Perfect Gaussian Integer Sequences of Arbitrary ... - IEEE Xplore

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 61, NO. 7, JULY 2015

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Perfect Gaussian Integer Sequences of Arbitrary Composite Length Ho-Hsuan Chang, Chih-Peng Li, Senior Member, IEEE, Chong-Dao Lee, Member, IEEE, Sen-Hung Wang, Member, IEEE, and Tsung-Cheng Wu, Member, IEEE

Abstract— A composite number can be factored into either N = mp or N = 2n , where p is an odd prime and m, n ≥ 2 are integers. This paper proposes a method for constructing degree-3 and degree-4 perfect Gaussian integer sequences (PGISs) of an arbitrary composite length utilizing an upsampling technique and the base sequence concept proposed by Hu, Wang, and Li. In constructing the PGISs, the degree of the sequence is defined as the number of distinct nonzero elements within one period of the sequence. This paper commences by constructing degree-3 PGISs of odd prime length, followed by degree-2 PGISs of odd prime length. The proposed method is then extended to the construction of degree-3 and degree-4 PGISs of composite length N = mp. Finally, degree-3 and degree-4 PGISs of length N = 4 are built to facilitate the construction of degree-3 and degree-4 PGISs of length N = 2n , where n ≥ 3. Index Terms— Gaussian integer, perfect sequence, periodic auto-correlation function (PACF).

I. I NTRODUCTION EQUENCES with an ideal periodic autocorrelation function (PACF) [1]–[5] are widely used in modern communication systems for such applications as channel estimation [6]–[8], synchronization [2], [9], peak-to-average power ratio reduction [10], [11], and modulation [12]–[16]. A sequence is regarded as perfect if it has an ideal PACF. In practical systems, perfect binary or quadri-phase sequences are preferred due to their simple implementation. However, perfect binary sequences of length N > 4 and perfect quadriphase sequences of length N > 16 have yet to be found [9]. For a large alphabet, various perfect polyphase sequences have been proposed, including Frank-Zadoff-Chu (FZC) sequences [2], [3], [17], Milewski sequences [6], and their various modifications and combinations [18], [19]. Among these sequences, FZC sequences have attracted particular attention for synchronization and random access in long-term evolution systems. However, both the real and the imaginary parts of the elements in FZC sequences are real valued,

S

Manuscript received July 22, 2014; revised April 25, 2015; accepted May 24, 2015. Date of publication June 1, 2015; date of current version June 12, 2015. (Corresponding author: Chih-Peng Li.) H.-H. Chang, C.-D. Lee, and T.-C. Wu are with the Department of Communication Engineering, I-Shou University, Kaohsiung 84001, Taiwan (e-mail: [email protected]; [email protected]; [email protected]). C.-P. Li is with the Department of Electrical Engineering and the Institute of Communications Engineering, National Sun Yat-sen University, Kaohsiung 804, Taiwan (e-mail: [email protected]). S.-H. Wang is with the Institute of Communications Engineering, National Sun Yat-sen University, Kaohsiung 804, Taiwan (e-mail: [email protected]). Communicated by K. Yang, Associate Editor for Sequences. Digital Object Identifier 10.1109/TIT.2015.2438828

and hence their computational complexity is significantly increased. Consequently, (complex) integer-valued sequences with an ideal PACF are of great interest. A Gaussian integer sequence (GIS) is a sequence in which all of the elements have the form of complex numbers, that is, a + bj , where a and b are both integers. Although their implementation is simple, perfect GISs (PGISs) have attracted only limited attention in the literature since their design has remained unknown until recently. A general form of even-length PGISs was recently presented in [1], in which the sequences were constructed by linearly combining four base sequences (or their cyclic shift equivalents) using Gaussian integer coefficients of equal magnitudes. In addition, PGISs of odd prime length were constructed in [20] by utilizing cyclotomic classes of orders 2 and 4 with respect to GF( p), where p is an odd prime. Finally, PGISs with length p · q, where p and q are twin primes, were presented in [21] based on Whiteman’s generalized cyclotomy of order 2. Existing methods are limited to the construction of PGISs of even length, odd prime length, or odd composite length. However, in practice, various applications require different sequence lengths. Accordingly, systematic methods for constructing PGISs of arbitrary composite length are of substantial interests. This study proposes a method for constructing PGISs of arbitrary composite length and various degrees. Note that the degree of a GIS is defined as the number of distinct nonzero Gaussian integers within one period of the sequence. The study commences by constructing degree-3 and degree-2 PGISs of odd prime length utilizing the base sequence concept proposed in [1]. An upsampling technique is then applied to construct two classes of degree-3 PGISs. The first class comprises sparse PGISs (that is, most of the sequence elements are zero) obtained by upsampling degree-3 PGISs of odd prime length. The second class comprises general degree-3 PGISs of composite length obtained by combining upsampled degree-2 PGISs of odd prime length and a base sequence. Finally, two classes of degree-4 PGISs of composite length are constructed. The first class consists of general degree-4 PGISs of composite length obtained by combining upsampled degree-3 PGISs of odd prime length and a base sequence. The second class consists of degree-4 PGISs of length N = 2n , where n ≥ 2. To the best of the authors’ knowledge, this study not only represents the first reported use of an upsampling technique to construct PGISs, but is also the first study to consider

