definitions of almost perfect sequences and an interleaved sequence, and the expression of 16-QAM constellation. 2.1 Almost Perfect Sequences Let vr = {vr (t)} = (vr (0), vr (1), vr (2), · · · , vr (N − 1)) and v s = {v s (t)} = (v s (0), v s (1), v s (2), · · · , v s (N − 1)) be two complex sequences with each of length N. We define the correlation function between the sequences {vr (t)} and {v s (t)} as follows. Rvr ,vs (τ) =
N−1
vr (t)v s (t + τ),
(3)
t=0
where v s (t) denotes the complex-conjugate of v s (t) and the addition t + τ is counted modulo N. If r = s, Rvr ,vr (τ) is referred to as an autocorrelation function, otherwise, a crosscorrelation function. If the autocorrelation of a sequence {vr (t)} with period N satisfies N τ = 0 (mod N) (4) Rvr ,vr (τ) = 0 other τ (except a few values), the sequence {vr (t)} is referred to as an almost perfect sequence. Apparently, when the condition “except a few values” in Eq. (4) is deleted, the sequence {vr (t)} is called a perfect sequence, whose many constructions are given in Refs. [1] and [17]–[19]. In communication applications, the nonzero out-ofphase autocorrelation values result in increase of error probability, therefore the almost perfect sequences with only a nonzero nontrivial autocorrelation value are preferred. 2.2 Periodic Complementary Sequence Pair Let (vr , vs ) consist of two sub-sequences with each of length N. If we have > 0 τ ≡ 0 (mod N) (5) Rvr ,vr (τ) + Rvs ,vs (τ) = 0 τ 0 (mod N), we refer to the sequence set (vr , v s ) as a periodic complementary sequence (PCS) pair. Let us have two PCS pairs (vr , v s ) and (vr , v s ). If we have Rvr ,vr (τ) + Rvs ,vs (τ) = 0
(∀ τ),
(6)
I {vr,s (t)} = {vr (k)} {v s (k)} = (vr (0), v s (0), vr (1), v s (1), · · · , vr (N − 1), v s (N − 1)),
(7)
where 0 ≤ t ≤ 2N − 1 and 0 ≤ k ≤ N − 1, which implies that an interleaved sequence has a length 2N. 2.4 16-QAM Constellation The M 2 -QAM constellation is the set {a + bi| − M + 1 ≤ a, b ≤ M − 1, and a, b odd},
(8)
where the symbol “i” denotes the imaginary unit, that is, i2 = −1. When M = 2m , the M 2 -QAM constellation can be driven by the quaternary phase-shift keying (QPSK) constellation [20]. More clearly, the M 2 -QAM constellation is equivalent to ⎞ ⎛m−1 ⎧ ⎫ ⎪ ⎪ ⎟⎟⎟ ⎜⎜⎜ ⎪ ⎪ ⎨ ⎬ k a k ⎟ ⎜ ⎟ ⎜ 2 i ∈ Z (1 + i) , (9) |a ⎪ ⎪ ⎟ ⎜ k 4 ⎪ ⎪ ⎠ ⎝ ⎩ ⎭ k=0
where Z4 = {0, 1, 2, 3}. In particular, the 16-QAM constellation has {(1 + i)(ia0 + 2ia1 )|a0 , a1 ∈ Z4 }.
(10)
Apart from the expression referred to above, the 16QAM constellation can be equivalently described as [13], [21] {(1 − i)(ia0 − 2ia1 )|a0 , a1 ∈ Z4 }. 3.
(11)
Constructions of Almost Perfect 16-QAM Sequences
In this section, one of the main results in this letter will be given, and three almost perfect 16-QAM sequence families are proposed from quadriphase perfect sequences. Throughout Sects. 3 and 4, let the sequence {ur (t)} be a quadriphase perfect sequence with length N, that is, Rur ,ur (τ) =
N−1
ur (t)−ur (t+τ)
i
t=0
=
N 0
τ=0 τ 0,
(12)
and δ1 and δ2 be two integers with 0 ≤ δ1 , δ2 < N and δ1 δ2 . Note that the period N of the known quadriphase perfect sequences must be even.
we say that those two PCS pairs are the mate to each other.
