Two-Dimensional Golay Complementary Array Sets from Generalized Boolean Functions
aa r X i v : . [ c s . I T ] F e b Two-Dimensional Golay Complementary ArraySets from Generalized Boolean Functions
Cheng-Yu Pai and Chao-Yu Chen,
Member, IEEE
Abstract
The one-dimensional (1-D) Golay complementary set (GCS) has many well-known properties andhas been widely employed in engineering. The concept of 1-D GCS can be extended to the two-dimensional (2-D) Golay complementary array set (GCAS) where the 2-D aperiodic autocorrelationsof constituent arrays sum to zero except for the 2-D zero shift. The 2-D GCAS includes the 2-DGolay complementary array pair (GCAP) as a special case when the set size is 2. In this paper, 2-Dgeneralized Boolean functions are introduced and novel constructions of 2-D GCAPs, 2-D GCASs, and2-D Golay complementary array mates based on generalized Boolean functions are proposed. Explicitexpressions of 2-D Boolean functions for 2-D GCAPs and 2-D GCASs are given. Therefore, they are alldirect constructions without the aid of other existing 1-D or 2-D sequences. Moreover, for the columnsequences and row sequences of the constructed 2-D GCAPs, their peak-to-average power ratio (PAPR)properties are also investigated.
Index Terms
Golay complementary pair (GCP), Golay complementary array pair (GCAP), Golay complementaryarray mate, Golay complementary array set (GCAS), peak-to-average power ratio (PAPR).
I. I
NTRODUCTION
One-dimensional (1-D) Golay complementary pair (GCP) [1] and its extension, Golay com-plementary set (GCS) [2], have zero autocorrelation sums for non-zero shifts and hence havebeen found many applications, such as channel estimation [3], synchronization [4], interference
The material in this paper was presented in part at the IEEE International Symposium on Information Theory (ISIT), June2020. This work was supported the Ministry of Science and Technology, Taiwan, R.O.C., under Grants MOST 109–2628–E–006–008–MY3.The authors are with the Department of Engineering Science, National Cheng Kung University, Tainan 701, Taiwan, R.O.C.(e-mail: {n98081505, super}@mail.ncku.edu.tw).
February 9, 2021 DRAFT mitigation for multi-carrier code division multiple access (MC-CDMA) [5], and peak-to-averagepower ratio (PAPR) control in orthogonal frequency division multiplexing (OFDM) [6]–[9].Such special 1-D sequence pairs and sets can be extended to two-dimensional (2-D) arraypairs and sets, called 2-D Golay complementary array pairs (GCAPs) [10]–[13] and 2-D Golaycomplementary array sets (GCASs) [14], [15], respectively. For 2-D GCAPs and 2-D GCASs,the aperiodic autocorrelations of constituent arrays sum up to zero except for the 2-D zeroshift. Owing to their good autocorrelation properties, they have applications in radar [16],synchronization [17], [18], multiple-input multiple-output (MIMO) [15], and can be used asspreading sequences in the 2-D MC-CDMA system [19], [20].In 1999, Davis and Jedweb first proposed a direct construction of 1-D GCPs based ongeneralized Boolean functions [8]. The construction from generalized Boolean functions hasalgebraic structure and hence can be friendly for efficient hardware generations. Since then,there have been a number of literature investigating constructions of 1-D sequences from Booleanfunctions, including GCSs [9], [21]–[25], complete complementary codes (CCCs) [26]–[29]. Z-complementary pairs (ZCPs) [30]–[35], and Z-complementary sets (ZCSs) [36], [37].In [10], 2-D binary GCAPs can be obtained from existing 1-D GCPs or 2-D GCAPs by usingKronecker product. Later in [13], Fiedler et al. applied the method given in [10] recursivelyto construct multi-dimensional GCAPs and then 2-D GCAPs can be obtained via projection. In[11], via concatenating existing 1-D GCPs or interleaving existing 2-D GCAPs, 2-D GCAPs canbe constructed. In [14], Zeng and Zhang proposed a construction of 2-D GCASs based on 2-Dperfect arrays and only periodic GCASs were considered. Recently, [15] provided constructionsof 2-D GCASs from existing 1-D GCSs or 1-D CCCs. In addition to 2-D GCAPs and GCASs,2-D CCCs were studied in [19], [20], [38], [39]. 