LETTER Almost Perfect Sequences and Periodic Complementary Sequence Pairs over the 16-QAM Constellation

Transcription

1 400 IEICE TRANS. FUNDAMENTALS VOL.E95 A NO.1 JANUARY 2012 LETTER Almost Perfect Sequences Periodic Complementary Sequence Pairs over the 16-QAM Constellation Fanxin ZENG a) Member Xiaoping ZENG Nonmember Zhenyu ZHANG Member Guixin XUAN Nonmember SUMMARY Based on quadriphase perfect sequences 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 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 v r (t)} ( ) Manuscript received June Manuscript revised August The authors are with College of Communication Engineering Chongqing University Chongqing China. The authors are with the Chongqing Key Laboratory of Emergency Communication Chongqing Communication Institute Chongqing China. The work is supported by the National Natural Science Foundation of China (NSFC) under Grants the Ministry of Industry Information Technology of China (No.Equipment[2010]307) the Natural Science Project of CQ (CSTC 2009BA DA AB BB2203) the Open Research Foundation of Chongqing Key Laboratory of Signal Information Processing under Grant CQSIP a) fzengx@yahoo.com.cn DOI: /transfun.E95.A.400 (1) whose autocorrelation is R vr v r (0 τ 15) (2) ( ). Subsequently such sequences are generalized to almost perfect polyphase sequences by Lüke the number of the nonzero nontrivial autocorrelation values is enlarged to more than one [7]. An example given by Lüke 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 were firstly investigated by Golay [8]. Quickly a complementary sequence pair is generalized to the periodic case from the original aperiodic case [9] to polyphase case from binary case [10]. For more messages on complementary sequences the reader is recommended to refer to Refs. [1] [11]. In present communications the signals over the quadrature amplitude modulation (QAM) constellation have been widely used such as in 3GPP stard [12] where 16- QAM 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) high TDR 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 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 considered 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 h the periodic complementary sequence (PCS) pairs over the 16-QAM constellation are constructed as well. Copyright c 2012 The Institute of Electronics Information Communication Engineers

2 LETTER Preliminaries In this section for convenience of the reader we will review the related definitions of almost perfect sequences an interleaved sequence the expression of 16-QAM constellation. 2.1 Almost Perfect Sequences Let v r v r (t)} (v r (0)v r (1)v r (2) v r (N 1)) 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 v r (t)} v s (t)} as follows. R vr v s (τ) v r (t)v s (t + τ) (3) where v s (t) denotes the complex-conjugate of v s (t) the addition t + τ is counted modulo N. Ifr s R vr v r (τ) isreferred to as an autocorrelation function otherwise a crosscorrelation function. If the autocorrelation of a sequence v r (t)} with period N satisfies N τ 0 (mod N) R vr v r (τ) (4) 0 other τ (except a few values) the sequence v r (t)} is referred to as an almost perfect sequence. Apparently when the condition except a few values in Eq. (4) is deleted the sequence v r (t)} is