Perfect Gaussian Integer Sequences of Period p With Degrees Equal

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 65, NO. 9, SEPTEMBER 2017

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Perfect Gaussian Integer Sequences of Period p k With Degrees Equal to or Less Than k + 1 Kuo-Jen Chang and Ho-Hsuan Chang

Abstract— This paper presents the construction of perfect Gaussian integer sequence (PGIS) of period N = p k with degrees equal to or less than k + 1, where p is a prime number and k ≥ 2. The study begins with the partitioning of Z N into k + 1 subsets, from which k +1 base sequences are defined to construct degree-(k + 1) PGISs. The k constraint equations derived from matching the ideal periodic autocorrelation function criterion in the time domain are nonlinear. We propose a decomposition method that can transform these k nonlinear equations into 2k−2 linear equations with 2k − 2 variables. The number of nonlinear constraint equations derived from the frequency domain is k + 1, where the k + 1 sequence coefficients can be derived sequentially by an iteration algorithm. These two proposed schemes simplify the construction of the degree-(k + 1) PGIS. PGISs having less than k + 1 degrees can be constructed in two steps. First, some sequence coefficients are assigned a zero value or one or more pairs of two consecutive base sequences are combined into one. The second step involves the adjustment of new sequence coefficients for the associated sequence to be perfect. We show that there exist at least 2k + ((k(k − 1)(k − 2))/6) distinct PGIS patterns of period N = p k with degrees equal to or less than k+1. Index Terms— Gaussian integers, PACF, perfect sequence, PGIS.

I. I NTRODUCTION

A

GAUSSIAN integer sequence (GIS) is a sequence with coefficients that are complex numbers a + bj , where √ j = −1 and a and b are integers. The construction of perfect Gaussian integer sequence (PGIS) has become an important research topic [1]–[8] because integers require less memory; also, the implementation of a PGIS is simpler than those of other perfect sequences (PSs) with real or complex coefficients, in which a sequence is perfect if and only if it has an ideal periodic autocorrelation function (PACF). Regarding the applications of PGISs, the PGISs were applied to the OFDM systems for peak to average power ratio (PAPR) reduction [9]. Then, PGISs were used to construct a transform matrix for the precoded OFDM systems to

Manuscript received March 7, 2017; revised May 15, 2017; accepted June 6, 2017. Date of publication June 12, 2017; date of current version September 14, 2017. This work was supported by the Ministry of Science and Technology, Taiwan, ROC, under grants MOST 104-2221-E-214-008, MOST 105-2221-E-214-006, and MOST 106-2119-M-027-003. This paper was presented in the 2017 IEEE International Symposium on Information Theory. The associate editor coordinating the review of this paper and approving it for publication was C. Tian. (Corresponding author: Ho-Hsuan Chang.) K.-J. Chang is with the Civil Engineering Department, National Taipei University of Technology, Taipei 10608, Taiwan (e-mail: [email protected]). H.-H. Chang is with the Communication Engineering Department, I-SHOU University, Kaohsiung 84001, Taiwan (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TCOMM.2017.2714702

achieve full frequency diversity and obtain optimal bit error rate (BER) [10]. PGISs were also adopted as the frequencydomain comb-spectrum (CS) codes for a novel CS-CDMA system [11]. Recently, a CDMA scheme based on PGISs, called PGIS-CDMA system, was developed by Chang et al. [12], where a set of PGISs can be functioned as the pseudonoise (PN) codes, e.g., m-sequences, Gold sequences, Kasami sequences, and bent sequences, to a direct sequence (DS)-CDMA system. When a set of PGIGs is applied for channelization to a PGIS-CDMA scheme, PGIS-CDMA can outperform DS-CDMA from carrier-to-interference ratio ( CI ) point of view [12]; however, the value of CI is determined by both the number of distinct PGIS patterns and sequence degrees of these PGISs [12], where both pattern and degree are defined in Section II. Finally, we would mention that some applications, such as channel estimation, equalization, synchronization, and CDMA systems, currently being realized using PSs [13]–[18] should be substituted by PGISs in the future by reasons of the same ideal PACF property and implementation simplicity. By tracing the construction of PGISs, a general form of even-period PGISs was presented in [1], in which the PGISs were constructed by linearly combining four base sequences or their cyclic shift equivalents using Gaussian integer coefficients of equal magnitude. Yang et al. [2] constructed PGISs of odd prime period p by using cyclotomic classes with respect to the multiplicative group of GF( p). Ma et al. [3] later presented PGISs with a period of p( p+2) based on Whiteman’s generalized cyclotomy of order two over Z p( p+2) , where p and ( p+2) are twin primes. Chang et al. [4] introduced the concept of the degree of a sequence and constructed degree-2 and degree-3 PGISs of prime period p. Then, they up-sampled these PGISs by a factor of m and filled them with new coefficients to build degree-3 and degree-4 PGISs of arbitrary composite period N = mp. Lee et al. focused on constructing degree-2 PGISs of various periods using two-tuple-balanced sequences and cyclic difference sets [5], [6]. Algorithms that could generate PGISs of arbitrary period were developed by Pei and Chang [7], and one of these algorithms could be applied to construct degree-5 PGISs of prime period p ≡ 1(mod 4) by applying the generalized Legendre sequences (GLS). A systematic method for constructing sparse PGISs in which most of the elements are zero was recently proposed [8]. This paper presents a new construction of PGISs of period N = pk with degrees equal to or less than k + 1, where p is a prime number and k ≥ 2. The periodic PGISs of period pk with p > 2 first appeared in [7], in which Pei constructed

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these PGISs in four steps. 1) The Legendre sequence (LS) of the prime period p was duplicated pk−1 times to generate a GLS of length pk . 2) The elements of GLS were subjected to adjustments such that the spectrum of the sequence was almost the same magnitude. 3) It required k times iteration to eliminate the remaining zeros of the sequence obtained from the second step. 4) Taking discrete Fourier transform (DFT) upon the resultant sequence generated a PGIS. In [7], PGISs of period pk can be treated as a special case of arbitrary length PGISs, in which this topic is only addressed briefly; however, we explore the theoretical study with new approaches of constructing PGISs of period pk in this paper. The study begins with the partitioning of the set Z N = {0, 1, 2, . . . , N − 1} into k + 1 subsets, where N = pk , from which k + 1 base sequences are defined for constructing PGISs with degree-(k + 1). We apply both the time and frequency domains two different approaches for the construction of PGISs. In the time domain approach, the k + 1 sequence coefficients are designed to match the ideal PACF criterion for the associated sequence to be perfect. It results in k constraint equations, which are all nonlinear. It is critical to derive integer solutions simultaneously from a set of k nonlinear equations, especially when k is greater than three. We introduce a method that can decompose each nonlinear equation into an equivalent linear system of two equations. By means of decomposition and transformation, the integer solutions of a set of k nonlinear constraint equations can be obtained by solving an equivalent system of 2k − 2 linear equations with 2k − 2 variables. This scheme simplifies the construction of a degree-(k + 1) PGIS. The frequency domain approach aims at matching the spectrum of a sequence with the flat magnitude, in which the number of nonlinear constraint equations that govern k + 1 sequence coefficients for the associated sequence to be perfect becomes k + 1. We derive an iteration algorithm wherein the values of k + 1 sequence coefficients can be obtained sequentially without solving a set of k + 1 nonlinear equations. The frequency domain approach for constructing degree-(k + 1) PGISs is characterized by simplicity. The construction of degree-(k + 1) PGISs is based on the same number of k + 1 base sequences; therefore, PGISs with degrees less than k + 1 can be spanned by fewer base sequences. The first step for constructing PGISs with lower degrees aims at decreasing the number of base sequences. This task can be achieved by either combining one or more pairs of two consecutive base sequences into one or setting some base sequences with zero-valued coefficients. When more base sequences are merged or more zero-valued coefficients are assigned, the degrees of the associated PGISs can be gradually decreased. The second step involves the adjustment of new sequence coefficients to match the criteria for the associated sequences to be perfect. The paper is organized as follows. Preliminaries and definitions are introduced in Section II. The partitioning of Z N is addressed in Section III. Sections IV and V present the construction of degree-(k +1) PGISs from the time and frequency domains, respectively. The study of constructing PGISs with degrees less than k +1 is in Section VI. Conclusions are drawn in Section VII.