2-D CCCs can be seen as a collection of 2-DGCASs, where any two different 2-D GCAS are mutually orthogonal. The existing constructionsof 2-D CCCs need the help of special 1-D sequences, such as Welti codes [38] and 1-D CCCs[20].So far, most constructions of 2-D GCAPs or GCASs still require existing sequences or arraysas kernels. Motivated by this, in this paper, novel constructions of 2-D GCAPs, 2-D GCASs,and 2-D Golay complementary array mate based on generalized Boolean functions are proposed.Our proposed constructions are direct constructions and do not require the aid of any existingarrays or specific 1-D sequences. The newly proposed 2-D GCAPs and Golay complementary
February 9, 2021 DRAFT array mates can include our previous results [40, Th.6] and [40, Th.7] as special cases * . Besides2-D GCAPs and 2-D Golay complementary array mate, we further propose a construction of2-D GCASs from Boolean functions as well. To the best of authors’ knowledge, this is the firstwork to directly construct GCASs without the aid of other special sequences. Furthermore, weanalyze the column sequence PAPR and the row sequence PAPR for our proposed 2-D GCAPsand their PAPR upper bounds are derived, respectively, in this paper. Note that the columnsequence PAPR is concerned in the MC-CDMA system [28], [41].The rest of this paper is organized as follows. Section II gives some notations and definitions.New constructions of 2-D GCAPs, GCASs, and Golay complementary array mates are presentedin Section III. The column sequence PAPRs and row sequence PAPRs are also discussed. Finally,we conclude our paper in Section IV.II. P RELIMINARIES AND N OTATIONS
The following notations will be used throughout this paper: • ( · ) ∗ denotes the complex conjugation. • ( · ) T denotes the transpose. • is an all-one vector. • Z q = { , , · · · , q − } is the ring of integers modulo q . • Let ξ = e π √− /q . • We consider even integer q in this paper.A complex-valued array C of size L × L can be expressed as C = ( C g,i ) , ≤ g < L , ≤ i < L . (1) Definition 1:
The 2-D aperiodic cross-correlation function of arrays C and D at shift ( u , u ) is defined as ρ ( C , D ; u , u ) = L − X g =0 L − X i =0 D g + u ,i + u C ∗ g,i , (2) * In our previous conference paper [40], we provided constructions of 2-D GCAPs and 2-D Golay complementary array matesfrom Boolean functions which can be found in [40, Th.6] and [40, Th.7], respectively. The result from [40, Th.6] will bedescribed in Theorem 9 in this paper. Then, we provide more general constructions of 2-D GCAPs and Golay complementaryarray mates in Theorem 12 and Theorem 16, respectively.
February 9, 2021 DRAFT where D g + u ,i + u = 0 when ( g + u )
6∈ { , , · · · , L − } or ( i + u )
6∈ { , , · · · , L − } .When C = D , ρ ( C , C ; u , u ) is called the 2-D aperiodic autocorrelation function of C anddenoted by ρ ( C ; u , u ) . Note that ρ ( C ; u , − u ) = ρ ∗ ( C ; − u , u ) .If we take L = 1 , then the array C can be reduced to a 1-D sequence C = C i for i =0 , , . . . , L − . Therefore, the corresponding 1-D autocorrelation can be given by ρ ( C ; u ) = L − X i =0 C i + u C ∗ i , (3)where C i + u = 0 when ( i + u )
6∈ { , , · · · , L − } . In this paper, we only consider q -PSKmodulation. Thus, we define a q -ary array c and (1) can be rewritten as c = ( c g,i ) and C = ( C g,i ) = ( ξ c g,i ) = ξ c , (4)where ≤ g < L and ≤ i < L . The 2-D aperiodic cross-correlation function given in (2)can also be expressed as ρ ( C , D ; u , u ) = L − X g =0 L − X i =0 ξ d g + u ,i + u − c g,i . (5)If taking L = 1 , the corresponding 1-D autocorrelation can be modified as ρ ( C ; u ) = L − X i =0 ξ c i + u − c i . (6) Definition 2 (Golay Complementary Set): [2] A set of N sequences C , C , · · · , C N − oflength L is a 1-D GCS, denoted by ( N, L ) -GCS, if and only if N − X l =0 ρ ( C l ; u ) = , u = 0 N L, u = 0 . (7)Note that a GCS is reduced to a GCP by taking N = 2 and each sequence in a GCP is a Golaysequence. Definition 3 (Golay Complementary Mate): [2] For a GCP ( C , C ) , if another GCP ( D , D ) meets the following condition: ρ ( C , D ; u ) + ρ ( C , D ; u ) = 0 for all u, (8)then they are called the Golay complementary mate of each other.