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TABLE I S UMMARY OF K NOWN PGISs

the construction of PGISs of arbitrary composite length. Table I summarizes the main results obtained in this study, together with those previously presented in the literature. II. P RELIMINARIES N−1 Let the bold lowercase notation s = {s [n]}n=0 indicate a sequence of length N, where s [n] is the nth component of s. N−1 In addition, define s−1 ≡ s (−n) N n=0 , where (·) N is the modulo N operation. Finally, let Rs ≡ s ⊗ N s∗−1 = {R [τ ]}τN−1 =0 denote the periodic autocorrelation function of s, that is,

R [τ ] =

N−1

s [n] · s (n − τ ) N , ∗

N−1 be a PGIS of finite degree. Theorem 1: Let w = {w [n]}n=0

The upsampled w in (2) is also a PGIS with the same degree. Proof: The number of different nonzero elements in w

is the same as that in w. Hence, w and the same w have m N−1 N−1 degree. Let W = {w [n]}n=0 and W = W [n] n=0 denote the DFTs of w and w , respectively. Then, it follows that

W [l N + k] = =

m N−1

w [n] e− j 2πn(l N+k)/m N

n=0 N−1

w [i ] e− j 2πik/N = W [k],

i=0

(1)

n=0

where ⊗ N denotes N-point circular convolution and the superscript ∗ indicates complex conjugation. N−1 Let S = {S [n]}n=0 denote the discrete Fourier transform (DFT) of s. The DFT of Rs is then given as N−1 S◦S∗ = {|S [n]|2 }n=0 , where ◦ and |·| denote the componentwise product operation and the Euclidean norm, respectively. The sequence s is regarded as perfect if R [τ ] = E · δ [τ ], where E is the energy of s and δ [τ ] is a Kronecker delta sequence of length N. The DFT pair Rs and S ◦ S∗ = |S|2 indicates that s is perfect if and only √ if the magnitude spectrum E, ∀ 0 ≤ n ≤ N − 1. of s is flat, that is, |S [n]| = In this study, PGISs are constructed using a frequency-domain approach, that is, the DFTs of the PGISs have a flat magnitude spectrum. N−1 be a GIS of length N. In addition, let Let w = {w [n]}n=0 m N−1

a new GIS w = w [n] n=0 of length m · N be constructed by upsampling w, that is, n , n = 0, m, . . . , (N − 1)m w m (2) w [n] = 0, otherwise.

where k = 0, 1, . . . , N − 1 and l = 0, 1, . . . , m − 1. Since w is a PGIS, the condition |W [0]| = |W [1]| = · · · = |W Thus, the condition W [0] =

[N −

1]| holds.

W [1] = · · · = W [m N − 1] is also true. In other words, w is confirmed as a PGIS with the same degree as that of w. III. C ONSTRUCTION OF PGISs OF O DD P RIME L ENGTH U SING F REQUENCY-D OMAIN A PPROACH This section presents the construction of degree-2 and degree-3 PGISs of odd prime length. Let p = e f +1 be an odd prime, where e and f are both positive integers. In addition, let α be the generator of multiplicative group of GF( p). The cyclotomic classes of order e with respect to GF( p) (e) are defined as Cm = α m+en : 0 ≤ n < f , 0 ≤ m < e. Set e = 2 and obtain C0(2) and C1(2) . The base sequences x0 and x1 obtained from C0(2) and C1(2) are defined as follows: 1, n ∈ C0(2) x 0 [n] = (3) 0, otherwise, 1, n ∈ C1(2) x 1 [n] = (4) 0, otherwise.

CHANG et al.: PGISs OF ARBITRARY COMPOSITE LENGTH

p−1

4109

p−1

Let X0 = {X 0 [n]}n=0 and X1 = {X 1 [n]}n=0 denote the DFTs of x0 and x1 , respectively. Theorem 2: Let p = 2 f + 1 be an odd prime. If f is odd, then the DFTs of x0 and x1 are given as ⎧ ⎪ n=0 ⎪ f, ⎨ √ (2) X 0 [n] = − 1 + j p /2, n ∈ C0 ⎪ ⎪ ⎩− 1 − j √ p /2, n ∈ C (2) , 1 ⎧ ⎪ n=0 ⎪ ⎨ f, √ (2) X 1 [n] = − 1 − j p /2, n ∈ C0 ⎪ ⎪ ⎩− 1 + j √ p /2, n ∈ C (2) . 1

Theorem 3: Let p = 2 f + 1 be an odd prime. If f is even, then the DFTs of x0 and x1 are given as ⎧ ⎪ n=0 ⎪ ⎨ f, √ (2) X 0 [n] = − 1 − p /2, n ∈ C0 ⎪ ⎪ ⎩− 1 + √ p /2, n ∈ C (2) , 1

⎧ ⎪ n=0 ⎪ ⎨ f, √ p /2, n ∈ C0(2) − 1 + X 1 [n] = ⎪ ⎪ ⎩− 1 − √ p /2, n ∈ C (2) . 1 The proofs of Theorems 2 and 3 are provided in Appendix. A. Degree-3 PGISs of Odd Prime Length To construct PGISs of odd prime length, three base sequences are required. Two of these sequences are taken as x0 and x1 defined in (3) and (4), respectively, while the third sequence is selected from either

p−1

In the present study, x2 is selected for further investigation. However, the derivations for x3 are similar. Note that the DFTs of x2 and x3 are given respectively as X2 = [1, . . . , 1] and X3 = [ p, 0, . . . , 0]. p

p−1

Theorem 4: Let p = 2 f + 1 be an odd prime. If f is odd, then the sequence s = a · x2 + b · x0 + c · x1 with nonzero Gaussian integers a, b, and c is a degree-3 PGIS of length p under the following constraints:

√

p (b − c)

(b + c)

|a + (b + c) · f | = a − +j·

2 2

√

p (b − c)

(b + c)

−j· (5) = a −

. 2 2 If f is even, the constraints are given by

√

p (b − c)

(b + c)

|a + (b + c) · f | = a − +

2 2

√

p (b − c)

(b + c)

− = a −

. 2 2

√

√

} for odd f and {a + (b + c)· f, a − (b+c) } j 2 2 ± 2 for even f . The flat magnitude spectrum requirement for s to be a PGIS results in the constraints in (5) and (6). Let a = a R + j a I , b = b R + j b I , and c = c R + j c I , where a R , a I , b R , b I , c R , and c I are integers. The condition in (5) leads to the following constraints: p(b−c)

p(b−c)

b I c R − c I b R = a I (c R − b R ) + a R (b I − c I ), − ( + b R c R + b I c I ) = a R (c R + b R ) + a I (c I + b I ),

(7) (8)

2 2 where = f −1 2 · [(b R + c R ) + (b I + c I ) ]. If b R , b I , c R , and c I are known, and the determinant of the augmented matrix is nonzero, that is,

bI − cI cR − b R

2 2 2 2

c R + b R b I + c I = b R + b I − c R + c I = 0,

then the constraints in (7) and (8) lead to a unique solution pair (a R , a I ) = (P0 /Q 0 , P1 /Q 1 ), where P0 , Q 0 , P1 , and Q 1 are all integers. Let Q be the least common multiple of Q 0 and Q 1 . Substituting Qb R , Qb I , Qc R , and Qc I into (7) and (8) yields the integer solution pair (a R , a I ) = (Q P0 /Q 0 , Q P1 /Q 1 ). Similarly, the condition in (6) leads to the following constraints: c2 − b2I c2R − b 2R + I = a R (c R − b R ) + a I (c I − b I ), (9) 2 2 1 + 2 = − p · a R (c R + b R ) − p · a I (c I + b I ), (10) where ( p + 1) (b R − c R )2 − bR cR, 4 ( p + 1) (b I − c I )2 − bI cI . 2 = (b I + c I )2 f 2 − 4 1 = (b R + c R )2 f 2 −

x2 = [1, 0, . . . , 0] or x3 = [1, 1, . . . , 1]. p−1

Proof: Theorems 2 and 3 indicate that the DFT of s consists of three elements, namely {a + (b + c) · f, a − (b+c) 2 ±

(11) (12)

Again, if b R , b I , c R , and c I are known, and the augmented matrix is given by

cR − bR cI − bI

− p · (c R + b R ) − p · (b I + c I ) = b R b I − b R c I = 0, the constraints in (9) and (10) lead to a unique solution pair a R and a I . Consider the following illustrative examples for odd and even f , respectively. Example 1: Let f = 9. The sequence length is p = 2 · 9 + 1 = 19. If b = −30 − 30 j and c = 20 + 20 j are selected, then a = −20 − 20 j is obtained. Therefore, a degree-3 PGIS of length 19 is given by s = [a, b, c, c, b, b, b, b, c, b, c, b, c, c, c, c, b, b, c]. Example 2: Let f = 6 and p = 2·6+1 = 13. If b = 4+2 j and c = −2 − 4 j are selected, then a = −1 + j is obtained. A degree-3 PGIS of length 13 is then given by

(6)

s = [a, b, c, b, b, c, c, c, c, b, b, c, b].

4110


B. Degree-2 PGISs of Odd Prime Length

Let x4 and X4 be the base sequence and its associated DFT given by

Consider the case where f is odd and p = 2 f + 1 is an odd prime. To construct a degree-2 PGIS of odd prime length, select {x0 + x2 , x1 } as the base sequence set. Theorem 5: Let p = 2 f + 1 be an odd prime and f be an odd integer. The GIS s = a(x0 + x2 ) + bx1 with nonzero Gaussian integers a and b is a degree-2 PGIS if

(a − b) 1 + j √ p

|a ( f + 1) + b f | =

2

(a − b) 1 − j √ p

(13) =

.

2 Proof: Consider Theorem 2 where f is odd. The DFT of s = a(x0 + x2 ) + bx1 consists of three different elements, √ namely {a( f + 1) + b f, (a − b)(1 ± j p)/2}. The flat magnitude spectrum requirement gives the constraint in (13). The first equality in (13) leads to the following constraint: b 2R + b 2I (a R + b R ) + (a I + b I ) 2

2

=

f +1 . 2

(14)