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 65, NO. 9, SEPTEMBER 2017

II. P RELIMINARIES Let s = denote a sequence of period N = pk , where s[n] is the nth component of s. Rs = {Rs [τ ]}τN−1 =0 is the PACF of s expressed as, N−1 {s[n]}n=0

Rs [τ ] =

N−1 

s[n]s ∗ [(n − τ ) N ],

(1)

n=0

where the superscript ∗ denotes the complex conjugate operation, and (·) N is the modulo N operation. Define N−1 s−1 = {s[(−n) N ]}n=0 . As can be easily shown, Rs = s ⊗ s∗−1 , where ⊗ denotes the circular convolution operation. Let N−1 S = {S[n]}n=0 denote the DFT of s. The DFT of Rs is then given by S ◦ S∗ =|S|2 , where ◦ and | · | denote the componentwise product operation and the Euclidean norm, respectively. The sequence s is said to be perfect if and  only if it has N−1 |s[n]|2 an ideal PACF, i.e., Rs = E · δ N , where E = n=0 is the energy of sequence s, and δ N is a delta sequence of period N. The DFT pair-relationship between Rs = E · δ N and S ◦ S∗ = |S|2 indicates that a sequence s is perfect if and only √ if the spectrum magnitude of s is flat, i.e., |S [n]| = E, 0 ≤ n ≤ N − 1. Definition 1: The degree of a sequence is defined as the number of distinct nonzero sequence elements within one period of the sequence. Definition 2: The pattern of a sequence is defined as the distribution of distinct nonzero elements within one period of the sequence. Note that for a set of PGISs, a sequence of period N has distinct N circular shifts, and the sequence and its circular shifts are considered to belong to the same sequence pattern in this paper; however, two PGISs of the same period and degree can have different patterns, which some examples are shown in TABLE II. For a set of sequences of the same period, the larger size of this sequence family is the more applications and advantages of this set should be, where the size of a sequence family is bounded by the number of available distinct patterns and degrees. With the definitions of sequence degree and pattern, we would mention that numerous different PGISs belonging to the same sequence family can be uniquely defined or categorized in terms of the pattern, period and degree three parameters. III. T HE PARTITION OF S ET Z N Let N = where p is a prime number and k ≥ 2. The set { p n }kn=0 consists of k +1 distinct positive factors of the integer N = pk . For i = 0, 1, . . . , k − 1, Z+ = {1, 2, . . . , pk−i − 1} p k−i denotes a set consisting of pk−i − 1 positive integers less than p k−i , and m + Z+ = {m + a|a ∈ Z+ }, where m is an p k−i p k−i integer. We define S1, pi = {a|gcd(a, pi ) = 1, a ∈ Z+ }, where pi i gcd(a, p ) denotes the greatest common divisor of a and pi . p i−1 −1 Lemma 1: S1, pi = {Z+ and |S1, pi | = pi−1 p + np}n=0 ( p − 1), where i ≥ 1. Proof: First, since the i + 1 positive factors of the integer p i are 1, p, p2 , . . . , pi , any integer mp will not belong to S1, pi . Second, besides the elements Z+ p = {1, 2, . . . , p − 1} pk ,

CHANG AND CHANG: PGISs OF PERIOD P K WITH DEGREES EQUAL TO OR LESS THAN K + 1

which belong to S1, pi , other elements belonging to Z+ p + np are also the elements of S1, pi , where n = 1,

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TABLE I + T HE Z+ p − Z p C AYLEY TABLE [19]

p i−1 −1

2, . . . , pi−1 − 1. This proves S1, pi = {Z+ . p + np}n=0 | = p − 1, the cardinality of S is Finally, given that |Z+ i 1, p p |S1, pi | = p i−1 ( p − 1). Let S0 = {0} and define S pi = {api |gcd(a, pk−i ) = 1, a ∈ + Z pk−i }, i = 1, 2 . . . , k − 1. Lemma 2: |S pi | = pk−i−1 ( p − 1). = Proof: The cardinality of the set S pi + i k−i {ap |gcd(a, p ) = 1, a ∈ Z pk−i } is the same as that of S1, pk−i = {a|gcd(a, pk−i ) = 1, a ∈ Z+ }. By Lemma 1, p k−i k−i−1 ( p − 1). we can derive that |S pi | = p Theorem 1: Let N = pk . The set Z N = {0, 1, 2, . . . , N − 1} = S0 ∪ Z+ can be partitioned into k + 1 subsets, pk + such that Z pk = S p0 ∪ S p1 ∪ · · · ∪ S pk−1 , where ∪ denotes the disjoint union. Proof: By Lemmas 1 and 2, it follows that S p0 = } = {a|a = mp, S1 = {a|gcd(a, pk ) = 1, a ∈ Z+ pk + k−1 m = 1, . . . , p − 1, a ∈ Z pk }, where |S p0 | = pk−1 ( p − 1). k−1 − 1 elements which belong to Z+ are the eleThe other p pk + + ments of { pa|a ∈ Z pk−1 }. Z pk−1 can be partitioned in the same to obtain S p = {ap|gcd(a, pk−1) = 1, a ∈ Z+ } way as Z+ pk p k−1 k−2 k−2 − 1 elements which with |S p | = p ( p − 1). The other p 2 a|a ∈ Z+ }, where are the elements of { p belong to Z+ pk p k−2 + can be partitioned as Z , etc. This iteration continues Z+ k−2 k−1 p p k − 1 times to obtain S pk−1 = {apk−1 |gcd(a, p) = 1, a ∈ Z+ p} k−1 with |S k−1 | = p − 1. It is obvious that i=0 |S p i | = k−1 pk−i−1 k − 1 = |Z+ |. p ( p − 1) = p i=0 pk IV. C ONSTRUCTION OF D EGREE -(k + 1) PGISs F ROM THE T IME D OMAIN A. PACF of Degree-(k + 1) PGIS Let c0 = δ N be the first base sequence, where δ N is a delta sequence of period N = pk . The other k base sequences N−1 , i = 1, 2, . . . , k, are defined based on the ci = {ci [n]}n=0  = ki=1 S pi−1 expressed as follows: partitioning of Z+ pk  1, n ∈ S pi−1 , ci [n] = (2) 0, otherwise. Lemma 3: Base sequence c0 = δ N and other base sequences defined in (2) are even, i.e., ci [n] = ci [(−n) N ] = ci [N − n], i = 1, 2, . . . , k, where (·) N denotes the modulo N operation. Proof: First, δ N is even. Next, the elements of S1 can be expressed as S1 = {a|a = (mp) N , a ∈ Z+ N , ∀ m}. When n = mp, then −n = (mp) N is true as well. Since both n and (−n) N belong to S1 , it has c1 [n] = ci [(−n) N ] = ci [N − n] = 1. When n = mp, then −n = (mp) N is also true. This implies c1 [n] = ci [(−n) N ] = ci [N − n] = 0. }. Third, let S1, pk−i+1 = {a|gcd(a, pk−i+1 ) = 1, a ∈ Z+ p k−i+1 Similar to the result for S1 , n ∈ S1, pk−i+1 ⇔ (−n) pk−i+1 ∈ S1, pk−i+1 . Since S pi−1 = {api−1 |gcd(a, pk−i+1 ) = 1, } = {api−1 |a ∈ S1, pk−i+1 }, it follows that a ∈ Z+ p k−i+1

n ∈ S pi−1 ⇔ (−n) N ∈ S pi−1 . We conclude that either both n and (−n) N belong to S pi−1 or neither of them belongs to S pi−1 . This result proves that ci [n] = ci [(−n) N ] = ci [N − n] is true, for i = 2, . . . , k. + We define mp+Z+ p = {mp+a|a ∈ Z p }. With this definition, + + it is easy to show that S1 = {Z p } ∪ { p + Z+ p } ∪ {2 p + Z p } ∪ k + + + · · ·∪{( p − p)+Z p }. We define Z p −Z p = {a −b|a, b ∈ Z+ p} + in TABLE I. and make a list of ( p − 1)2 elements of Z+ − Z p p In each row of this table, except for the “0” element, all other p − 2 elements belong to S1 . It is obvious that (mp + Z+ p)− + −Z+ ). This implies that except the ) = (m−n) p+(Z (np+Z+ p p p + “0” element of the Z+ p −Z p table is replaced by “(m−n) p,” all + other p−2 elements in each row of the (mp+Z+ p )−(np+Z p ) table belong to S1 . These results can be summarized to prove Lemmas 4 and 5. Note that Lemmas 4, 5, and 6 are essential to prove Theorem 2. Lemma 4: Define a set A = {(n − τ ) pk |τ, n ∈ S1 }. For any fixed τ = τ0 ∈ S1 , the number of elements in the subset {(n−τ0 ) pk } is pk−1 ( p−1). Among these p k−1 ( p−1) elements, one of which is “0,” the number of elements belonging to S1 is ( p − 2) pk−1 , and the number of other elements which belong to S pi is ( p − 1) p k−(i+1) , i = 1, 2, . . . , k − 1. Lemma 5: Define k − 1 sets Ai = {(n − τ ) pk |τ, n ∈ S pi }, i = 1, 2, . . . , k − 1. For any fixed τ = τ1 ∈ S pi , the number of elements in the subset {(n − τ1 ) pk } is p k−(i+1) ( p − 1). Among these pk−(i+1) ( p − 1) elements, one of which is “0,” the number of elements belonging to S pi is ( p − 2) pk−(i+1) , and the number of other elements that belong to S pi+n is ( p − 1) pk−(i+n+1) , n = 1, 2, . . . , k − i − 1. Lemma 6: Define a set A = {(n − τ ) pk |τ ∈ S pm , n ∈ S pl }. For any fixed τ = τ2 ∈ S pm , the number of elements in the subset {(n − τ2 ) pk } is pk−(l+1) ( p − 1). These pk−(l+1) ( p − 1) elements belong to S1 . Proof: Since there has no common element between S pm and S pl two sets, for two arbitrary τ ∈ S pm and n ∈ S pl , we can derive (n − τ ) pk ∈ S1 . N−1 Sequence s = {s[n]}n=0 of period N = p k is defined as ⎧ ⎪ ⎨a0 , n = 0, (3) s[n] = a1 , n ∈ S1 , ⎪ ⎩ ai , n ∈ S pi−1 , i = 2, . . . , k. In (3), ai , i = 0, 1, . . . , k, are k +1 nonzero Gaussian integers. N−1 Sequence s = {s[n]}n=0 = ki=0 ai ci is a linear combination of base sequences {ci }ki=0 . Theorem 2: For the sequence s defined in (3), the N elements of PACF Rs = {Rs [τ ]}τN−1 =0 , defined in (1), have at most