February 9, 2021 DRAFT
Definition 4 (Golay Complementary Array Pair):
A pair of arrays C and D of size L × L is called a GCAP, if ρ ( C ; u , u ) + ρ ( D ; u , u ) = L L , ( u , u ) = (0 , , ( u , u ) = (0 , . (9)If C = ( ξ c g,i ) and D = ( ξ d g,i ) where c = ( c g,i ) and d = ( d g,i ) over Z q for ≤ g < L , ≤ i Given two GCAPs ( A , B ) and ( C , D ) ,they are called the Golay complementary array mate of each other if ρ ( A , C ; u , u ) + ρ ( B , D ; u , u ) = 0 for all ( u , u ) . (10) Definition 6 (Golay Complementary Array Set): Given a set of array G = { C l | l = 0 , · · · , N − } , where each array C l is of size L × L , G is called an ( N, L , L ) -GCAS if N − X l =0 ρ ( C l ; u , u ) = N L L , ( u , u ) = (0 , , ( u , u ) = (0 , . (11)If C l = ( ξ c l ) where c l is a q -ary array over Z q for l = 0 , , · · · , N − , then we call the arrayset { c , c , · · · , c N − } a q -ary GCAS. Actually, a GCAP is a (2 , L , L ) -GCAS. A. Peak-to-Average Power Ratio For a q -PSK modulated array C given in (4), we define C g and C Ti as the g -th row sequenceand the i -th column sequence, respectively. That is, C g = ( C g, , C g, , · · · , C g,L − ) , C Ti = ( C ,i , C ,i , · · · , C L − ,i ) T . (12)For the row sequence C g , the complex baseband OFDM signal is given by S C g ( t ) = L − X i =0 C g,i e π √− it = L − X i =0 ξ c g,i e π √− it , ≤ t ≤ (13)where L equals to the number of subcarriers. The PAPR of a row sequence C g is defined asPAPR ( C g ) = max ≤ t ≤ | S C g ( t ) | L (14)where L is the average power for q -PSK modulated sequences. Similarly, the PAPR of a columnsequence C Ti can also be given byPAPR ( C Ti ) = max ≤ t ≤ | S C Ti ( t ) | L , (15) February 9, 2021 DRAFT where S C Ti ( t ) = L − X g =0 C g,i e π √− gt = L − X g =0 ξ c g,i e π √− gt , ≤ t ≤ . (16)Note that for an MC-CDMA system, the column PAPR of C Ti is concerned since C Ti is spreadin the i -th chip-slot over the L subcarriers [28]. B. Generalized Boolean Functions Here, we will introduce the 2-D generalized Boolean function. A 2-D generalized Booleanfunction f : Z n + m → Z q comprises n + m variables y , y , . . . , y n , x , x , . . . , x m , where x i , y g ∈{ , } for g = 1 , , . . . , n and i = 1 , , . . . , m . We define the monomial of degree r as a productof r distinct variables. For example, x x x is a monomial of degree 3 and x x x y y is amonomial of degree 5. For simplicity, we define the variables z , z , · · · , z n + m as z l = y l for ≤ l ≤ n ; x l − n for n < l ≤ m + n, (17)which will be very useful in our proposed construction methods. For a q -ary generalized Booleanfunction with n + m variables, we define the associated array as f = f , f , · · · f , m − f , f , · · · f , m − ... ... . . . ... f n − , f n − , · · · f n − , m − (18)by letting f g,i = f (( g , g , · · · , g n ) , ( i , i , · · · , i m )) (19)where ( g , g , · · · , g n ) and ( i , i , · · · , i m ) are binary representations of the integers g = P nh =1 g h h − and i = P mj =1 i j j − , respectively. Example 1: If q = 4 , n = 2 , and m = 3 , the associated array to the generalized Booleanfunction f = 2 z + z + 3 z z + 2 z is given by f = . This generalized Boolean function f can also be stated as f = 2 y + y + 3 x x + 2 x accordingto (17). February 9, 2021 DRAFT III. C ONSTRUCTIONS OF GCAP S AND GCAS S In this section, constructions of 2-D GCAPs and 2-D GCASs based on generalized Booleanfunctions will be proposed by each subsection. In addition, a construction of 2-D Golay comple-mentary array mates will be provided as well. Based on our proposed 2-D GCAPs, the columnsequence PAPR and row sequence PAPR will be investigated and their upper bounds on PAPRswill also be given, respectively. A. GCAPs Based on Generalized Boolean Functions In this subsection, we will first restate a basic construction of 2-D GCAPs in [40, Th. 6] andthen extend it to new constructions of 2-D GCAPs and 2-D Golay complementary array mates.The