3.1 Construction Methods
2.3 An Interleaved Sequence
Construction I: In accordance with Eq. (10), a 16-QAM sequence can be given by
For sequences {vr (t)} and {v s (t)} with the same period N, we
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q1 (t) = (1 + i)[iur (t+δ1 ) + 2iur (t+δ2 ) ],
(13)
where
whose properties are given by Theorem 1: The sequence {q1 (t)} is an almost perfect 16QAM sequence, whose out-of-phase autocorrelation functions have at most two nonzero values. When δ1 = 0 and δ2 = N/2 or vice versa, only one nonzero value occurs. Proof : For the sake of convenience, let δ2 > δ1 . In accordance with Eq. (3), we have Rq1 ,q1 (τ) =
N−1
q1 (t)q1 (t + τ)
t=0
= (1 + i)(1 − i)
N−1 [iur (t+δ1 ) + 2iur (t+δ2 ) ][i−ur (t+δ1 +τ) t=0
+2i−ur (t+δ2 +τ) ] = 10Rur ,ur (τ)+4Rur ,ur (τ+δ2 −δ1 )+4Rur ,ur (τ+δ1 −δ2 ) ⎧ 10N τ = 0 ⎪ ⎪ ⎪ ⎪ ⎪ τ = δ2 − δ1 ⎨ 4N (14) =⎪ ⎪ 4N τ = N − δ2 + δ1 ⎪ ⎪ ⎪ ⎩ 0 other.
q2 (t) = (1 + i)[iur (t+δ2 ) + 2iur (t+δ1 ) ] p2 (t) = (1 − i)[iur (t+δ2 ) − 2iur (t+δ1 ) ],
Theorem 3: The interleaved sequences in (18) are almost perfect 16-QAM sequences with at most four nonzero outof-phase autocorrelation values. In particular, when δ1 = 0 and δ2 = N/2 or vice versa, only two nonzero out-of-phase autocorrelation values occur. Proof : For the sake of convenience and simplicity, we consider only q p and δ2 > δ1 , and the others are omitted 2 1 due to mostly similar derivation. Subsequently, we investigate the autocorrelation to be coped with by the time shifts τ even and odd, respectively. Case 1: The time shift τ = 2η. In this case, the relationship between the interleaved sequences uqI 1 ,p2 (t) and its cyclical shift version is as follows. p2 (1) ··· q1 (0) p2 (0) q1 (1) . q1 (η) p2 (η) q1 (η + 1) p2 (η + 1) · · ·
RuqI
whose properties are given by
+
R p1 ,p1 (τ) = 10Rur ,ur (τ) − 4Rur ,ur (τ + δ2 − δ1 ) −4Rur ,ur (τ + δ1 − δ2 ).
1
2
or uqI 2 ,p1 (t) = q p , 2
1
N−1
q1 (k)q1 (k + η)
N−1
p2 (k)p2 (k + η)
= 2[Rur ,ur (η)+2Rur ,ur (η+δ2 −δ1 )+2Rur ,ur (η+δ1 −δ2 ) +4Rur ,ur (η)] + 2[Rur ,ur (η) − 2Rur ,ur (η + δ1 − δ2 ) −2Rur ,ur (η + δ2 − δ1 ) + 4Rur ,ur (η)] 20N η = 0 (i.e., τ = 0) = 20Rur ,ur (η) = . (22) 0 other Case 2: The shift time τ = 2η + 1. In this case, the relationship between the interleaved sequences uqI 1 ,p2 (t) and its cyclical shift version is as follows. q1 (1) p2 (1) ··· q1 (0) p2 (0) . p2 (η) q1 (η + 1) p2 (η + 1) q1 (η + 2) · · ·
(23)
Hence, we have RuqI
(τ) ,uI 1 ,p2 q1 ,p2
=
N−1
q1 (k)p2 (k + η)
k=0
(17)
Construction III: In this construction, an almost perfect 16-QAM sequence is produced by using an interleaved technique. Hence, by employing (7) we have uqI 1 ,p2 (t) = q p
(τ) =
k=0
Theorem 2: The sequence {p1 (t)} is an almost perfect 16QAM sequence, whose out-of-phase autocorrelation functions have at most two nonzero values. When δ1 = 0 and δ2 = N/2 or vice versa, only one nonzero value appears. Proof : With the same argumentations in Theorem 1, nothing needs to be stated apart from that Eq. (14) is substituted into
,uI 1 ,p2 q1 ,p2
k=0
which results from τ − N/2 = τ + N/2 (mod N). Therefore, only a nonzero nontrivial autocorrelation value occurs in this case. Construction II: In accordance with Eq. (11), a 16-QAM sequence can be given by (16)