Note that the second equality in (13) is always true for any a and b. The constraint in (14) has numerous solutions. Let f = 9 and p = 2 · 9 + 1 = 19 be used for illustration purposes. Example 3: When f = 9 and p = 19, one solution pair of (14) is given by a = −7 − j and b = 10 + 5 j . A degree-2 PGIS of length 19 thus has the form s = [a, a, b, b, a, a, a, a, b, a, b, a, b, b, b, b, a, a, b]. However, as demonstrated in the following theorem, the sequence s = a (x0 + x2 ) + bx1 cannot be used to construct degree-2 PGISs of odd prime length p = 2 f + 1 in the case of even f . Theorem 6: Let f be an even integer and p = 2 f + 1 be an odd prime. There exists no degree-2 PGIS s = a (x0 + x2 ) + bx1, where a and b are nonzero Gaussian integers. Proof: For even f and odd prime p, the DFTs of x0 and x1 are given in Theorem 3, and the DFT of x2 is shown in Section III-A. It is easily shown that the DFT of of three elements, namely s = a (x0 + x2 ) + bx1 consists √ a ( f + 1) + b f, (a − b) 1 ± p /2 . The flat magnitude spectrum requirement for s to be a PGIS is given by

(a − b) 1 + √ p

|a ( f + 1) + b f | =

2

√

(a − b) 1 − p

=

(15)

.

2 The second equality results in a = b, and hence s contains only one nonzero element. The following discussions present the design of degree-2 PGISs of odd prime length for the case of even f .

x4 = [( p − 1) , −1, . . . , −1],

(16)

p−1

X4 = [0, p, p, . . . , p].

(17)

p−1

One can apply Theorem 7 to construct degree-2 PGISs of prime length p = 2 f + 1 for even f . Note that degree-2 PGISs of arbitrary length are also determined by Theorem 7. Theorem 7: The GIS s = ax3 + bx4 with nonzero Gaussian integers a and b is a degree-2 PGIS if |a| = |b|. Proof: The DFT of s = ax3 + bx4 is given by [ap, bp, bp, . . . , bp]. p−1

Therefore, the flat magnitude spectrum requirement for s to be a PGIS yields |a| = |b|. Example 4: Two Gaussian integers a = 1 + 2 j and b = 2 − j satisfy the requirement |a| = |b|. Three degree-2 PGISs, namely s2 , s5 , and s19 , of length N = 2, 5, and 19, respectively, are presented as follows: cs2 = [3 + j, −1 + 3 j ], s5 = [9 − 2 j, −1 + 3 j, −1 + 3 j, −1 + 3 j, −1 + 3 j ], s19 = [37 − 16 j, −1 + 3 j, −1 + 3 j, . . . , −1 + 3 j ]. 18

IV. D EGREE -3 PGISs OF C OMPOSITE L ENGTH This section proposes two classes of degree-3 PGISs of composite length. The first class comprises degree-3 PGISs obtained by upsampling degree-3 PGISs of odd prime length. Most of the elements in the upsampled sequences are zeros. In other words, the PGISs have a sparse form. The second class consists of general degree-3 PGISs obtained by combining upsampled degree-2 PGISs of odd prime length and a base sequence. In this case, all of the elements in the PGISs are nonzeros. A. Degree-3 Sparse PGISs of Composite Length In certain applications, e.g., that in [10], the computational complexity is reduced if most of the elements in the PGISs are zeros, that is, the PGISs have a sparse form. In accordance with Theorem 1, a degree-3 sparse PGIS of composite length N = p · m can be constructed by upsampling a degree-3 PGIS of prime length p. The degree-3 sparse PGIS contains p · (m − 1) zero elements and p nonzero elements. B. General Degree-3 PGISs of Composite Length This subsection designs general degree-3 PGISs of composite length using upsampled degree-2 PGISs of prime length. Let the base sequences x0 + x2 , x1 , x3 , and x4 of length p be upsampled to form the sequences x0E + x2E , x1E , x3E , and x4E of length p · m, respectively. Moreover, let an additional


4111

base sequence x5E and its associated DFT X5E be defined as follows: 1st

pth

2nd

x5E = [0, 1, . . . , 1, 0, 1, . . . , 1, . . . , 0, 1, . . . , 1], m

m−1 1st

(18)

m−1 2nd

mth

X5E = [ p(m − 1), 0, . . . , 0, − p, 0, . . . , 0, . . . , − p, 0, . . . , 0]. p−1

p−1

p−1

(19) Theorem 8: Let p = 2 f + 1 be an odd prime. When f is odd, the sequence s = ax5E + b (x0E + x2E ) + cx1E with nonzero Gaussian integers a, b, and c is a degree-3 PGIS of length N = p · m if the following constraint holds:

(b − c) 1 + j √ p (b − c) 1 − j √ p

=

2 2 = | pa (m − 1) + b (1 + f ) + c f | = |− pa + b (1 + f ) + c f |. (20) Note that this degree-3 PGIS is formed by combining an upsampled degree-2 PGIS and the base sequence x5E . Proof: The DFT of s = ax5E + b (x0E + x2E ) + + cx1E consists of four elements, namely { pa (m − 1) √ b (1 + f ) + c f, − pa + b (1 + f ) + c f, (b − c) 1 ± j p /2}. The flat magnitude spectrum requirement gives the constraint in (20). The last two equalities in (20) lead to the following constraints: ( f + 1) (b R + c R )2 + (b I + c I )2 − 2 c2R + c2I (21) = 2 p (m − 1) a 2R + a 2I , 2 2 p (m − 2) a R + a I + 2 ( f + 1) (a R b R + a I b I ) = −2 f (a R c R + a I c I ). (22) It should be noted that the first equality in (20) is always true for arbitrary b and c. Example 5: If f = 1, then p = 2 f + 1 = 3. For the case of m = 3, a = −4 −4 j , b = −4 +8 j , and c = 14 −10 j fulfill both constraints in (21) and (22). Thus, a degree-3 PGIS of composite length 9 is given by s = [b, a, a, b, a, a, c, a, a]. For the case of even f , degree-3 PGISs of composite length N = p · m do not exit for s = ax5E + b (x0E + x2E ) + cx1E . (Note that the corresponding proof is omitted for brevity.) The following discussions present the design for degree-3 PGISs of composite length for the case of even f based on the degree-2 PGIS of odd prime length constructed in Theorem 7. Note, however, that the following theorem demonstrates the construction of degree-3 PGISs of composite length not only for the case of even f, but also for any arbitrary sequence length greater than 1. Theorem 9: The sequence s = ax5E + bx3E + cx4E with nonzero Gaussian integers a, b, and c is a degree-3 PGIS of length N = p · m if |a (m − 1) + b| = |b − a| = |c|.