called a perfect sequence whose many constructions are given in Refs. [1] [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 (v r v s ) consist of two sub-sequences with each of length N. Ifwehave > 0 τ 0 (mod N) R vr v r (τ) + R vs v s (τ) (5) 0 τ 0 (mod N) we refer to the sequence set (v r v s ) as a periodic complementary sequence (PCS) pair. Let us have two PCS pairs (v r v s )(v r v s ). If we have R vr v r (τ) + R vs v s (τ) 0 ( τ) (6) we say that those two PCS pairs are the mate to each other. 2.3 An Interleaved Sequence For sequences v r (t)} v s (t)} with the same period Nwe construct a new sequence called an interleaved sequence which is denoted by v I rs(t)} v r (k)} v s (k)} or v I rs v r v s as follows. vrs(t)} I v r (k)} v s (k)} (v r (0)v s (0)v r (1) v s (1) v r (N 1)v s (N 1)) (7) where 0 t 2N 10 k N 1 which implies that an interleaved sequence has a length 2N QAM Constellation The M 2 -QAM constellation is the set a + bi M + 1 a b M 1 a b odd} (8) where the symbol i denotes the imaginary unit that is i 2 1. When M 2 m them 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 (1 + i) 2 k i a k a k Z 4 (9) where Z }. In particular the 16-QAM constellation has (1 + i)(i a 0 + 2i a 1 ) a 0 a 1 Z 4 }. (10) Apart from the expression referred to above the 16- QAM constellation can be equivalently described as [13] [21] (1 i)(i a 0 2i a 1 ) a 0 a 1 Z 4 }. (11) 3. Constructions of Almost Perfect 16-QAM Sequences In this section one of the main results in this letter will be given three almost perfect 16-QAM sequence families are proposed from quadriphase perfect sequences. Throughout Sects. 3 4 let the sequence u r (t)} be a quadriphase perfect sequence with length Nthatis R ur u r (τ) i u r(t) u r (t+τ) N τ 0 0 τ 0 (12) δ 1 δ 2 be two integers with 0 δ 1 δ 2 < N δ 1 δ 2. Note that the period N of the known quadriphase perfect sequences must be even. 3.1 Construction Methods Construction I: In accordance with Eq. (10) a 16-QAM sequence can be given by

3 402 IEICE TRANS. FUNDAMENTALS VOL.E95 A NO.1 JANUARY 2012 q 1 (t) (1 + i)[i u r(t+δ 1 ) + 2i u r(t+δ 2 ) ] (13) whose properties are given by Theorem 1: The sequence q 1 (t)} is an almost perfect 16- QAM sequence whose out-of-phase autocorrelation functions have at most two nonzero values. When δ 1 0 δ 2 N/2 or vice versa only one nonzero value occurs. Proof : For the sake of convenience let δ 2 >δ 1. In accordance with Eq. (3) we have R q1 q 1 (τ) q 1 (t)q 1 (t + τ) (1 + i)(1 i) [i u r(t+δ 1 ) + 2i u r(t+δ 2 ) ][i u r(t+δ 1 +τ) +2i u r(t+δ 2 +τ) ] 10R ur u r (τ)+4r ur u r (τ+δ 2 δ 1 )+4R ur u r (τ+δ 1 δ 2 ) 10N τ 0 4N τ δ 2 δ 1 (14) 4N τ N δ 2 + δ 1 0 other. It is apparent that apart from δ 1 0δ 2 N/2 R q1 q 1 (τ) has two nonzero values except the center shift. Whereas when δ 1 0δ 2 N/2 we have R q1 q 1 (τ) 10N τ 0 8N τ N/2 0 other (15) which results from τ N/2 τ + N/2 (modn). 