(21)
Hence, we have
It is apparent that apart from δ1 = 0 and δ2 = N/2, Rq1 ,q1 (τ) has two nonzero values except the center shift. Whereas when δ1 = 0 and δ2 = N/2, we have ⎧ ⎪ 10N τ = 0 ⎪ ⎪ ⎨ 8N τ = N/2 , Rq1 ,q1 (τ) = ⎪ (15) ⎪ ⎪ ⎩ 0 other
p1 (t) = (1 − i)[iur (t+δ1 ) − 2iur (t+δ2 ) ],
(19) (20)
(18)
+
N−1
p2 (k)q1 (k + η + 1)
k=0
= 2i[Rur ,ur (η + δ2 − δ1 ) − 2Rur ,ur (η) + 2Rur ,ur (η) −4Rur ,ur (η + δ1 − δ2 )] − 2i[Rur ,ur (η + δ1 − δ2 + 1) +2Rur ,ur (η+1)−2Rur ,ur (η+1)−4Rur ,ur (η+δ2 −δ1 +1)] = 2i[Rur ,ur (η + δ2 − δ1 ) − 4Rur ,ur (η + δ1 − δ2 ) −Rur ,ur (η + δ1 − δ2 + 1) + 4Rur ,ur (η + δ2 − δ1 + 1)]
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⎧ ⎪ 2Ni ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ −8Ni ⎪ ⎪ ⎨ −2Ni =⎪ ⎪ ⎪ ⎪ ⎪ 8Ni ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩0
η = N − δ2 + δ1 (i.e., τ = 2(N − δ2 + δ1 ) + 1) η = δ2 − δ1 (i.e., τ = 2(δ2 − δ1 ) + 1) η = δ2 − δ1 − 1 (i.e., τ = 2(δ2 − δ1 ) − 1) , (24) η = N − δ2 + δ1 − 1 (i.e., τ = 2(N − δ2 + δ1 ) − 1) other
which is apparently with four nonzero out-of-phase autocorrelation values apart from the following cases. Case: δ1 = 0 and δ2 = N/2. Note that “η − N/2 = η + N/2 (mod N)” and “η − N/2 + 1 = η + N/2 + 1”. Hence, we have RuqI
1 ,p2
,uqI 1 ,p2 (τ)
= 2i[Rur ,ur (η + N/2) − 4Rur ,ur (η − N/2)
−Rur ,ur (η − N/2 + 1) + 4Rur ,ur (η + N/2 + 1)] = 6i[−Rur ,ur (η + N/2) + Rur ,ur (η + N/2 + 1)] ⎧ ⎪ −6Ni η = N/2 (i.e., τ = N + 1) ⎪ ⎪ ⎨ 6Ni η = N/2 − 1 (i.e., τ = N − 1) . (25) =⎪ ⎪ ⎪ ⎩ 0 other Obviously, only two nonzero nontrivial autocorrelation values occur in this case. Example 1: In order to illuminate the proposed methods’ validity, a simple example is given due to space limitation. We choose arbitrarily a quadriphase perfect sequence from Example 2 in Ref. [19] as follows. {ur (t)} = (0000103220203012). Take δ1 = 0 and δ2 = N/2 = 8. As a consequence, we have
−96i, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), with the number 2 of the nonzero out-of-phase a ··· autocorrelation values exactly, as predicted, where ··· ··· b ··· denotes the element a + bi in the 16-QAM constellation. 4.
16-QAM Periodic Complementary Sequence Pairs
In this section, another main result will be stated, and the PCS pairs over the 16-QAM constellation will be given. 4.1 Construction Methods Construction IV: Theorem 4: Let {q1 (t)}, {p1 (t)}, {q2 (t)}, and {p2 (t)} be produced by Eqs. (13), (16), (19), and (20), respectively. Hence, (q , p ) is a PCS pair, and so is (p , q ). 1
Rq1 ,q1 (τ) = (160, 0, 0, 0, 0, 0, 0, 0, 128, 0, 0, 0, 0, 0, 0, 0) with a nonzero out-of-phase autocorrelation value exactly as predicted, {p2 (t)} =
−3 −1 −3 −1 −3 −1 3 1 3 −1 3 −1 3 −1 −3 1 , 3 1 3 1 −3 1 3 −1 −3 1 −3 1 3 1 −3 −1
whose autocorrelation function to 0 ≤ τ ≤ 15 is R p2 ,p2 (τ) = (160, 0, 0, 0, 0, 0, 0, 0, −128, 0, 0, 0, 0, 0, 0, 0), and q1 p2 = −1 −3 3 −1 −1 −3 3 −1 1 −3 3 −1 −1 3 −3 1 −1 3 3 1 −1 3 3 1 −1 −3 3 1 1 3 −3 −1 1 3 3 −1 1 3 3 −1 −1 3 3 −1 1 −3 −3 1 , 1 −3 3 1 1 −3 3 1 1 3 3 1 −1 −3 −3 −1
2
2
Proof : In accordance with the definition in (5), and Eqs. (14) and (17), we have 20N τ = 0 (26) Rq1 ,q1 (τ)+R p1 ,p1 (τ) = 20Rur ,ur (τ) = 0 τ 0, due to Eq. (12). This theorem follows immediately.