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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 65, NO. 9, SEPTEMBER 2017

k + 1 distinct values, Rk , k = 0, 1, . . . , k, which their values are given by ⎧ ⎪ ⎨ R0 , τ = 0, (4) Rs [τ ] = R1 , τ ∈ S1 , ⎪ ⎩ R , τ ∈ S i−1 , i = 2, . . . , k, i p where ⎧ k ⎪ |ai |2 pk−i , ⎪ R0 = |a0 |2 + ( p − 1) ⎪ i=1 ⎪ ⎪ ⎪ ⎪ R1 = (a0 a1∗ + a1 a0∗ ) + |a1 |2 ( p − 2) p k−1 ⎪ ⎪ k ⎪ ⎪ ⎪ + ( p − 1) (ai a1∗ + a1 ai∗ ) p k−i , ⎪ ⎪ i=2 ⎪ ⎪ ∗ ∗ ⎪ a0 ) + |ai |2 ( p − 2) pk−i ⎪ ⎨ Ri = (a0 ai + ai i−1 + ( p − 1) (an ai∗ + ai an∗ ) p k−n n=1 ⎪ ⎪ k ⎪ ⎪ ⎪ ⎪ + ( p − 1) (an ai∗ + ai an∗ ) p k−n , ⎪ ⎪ n=i+1 ⎪ ⎪ ⎪ i = 2, . . . , k − 1, ⎪ ⎪ ⎪ ∗ ∗ ⎪ Rk = (a0 ak + ak a0 ) + |ak |2 ( p − 2) ⎪ ⎪ k−1 ⎪ ⎩ (a a ∗ + a a ∗ ) p k−i . + ( p − 1) i 1

i=1

Rs [τ ] =

(5)

n=0



+ ···+



1 i



+ · · · + ai + ak



s ∗ [(n−τ ) N ] term

n∈S pi−1

of (6), there is no common element between τ and n; therefore, ∗ k−i by Lemma 6. the number of s ∗ [(n − τ ) N ]∗= a1 is ( p − 1) p This results in ai s [(n − τ ) N ] = ( p − 1) pk−i ai a1∗ , n∈S pi−1

i = 2, 3, . . . , k. The summation of all these resultsimplies that R1 = (a0 a1∗ +a1 a0∗ )+|a1 |2 ( p −2) pk−1 +( p −1) ki=2 (ai a1∗ + a1 ai∗ ) pk−i . 2, . . . , k − 1, first, a0 s ∗ [(−τ ) N ] = 3) When τ ∈ S pi−1 , i = ∗ a0 ai . Second, for the ai s ∗ [(n−τ ) N ] term of (6), where both τ and n belong to the same set S pi−1 , the number of elements in this term is pk−(i+1) ( p − 1), among which there is a single term s ∗ [(n − τ ) N ] = s ∗ [0] = a0∗ , the number of elements belonging to S pi−1 is ( p − 2) pk−i which results in  ∗ s [(n − τ ) N ]=( p − 2) pk−i |ai |2 , and the number of ai

n∈S pi+n−1

n∈S pm−1

m = i , because there is one common element between τ and  s ∗ [(n − τ ) N ] = n, it has s ∗ [(n − τ ) N ] = a1∗ and ai

s[n]s ∗ [(n − τ ) N ]

n∈S pm−1 ∗ k−m . This implies that Ri = (a0 ai∗ + ai a1 ( p − 1) p  ∗ 2 ∗ ∗ k−n + ai a0 ) + |ai | ( p − 2) pk−i + ( p − 1) i−1 n=1 (an ai + ai an ) p k ∗ ∗ k−n ( p − 1) n=i+1 (an ai + ai an ) p , i = 2, . . . , k − 1. 4) Finally, when τ ∈ S pk−1 it has a0 s ∗ [(−τ ) N ] = , first, ∗ ∗ a0 ak . For the last term ak s [(n − τ ) N ] of (6), where n∈S pk−1



s ∗ [(n − τ ) N ]

n∈S1 ∗

both τ and n belong to the same  set S pk−1 and the number of elements is ( p − 1), it has ak s ∗ [(n − τ ) N ]=ak a0∗ +

s [(n − τ ) N ] + · · ·

n∈S pi−1

s ∗ [(n − τ ) N ].

(6)

n∈S pk−1

For a single τ , the value of Rs [τ ] is calculated over the entire domain of variable n through (6), where the final expression of (6) consists of k + 1 terms. For variable n, the numbers of n belonging to these k + 1 terms of (6) are 1 and |S pi | = pk−i−1 ( p − 1), i = 0, 1, . . . , k − 1, respectively. The details of the derivation of Rs [τ ] are given below: 1) When can be easily shown that Rs [0]=|a0 |2 + k τ = 0,2 itk−i ( p − 1) i=1 |ai | p =R0 . 2) When τ ∈ S1 , first, a0 s ∗ [(−τ ) N ] = a0 a1∗ , which is the n = 0 term of (6). Then, for the a1 s ∗ [(n − τ ) N ] term n∈S1

of (6), where both τ and n belong to the same set S1 , there are ( p − 1) pk−1 numbers of n in s ∗ [(n − τ ) N ] for a n∈S1

fixed τ ∈ S1 . By Lemma 4, there is exactly one n(= τ ) that results in s ∗ [(n − τ ) N ] = s ∗ [0], and the number of elements belonging to S1 is ( p − 2) p k−1 , which results in the same number of s ∗ [(n − τ ) N ] = a1∗ . Also, the number of elements that belong to S pi is ( p − 1) p k−(i+1) , which results in the ∗ ∗ same number . . . , k. Thus, of ∗s [(n − τ ) N ] = a∗i , i = 2, 3, k−1 we have a1 s [(n − τ ) N ] = a1 a0 + ( p − 2) p |a1 |2 + a1 n∈S1



n = 2, 3, . . . , k −1, n = i − 1. Third, for other k − 1 s ∗ [(n − τ ) N ], m ∈ {2, 3, . . . , k − 1}, terms of (6), ai

s[n]s [(n − τ ) N ]

= a0 s ∗ [(−τ ) N ] + a1

Next, for the ai

n∈S pi−1

n∈S1 ∗

n∈S pk−1

∗ k−i . i=2 ai p

other elements that belong to S pi+n−1 is ( p−1) p k−(i+n) , which  results in ai s ∗ [(n − τ ) N ] = ai an∗ ( p − 1) p k−(i+n) ,

s[n]s ∗ [(n − τ ) N ]

= s[0]s ∗ [(−τ ) N ] +

k

n∈S pi−1

Proof: The PACF of s is given by (1), where N−1 

( p−1)

n∈S pk−1

− 2). For the other k − 1 terms of (6), the derivation |ak |2 ( p  of ai s ∗ [(n − τ ) N ] is similar, and is omitted for brevity. n∈S pi−1

We conclude that Rk = (a0 ak∗ + ak a0∗ ) + |ak |2 ( p − 2)+  ∗ ∗ k−i . ( p − 1) k−1 i=1 (ai a1 + a1 ai ) p Corollary 1: The sequence s defined in (3) is a degree-(k + 1) PGIS if and only if the following k constraint equations can be satisfied simultaneously: ⎧ ⎪ (a0 a1∗ + a1 a0∗ ) + |a1 |2 ( p − 2) p k−1 ⎪ ⎪ ⎪ k ⎪ ⎪ ⎪ + ( p − 1) (ai a1∗ + a1 ai∗ ) p k−i = 0, ⎪ ⎪ i=2 ⎪ ⎪ ⎪ ⎪ (a0 ai∗ + ai a0∗ ) + |ai |2 ( p − 2) p k−i ⎪ ⎪ ⎪ i−1 ⎪ ⎪ ⎨ + ( p − 1) (an ai∗ + ai an∗ ) pk−n n=1 (7) k ⎪ ∗ ∗ k−n ⎪ ⎪ + ( p − 1) (an ai + ai an ) p = 0, ⎪ n=i+1 ⎪ ⎪ ⎪ ⎪ i = 2, . . . , k − 1, ⎪ ⎪ ⎪ ⎪ ∗ ∗ 2 ⎪ ⎪ (a0 ak + ak a0 ) + |ak | ( p − 2) ⎪ ⎪ ⎪ k−1 ⎪ ⎩ (ai a1∗ + a1 ai∗ ) pk−i = 0. + ( p − 1) i=1

Proof: First, s is a PGIS if and only if Rs [τ ] = E · δ N , which implies that Ri = 0, i = 1, 2, . . . , k, according to