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 p 1 (t) (1 i)[i u r(t+δ 1 ) 2i u r(t+δ 2 ) ] (16) whose properties are given by Theorem 2: The sequence p 1 (t)} is an almost perfect 16- QAM sequence whose out-of-phase autocorrelation functions have at most two nonzero values. When δ 1 0 δ 2 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 R p1 p 1 (τ) 10R ur u r (τ) 4R ur u r (τ + δ 2 δ 1 ) (17) 4R ur u r (τ + δ 1 δ 2 ). Construction III: In this construction an almost perfect 16-QAM sequence is produced by using an interleaved technique. Hence by employing (7) we have u I q 1 p 2 (t) q 1 or u I q 2 p 1 (t) q 2 p 1 (18) where q 2 (t) (1 + i)[i u r(t+δ 2 ) + 2i u r(t+δ 1 ) ] (19) p 2 (t) (1 i)[i u r(t+δ 2 ) 2i u r(t+δ 1 ) ] (20) 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 δ 1 0 δ 2 N/2 or vice versa only two nonzero out-of-phase autocorrelation values occur. Proof : For the sake of convenience simplicity we consider only q 1 δ 2 >δ 1 the others are omitted due to mostly similar derivation. Subsequently we investigate the autocorrelation to be coped with by the time shifts τ even odd respectively. Case 1: The time shift τ 2η. In this case the relationship between the interleaved sequences u I q 1 p 2 (t) its cyclical shift version is as follows. q 1 (0) p 2 (0) q 1 (1) p 2 (1) q 1 (η) p 2 (η) q 1 (η + 1) p 2 (η + 1). (21) Hence we have R u I q1 p 2 uq I (τ) q 1 p 1 (k)q 1 (k + η) 2 N 1 + p 2 (k)p 2 (k + η) 2[R ur u r (η)+2r ur u r (η+δ 2 δ 1 )+2R ur u r (η+δ 1 δ 2 ) +4R ur u r (η)] + 2[R ur u r (η) 2R ur u r (η + δ 1 δ 2 ) 2R ur u r (η + δ 2 δ 1 ) + 4R ur u r (η)] 20N η 0(i.e.τ 0) 20R ur u r (η). (22) 0 other Case 2: The shift time τ 2η + 1. In this case the relationship between the interleaved sequences u I q 1 p 2 (t) its cyclical shift version is as follows. q 1 (0) p 2 (0) q 1 (1) p 2 (1) p 2 (η) q 1 (η + 1) p 2 (η + 1) q 1 (η + 2). (23) Hence we have R u I q1 p 2 uq I (τ) q 1 p 1 (k)p 2 (k + η) 2 N 1 + p 2 (k)q 1 (k + η + 1) 2i[R ur u r (η + δ 2 δ 1 ) 2R ur u r (η) + 2R ur u r (η) 4R ur u r (η + δ 1 δ 2 )] 2i[R ur u r (η + δ 1 δ 2 + 1) +2R ur u r (η+1) 2R ur u r (η+1) 4R ur u r (η+δ 2 δ 1 +1)] 2i[R ur u r (η + δ 2 δ 1 ) 4R ur u r (η + δ 1 δ 2 ) R ur u r (η + δ 1 δ 2 + 1) + 4R ur u r (η + δ 2 δ 1 + 1)]

4 LETTER 2Ni η N δ 2 + δ 1 (i.e. τ 2(N δ 2 + δ 1 ) + 1) 8Ni η δ 2 δ 1 (i.e. τ 2(δ 2 δ 1 ) + 1) 2Ni η δ 2 δ 1 1(i.e.τ 2(δ 2 δ 1 ) 1) (24) 8Ni 0 other η N δ 2 + δ 1 1(i.e. τ 2(N δ 2 + δ 1 ) 1) which is apparently with four nonzero out-of-phase autocorrelation values apart from the following cases. Case: δ 1 0δ 2 N/2. Note that η N/2 η+ N/2(modN) η N/2+ 1 η + N/ Hence we have R u I q1 p 2 uq I (τ) 2i[R 1 p ur u r (η + N/2) 4R ur u r (η N/2) 2 R ur u r (η N/2 + 1) + 4R ur u r (η + N/2 + 1)] 6i[ R ur u r (η + N/2) + R ur u r (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. u r (t)} ( ). Take δ 1 0δ 2 N/2 8. As a consequence we have q 1 (t)} R q1 q 1 (τ) ( ) with a nonzero out-of-phase autocorrelation value exactly as predicted p 2 (t)} R p2 p 2 (τ) ( ) q 1 ( whose autocorrelation function to 0 τ 31 is R q1 p 2 (τ) ( i 0 ) 96i ) with the number 2 of the nonzero out-of-phase autocorrelation values exactly as predicted where ( a b ) denotes the element a + bi in the 16-QAM constellation QAM Periodic Complementary Sequence Pairs In this section another main result will be stated the PCS pairs over the 16-QAM constellation will be given. 