Theorem 5: Let {q1 (t)}, {p1 (t)}, {q2 (t)}, and {p2 (t)} be produced by Eqs. (13),(16), (19), and (20), respectively. Hence, (q , p ) and (p , q ) are the mate to each other. 1
1
2
2
Proof : In accordance with the definition in (6) and Theorem 4, we only need to prove
{q1 (t)} = −1 3 −1 3 1 3 −1 −3 1 3 1 3 −1 3 1 −3 , −1 3 −1 3 −1 3 1 −3 1 3 1 3 1 3 −1 −3 whose autocorrelation function to 0 ≤ τ ≤ 15 is
1
Rq1 ,p2 (τ) + R p1 ,q2 (τ) = 0
(∀ τ).
(27)
Since we have Rq1 ,p2 (τ) + R p1 ,q2 (τ) =
N−1
q1 (t)p2 (t + τ)
t=0
+
N−1
p1 (t)q2 (t + τ)
t=0
= (1 + i)2 iur (t+δ1 ) + 2iur (t+δ2 ) i−ur (t+δ2 +τ) −2i−ur (t+δ1 +τ) + (1 − i)2 iur (t+δ1 ) − 2iur (t+δ2 ) −ur (t+δ2 +τ) + 2i−ur (t+δ1 +τ) ·i = 2i Rur ,ur (τ + δ2 − δ1 ) − 2Rur ,ur (τ) + 2Rur ,ur (τ) −4Rur ,ur (τ + δ1 − δ2 ) − 2i Rur ,ur (τ + δ2 − δ1 ) + 2Rur ,ur (τ) −2Rur ,ur (τ) − 4Rur ,ur (τ + δ1 − δ2 ) = 0. (28)
whose autocorrelation function to 0 ≤ τ ≤ 31 is
This concludes our proof. Example 2: The quadriphase perfect sequence {ur (t)} is the same as the one in Example 1, and δ1 = 0 and δ2 = 8. Therefore, we have 1 3 1 3 −1 3 1 −3 −1 3 −1 3 1 3 −1 −3 {q2 (t)} = , 1 3 1 3 1 3 −1 −3 −1 3 −1 3 −1 3 1 −3
Rq1 ,p2 (τ) = (320, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 96i, 0,
whose autocorrelation function to 0 ≤ τ ≤ 15 is
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Rq2 ,q2 (τ) = (160, 0, 0, 0, 0, 0, 0, 0, 128, 0, 0, 0, 0, 0, 0, 0),
employed is unaltered, and δ1 = 15 and δ2 = 3. Therefore, the interleaved sequence q p is 2
1
and
whose autocorrelation function to 0 ≤ τ ≤ 15 is
{q p } 1 2 1 3 −1 −1 3 −1 3 −3 −1 −3 −3 = 1 −3 3 3 3 1 −1 1 −1 3 −1 1 3 1 1 3 −1 −3 3 −1 −3 3 3 1 −3 −3 −3 3 1 1 −1 −1 3 1 1
R p1 ,p1 (τ) = (160, 0, 0, 0, 0, 0, 0, 0, −128, 0, 0, 0, 0, 0, 0, 0)
whose periodic autocorrelation to 0 ≤ τ ≤ 31 is
{p1 (t)} = 3 −1 3 −1 3 −1 −3 1 −3 −1 −3 −1 −3 −1 3 1 , −3 1 −3 1 3 1 −3 −1 3 1 3 1 −3 1 3 −1
and the interleaved sequence p q is given as 1
It is apparent that Eqs. (26) and (27) hold, which implies that (q , p ) and (p , q ) are the mate to each other. 1 1 2 2 Construction V: Theorem 6: Let {q1 (t)}, {p1 (t)}, {q2 (t)}, and {p2 (t)} be produced by Eqs. (13), (16), (19), and (20), respectively. Hence, (q p , p q ) is a PCS pair. 1
2
1
2
1
p ,q p 2
1
2
(2η) = Rq1 ,q1 (η) + R p2 ,p2 (η)
2
{p q } 1 2 −3 −1 −1 1 −1 3 3 3 3 1 3 −3 −1 3 1 1 = 3 −1 −3 3 1 3 1 1 −3 1 −1 1 1 3 −3 −3 −3 −1 1 −1 −1 3 −3 −3 3 1 −3 3 −1 3 −1 −1 . 3 −1 3 −3 1 3 −1 −1 −3 1 1 −1 1 3 3 3 whose periodic autocorrelation to 0 ≤ τ ≤ 31 is R p q ,p q (τ) = (320, 0, 0, 0, 0, 0, 0, 32i, 0, 1 2 1 2 128i, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, −128i, 0, −32i, 0, 0, 0, 0, 0, 0).