CHANG AND CHANG: PGISs OF PERIOD P K WITH DEGREES EQUAL TO OR LESS THAN K + 1

Theorem 2. Second, the degree of s is k + 1, because there are k + 1 distinct nonzero coefficients ai , i = 0, 1, . . . , k. B. Integer Solutions of the Constraint Equations Let ai = x i + j yi , i = 0, 1, 2, . . . , k, be k + 1 nonzero Gaussian integers. Inserting these coefficients into the constraint equations of (7), we derive a system of k nonlinear equations as shown in (8), as shown at the top of the next page. In (8), the number of equations is k, but the number of unknown parameters is 2(k+1); there exist numerous solutions for this system of k nonlinear equations. Before deriving the solutions, we modify (8) by replacing equation Ri = 0 with Ri = Ri − Ri−1 = 0, i = 3, 4, . . . , k, which results in an equivalent system of k nonlinear equations, where Ri = 0 and Rk = 0 equations are given by: Ri = Ri − Ri−1 = (x i − x i−1 )(2x 0 − pk−i+1 x i−1 + pk−i ( p − 2)x i k + 2( p − 1) x n pk−n ) n=i+1

+ (yi − yi−1 )(2y0 − pk−i+1 yi−1 + pk−i ( p − 2)yi k + 2( p − 1) yn pk−n ) = 0, (9) n=i+1

and Rk = Rk − Rk−1 = (x k − x k−1 )(2x 0 − px k−1 + ( p − 2)x k ) + (yk − yk−1 )(2y0 − pyk−1 + ( p − 2)yk ) = 0. (10) Both (9) and (10) two nonlinear equations can be decomposed into two equivalent linear systems of two equations, respectively, expressed as follows: ⎧ ⎪ x i − x i−1 + yi − yi−1 = 0, ⎪ ⎪ ⎨ 2x 0 − 2y0 − pk−i+1 x i−1 + pk−i ( p − 2)x i  (11) ⎪ + 2( p − 1) kn=i+1 x n pk−n + pk−i+1 yi−1 ⎪ ⎪ ⎩ − p k−i ( p − 2)y − 2( p − 1) k k−n = 0. i n=i+1 yn p ⎧ ⎪ ⎨x k − x k−1 + yk − yk−1 = 0, (12) 2x 0 − px k−1 + ( p − 2)x i ⎪ ⎩ − 2y + py − ( p − 2)y = 0. 0 k−1 k By substituting equations Rk = 0 and Ri = 0, i = 3, 4, . . . , k − 1, in (8) with (11) and (12), the original system of k nonlinear equations can be transformed into an equivalent system of 2k − 2 equations given by ⎧ ⎪ 2(x 0 x 1 + y1 y0 ) + (x 12 + y12 )( p − 2) p k−1 ⎪ ⎪ k ⎪ ⎪ ⎪ + 2( p − 1) (x i x 1 + y1 yi ) p k−i = 0, ⎪ ⎪ i=2 ⎪ ⎪ ⎪(x 22 + y22 )( p − 2) p k−2 + (x 12 + y12 )( p − 1) pk−1 ⎪ ⎪  ⎪ ⎪ + 2(x x + y y )+2( p − 1) k (x x + y y ) p k−i = 0, ⎪ 0 2 2 0 i 1 1 i ⎪ ⎪ i=3 ⎪ ⎪ ⎪ x − x + y − y = 0, i−1 i i−1 ⎪ ⎪ i ⎨ 2x 0 − pk−i+1 x i−1 + p k−i ( p − 2)x i k ⎪ + 2( p − 1) x n pk−n − 2y0 + p k−i+1 yi−1 ⎪ ⎪ n=i+1 ⎪ k ⎪ ⎪ ⎪ − pk−i ( p − 2)yi − 2( p − 1) yn pk−n = 0, ⎪ ⎪ n=i+1 ⎪ ⎪ ⎪ i = 3, . . . , k − 1, ⎪ ⎪ ⎪ ⎪ ⎪ x − x + y − y k k−1 k k−1 = 0, ⎪ ⎪ ⎪ ⎪ − px + ( p − 2)x k 2x 0 k−1 ⎪ ⎪ ⎩ − 2y + py − ( p − 2)y = 0. 0

k−1

k

(13)

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In (13) above, if we treat x 1 , x 2 , y1 , and y2 as four constants and all other 2k − 2 x i , and yi as variables, then (13) can be considered a system of 2k − 2 linear equations with 2k − 2 variables. Given four appropriately chosen integers x 1 , x 2 , y1 , and y2 , the unique set of integer solutions for the 2k − 2 unknown variables of x i , and yi can be obtained. We present the construction of two PGISs of p4 and p6 period examples, respectively, for demonstration. When k = 4, a system of six linear equations obtained from (13) is presented in matrix form Ax = b. In this expression, the coefficient matrix A of size 6 × 6 is given by ⎤ ⎡ 0 1 0 0 1 0 ⎢2 p( p−2) 2( p−1) − 2 p(2− p) 2(1− p) ⎥ ⎥ ⎢ ⎥ ⎢0 − 1 1 0 − 1 1 ⎥, ⎢ ⎥ ⎢2 − p p−2 − 2 p 2− p ⎥ ⎢ ⎣2x 1 2( p−1) px 1 2( p−1)x 1 2y1 2( p−1) py1 2( p−1)y1⎦ 2x 2 2( p−1) px 2 2( p−1)x 2 2y2 2( p−1) py2 2( p−1)y2 (14) x = [x 0 x 3 x 4 y0 y3 y4 ]T is a 6 × 1 vector of variables, and b = [x 2 + y2 p2 (x 2 − y2 ) 0 0 (2 − p) p3 (x 12 + y12 ) + 2(1 − p) p2 (x 1 x 2 + y1 y2 ) (1− p) p3 (x 12 + y12 )+(2− p) p2 (x 22 + y22 )]T . The integer solution of linear system Ax = b can be obtained from evaluating x = A−1 b, given that x 1 , x 2 , y1 , and y2 four parameters are chosen appropriately. Example 1: When p = 3, five positive factors of N = 34 are 1, 3, 9, 27, and 81. Z81 = S0 ∪ S1 ∪ · · · ∪S27 , where S0 = {0}, S3 = {3, 6, 12, 15, 21, 24, 30, 33, 39, 42, 48, 51, 57, 60, 66, 69, 75, 78}, S9 = {9, 18, 36, 45, 63, 72}, S27 = {27, 54}, and S1 consists of the rest of other 54(= 33 · (3 − 1)) positive integer numbers less than 34 . When x 1 = 2, x 2 = 2, y1 = 2, and y2 = 4 are chosen, we derive x 0 = −151, y0 = −590, x 3 = −16, y3 = 22, x 4 = 38, and y4 = −32 from solving Ax = b based on (14), and a degree-5 PGIS is given by s = (a0 , a1 , a1 , a2 , a1 , a1 , a2 , a1 , a1 , a3 , a1 , a1 , a2 , a1 , a1 , a2 , a1 , a1 , a3 , a1 , a1 , a2 , a1 , a1 , a2 , a1 , a1 , a4 , a1 , a1 , a2 , a1 , a1 , a2 , a1 , a1 , a3 , a1 , a1 , a2 , a1 , a1 , a2 , a1 , a1 , a3 , a1 , a1 , a2 , a1 , a1 , a2 , a1 , a1 , a4 , a1 , a1 , a2 , a1 , a1 , a2 , a1 , a1 , a3 , a1 , a1 , a2 , a1 , a1 , a2 , a1 , a1 , a3 , a1 , a1 , a2 , a1 , a1 , a2 , a1 , a1 ), where ai = x i + j yi , i = 0, 1, 2, 3, 4. Example 2: When N = 26 = 64, the seven positive factors of 64 are 1, 2, 22 , 23 , 24 , 25 , and26 . We can make partition the set Z64 into seven subsets S0 ∪ 5n=0 S2(5−n) , where S0 = {0}, S4 = {4, 12, 20, 28, 36, 40, 44, 52}, S8 = {8, 24, 40, 56}, S16 = {16, 48}, S32 = {32}, S2 consists of the rest of other 16 positive even integers less than 64, and S1 consists of 32 positive odd integers less than 64. In this case, a system of ten linear equations with ten variables is obtained from (13). Since the construction procedures are similar to that of Example 1, we do not present this linear system for brevity. Given that x 1 = 1, x 2 = 1, y1 = 1, and y2 = 2 are chosen, we can derive the integer solutions for ten variables as follows: x 0 = −59, y0 = −34, x 3 = −3, y3 = 6, x 4 = 5,

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⎧ k ⎪ ⎪ R1 = 2(x 0 x 1 + y1 y0 ) + (x 12 + y12 )( p − 2) pk−1 + 2( p − 1) (x i x 1 + y1 yi ) pk−i = 0, ⎪ ⎪ i=2 ⎪  ⎪ i−1 ⎪ ⎪ y0 ) + (x i2 + yi2 )( p − 2) pk−i + 2( p − 1) (x n x i + yi yn ) pk−n ⎪ ⎨ Ri = 2(x 0 x i + yi  n=1 k + 2( p − 1) (x n x i + yi yn ) p k−n = 0, ⎪ n=i+1 ⎪ ⎪ ⎪ ⎪ i = 2, . . . , k − 1, ⎪ ⎪ k−1 ⎪ ⎪ ⎩ Rk = 2(x 0 x k + yk y0 ) + (x 2 + y 2 )( p − 2) + 2( p − 1) (x i x 1 + y1 yi ) pk−i = 0. k k