4.1 Construction Methods Construction IV: Theorem 4: Let q 1 (t)} p 1 (t)} q 2 (t)} p 2 (t)} be produced by Eqs. (13) (16) (19) (20) respectively. Hence (q p ) is a PCS pair so is (p q ) Proof : In accordance with the definition in (5) Eqs. (14) (17) we have 20N τ0 R q1 q 1 (τ)+r p1 p 1 (τ)20r ur u r (τ) (26) 0 τ 0 due to Eq. (12). This theorem follows immediately. Theorem 5: Let q 1 (t)} p 1 (t)} q 2 (t)} p 2 (t)} be produced by Eqs. (13)(16) (19) (20) respectively. Hence (q p )(p q ) are the mate to each other Proof : In accordance with the definition in (6) Theorem 4 we only need to prove 403 R q1 p 2 (τ) + R p1 q 2 (τ) 0 ( τ). (27) Sincewehave R q1 p 2 (τ) + R p1 q 2 (τ) q 1 (t)p 2 (t + τ) p 1 (t)q 2 (t + τ) N 1 + (1 + i) 2[ i u r(t+δ 1 ) + 2i u r(t+δ 2 ) ][ i u r(t+δ 2 +τ) 2i u r(t+δ 1 +τ) ] + (1 i) 2[ i u r(t+δ 1 ) 2i u r(t+δ 2 ) ] [i u r(t+δ 2 +τ) + 2i u r(t+δ 1 +τ) ] 2i [ R ur u r (τ + δ 2 δ 1 ) 2R ur u r (τ) + 2R ur u r (τ) 4R ur u r (τ + δ 1 δ 2 ) ] 2i [ R ur u r (τ + δ 2 δ 1 ) + 2R ur u r (τ) 2R ur u r (τ) 4R ur u r (τ + δ 1 δ 2 ) ] 0. (28) This concludes our proof. Example 2: The quadriphase perfect sequence u r (t)} is the same as the one in Example 1 δ 1 0δ 2 8. Therefore we have q 2 (t)}

5 404 IEICE TRANS. FUNDAMENTALS VOL.E95 A NO.1 JANUARY 2012 R q2 q 2 (τ) ( ) p 1 (t)} R p1 p 1 (τ) ( ) with only a nonzero nontrivial autocorrelation value as predicted. After calculating crosscorrelation function to 0 τ 15 we have R q1 p 2 (τ) ( i ) R p1 q 2 (τ) ( i ). It is apparent that Eqs. (26) (27) hold which implies that (q p )(p q ) are the mate to each other Construction V: Theorem 6: Let q 1 (t)} p 1 (t)} q 2 (t)} p 2 (t)} be produced by Eqs. (13) (16) (19) (20) respectively. Hence (q p p q )isapcspair Prrof : Similar to the proof of Theorem 3 by the relationships in (21) (23) we have R q1 p 2 q 1 p 2 (2η) R q1 q 1 (η) + R p2 p 2 (η) (29) R q1 p 2 q 1 p 2 (2η + 1) R q1 p 2 (η) + R p2 q 1 (η + 1). (30) With the same argumentations we have R p1 q 2 p 1 q 2 (2η) R p1 p 1 (η) + R q2 q 2 (η) (31) R p1 q 2 p 1 q 2 (2η + 1) R p1 q 2 (η) + R q2 p 1 (η + 1). (32) Hence the sum of periodic autocorrelation functions of the two interleaved sequences q 1 p 1 q 2 can be given by R q1 p q p (2η) + R p1 q p q (2η) [R q1 q 1 (η) + R p1 p 1 (η)] + [R p2 p 2 (η) + R q2 q 2 (η)] 40N η 0 40R ur u r (η) (33) 0 η 0 which results from Theorem 4 R q1 p q p (2η + 1) + R p1 q p q (2η + 1) [R q1 p 2 (η)+r p1 q 2 (η)]+[r p2 q 1 (η+1)+r q2 p 1 (η+1)] 0 (34) which is due to Theorem 5. This theorem is true from (33) (34). Example 3: The quadriphase perfect sequence u r (t)} employed is unaltered δ 1 15 δ 2 3. Therefore the interleaved sequence q 1 is q p } 1 ( ) whose periodic autocorrelation to 0 τ 31 is R q1 p 2 q 1 p 2 (τ) ( i 0 128i i 0 32i ) the interleaved sequence p 1 q 2 is given as p q } 1 ( ) whose periodic autocorrelation to 0 τ 31 is R p1 q 2 p 1 q 2 (τ) ( i 0 128i i 0 32i ). It is apparent that we have R q1 p q p (τ) + R p1 q p q (τ) τ 0 (mod N) 0 other which indicates that (q 1 p 1 q 2 )isapcspair. 4.2 Discussion The simulation results by a computer show that if one of δ 1 δ 2 is given 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 2N respectively to a given N wherem 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 distinct 16-QAM PCS pairs with lengths respectively. 5. Conclusion This letter presents three families of almost perfect 16-QAM sequences the PCS pairs over the 16-QAM constellation 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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