Prrof : Similar to the proof of Theorem 3, by the relationships in (21) and (23) we have Rq
−1 −1 1 1 −3 3 1 −1 . 3 −3
Rq p ,q p (τ) = (320, 0, 0, 0, 0, 0, 0, −32i, 0, 2 1 2 1 −128i, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 128i, 0, 32i, 0, 0, 0, 0, 0, 0),
with only a nonzero nontrivial autocorrelation value, as predicted. After calculating crosscorrelation function to 0 ≤ τ ≤ 15, we have Rq1 ,p2 (τ) = (0, 0, 0, 0, 0, 0, 0, 0, 0, −96i, 0, 0, 0, 0, 0, 0, 0) R p1 ,q2 (τ) = (0, 0, 0, 0, 0, 0, 0, 0, 0, 96i, 0, 0, 0, 0, 0, 0, 0).
−3 3 −1 3 3 −1 3 1
(29)
It is apparent that we have
and Rq
1
p ,q p 2
1
2
(2η + 1) = Rq1 ,p2 (η) + R p2 ,q1 (η + 1).
Rq
(30)
1
q ,p q 2
1
2
(2η) = R p1 ,p1 (η) + Rq2 ,q2 (η)
p ,q p
=
With the same argumentations, we have Rp
1
(31)
and
2
1
2
(τ) + R p
640 0
1
q ,p q 1
2
2
(τ)
τ ≡ 0 (mod N) other,
which indicates that (q p , p q ) is a PCS pair. 1
2
1
2
4.2 Discussion Rp
1
q ,p q 2
1
2
(2η + 1) = R p1 ,q2 (η) + Rq2 ,p1 (η + 1).
(32)
Hence, the sum of periodic autocorrelation functions of the two interleaved sequences q p and p q can be 2 1 1 2 given by Rq
1
p ,q p 2
1
2
(2η) + R p
1
q ,p q 2
1
2
(2η)
= [Rq1 ,q1 (η) + R p1 ,p1 (η)] + [R p2 ,p2 (η) + Rq2 ,q2 (η)] 40N η = 0 (33) = 40Rur ,ur (η) = 0 η 0, which results from Theorem 4, and Rq
1
p ,q p 2
1
2
(2η + 1) + R p
1
q ,p q 2
1
2
(2η + 1)
= [Rq1 ,p2 (η)+R p1 ,q2 (η)]+[R p2 ,q1 (η+1)+Rq2 ,p1 (η+1)] = 0, (34) which is due to Theorem 5. This theorem is true from (33) and (34). Example 3: The quadriphase perfect sequence {ur (t)}
The simulation results by a computer show that if one of δ1 and δ2 is given, and the other varies from 0 to N − 1 apart from a given value, the resultant 16-QAM PCS pairs are distinct in terms of cyclical shift equivalence. Hence, we can obtain (N − 1) × M distinct 16-QAM PCS pairs with lengths N and 2N, respectively to a given N, where M denotes the family size of quaternary perfect sequences employed. For example, in Ref. [19], 18 distinct quadriphase perfect sequences with length N = 16 are proposed, therefore, we have 15 × 18 = 270 distinct 16-QAM PCS pairs with lengths 16 and 32, respectively. 5.