(8)

i=1

y4 = −2, x 5 = −11, y5 = 14, x 6 = 21, and y6 = −18. A degree-7 PGIS of period N = 64 is given by s = (a0 , a1 , a2 , a1 , a3 , a1 , a2 , a1 , a4 , a1 , a2 , a1 , a3 , a1 , a2 , a1 , a5 , a1 , a2 , a1 , a3 , a1 , a2 , a1 , a4 , a1 , a2 , a1 , a3 , a1 , a2 , a1 , a6 , a1 , a2 , a1 , a3 , a1 , a2 , a1 , a4 , a1 ,

Let Ci denote the DFT of ci , i = 0, 1, 2, . . . , k. Lemma 7: The DFT of c1 is given by C1 = pk−1 · ( p − 1, 0, . . . , 0, −1, 0, . . . , 0, . . . , −1, 0, . . . , 0).          p k−1

(15) Proof: The DFT of a sequence d p = (0, 1, . . . , 1) is    p

( p − 1, −1, . . . , −1). c1 can be obtained from the sequence    p

d p by duplicating it pk−1 times; therefore, the DFT of c1 is a zero padding spectrum multiplied by pk−1 according to the theory of DFT. We can apply the result of Lemma 7 to derive Lemmas 8 and 9. Lemma 8: The DFT of ci , i = 2, 3, . . . , k − 1, is given by (16)

p i−1

where p k−i+1

   Ci = ( p − 1, 0, . . . , 0, −1, 0, . . . , 0, . . . , −1, 0, . . . , 0).          p k−i

p k−i

p k−i

Lemma 9: The DFT of ck is given by Ck = ( p − 1, −1, . . . , −1, p − 1, −1, . . . , −1,       p

n=1 k 

an pk−n |,

n=1

V. C ONSTRUCTION OF D EGREE -(k + 1) PGISs F ROM THE F REQUENCY D OMAIN

Ci = pk−i · (Ci , Ci , . . . , Ci )   

an pk−n | = · · ·

i = 1, 2, . . . , k − 1, (18)

where ai = x i + j yi , i = 0, 1, . . . , 6.

p k−1

|a0 − ak | = |a0 + ( p − 1)

k 

= |a0 − ai pk−i + ( p − 1)

a2 , a1 , a3 , a1 , a2 , a1 , a5 , a1 , a2 , a1 , a3 , a1 , a2 , a1 , a4 , a1 , a2 , a1 , a3 , a1 , a2 , a1 ),

p k−1

terms can be satisfied:

p

. . . , p − 1, −1, . . . , −1). (17)    p

 Theorem 3: Sequence s = ki=0 ai ci is a degree-(k + 1) PGIS if and only if the following constraint equation of k + 1

Gaussian integers. where ai ’s are k + 1 distinct nonzero Proof: 1) First, the DFT of s = ki=0 ai ci is denoted by  S = ki=0 ai Ci . Except that C0 is a vector with all elements are 1, all other Ci are three-valued vectors and these values belong to k sets {0, pk−i ( p − 1), − pk−i }, i = 1, 2, . . . , k, respectively, by the results of Lemmas 7, 8 and 9. 2) Second, from Lemma 8, the pi nonzero elements of p i −1 vector Ci are located at the entries of {npk−i }n=0 , where i = 2, 3, . . . , k − 1. By analyzing and aligning all nonzero entries of all Ci , i =0, 1, . . . , k, it shows that the vector k N−1 ) = 1 distinct values, S(= {S[n]}n=0 i=0 ai Ci has k +  which are a0 − ak , a0 − ai pk−i + ( p − 1) kn=1 an pk−n and  a0 + ( p − 1) kn=1 an pk−n , i = 1, 2, . . . , k − 1, as shown in (18). 3) Third, the spectral magnitude flat criterion, which |S[n]| = |S[k]| ∀n, k ∈ Z N , for the sequence s to be perfect indicates  that the constraint equation |a0 − ak | = |a0 + ( p − 1) kn=1 an pk−n | = · · · = |a0 − ai pk−i + ( p − 1) k k−n | should be satisfied. n=1 an p 4) Finally, the degree of s is k + 1 because there are exactly k + 1 nonzero sequence coefficients. The k + 1 constraint equations of (18) are nonlinear with 2(k + 1) unknown variables, where ai =x i + j yi , i = 0, 1, 2, . . . , K . To derive a systematic method that can obtain a complete set of integer solutions for (18) is challenging. However, we can provide two sufficient conditions to easily derive the sequence coefficients instead of solving these complex nonlinear constraint equations. The first condition guarantees that the sequence using the derived sequence coefficients is PGIS, and the second condition ensures that the degree of the associated PGIS is k + 1. These two schemes are addressed in Theorems 4 and 5. Theorem 4: A linear system of k + 1 equations is defined as follows: ⎧ k ⎪ ⎪ a0 + ( p − 1) an pk−n = b0 , ⎪ ⎪ n=1 ⎪ k ⎨ an pk−n = bi , a0 − ai pk−i + ( p − 1) (19) n=i+1 ⎪ ⎪ i = 1, 2, . . . , k − 1, ⎪ ⎪ ⎪ ⎩a − a = b . 0 k k

CHANG AND CHANG: PGISs OF PERIOD P K WITH DEGREES EQUAL TO OR LESS THAN K + 1

When the conditions |b0 | = |b1 | = · · · = |bk | hold, the solutions of this linear system of k + 1 equations match the spectral flat criterion for the associated sequence to be a PGIS. Proof: When the conditions |b0 | = |b 1 | = · · · = |bk | hold, the conditions |a0 − ak |=|a0 + ( p − 1) kn=1 an pk−n | =  · · · = |a0 − ai pk−i + ( p − 1) kn=1 an pk−n | of (18) are well satisfied. This implies that the solutions of (19) are also the solutions of (18). It follows that the sequence coefficients {ai }ki=0 obtained from (19) can construct a PGIS. The linear system of equations (19) can be expressed in matrix form Ax = b, where x = [a0 a1 · · · ak ]T is a (k+1)×1 vector of variables, b = [b0 b1 · · · bk ]T is a (k + 1) × 1 data vector, and the coefficient matrix A of size (k + 1) × (k + 1) is presented in (20), as shown at the top of the next page. Note that the inverse matrix of A, denoted by A−1 , has a closed form with explicit entries, which is derived and presented in (21), as shown at the top of the next page. When matrix A−1 is multiplied with pk , the values of all entries of pk · A−1 become integers. Since the entries of A−1 are known and the data vector b = [b0 b1 · · · bk ]T has flat magnitude, where |bi | = c, ∀i , the sequence coefficients of PGISs can be obtained directly from a matrix multiplication operation, that is, x = pk · A−1 b. Compared with the time domain approach, the frequency domain PGIS construction is simpler, where the computing load from solving a linear system of k + 1 equations can be avoided. Note that all base applied for the construction sequences k a c are even functions, described of sequence s = i i i=0 in Lemma 3; therefore, it follows that the sequence s is also even. In addition, the elements of all Ci are integers, as shown in Lemmas 7, 8, and 9. Thus, when the elements of b = [b0 b1 · · · bk ]T are assigned as integers, the values of the sequence coefficients of the associated sequences are also integers. This property implies that both the real and imaginary parts of b = br + j bi can be applied to simultaneously construct two different PGISs if both parts also have flat magnitudes. Thus, we can construct three different PGISs in one step, which is demonstrated in the following example. When p = 3 and k = 4 are substituted into (21), a 5 × 5 coefficient matrix A and its inverse A−1 are given as follows ⎡ ⎤ 1 54 18 6 2 ⎢ 1 −27 18 6 2 ⎥ ⎢ ⎥ ⎢ A = ⎢1 0 −9 6 2 ⎥ ⎥, ⎣1 0 0 −3 2 ⎦ 1 0 0 0 −1 ⎡ ⎤ 1 2 6 18 54 ⎢ 1 −1 0 0 0 ⎥ ⎥ 1 ⎢ ⎥. 1 2 −3 0 0 A−1 = 4 · ⎢ (22) ⎢ ⎥ 3 ⎣1 2 6 −9 0 ⎦ 1 2 6 18 −27 Example 3: For N = 34 = 81, we can assign br = [−1 1 − 1 1 − 1]T and bi = [2 2 2 − 2 2]T . Let b = [b0 b1 b2 b3 b4 ]T = br + j bi . Three vectors b, br , and bi have flat magnitudes. We can simultaneously construct three PGISs sr , si , and s = sr + j si , associated with br , bi , and b = br + j bi , respectively. The values of sequence coefficients x = [a0 a1 a2 a3 a4 ]T can be obtained