proposed constructions can include [40, Th. 6] and [40, Th. 7] as special cases.Let us first introduce the well-known constructions of 1-D GCPs [8] and 1-D GCSs [27] inthe following Lemmas, which will be used hereinafter. Lemma 7: [8], [9] For any integer m ≥ , let π be a permutation of the set { , , · · · , m } .Let the generalized Boolean function be given by f = q m − X l =1 x π ( l ) x π ( l +1) + m X l =1 p l x l + p (20)where p l ∈ Z q for l = 0 , , · · · , m . The pair ( f , f ′ ) = (cid:16) f , f + q x π (1) (cid:17) (21)is a q -ary GCP of length m . Note that such constructed GCP is also called a Golay-Davis-Jedwab(GDJ) pair and its PAPR is at most 2. Lemma 8: [27] For any integer m ≥ and any positive integer k ≤ m , let { , , · · · , m } bedivided into a partition I , I , · · · , I k and π α be a bijection mapping from { , , · · · , m α } to I α where m α = | I α | for α = 1 , , · · · , k . For the generalized Boolean function given by f = q k X α =1 m α − X β =1 x π α ( β ) x π α ( β +1) + m X l =1 p l x l + p (22)where p l ’s ∈ Z q , the set G = ( f + q k X α =1 λ α x π α (1) : λ α ∈ { , } ) (23)is a (2 k , m ) -GCS. It has been proved that a GCS of size k has PAPR at most k [9]. February 9, 2021 DRAFT In what follows, we will provide a theorem to construct 2-D GCAPs based on generalizedBoolean functions. Theorem 9: [40, Th. 6] For a q -ary array c , let π be a permutation of { , , · · · , m } and π be a permutation of { , , · · · , n } . Let the generalized Boolean function f = q m − X l =1 x π ( l ) x π ( l +1) + n − X s =1 y π ( s ) y π ( s +1) + x π ( m ) y π (1) ! + m X l =1 p l x l + n X s =1 λ s y s + p (24)where p l , λ s ∈ Z q . Then, the array pair ( c , d ) = (cid:16) f , f + q x π (1) (cid:17) (25)is a q -ary GCAP of size n × m . Proof: Taking a 2-D GCAP ( C , D ) of size n × m , we need to show that ρ ( C ; u , u ) + ρ ( D ; u , u ) = 0 , for ( u , u ) = (0 , . (26)Let the array C = ξ c = ξ c , ξ c , · · · ξ c , m − ξ c , ξ c , · · · ξ c , m − ... ... . . . ... ξ c n − , ξ c n − , · · · ξ c n − , m − , (27)where c is expressed as c = q m − X l =1 x π ( l ) x π ( l +1) + n − X s =1 y π ( s ) y π ( s +1) + q x π ( m ) y π (1) ! + m X l =1 p l x l + n X s =1 λ s y s + p · . (28)Therefore, (26) is equivalent to ρ ( C ; u , u ) + ρ ( D ; u , u ) = n − X g =0 2 m − X i =0 (cid:0) ξ c g + u ,i + u − c g,i + ξ d g + u ,i + u − d g,i (cid:1) = m − X i =0 2 n − X g =0 (cid:0) ξ c g + u ,i + u − c g,i + ξ d g + u ,i + u − d g,i (cid:1) = 0 (29) February 9, 2021 DRAFT for − n < u < n , − m < u < m and ( u , u ) = (0 , . For given integers g, i , let h = g + u , j = i + u . We also let ( g , g , · · · , g n ) , ( h , h , · · · , h n ) , ( i , i , · · · , i m ) , and ( j , j , · · · , j m ) be the binary representations of g, h, i , and j , respectively. Then, four cases are considered asfollows to prove (29). Case 1: We suppose i π (1) = j π (1) and u = 0 . We can obtain c h,j − c g,i − d h,j + d g,i = q i π (1) − j π (1) ) ≡ q q ) (30)implying ξ c h,j − c g,i /ξ d h,j − d g,i = − for all g = 0 , , · · · , n − . Therefore, n − X g =0 (cid:0) ξ c h,j − c g,i + ξ d h,j − d g,i (cid:1) = 0 . (31) Case 2: Suppose i π (1) = j π (1) and u = 0 . We assume t is the smallest number such that i π ( t ) = j π ( t ) . Then, we let i ′ and j ′ be integers different from i and j , respectively, only in oneposition, i.e., i ′ π ( t − = 1 − i π ( t − , j ′ π ( t − = 1 − j π ( t − . Hence, we have c g,i ′ − c g,i = q (cid:0) i π ( t − i ′ π ( t − − i π ( t − i π ( t − + i ′ π ( t − i π ( t ) − i π ( t − i π ( t ) (cid:1) + p π ( t − i ′ π ( t − − p π ( t − i π ( t − ≡ q (cid:0) i π ( t − + i π ( t ) (cid:1) + p π ( t − (1 − i π ( t − ) (mod q ) . (32)Due to the fact that i π ( t − = j π ( t − and i π ( t − = i π ( t − , we have c h,j − c g,i − c h,j ′ + c g,i ′ ≡ q (cid:0) i π ( t − − j π ( t − + i π ( t ) − j π ( t ) (cid:1) + p π ( t − (2 j π ( t − − i π ( t − ) ≡ q (cid:0) i π ( t ) − j π ( t ) (cid:1) ≡ q q ) (33)which means ξ c h,j − c g,i + ξ c h,j ′ − c g,i ′ = 0 . Similarly, we can also obtain ξ d h,j − d g,i + ξ d h,j ′ − d g,i ′ = 0 implying n − X g =0 (cid:0) ξ c h,j − c g,i + ξ c h,j ′ − c g,i ′ + ξ d h,j − d g,i + ξ d h,j ′ − d g,i ′ (cid:1) = 0 . (34) Case 3: We suppose h π (1) = g π (1) and u = 0 implying j = i . For simplicity, we let a = q m − X l =1 x π ( l ) x π ( l +1) and b = q n − X s =1 y π ( s ) y π ( s +1) . (35)Then, (28) can be rewritten as c = a + b + q x π ( m ) y π (1) + m X l =1 p l x l + n X s =1 λ s y s + p · . (36) February 9, 2021 DRAFT0 Thus, we have m − X i =0 ξ c h,i − c g,i = m − X i =0 ξ b h − b g + q i π m ) + n P s =1 λ s ( h s − g s ) ! = ξ b h − b g + n P s =1 λ s ( h s − g s ) ! m − X i =0 ξ q i π m ) = 0 (37)where the last equality comes from m − P i =0 ξ q i π m ) = 0 . Similarly, we can also obtain m − P i =0 ξ d h,i − d g,i =0 which means m − X i =0 (cid:0) ξ c h,i − c g,i + ξ d h,i − d g,i (cid:1) = 0 . (38) Case 4: Suppose h π (1) = g π (1) and u = 0 . Assuming t is the smallest integer with g π ( t ) = h π ( t ) , we let g ′ and h ′ be the integers different from g and h , respectively, only in position π ( t − . That is, g ′ π ( t − = 1 − g π ( t − , h ′ π ( t − = 1 − h π ( t − . Using a similar argument asin Case 2, we can obtain m − X i =0 (cid:0) ξ c h,i − c g,i + ξ c h ′ ,i − c g ′ ,i + ξ d h,i − d g,i + ξ d h ′ ,i − d g ′ ,i (cid:1) = 0 . (39)Combining these four cases, we can prove that ( C , D ) is a 2-D GCAP of size n × m . Corollary 10: The PAPR of any row sequence of the constructed 2-D GCAPs from Theorem9 is upper bounded by 2. Proof: We let c g be the g -th row sequence of the array c in (28). For a given g , c g can beexpressed as c g = q m − X l =1 x π ( l ) x π ( l +1) + m X l =1 p l x l + q x π ( m ) g π (1) + ( p + κ ) (40)where κ = q n − X s =1 g π ( s ) g π ( s +1) + n X s =1 λ s g s (41)and g = P ns =1 g s s − . It can be seen that c g is a sequence of a GDJ pair by Lemma 7; henceits PAPR is at most 2. Corollary 11: The constructed 2-D GCAPs from Theorem 9 also have column sequence PAPRsat most 2. Proof: Similar to the proof of Corollary 10, the i -th column sequence c Ti is shown as c Ti = q n − X s =1 y π ( s ) y π ( s +1) + n X s =1 λ s y s + q i π ( m ) y π (1) + ( p + κ ′ ) (42) February 9, 2021 DRAFT1 where κ ′ = q m − X l =1 i π ( l ) i π ( l +1) + m X l =1 p l i l (43)and i = P ml =1 i l l − . Clearly, each column sequence can be viewed as a sequence of a GDJ pairand therefore has PAPR upper bounded by 2. Example 2: For q = 4 , m = 3 , and n = 2 , we let π = (3 , , , π = (1 , , and thegeneralized Boolean function f = 2( x x + x x + y y + x y ) + x by taking p = 1 . Thearray pair ( c , d ) = ( f , f + 2 x ) given by c = and d = is a GCAP of size × . According to Corollary 10 and Corollary 11, we know that both themaximum row sequence PAPR and the maximum column sequence PAPR are bounded by 2.Actually, each row sequence PAPR of c and d is exact 2 and each column sequence PAPR of c and d is 1.7698.Next, we extend Theorem 9 to a general construction of 2-D GCAPs. Theorem 12: Let π be a permutation of { , , · · · , n + m } and the generalized Boolean functioncan be given by f = q n + m − X l =1 z π ( l ) z π ( l +1) + n + m X l =1 p l z l + p (44)where z is defined in (17) and p l , p ∈ Z q . The array pair ( c , d ) = (cid:16) f , f + q z π (1) (cid:17) (45)forms a q -ary GCAP of size n × m . Proof: Similarly, we need to prove that ρ ( C ; u , u ) + ρ ( D ; u , u ) = n − X g =0 2 m − X i =0 (cid:0) ξ c g + u ,i + u − c g,i + ξ d g + u ,i + u − d g,i (cid:1) = 0 (46)for − n < u < n , − m < u < m and ( u , u ) = (0 , . From (4), we know that C = ξ c ,where c is expressed as c = q m + n − X l =1 z π ( l ) z π ( l +1) + n + m X l =1 p l z l + p · . (47) February 9, 2021 DRAFT2 Then we let h = g + u , j = i + u for any integers g and i . For the sake of easy presentation,we define a l = g l for ≤ l ≤ n ; i