Conclusion
This letter presents three families of almost perfect 16-QAM sequences and the PCS pairs over the 16-QAM constellation, and the simulation results by a computer illuminate the validity of the proposed methods. Therefore, the resulting sequences over the 16-QAM constellation provide potential
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candidates for applications in the future. Acknowledgments The authors would like to thank anonymous referees for helpful suggestions that greatly improve the presentation quality of this letter. References [1] P.Z. Fan and M. Darnell, “Sequence design for communications applications,” John Wilet & Sons, 1996. [2] R.F. Brown and G.C. Goodwin, “New class of pseudorandom binary sequences,” Electron. Lett., vol.3, pp.198–199, May 1967. [3] J. Wolfmann, “Almost perfect autocorrelation sequences,” IEEE Trans. Inf. Theory, vol.38, pp.1412–1418, July 1992. [4] P. Langevin, “Almost perfect binary functions,” Appl. Algebra in Eng., Commun. Comput., vol.4, pp.95–102, 1993. [5] A. Pott and S.P. Bradley, “Existence and nonexistence of almostperfect autocorrelation sequences,” IEEE Trans. Inf. Theory, vol.41, pp.301–304, Jan. 1995. [6] H.D. L¨uke and H.D. Schotten, “Odd-perfect almost binary correlation sequences,” IEEE Trans. Aerosp. Electron. Syst., vol.31, pp.495–498, Jan. 1995. [7] H.D. L¨uke, “Almost-perfect polyphase sequences with small phase alphabet,” IEEE Trans. Inf. Theory, vol.43, no.1, pp.361–363, Jan. 1997. [8] M.J.E. Golay, “Complementary series,” IRE Trans. Inf. Theory, vol.IT-7, pp.82–87, April 1960. [9] L. B¨omer and M. Antweiler, “Periodic complementary binary sequences,” IEEE Trans. Inf. Theory, vol.36, no.6, pp.1487–1497, Nov. 1990.
[10] C.C. Tseng and C.L. Liu, “Complementary sets of sequences,” IEEE Trans. Inf. Theory, vol.IT-18, no.5, pp.644–652, Sept. 1972. [11] M.G. Parker, C. Tellambura, and K.G. Paterson, “Golay complementary sequences,” in Wiley Encyclopedia of Telecommunicatins, ed. J.G. Proakis, Wiley, 2003. [12] 3GPP TS 36 2.11, v.8.2.0, Release 8, March 2008. [13] M. Anand and P.V. Kumar, “Low-correlation sequences over the QAM constellation,” IEEE Trans. Inf. Theory, vol.54, no.2, pp.791– 810, Feb. 2008. [14] F.X. Zeng and Z.Y. Zhang, “16-QAM sequences with zero correlation zone from the known quadriphase ZCZ sequences,” IEICE Trans. Fundamentals, vol.E94-A, no.3, pp.1023–1028, March 2011. [15] C.V. Chong, R. Venkataramani, and V. Tarokh, A new construction of 16-QAM Golay complementary sequences, IEEE Trans. Inf. Theory, vol.49, no.11, pp.2953–2959, Nov. 2003. [16] S. Boztas and U. Parampalli, “Nonbinary sequences with perfect and nearly perfect autocorrelations.” Proc. of IEEE ISIT2010, pp.1300– 1304, Austin, Texas, U.S.A., June 2010. [17] D.C. Chu, “Polyphase codes with good periodic correlation properties,” IEEE Trans. Inf. Theory, vol.18, no.4, pp.531–532, July 1972. [18] R.L. Frank and S.A. Zadoff, “Phase shift pulse codes with good periodic correlation properties,” IRE Trans. Inf. Theory (Corresp.), vol.IT-8, pp.381–382, Oct. 1962. [19] F.X. Zeng, “New perfect polyphase sequences and mutually orthogonal ZCZ polyphase sequence sets,” IEICE Trans. Fundamentals, vol.E92-A, no.7, pp.1731–1736, July 2009. [20] B. Tarokh and H.R. Sadjadpour, “Construction of OFDM-QAM sequences with low peak to average power ratio,” IEEE Trans. Commun., vol.51, no.1, pp.25–28, Jan. 2003. [21] G. Garg, P.V. Kumar, and C.E. Veni Madhavan, “Low correlation interleaved QAM sequences,” Proc. IEEE ISIT2008, pp.1935–1939, Toronto, Canada, July 2008.