3729

directly by using formula 34 · A−1 b, where A−1 is shown in (22). The integer solutions for these sequence coefficients are x 0 = −41, x 1 = −2, x 2 = 4, x 3 = −14, x 4 = 40, y0 = 90, y1 = y2 = 0, y3 = 36, and y4 = −72. Applying five sets S0 and S3n , n = 0, 1, 2, 3, defined in Example 1, two PGISs sr and si are given by sr = (x 0 , x 1 , x 1 , x 2 , x 1 , x 1 , x 2 , x 1 , x 1 , x 3 , x 1 , x 1 , x 2 , x1, x1 , x2, x1 , x1 , x3, x1 , x1, x2 , x1, x1 , x2 , x1, x1 , x4, x1 , x1, x2 , x1 , x1, x2 , x1 , x1, x3 , x1, x1 , x2 , x1, x1, x2 , x1, x1 , x3 , x1, x1 , x2, x1 , x1, x2 , x1 , x1, x4 , x1, x1 , x2, x1 , x1 , x2, x1 , x1, x3 , x1, x1 , x2 , x1, x1 , x 2 , x 1 , x 1 , x 3 , x 1 , x 1 , x 2 , x 1 , x 1 , x 2 , x 1 , x 1 ), and si = (y0 , 0, 0, 0, 0, 0, 0, 0, 0, y3 , 0, 0, 0, 0, 0, 0, 0, 0, y3 , 0, 0, 0, 0, 0, 0, 0, 0, y4 , 0, 0, 0, 0, 0, 0, 0, 0, y3 , 0, 0, 0, 0, 0, 0, 0, 0, y3 , 0, 0, 0, 0, 0, 0, 0, 0, y4 , 0, 0, 0, 0, 0, 0, 0, 0, y3 , 0, 0, 0, 0, 0, 0, 0, 0, y3 , 0, 0, 0, 0, 0, 0, 0, 0).

(23)

The degrees of PGISs sr , si and s = sr + j si are 5, 3, and 5, respectively, where the former two PGISs have integer coefficients and the coefficients of s are Gaussian integers. These examples demonstrate that the conditions of |b0 | = |b1 | = · · · = |bk | applied to the linear system of (19) can only guarantee that the derived sequences are PGISs; however, the degree of these PGISs may not always be k + 1. Additional conditions should be applied to these bi ’s from which the derived sequence coefficients can exactly construct PGIS with degree-(k+1). In Theorem 5, an iteration algorithm is derived to calculate the sequence coefficients directly instead of solving the complex constraint equations. Theorem 5: Let {bi }ki=0 be k + 1 nonzero Gaussian integers with bi − bi−1 = 0 and |bi | = |bn |, i, n ∈ {0, 1, 2, . . . , k}. Sequence s = pk · ki=0 ai ci is a degree-(k + 1) PGIS of period N = pk , in which the values of sequence coefficients {ai }ki=0 of s are assigned by the following iteration algorithm: ⎧ 1 ⎪ ⎪a1 = k (b0 − b1 ), ⎪ ⎪ p ⎪ ⎪ ⎨ 1 ai = ai−1 + k−i+1 (bi−1 − bi ), (24) p ⎪ ⎪ ⎪ ⎪ i = 2, 3, . . . , k, ⎪ ⎪ ⎩ a0 = ak + bk . Proof: First, the augmented matrix [A|b] in (25), as shown at the top of the next page, of the linear system of  equations (19) is equivalent to the augmented matrix A |b in (26), as shown at the top of the next page. The element ai of vector x = [a0 a1 · · · ak ]T can be easily derived from the equivalent linear system A x = b , and the result is exactly the same as the above algorithm (24). Second, when the two adjacent Gaussian integers are not the same, bi − bi−1 = 0 to all i , all sequence coefficients ai are distinct. This implies  that the sequence s = pk · ki=0 ai ci is a degree-(k + 1) PGIS of period N = pk .

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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 65, NO. 9, SEPTEMBER 2017

⎡

1 ⎢1 ⎢ ⎢ ⎢1 A=⎢ ⎢. ⎢ .. ⎢ ⎣1 1

A−1

( p − 1) pk−1 − pk−1

( p − 1) p k−2 ( p − 1) p k−2

··· ···

··· ···

0 .. .

− pk−2 .. .

( p − 1) pk−3 .. .

0 0

0 0

··· ···

··· .. . 0 ···

··· ··· ··· .. .

( p − 1) p k−2 0 0 .. .

. ( p − 1) p k−3 ···

0 − pk−2 ( p − 1) pk−2

⎡

1 ⎢1 ⎢ ⎢1 ⎢ 1 ⎢ 1 = k⎢ p ⎢ ⎢ .. ⎢. ⎢ ⎣1 1 ⎡

1 ⎢1 ⎢ ⎢ ⎢ [A|b] = ⎢ 1. ⎢. ⎢. ⎣1 1 ⎡ 0 ⎢0 ⎢ ⎢0    ⎢ . A |b = ⎢ ⎢ .. ⎢ ⎢0 ⎣0 1

p−1 −1 p−1

( p − 1) p 0 −p

··· 0 0

p−1 .. .

( p − 1) p .. .

− p2 .. . ··· ···

p−1 p−1

··· ( p − 1) p

..

( p − 1) pk−1 − p k−1

( p − 1) pk−2 ( p − 1) pk−2

··· ···

··· ···

0 .. .

− p k−2 .. .

( p − 1) p k−3 .. .

0 0

0 0 ··· 0 pk−2 .. .

··· .. . 0 ··· | | |

pk − pk−1 0 .. .

0 pk−1 − p k−2 .. .

0 0 0

··· 0 0

0 ··· ···

··· ··· 0 .. .

··· ··· 0 0 ··· .. .

0 0 0 .. .

0 ···

−p 0

0 p −1

− p2

VI. C ONSTRUCTION OF PGISs W ITH D EGREES L ESS T HAN k + 1 A. Construction of Degree-k PGISs To derive PGISs with degrees less than k+1, in the first step, we will replace equation R2 = 0 of (8) by R2 = R2 − R1 = 0. In the second step, both R1 = 0 and R2 = 0 two equations are decomposed into two equivalent linear systems of two equations, which this process is similar to that of decomposing other Ri = 0 equations, as shown in (11) and (12). Here we have R2 = R2 − R1 = (x 2 − x 1 )(2x 0 − p k−1 x 1 + pk−2 ( p − 2)x 2 + 2( p − 1)

k 

x n pk−n )

n=3

+ (y2 − y1 )(2y0 − pk−1 y1 + pk−2 ( p − 2)y2 + 2( p − 1)

k  n=3

yn pk−n ) = 0,

p2

| | | |

⎤ p−1 p − 1⎥ ⎥ ⎥ p − 1⎥ ⎥ .. ⎥ . ⎥ ⎥ p − 1⎦ −1

( p − 1) p ( p − 1) p .. . ( p − 1) p −p 0

( p − 1) p ( p − 1) p .. . ( p − 1) p −p 0 b0 − b1 b1 − b2 b2 − b3 .. .

(20)

⎤ ( p − 1) p k−1 ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ ⎥ 0 ⎥ ⎥ .. ⎥ . ⎥ ⎦ 0 − pk−1 p−1 p−1

| |

p−1 .. .

|

p−1 −1 ⎤

| | |

(21)

⎤ b0 b1 ⎥ ⎥ ⎥ b2 ⎥ .. ⎥ ⎥ . ⎥ bk−1 ⎦ bk

(25)

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ bk−2 − bk−1 ⎥ b −b ⎦ k−1

(26)

k

bk

and R1 = x 1 (2x 0 + pk−1 ( p − 2)x 1 + 2( p − 1)

k 

x n pk−n )

n=2

+ y1 (2y0 + pk−1 ( p − 2)y1 + 2( p − 1) = 0.

k 

yn pk−n )

n=2

With these modifications, (13) becomes ⎧ k ⎪ ⎪ −2y0 + pk−1 ( p − 2)y1 − 2( p − 1) yn pk−n ) = x 1 ⎪ ⎪ n=2 ⎪  k ⎪ ⎪ ⎪ 2x 0 + pk−1 ( p − 2)x 1 + 2( p − 1) x n pk−n ) = y1 ⎪ ⎪ n=2 ⎪ ⎪ ⎪ x i − x i−1 + yi − yi−1 = 0, ⎪ ⎪ ⎪ ⎪ ⎪ 2x 0 − pk−i+1 x i−1 + pk−i ( p − 2)x i ⎪ ⎪ ⎪ k ⎨ + 2( p − 1) x n pk−n − 2y0 + pk−i+1 yi−1 n=i+1 k ⎪ ⎪ ⎪ − p k−i ( p − 2)yi − 2( p − 1) yn pk−n = 0, ⎪ ⎪ n=i+1 ⎪ ⎪ ⎪ ⎪ i = 2, . . . , k − 1, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪x k − x k−1 + yk − yk−1 = 0, ⎪ ⎪ ⎪ 2x 0 − px k−1 + ( p − 2)x k ⎪ ⎪ ⎪ ⎩ − 2y + py 0 k−1 − ( p − 2)yk = 0. (27)