(23)

Proof: The DFT of s = ax5E + bx3E + cx4E consists of three elements, namely {a (m − 1) p + bp, (b − a) p, cp}, which yields the constraint in (23). The condition in (23) leads to the following constraints: (24) (m − 2) a 2R + a 2I + 2 (a R b R − a I b I ) = 0, 2 2 2 2 (25) (a R − b R ) + (a I − b I ) − c R + c I = 0. One of the solutions of given by ⎧ aR = ⎪ ⎪ ⎨ aI = cR = ⎪ ⎪ ⎩ cI =

the constraints in (24) and (25) is −2b R / (m − 2) −2b I / (m − 2) mb R / (m − 2) −mb I / (m − 2).

If b R = b I = m − 2 is selected, then a = −2 − 2 j , b = (m − 2) + (m − 2) j , and c = m − m j are obtained. Example 6: For p = 5 and m = 3, a = −2−2 j , b = 1+ j , and c = 3−3 j , a degree-3 PGIS of length N = 15 is given by s = [13 − 11 j, −2 − 2 j, −2 − 2 j, −2 + 4 j, −2 − 2 j, − 2 − 2 j, −2 + 4 j, −2 − 2 j, −2 − 2 j, −2 + 4 j, − 2 − 2 j, −2 − 2 j, −2 + 4 j, −2 − 2 j, −2 − 2 j ]. V. D EGREE -4 PGISs OF C OMPOSITE L ENGTH This section constructs two classes of degree-4 PGISs of composite length. The first class comprises general degree-4 PGISs obtained by combining upsampled degree-3 PGISs of odd prime length and a base sequence, while the second class comprises degree-4 PGISs of length N = 2n , where n ≥ 2. A. General Degree-4 PGISs of Composite Length Let general degree-4 PGISs of composite length be constructed based on degree-3 PGISs of odd prime length. The four base sequences are either {x0E , x1E , x2E , x5E } or {x0E , x1E , x3E , x5E }, where x3E is a sequence of length N = m · p in which all of the elements have a value of one. The following discussions arbitrarily choose the latter sequence set for further investigation. However, the derivations for the former set are similar. Theorem 10: Let p = 2 f + 1 be an odd prime. When f is even, the sequence s = a · x3E + b · x0E + c · x1E + d · x5E with nonzero Gaussian integers a, b, c, and d is a degree-4 PGIS of length N = p · m if the following constraint holds:

√ √

(b + c) p(b − c)

(b + c) p(b − c)

− − + = −

2 2 2 2 = | pma + (b + c) f + p(m − 1)d| = |(b + c) f − pd|. (26) When f is odd, the constraint is given by

√ √

(b + c) j p(b − c)

(b + c) j p(b − c)

− − +

= − 2

2 2 2 = | pma + (b+ c) f + p(m − 1)d| = |(b + c) f − pd|. (27)

4112


Proof: For the case of even f, the DFT of s = a ·x3E +b · x0E +c·x1E +d ·x5E consists of four different elements, namely √ p(c−b) ± , (b+c) f − pd}. { pma+(b +c) f + p(m−1)d, − (b+c) 2 2 Hence, the constraint in (26) is obtained. The proof of the constraint in (27) for the case of odd f is similar. The condition in (26) leads to the following constraints:

Theorem 11: The sequence s = a · x6 + b · x7 + c · x8 with nonzero Gaussian integers a, b, and c is a degree-3 PGIS of length N = 4 if the following constraint holds:

b2R + b 2I = c2R + c2I , 2 4 p d R + d I2 + ( p − 3) (b R + c R )2 + (b I + c I )2

(28)

|a + b + 2c| = |a + b − 2c| = |a − b| .

Proof: The DFT of s = a·x6 +b·x7 +c·x8 consists of three elements, namely {a + b ± 2c, a − b.}. Thus, the constraint in (34) is easily obtained. The condition in (34) leads to the following constraints:

+ 4 (b R c R + b I c I ) − 4( p − 1) × [(b R + c R ) d R + (b I + c I ) d I ] = 0, pm a 2R + a 2I + ( p − 1) [(b R + c R ) (a R + d R ) + (b I + c I ) (a I + d I )] + p (m − 2) d R2 + d I2 + 2 p(m − 1)(a R d R + a I d I ) = 0.

c R a R + c I a I = −b R c R − b I c I ,

Similarly, the condition in (27) leads to the following constraints: b R cI = cR bI , 2 2 4 p d R + d I + ( p − 3) (b R + c R )2 + (b I + c I )2 + 4 (b R c R + b I c I ) − 4( p − 1) × [(b R + c R ) d R + (b I + c I ) d I ] = 0, pm a 2R + a 2I + ( p − 1) [(b R + c R ) (a R + d R ) + (b I + c I ) (a I + d I )] + p (m − 2) d R2 + d I2 + 2 p(m − 1) (a R d R + a I d I ) = 0.

bR aR + bI aI =

(29)

(30)

(31)

(34)

−c2R

− c2I .