l − n for n < l ≤ m + n,b l = h l for ≤ l ≤ n ; j l − n for n < l ≤ m + n. (48)Therefore, (19) can be rewritten as f g,i = f ( a , a , · · · a m + n ) and f h,j = f ( b , b , · · · b m + n ) , (49)respectively. Then, we consider two cases to show that (46) holds. Case 1: Suppose a π (1) = b π (1) . We can obtain c h,j − c g,i − d h,j + d g,i = q a π (1) − b π (1) ) ≡ q q ) (50)implying ξ c h,j − c g,i + ξ d h,j − d g,i = 0 . (51) Case 2: Suppose a π (1) = b π (1) . We assume t is the smallest number such that a π ( t ) = b π ( t ) .Let a ′ and b ′ be integers different from a and b , respectively, only in one position. That is, a ′ π ( t − = 1 − a π ( t − , b ′ π ( t − = 1 − b π ( t − . If ≤ π ( t − ≤ n , by using (48), we have c g ′ ,i − c g,i = q (cid:0) a π ( t − g ′ π ( t − − a π ( t − g π ( t − + g ′ π ( t − a π ( t ) − g π ( t − a π ( t ) (cid:1) + p π ( t − g ′ π ( t − − p π ( t − g π ( t − ≡ q (cid:0) a π ( t − + a π ( t ) (cid:1) + p π ( t − (1 − g π ( t − ) (mod q ) (52)where a ′ π ( t − = g ′ π ( t − and a π ( t − = g π ( t − . Since a π ( t − = b π ( t − and a π ( t − = b π ( t − , from(48), we can obtain c h,j − c g,i − c h ′ ,j + c g ′ ,i ≡ q (cid:0) a π ( t − − b π ( t − + a π ( t ) − b π ( t ) (cid:1) + p π ( t − (2 h π ( t − − g π ( t − ) ≡ q (cid:0) a π ( t ) − b π ( t ) (cid:1) ≡ q q ) (53)implying ξ c h,j − c g,i + ξ c h ′ ,j − c g ′ ,i = 0 . Similarly, we can also obtain ξ d h,j − d g,i + ξ d h ′ ,j − d g ′ ,i = 0 . If n < π ( t − ≤ n + m , note that a ′ π ( t − = i ′ π ( t − − n and a π ( t − = i π ( t − − n according to (48).By following the similar argument as mentioned above, we can get ξ c h,j − c g,i + ξ c h,j ′ − c g,i ′ + ξ d h,j − d g,i + ξ d h,j ′ − d g,i ′ = 0 . (54) February 9, 2021 DRAFT3 From Case 1 and Case 2, we can say that the array pair ( C , D ) is indeed a 2-D GCAP. Remark 1: By setting { π (1) , π (2) , · · · , π ( n ) } = { , , · · · , n } and { π ( n +1) , π ( n +2) , · · · , π ( n + m ) } = { n + 1 , n + 2 , · · · , n + m } in Theorem 12, (44) can be rewritten as f = q n − X l =1 y π ( l ) y π ( l +1) + n + m − X l = n +1 x π ( l ) − n x π ( l +1) − n + y π ( n ) x π ( n +1) − n ! + n X l =1 p l y l + n + m X l = n +1 p l x l − n + p (55)according to (17). It can be observed that (55) is in the form of (24) and, therefore, Theorem12 includes Theorem 9 as a special case. Remark 2: When comparing (44) and (20), it can be found that Theorem 12 is a generalizationof 1-D GDJ pairs to 2-D GCAPs by applying the proper mapping in (17). Theorem 12 providesan explicit expression of 2-D Boolean functions for 2-D GCAPs.Like Corollaries 10 and 11, the PAPR properties for the constructed 2-D GCAPs from Theorem12 are described in the following Corollaries. Corollary 13: For a q -ary array pair ( c , d ) from Theorem 12, we define an index set W = { l | π ( l ) > n, l = 1 , , · · · , n + m } . If there exists an integer v and nonempty sets W , W , · · · , W v satisfying the following conditions, then the row sequences of c ( or d ) have PAPRs at most v .(C1) { W , W , · · · , W v } is a partition of the set W ;(C2) the elements in each W α are consecutive integers for ≤ α ≤ v . Proof: We let c g be the g -th row sequence of the array c . For the ease of presentation, weconsider the case for c and the other cases can be obtained by following the similar argument.For simplicity, we let σ α be a bijection from { , , · · · , m α } to the set { π ( l ) − n | l ∈ W α } with m α = | W α | and σ α ( i ) = π (min { W α } + i − − n for α = 1 , , · · · , v and i = 1 , , · · · , m α .The sequence c can be written as c = q v X α =1 m α − X β =1 x σ α ( β ) x σ α ( β +1) + m X l =1 p l x l + p · , p l ∈ Z q (56)which lies in a GCS in (23). Therefore, from Lemma 8, we can conclude that the maximumrow sequence PAPR of c and d is at most v . Corollary 14: Let W ′ = { l | ≤ π ( l ) ≤ n, l = 1 , , · · · , n + m } and follow the similarconditions (C1) and (C2) in Corollary 13 