CHANG AND CHANG: PGISs OF PERIOD P K WITH DEGREES EQUAL TO OR LESS THAN K + 1

There are 2k linear equations in (27), where the number of unknown variables is 2(k + 1). To derive the solutions of (27) for constructing a degree-k PGIS, we may assume that the two conditions x m = x m−1 and ym = ym−1 hold, where m ∈ {2, 3, . . . , k}. In this situation, two equations obtained from the  = R − R decomposition of equation Rm m m−1 = 0 no longer exists, and the number of equations of (27) becomes 2(k − 1) with 2k unknown variables. To solve this set of equations, we can assign arbitrary x i and yi as two constants and keep the remaining 2(k − 1) xl and yl as variables. The resultant system of 2(k − 1) linear equations with the same number of variables has a unique set of integer solutions. The available number of m ∈ {2, 3, . . . , k} is k−1; it follows that the number of available sequence patterns of PGISs with degree-k is also k − 1. Note that all sequence coefficients of these PGISs are not zero. When k = 4, let x 3 = x 4 and y3 = y4 , and taking x 1 and y1 are two constants. Under these conditions, the equation of (27) will derive a system of six linear equations Ax = b, where the coefficient matrix A is given by ⎤ 2( p2 −1) 0 0 0 2 2( p−1) p2 ⎥ ⎢−2 2 p2 (1− p) 2(1− p 2) 0 0 0 ⎥ ⎢ ⎥ ⎢0 1 0 0 1 0 ⎥ ⎢ ⎢ 2 p2 ( p−2) 2( p 2 −1) − 2 − p2 ( p−2) − 2( p 2 −1)⎥, ⎥ ⎢ ⎦ ⎣0 −1 1 0 −1 1 2 2 2 2 2 −p p −2 − 2 p − p +2 (28) ⎡

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Proof: Let di = pk ai , i = 0, 1, 2, . . . , k. From (24) of Theorem 5, we have ⎧ d1 = b0 − b1 , ⎪ ⎪ ⎪ ⎪ k ⎪ ⎪ ⎨d2 = p (b0 − b1 ) + p(b1 − b2 ), i−1 di = pk b0 + ( p − pk )b1 + ( p − 1) pn−1 bn (29) ⎪ n=2 ⎪ ⎪ ⎪ − pi−1 bi , i = 3, 4, . . . , k, ⎪ ⎪ ⎩ d0 = dk + bk . Given that all bi ’s have a flat magnitude constraint, |bi | = c is true for all i . From (29), di = 0 implies dn = 0, for all n < i . A degree-k PGIS has exactly k distinct nonzero coefficients; therefore, the single sparse PGIS with degree-k and period p k can be obtained by setting a1 = 0, when the rest of all coefficients cannot be zero, ai = 0, ∀ i = 1. Applying (27) to construct a degree-k PGIS of N = p6 is presented in Example 5, where p = 2. At first, we treat x 0 and y0 two parameters as constant. Then, the matrix form Ax = b of the associative system of ten linear equations is derived, where the coefficient matrix A is given in (30), as shown at the bottom of the next page, x = [x 2 x 3 x 4 x 5 x 6 y2 y3 y4 y5 y6 ]T , and b = [0 2(y0 − x 0 ) 0 2(y0 − x 0 ) 0 2(y0 − x 0 ) 0 2(y0 − x 0 ) 0 2(y0 − x 0 )]T . Example 5: Let N = 26 , a1 = 0, x 0 = −20, and y0 = 10. With these parameters, we can derive x 2 = 3, x 3 = −3, x 4 = 9, x 5 = −15, x 6 = 33, y2 = −3, y3 = 3, y4 = −9, y5 = 15, and y6 = −33 using (30). The resultant degree-6 PGIS is given by s = (a0 , 0, a2 , 0, a3 , 0, a2 , 0, a4 , 0, a2 , 0, a3 , 0,

x = [x 0 x 2 x 3 y0 y2 y3 ]T is a 6 × 1 vector of variables, and b = [−x 1 − ( p − 2) p3 y1 ( p − 2) p3 x 1 − y1 x 1 + y1 p3 (x 1 − y1) 0 0]T . The coefficient matrix (28) can be applied to construct a degree-4 PGIS of arbitrary period N = p4 , in which Example 4 demonstrates the case of N = 34 . Example 4: In the case of N = 34 = 81, let x 3 = x 4 and y3 = y4 . Given that x 1 = 1 and y1 = −1, we derive x 0 = −34, x 2 = −2, x 3 = 7, y0 = 33, y2 = 2, and y3 = −7 from x=A−1 b. A degree-4 PGIS of period N = 81 is given by

s = (a0 , a1 , a1 , a2 , a1 , a1 , a2 , a1 , a1 , a3 , a1 , a1 , a2 , a1 , a1 , a2 , a1 , a1 , a3 , a1 , a1 , a2 , a1 , a1 , a2 , a1 , a1 , a3 , a1 , a1 , a2 , a1 , a1 , a2 , a1 , a1 , a3 , a1 , a1 , a2 , a1 , a1 , a2 , a1 , a1 , a3 , a1 , a1 , a2 , a1 , a1 , a2 , a1 , a1 , a3 , a1 , a1 , a2 , a1 , a1 , a2 , a1 , a1 , a3 , a1 , a1 , a2 , a1 , a1 , a2 , a1 , a1 , a3 , a1 , a1 , a2 , a1 , a1 , a2 , a1 , a1 ),

where ai = x i + j yi , i = 0, 1, 2, 3. From (24) of Theorem 5, a1 = p1k (b0 −b1 ). Setting b0 = b1 results in a1 = 0, which constructs a sparse PGIS of degree-k; this PGIS has pk−1 ( p − 1) zero elements. Theorem 6: Except for a1 = 0, there is no other sparse PGIS with degree-k and period N = pk .

a2 , 0, a5 , 0, a2 , 0, a3 , 0, a2 , 0, a4 , 0, a2 , 0, a3 , 0, a2 , 0, a6 , 0, a2 , 0, a3 , 0, a2 , 0, a4 , 0, a2 , 0, a3 , 0, a2 , 0, a5 , 0, a2 , 0, a3 , 0, a2 , 0, a4 , 0, a2 , 0, a3 , 0, a2 , 0), where ai = x i + j yi , i = 0, 2, 3, . . . , 6. In summary, we constructed k distinct sequence patterns of PGISs with degree-k and period pk in this subsection; one of them is a sparse PGIS. B. Construction of PGISs With Degrees Less Than k From the previous subsection, the two conditions x m = x m−1 and ym = ym−1 are assumed. If two more conditions xl = xl−1 and yl = yl−1 are also held, the number of equations in (27) becomes 2(k − 2), and the number of unknown variables is 2(k − 1). Among these 2(k − 1) unknown variables, we can arbitrarily treat any two of them as constants, for example, x i and yi , and keep the remaining 2(k − 2) x n and yn as variables. Assigning these values makes (27) a system of 2(k − 2) linear equations with 2(k − 2) variables. A unique set of integer solutions for sequence coefficients can be derived to construct a degree-(k − 1) PGIS, which is similar to that of constructing a degree-k PGIS from solving a system of 2(k − 1) linear equations with 2(k − 1) variables addressed in the previous subsection. To construct a sparse degree-(k − 1) PGIS, the

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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 65, NO. 9, SEPTEMBER 2017

TABLE II T WELVE PGIS PATTERNS OF P ERIOD N = 24 W ITH D EGREES E QUAL TO OR L ESS T HAN 5

values of sequence coefficients of this sequence are governed by Corollary 2. Corollary 2: The sparse PGIS of degree-(k − i + 1) can be obtained by setting a1 = a2 = · · · = ai = 0, where i = 1, 2, . . . , k. Proof: It follows from (29) that ai = 0 implies an = 0, for all n < i . For example, the sequence si presented in (23) is a degree-3 sparse PGIS of period N = 34 , wherey1 = y2 = 0. Corollary 3: For sequence s = pk · ki=0 ai ci to be a PGIS, a0 = 0. Proof: The algorithm of (29) shows that a0 = ak + bk . Since bk = 0, a0 = 0 implies ak = −bk , where ak = ak−1 + 1 1 p (bk−1 − bk ) and ak−1 = ak−2 + p 2 (bk−2 − bk−1 ), and so on. Given that |bi | = |bn | = 0, i, n ∈ {0, 1, 2, . . . , k}, except all bi = 0, there is no other solution for ak = −bk . Theorem 5 and Corollary 2 show that there is a single degree-(k − i ) sparse PGIS, where i ∈ {1, 2, . . . , k − 2}. This PGIS can be obtained by assigning the zero value to i sequence coefficients, in which a1 = a2 = · · · = ai = 0. In the case of degree-(k − i ) PGISs for which all coefficients are not zero, i + 1 pairs of two consecutive coefficients should be assigned with the same values. From the first two equations of (27), we can not assume two conditions x 0 = x i and y0 = yi ,

⎡ ⎢p ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ A=⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

4

1 ( p − 2) −1 − p4 0 0 0 0 0 0

0 2( p − 1) p3 1 p3 ( p − 2) −1 − p3 0 0 0 0

0 2( p − 1) p2 0 2( p − 1) p2 1 p2 ( p − 2) −1 − p2 0 0

0 2( p − 1) p 0 2( p − 1) p 0 2( p − 1) p 1 p( p − 2) −1 −p

0 2( p − 1) 0 2( p − 1) 0 2( p − 1) 0 2( p − 1) 1 p−2

i ∈ {1, 2, . . . , k}, for the purpose of combining c0 with other base sequences and deriving integer solutions accordingly. This implies that the number of base sequences that can be combined to construct lower degree PGISs is also k. It is easy to show that the number of ways of choosing i + 1 pairs of consecutive two base sequences from a set of k base . Therefore, the available number of sequences is (k−i)(k−i−1) 2 distinct sequence patterns of degree-(k − i ) PGISs is also (k−i)(k−i−1) , where all elements of the sequence are not zero. 2 Finally, we can summarize the above analysis and results to form Theorem 7. distinct Theorem 7: There exist at least 2k + k(k−1)(k−2) 6 sequence patterns of PGISs of period N = pk with degrees equal to or less than k + 1. Proof: Let ai be the i t h sequence coefficient of the associated PGIS. First, there is a single sequence pattern of degree-(k + 1) PGIS. Second, the pattern of a degree(k − i ) sparse PGIS is given by assigning an = 0 for all n ≤ i + 1, n = 0, i ∈ {0, 1, 2, . . . , k − 1}; this set of sparse PGISs contributes k distinct sequence patterns to this sequence family. Third, there are k − 1 distinct patterns of degree-k PGISs in which all sequence coefficients are not zero. Fourth, , the number of distinct patterns of degree-(k−i ) is (k−i)(k−i−1) 2 when all coefficients are not zero, where i ∈ {1, 2, . . . , k −2}.