(35) (36)

If b R , b I , c R , and c I are given, then the constraints in (35) and (36) have only two unknown variables, namely a R and a I . If c R b I = c I b R , then the constraints in (35) and (36) result in a unique integer solution pair a R and a I . Example 9: The Gaussian integers a = −2 + j , b = 2 − j , and c = 1 − 2 j fulfill the constraints in (35) and (36). A degree-3 PGIS of length N = 4 is therefore given by s = [−2 + j, 1 − 2 j, 2 − j, 1 − 2 j ].

(32)

(33)

Two illustrative designs for even f = 2 and odd f = 1 are provided in the following. Example 7: For m = 3 and p = 2 ·2 +1 = 5, a = −9 −7 j , b = 5+15 j , c = 15+5 j , and d = 11+5 j fulfill the constraints in (28), (29), and (30). Therefore, a degree-4 PGIS of length N = 15 is given by

In this study, degree-4 PGISs of length N = 4 are designed using sequences x8 = [1, 0, 1, 0], x9 = [1, 0, −1, 0], x10 = [0, 1, 0, 1], and x11 = [0, 1, 0, −1], as demonstrated in Theorem 12. Theorem 12: The sequence s = a ·x8 +b·x9 +c·x10 +d ·x11 with nonzero Gaussian integers a, b, c, and d is a degree-4 PGIS of length 4 if the following constraint holds: |a + c| = |a − c| = |b − j d| = |b + j d|.

Proof: The DFT of s consists of four elements, namely {2a ± 2cj, 2b ± 2d j }. Thus, the constraint in (37) is easily obtained. The condition in (37) yields the following constraints:

s = [−9 − 7 j, 2 − 2 j, 2 − 2 j, −4 + 8 j, 2 − 2 j, 2 − 2 j, 6 − 2 j, 2 − 2 j, 2 − 2 j, 6 − 2 j, 2 − 2 j, 2 − 2 j, −4 + 8 j, 2 − 2 j, 2 − 2 j ]. Example 8: For m = 2 and p = 2 · 1 + 1 = 3, a = 1 + 5 j , b = 10 − 20 j , c = −6 + 12 j , and d = 6 − 12 j fulfill the constraints in (31), (32), and (33). Therefore, a degree-4 PGIS of length N = 6 is given by s = [1 + 5 j, 7 − 7 j, 11 − 15 j, 7 − 7 j, −5 + 17 j, 7 − 7 j ]. B. Construction of Degree-4 PGISs of Length N = 2n The odd-prime PGISs presented in Section III cannot be used to construct PGISs of length N = 2n , where n is a positive integer. Accordingly, this subsection proposes a method for designing degree-4 PGISs of length N = 2n , where n ≥ 2. The proposed method commences by designing degree-3 and degree-4 PGISs of length N = 4. Let degree-3 PGISs of length N = 4 be constructed using the base sequences x6 = [1, 0, 0, 0], x7 = [0, 0, 1, 0], and x8 = [0, 1, 0, 1], as shown in the following theorem.

(37)

a R c R = −a I c I , bR dI = bI dR , a 2R

+ a 2I

+

c2R

+ c2I = b2R + b 2I + d R2 + d I2 .

Example 10: Setting with a = 4+2 j , b = 2− j , c = 1−2 j , and d = −4 + 2 j , the constraints in (37) yield the following degree-4 PGIS of length N = 4: s = [6 + j, −3, 2 + 3 j, 5 − 4 j ]. To design degree-4 PGISs of length N = 2n , n ≥ 3, four base sequences are required. The first three base sequences are obtained by upsampling sequences x6 , x7 , x8 of length N = 4 into sequences x6E , x7E , x8E of length 4m. Meanwhile, the fourth base sequence x E and associated DFT X E are given respectively as 1st

2nd

4th

x E = [0, 1, . . . , 1, 0, 1, . . . , 1, . . . , 0, 1, . . . , 1], m

m−1 1st

m−1 2nd

mth

X E = [4(m − 1), 0, 0, 0, −4, 0, 0, 0, . . . , −4, 0, 0, 0].

(38)


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Theorem 13: The sequence s = ax E + bx6E + cx7E + dx8E with nonzero Gaussian integers a, b, c, and d is a degree-4 PGIS of composite length 4m if |−4a + b + c + 2d| = |4 (m − 1) a + b + c + 2d| = |b + c − 2d| = |b − c|. (39) Proof: The DFT of s = ax E +bx6E +cx7E +dx8E consists of four elements, namely {−4a + b + c + 2d, 4 (m − 1) a + b + c + 2d, b + c − 2d, b − c}. Thus, the constraint in (39) is easily obtained. According to Theorem 12, the sequence s = ax E + bx6E + cx7E +dx8E is a degree-4 PGIS of composite length 4m, where m is arbitrary integer. If m = 2n−2 , then degree-4 PGISs of length N = 2n exist for any n ≥ 3. The condition in (39) leads to the following constraints: 2 (m − 2) a 2R + a 2I + a R b R + a I b I + a R c R + a I c I + 2a R d R + 2a I d I = 0, (40) 2 2 d R + dI + b R cR + cI bI − b R d R − d I b I − c R d R + c I d I = 0, 2 2 a R + a 2I − a R b R − a I b I − a R c R − a I c I

(41)

+ b R d R − 2a R d R − 2a I d I + b I d I + c R d R + c I d I = 0. (42)

length by upsampling degree-2 and degree-3 PGISs of prime length, respectively. The construction of degree-3 and degree-4 PGISs of length N = 4 was then described. Finally, degree-4 PGISs of length N = 2n , where n ≥ 3, were obtained by upsampling degree-3 PGIS of length N = 4. A PPENDIX P ROOFS OF T HEOREMS 2 AND 3 Lemmas 1, 2, and 3 follow from classical cyclotomy theory [22], [23]. The three lemmas are essential to prove Theorems 2 and 3. Lemma 1: 1) If k, n ∈ C0(2) , then kn ∈ C0(2) . 2) If k ∈ C0(2) and n ∈ C1(2) , then kn ∈ C1(2) . (2) (2) 3) If k, n ∈ C1 , then kn ∈ C0 . (2) (2) 4) If k ∈ C1 and n ∈ C0 , then kn ∈ C1(2) . Lemma 2: Let p = 2 f + 1 be an odd prime. If f is odd, then √ 1+ j 2f +1 − j 2πk/ p e =− , (46) 2 (2) k∈C 0

e

k∈C 1(2)

The above constraints can be simplified as follows: a R (2 (m − 2) a R + (b R + c R + 2d R )) = −a I (2 (m − 2) a I + (b I + c I + 2d I )),

(43)

(d R − b R ) (d R − c R ) = − (d I − b I ) (d I − c I ),

(44)

k∈C 0

(45)

The constraints in (43), (44), and (45) contain eight variables. Therefore, numerous solutions exist. A set of integer solutions for the three constraints exist if ⎧ a R = (b R + c R ) /2 ⎪ ⎪ ⎪ ⎪ a ⎪ I = (b I + c I ) /2 ⎪ ⎨ dR = bR dI = cI ⎪ ⎪ ⎪ ⎪ b R + c R = −b I − c I ⎪ ⎪ ⎩ (m + 1) (b R − c I ) = (1 − m) (c R − b I ). and m = 2, Example 11: For the case of N = a = 3 − 3 j , b = 2 − 14 j , c = 4 + 8 j , and d = 2 + 8 j satisfy the constraints in (43), (44), and (45). Thus, a degree-4 PGIS of length N = 8 is given by s = [2 − 14 j, 3 − 3 j, 2 + 8 j, 3 − 3 j, 4 + 8 j, 3 − 3 j, 2 + 8 j, 3 − 3 j ]. 23

VI. C ONCLUSION This study has presented the construction of degree-3 and degree-4 PGISs of arbitrary composite length using an upsampling technique and the base sequence concept proposed in [1]. The study commenced by constructing degree-3 and degree-2 PGISs of odd prime length. The design was then extended to the construction of degree-3 and degree-4 PGISs of composite

√ 1− j 2f +1 . =− 2

(47)

Lemma 3: Let p = 2 f + 1 be an odd prime. If f is even, then √ 1− 2f +1 , (48) e− j 2πk/ p = − 2 (2)

(2a R − (b R + c R )) (a R − d R ) = (2a I − (b I + c I )) (a I − d I ).

− j 2πk/ p

e

− j 2πk/ p

=−

1+

(2)

√ 2f +1 . 2

(49)

k∈C 1

The proof of Theorem 2 is given as follows. Proof: From the constraint in (3), bk = 1 only when (2) k ∈ C0 . Thus, X 0 [n] =

p−1

bk e

− j 2πp kn

k=0

=

k∈C

e

− j 2πp kn

(2)

0 ⎧ f, ⎪ ⎪ √ ⎪ ⎪ − j 2π k 1+ j p ⎪ ⎪ , e p =− ⎨ 2 = k∈C0(2) √ ⎪ ⎪ − j 2π k 1− j p ⎪ ⎪ p , e =− ⎪ ⎪ ⎩ 2 (2)

n=0 (2)

n ∈ C0

(50) n∈

(2) C1 .

k∈C 1

In (50), X 0 [0]= When k, n ∈ Therefore,

C0(2) ,

X 0 [n] =

(2)

k∈C 0

(1) = f.

it follows from Lemma 1 that kn ∈ C0(2) .

e− j 2πkn/ p =

k∈C 0(2)

=

(2)

k∈C 0

kn∈C 0(2)

e

− j 2πk/ p

1+ j =− 2

e− j 2πkn/ p √

p

.

4114


When k ∈ C0(2) and n ∈ C1(2) , kn ∈ C1(2) by Lemma 1. Thus, X 0 [n] = e− j 2πkn/ p = e− j 2πkn/ p (2)

(2)

k∈C 0

=

nk∈C 1

e

− j 2πk/ p

(2)

1− j =− 2

√

p

.

k∈C 1

p−1 − j 2π kn − j 2πp kn Similarly, X 1 [n] = bk · e p = . (2) e k∈C 1 k=0 Therefore, X 1 [0] = k∈C (2) (1) = f. 1

When k, n ∈ C1(2) , kn ∈ C0(2) . Thus, e− j 2πkn/ p = e− j 2πkn/ p X 1 [n] = (2)

(2)

k∈C 1

=

kn∈C 1

e− j 2πk/ p = −

(2) k∈C 0

1+ j 2

√

p

.