1 − p ( p − 2) −1 p4 0 0 0 0 0 0 4

0 −2( p − 1) p3 1 − p3 ( p − 2) −1 p3 0 0 0 0

0 −2( p − 1) p2 0 −2( p − 1) p2 1 − p2 ( p − 2) −1 p2 0 0

0 −2( p − 1) p 0 −2( p − 1) p 0 −2( p − 1) p 1 − p( p − 2) −1 p

⎤ 0 −2( p − 1)⎥ ⎥ ⎥ 0 ⎥ −2( p − 1)⎥ ⎥ ⎥ 0 ⎥ −2( p − 1)⎥ ⎥ ⎥ 0 ⎥ −2( p − 1)⎥ ⎥ ⎦ 1 2− p

(30)

CHANG AND CHANG: PGISs OF PERIOD P K WITH DEGREES EQUAL TO OR LESS THAN K + 1

 (k−i)(k−i−1) It can be easily shown that k−2 = k(k−1)(k−2) n=1 2 6 is true. Summing up these four parts confirms that there exist distinct PGIS patterns of period N = pk with 2k + k(k−1)(k−2) 6 degrees equal to or less than k + 1. When N = 24 , 2 · 4 + 4·3·2 = 12 by Theorem 7. A list 6 of these 12 distinct sequence patterns of PGISs with degrees equal to or less than 5 is shown in TABLE II. VII. C ONCLUSIONS This paper addresses the construction of periodic PGISs with period N = pk and degrees equal to or less than k + 1. Based on the partitioning of the set Z N into k + 1 subsets, k + 1 base sequences can be defined to construct degree-(k + 1) PGISs in both the time and frequency domains. The k constraint equations derived from the time domain to match the ideal PACF criterion for a sequence to be perfect are nonlinear. Solving this set of k nonlinear equations is challenging especially when the degree of a PGIS is greater than three. We present a scheme to decompose and transform these nonlinear constraint equations into a linear system of equations for which it is easy to derive a unique set of integer solutions. The k + 1 nonlinear constraint equations derived from the frequency domain can be solved sequentially; this sequential solution is characterized by low computing load. These two proposed PGIS construction schemes are significant because they simplify the construction of degree-(k+1) PGISs. To construct PGISs with degrees less than k + 1, first, we either assign one or more pairs of two consecutive sequence coefficients with the same value or set the value of one or more sequence coefficients to zero; second, we adjust the new coefficients for the associated sequence to be perfect. It shows distinct PGIS patterns that there exist at least 2k + k(k−1)(k−2) 6 of period pk with degrees equal to or less than k + 1. In this paper, we derived the properties and theorems related to the partitioning of Z N and the construction of PGISs. ACKNOWLEDGMENT The authors would express their appreciation to the Editor for handling this paper and to the reviewers for their constructive and helpful comments and suggestions. R EFERENCES [1] W.-W. Hu, S.-H. Wang, and C.-P. Li, “Gaussian integer sequences with ideal periodic autocorrelation functions,” IEEE Trans. Signal Process., vol. 60, no. 11, pp. 6074–6079, Nov. 2012. [2] Y. Yang, X. Tang, and Z. Zhou, “Perfect Gaussian integer sequences of odd prime length,” IEEE Signal Process. Lett., vol. 19, no. 10, pp. 615–618, Oct. 2012. [3] X. Ma, Q. Wen, J. Zhang, and H. Zuo, “New perfect Gaussian integer sequences of period pq,” IEICE Trans. Fundam., vols. E96–A, no. 11, pp. 2290–2293, Nov. 2013. [4] H.-H. Chang, C.-P. Li, C.-D. Lee, S.-H. Wang, and T.-C. Wu, “Perfect Gaussian integer sequences of arbitrary composite length,” IEEE Trans. Inf. Theory, vol. 61, no. 7, pp. 4107–4115, Jul. 2015. [5] C.-D. Lee, Y.-P. Huang, Y. Chang, and H.-H. Chang, “Perfect Gaussian integer sequences of odd period 2m − 1,” IEEE Signal Process. Lett., vol. 12, no. 7, pp. 881–885, Jul. 2015.

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[6] C.-D. Lee, C.-P. Li, H.-H. Chang, and S.-H. Wang, “Further results on degree-2 Perfect Gaussian integer sequences,” IET Commun., vol. 10, no. 12, pp. 1542–1552, Mar. 2016. [7] S. C. Pei and K. W. Chang, “Perfect Gaussian integer sequences of arbitrary length,” IEEE Signal Process. Lett., vol. 22, no. 8, pp. 1040–1044, Aug. 2015. [8] S.-H. Wang, C.-P. Li, H.-H. Chang, and C.-D. Lee, “A systematic method for constructing sparse Gaussian integer sequences with ideal periodic autocorrelation functions,” IEEE Trans. Commun., vol. 64, no. 1, pp. 365–376, Jan. 2016. [9] C.-P. Li, S.-H. Wang, and C.-L. Wang, “Novel low-complexity SLM schemes for PAPR reduction in OFDM systems,” IEEE Trans. Signal Process., vol. 58, no. 5, pp. 2916–2921, May 2010. [10] S.-H. Wang, C.-P. Li, K.-C. Lee, and H.-J. Su, “A novel low-complexity precoded OFDM system with reduced PAPR,” IEEE Trans. Signal Process., vol. 63, no. 6, pp. 1368–1376, Mar. 2015. [11] S.-H. Wang and C.-P. Li, “Novel comb spectrum CDMA system using perfect Gaussian integer sequences,” in Proc. IEEE Global Commun. Conf. (IEEE Globecom), Dec. 2015, pp. 1–6. [12] H.-H. Chang, S.-C. Lin, and C.-D. Lee, “A CDMA scheme based on perfect Gaussian integer sequences,” Int. J. Electron. Commun., vol. 75, pp. 70–81, May 2017. [13] P. Z. Fan and M. Darnell, Sequences Design for Communication Applications. New York, NY, USA: Wiley, 1996. [14] S. H. Choi, J. S. Baek, J. S. Han, and J. S. Seo, “Channel estimations using orthogonal codes for AF multiple-relay networks over frequencyselective fading channels,” IEEE Trans. Veh, Technol., vol. 63, no. 1, pp. 417–423, Jan. 2014. [15] S. U. H. Qureshi, “Fast start-up equalization with period training sequences,” IEEE Trans. Inf. Theory, vol. 23, no. 4, pp. 553–563, Sep. 1977. [16] G. Gong, F. Huo, and Y. Yang, “Large zero autocorrelation zones of golay sequences and their applications,” IEEE Trans. Commun., vol. 61, no. 9, pp. 3967–3979, Sep. 2013. [17] A. Milewski, “Periodic sequences with optimal properties for channel estimation and fast start-up equalization,” IBM J. Res. Develop., vol. 27, no. 5, pp. 425–431, Sep. 1983. [18] D. C. Chu, “Polyphase codes with good periodic correlation properties,” IEEE Trans. Inf. Theory, vol. 18, no. 4, pp. 531–532, Jul. 1972. [19] W. K. Nicholson, Introduction to Abstract Algebra, 2nd ed. Hoboken, NJ, USA: Wiley, 1999.

Kuo-Jen Chang was born in Taiwan, in 1972. He received the B.S. and M.S. degrees from National Taiwan University, Taipei, Taiwan, in 1998 and 2000, respectively, and the Ph.D. degree from Université Montpellier 2, Montpellier, France, in 2005. His current research interests include unmanned aerial vehicle (UAV) system integration, light detection and ranging (Li-DAR) signal processing, and the UAV technology applying for natural hazards investigation and mitigation, e.g., active faults and landslides.

Ho-Hsuan Chang received the Ph.D. degree in electrical engineering from Syracuse University, Syracuse, NY, USA, in 1997. From 1997 to 2003, he joined the faculty of the Department of Electrical Engineering, Chinese Military Academy, Taiwan, as an Associate Professor. He is currently with the Department of Communication Engineering, I-Shou University, Kaohsiung, Taiwan. His research interests include wireless communication, signal processing, space-time coding, and sequence design.