When k ∈ C1(2) and n ∈ C0(2) , kn ∈ C1(2) by Lemma 1. Thus, e− j 2πkn/ p = e− j 2πkn/ p X 1 [n] = (2)

(2)

k∈C 1

=

kn∈C 1

e− j 2πk/ p = −

(2) k∈C 1

1− j 2

√

p

.

The desired result is then obtained as follows: X 1 [n] =

p−1

bk · e

=

e

− j 2πp kn

(2)

k=0

=

− j 2πp kn

k∈C 1

⎧ f, n=0 ⎪ ⎪ √ ⎪ − j 2π k 1− j p ⎪ (2) p ⎪ e = − 2 , n ∈ C0 ⎨ k∈C

(2)

1 √ ⎪ − j 2π k ⎪ 1+ j p ⎪ ⎪ e p = − 2 , n ∈ C1(2) . ⎪ ⎩ (2)

(51)

k∈C 0

The proof of Theorem 3 is similar to that of Theorem 2. The details are omitted here for brevity.

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Ho-Hsuan Chang received the Ph.D. degree in electrical engineering from Syracuse University, Syracuse, NY, 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 now with the Department of Communication Engineering, I-Shou University, Kaohsiung, Taiwan. His research interests include wireless communication, signal processing, spacetime coding and sequence design.


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Chih-Peng Li (SM’12) received the B.S. degree in Physics from National Tsing Hua University, Hsin Chu, Taiwan, in June 1989 and the Ph.D. degree in Electrical Engineering from Cornell University, Ithaca, NY, USA, in December 1997. From 1998 to 2000, Dr. Li was a Member of Technical Staff with the Lucent Technologies. From 2001 to 2002, he was a Manager of the Acer Mobile Networks. In 2002, he joined the faculty of the Institute of Communications Engineering, National Sun Yat-sen University, Kaohsiung, Taiwan, as an assistant professor. He has been promoted to Professor in 2010. Dr. Li is currently the Chairman of the Department of Electrical Engineering and the Director of the International Masters Program in Electric Power Engineering. His research interests include wireless communications, baseband signal processing, and data networks. Dr. Li is currently the Chair of the IEEE Communications Society Tainan Chapter, the Chair of the IEEE Broadcasting Technology Society Tainan Chapter, the Vice Chair of Chapter Coordination Committee for IEEE Asia Pacific Board, and the President of Taiwan Institute of Electrical and Electronics Engineering (TIEEE). Dr. Li also serves as the Editor of the IEEE T RANSACTIONS ON W IRELESS C OMMUNICATIONS, the Associate Editor of the IEEE T RANSACTIONS ON B ROADCASTING, the General Chair of 2014 IEEE Vehicular Technology Society Asia Pacific Wireless Communications Symposium (IEEE VTS APWCS 2014), and the General Co-chair of 2017 IEEE Information Theory Workshop (IEEE ITW 2017). Dr. Li was the lead guest editor of the Special Issue on Advances in Antenna Design and System Technologies for Next-Generation Cellular Systems, and International Journal of Antennas and Propagation. He was also the recipient of the 2014 Outstanding Professor Award of the Chinese Institute of Electrical Engineering Kaohsiung Branch and the 2015 Outstanding Professor Award of the Chinese Institute of Engineers Kaohsiung Branch.

Sen-Hung Wang (S’09–M’11) received the B.S. degree in electrical engineering from National Dong Hwa University, Hualien, Taiwan, in 2004, the M.S. degree in communications engineering and the Ph.D. degree in electrical engineering from National Sun Yat-sen University, Kaohsiung, Taiwan, in 2006 and 2010, respectively. From 2011 to 2014, Dr. Wang was a Postdoctoral Research Fellow with the Intel-NTU Connected Context Computing Center, National Taiwan University, Taipei, Taiwan. He is currently an Assistant Researcher with the Institute of Communications Engineering, National Sun Yat-sen University, Kaohsiung, Taiwan. His research interests include wireless communication, signal processing, sequence design, multiple access technology, M2M networks, and the fifth generation mobile communication technology.

Chong-Dao Lee (M’08) received the B.S. degree in Applied Mathematics, the M.E. and Ph.D. degrees in Information Engineering all from I-Shou University, Kaohsiung, Taiwan, R.O.C., in 1999 and 2001, 2006, respectively. He is currently an Associate Professor and a Chairman of the Department of Communication Engineering, I-Shou University, Kaohsiung, Taiwan, R.O.C. His research interests include error-correcting codes, finite fields, and sequence design.

Tsung-Cheng Wu (M’97) received the B.S. degree in communication engineering from National Chiao Tung University, the Ph.D. degree in electrical engineering from National Tsing Hua University, Hsinchu, Taiwan, R.O.C., in 1986 and 1997, respectively. Since September 2002, he has been an Assistant Professor with the Department of Communication Engineering, I-Shou University, Kaohsiung City, Taiwan, R.O.C. His research interests include cognitive radio networks, sensor networks, and compressed sensing.