with W replaced by W ′ . Then, the array c ( or d ) fromTheorem 12 has column sequence PAPR upper bounded by v . February 9, 2021 DRAFT4 Proof: Similarly, we let σ α be a bijection from { , , · · · , n α } to the set { π ( l ) | l ∈ W ′ α } with n α = | W ′ α | and σ α ( l ) = π (min { W ′ α } + l − for α = 1 , , · · · , v and l = 1 , , · · · , n α . Thecolumn sequences can be represented as q v X α =1 n α − X β =1 y σ α ( β ) y σ α ( β +1) + n X l =1 p ′ l y l + p ′ · , p ′ l ∈ Z q (57)implying that the maximum column sequence PAPR is bounded by v . Corollary 15: The number of distinct n × m array c obtained from Theorem 12 is ( n + m )!2 · q n + m +1 . (58) Proof: For a constructed array c from Theorem 12, it can be expressed as c = q n + m − X l =1 z π ( l ) z π ( l +1) + n + m X l =1 p l z l + p · . (59)We first calculate the number of the quadratic forms q n + m − X l =1 z π ( l ) z π ( l +1) . (60)Since π is a permutation of the set { , , · · · , n + m } , there exist ( n + m )!2 distinct quadratic formsin (60). Then, we have p l ∈ Z q for l = 0 , , · · · , n + m and hence we can determine ( n + m )!2 · q n + m +1 (61)different arrays c of size n × m of the form in (59). Example 3: Taking q = 2 , m = 3 , and n = 2 , we let ( π (1) , π (2) , π (3) , π (4) , π (5)) =(3 , , , , . Then, the generalized Boolean function is f = z z + z z + z z + z z by setting p l = 0 for all l in (44). Note that f can also be expressed as f = x x + x y + y y + y x according to (17). Then, the array pair ( c , d ) = ( f , f + z ) is a GCAP from Theorem 12 where c = and d = . For C = ( ξ c g,i ) and D = ( ξ d g,i ) with ≤ g < L , ≤ i < L , the aperiodic autocorrelationvalues of C and D are given in (62) and (63). We can see that their aperiodic autocorrelationssum to zero except for ( u , u ) = (0 , . From Corollary 13, we have W = { l | π ( l ) > } = { , , } and let W = { , } , W = { } be a partition of W with v = 2 implying that the row February 9, 2021 DRAFT5 ( ρ ( C ; u , u )) u = − ∼ ,u = − ∼ = − − − − − − 23 0 − − − − 10 0 0 0 0 0 0 32 0 0 0 0 0 0 0 − − − − − − − − − − , (62) ( ρ ( D ; u , u )) u = − ∼ ,u = − ∼ = − − − − − − − − − − − − − − − − − − 32 0 − − − − − − − − − − . (63)sequence PAPRs of c and d are at most v = 4 . Actually, the maximum row sequence PAPRof c and d is 3.4427. Also, we let W ′ = { l | ≤ π ( l ) ≤ } = { , } which yields v = 1 and thecolumn sequence PAPR is at most 2 according to Corollary 14. In fact, the column sequencePAPR of each column of c and d is 1.7698.Next, a construction of Golay complementary array mates is provided based on Theorem 12. Theorem 16: The array pair ( c ′ , d ′ ) is a Golay complementary array mate of the GCAP ( c , d ) given in (45) where ( c ′ , d ′ ) = (cid:16) f + q z π ( n + m ) , f + q z π (1) + q z π ( n + m ) (cid:17) (64)and f is the associated array to the Boolean function f in (44). Proof: For the 2-D GCAPs ( C , D ) = ( ξ c , ξ d ) and ( C ′ , D ′ ) = ( ξ c ′ , ξ d ′ ) of size n × m , February 9, 2021 DRAFT6 we would like to show that ρ ( C , C ′ ; u , u ) + ρ ( D , D ′ ; u , u ) = n − X g =0 2 m − X i =0 (cid:16) ξ c ′ g + u ,i + u − c g,i + ξ d ′ g + u ,i + u − d g,i (cid:17) = 0 (65)for − n < u < n and − m < u < m . It is noted that C = ξ c where c can be obtained from(47). We follow the same notations given in (48) in the proof of Theorem 12 and consider threecases below. Case 1: Assume a π (1) = b π (1) . Following a similar derivation in Case 1 in the proof of Theorem12, we can have ξ c ′ g + u ,i + u − c g,i + ξ d ′ g + u ,i + u − d g,i = 0 . (66) Case 2: We assume a π (1) = b π (1) and let t be the smallest integer with a π ( t ) = b π ( t ) . The similarresults can be obtained as provided in Case 2 in the proof of Theorem 12. When ≤ π ( t − ≤ n ,we have ξ c ′ h,j − c g,i + ξ c ′ h ′ ,j − c g ′ ,i + ξ d ′ h,j − d g,i + ξ d ′ h ′ ,j − d g ′ ,i = 0 , (67)and when n < π ( t − ≤ n + m , we also have ξ c ′ h,j − c g,i + ξ c ′ h,j ′ − c g,i ′ + ξ d ′ h,j − d g,i + ξ d ′ h,j ′ − d g,i ′ = 0 . (68) Case 3: Lastly, it only suffices to show that ρ ( C , C ′ ; 0 , 0) + ρ ( D , D ′ ; 0 , 0) = 0 . (69)For ≤ π ( n + m ) ≤ n , we have n − X g =0 (cid:16) ξ c ′ g,i − c g,i + ξ d ′ g,i − d g,i (cid:17) = 2 n − X g =0 ξ q g π ( n + m ) = 0 (70)where g π ( n + m ) is the π ( n + m ) -th bit of the binary representation vector ( g , g , · · · , g n ) of g .For n < π ( n + m ) ≤ m + n , we can also obtain m − X i =0 (cid:16) ξ c ′ g,i − c g,i + ξ d ′ g,i − d g,i (cid:17) = 2 m − X i =0 ξ q i π ( n + m ) − n = 0 (71)where i π ( n + m ) − n is the ( π ( n + m ) − n ) -th bit of the binary representation vector of i .Combining Case 1 to Case 3, we complete the proof. Remark 3: Theorem 16 includes the results in [40, Th.7] by letting { π ( l ) , π (2) , · · · , π ( n ) } = { , , · · · , n } and { π ( n + 1) , π ( n + 2) , · · · , π ( n + m ) } = { n + 1 , n + 2 , · · · , n + m } . Therefore,[40, Th. 7] is a special case of Theorem 16. February 9, 2021 DRAFT7 Example 4: Let us follow the same notations given in Example 3. Based on Theorem 16, wehave an array pair ( c ′ , d ′ ) = ( f + z , f + z + z ) given by c ′ = and d ′ = . Let C ′ = ( ξ c ′ g,i ) and D ′ = ( ξ d ′ g,i ) with ≤ g < and ≤ i < . Then, we list the aperiodiccross-correlations of ( C , C ′ ) and ( D , D ′ ) in (72) and (73). By observing the sum of the cross-correlations, ( C , D ) and ( C ′ , D ′ ) are indeed Golay complementary array mates of each other. ( ρ ( C , C ′ ; u , u )) u = − ∼ ,u = − ∼ = − − − − − − − − − − − − − − − 10 0 0 0 0 0 0 0 0 0 0 0 0 0 01 0 − − − − − − − − − − − , (72) ( ρ ( D , D ′ ; u , u )) u = − ∼ ,u = − ∼ = − − 12 0 2 0 6 0 − − − − − − − − − 13 0 − − − − − − 21 0 − − − − − . (73) February 9, 2021 DRAFT8 B. GCASs Based on Generalized Boolean Functions In this subsection, we extend Theorem 12 to propose a construction of ( N, L , L ) -GCASwith various set size N ≥ . Theorem 17: Considering nonnegative integers n, m, k with n + m ≥ and k ≤ n + m , welet nonempty sets I , I , · · · , I k be a partition of { , , · · · , n + m } . Let t α be the order of I α and π α be a bijection from { , , · · · , t α } to I α for α = 1 , , · · · , k . If the generalized Booleanfunction is given by f = q k X α =1 t α − X β =1 z π α ( β ) z π α ( β +1) + n + m X l =1 p l z l + p (74)where p l ’s ∈ Z q , then the array set G = ( f + q k X α =1 λ α z π α (1) : λ α ∈ { , } ) (75)forms a q -ary (2 k , n , m ) -GCAS. Proof: For any array c ∈ G , we need to demonstrate that X c ∈ G n − X g =0 2 m − X i =0 (cid:0) ξ c g + u ,i + u − c g,i (cid:1) = n − X g =0 2 m − X i =0 X c ∈ G (cid:0) ξ c g + u ,i + u − c g,i (cid:1) = 0 (76)for ( u , u ) = (0 , . Here, we let h = g + u , j = i + u and also let ( g , g , · · · , g n ) , ( h , h , · · · , h n ) , ( i , i , · · · , i m ) , and ( j , j , · · · , j m ) be the binary representation vectors of g, h, i , and j ,respectively. By combining the binary representations of g, h, i , and j as follows: a l = g l for ≤ l ≤ n ; i l − n for n < l ≤ m + n,b l = h l for ≤ l ≤ n ; j l − n for n < l ≤ m + n, (77)the proof of (76) will be concise and two cases are taken into account. Case 1: If a π α (1) = b π α (1) for some α ∈ { , , · · · , k } , then for any array c ∈ G , we can findan array c ′ = c + ( q/ z π α (1) ∈ G satisfying c h,j − c g,i − c ′ h,j + c ′ g,i = q a π α (1) − b π α (1) ) ≡ q q ) . (78) February 9, 2021 DRAFT9 Therefore, we have X c ∈ G ξ c h,j − c g,i = 0 . (79) Case 2: Suppose that a π α (1) = b π α (1) for all α = 1 , , · · · , k . We assume a π α ( β ) = b π α ( β ) for α = 1 , · · · , ˆ α − and β = 1 , , · · · , m α . Besides, we assume that ˆ β is the smallest numbersatisfying a π ˆ α ( ˆ β − = b π ˆ α ( ˆ β − . Let a ′ π ˆ α ( ˆ β − = 1 − a π ˆ α ( ˆ β − and b ′ π ˆ α ( ˆ β − = 1 − b π ˆ α ( ˆ β − . Followingthe similar argument as mentioned in Case 2 in the proof of Theorem 12, we have X c ∈ G ξ c h,j − c g,i + ξ c h ′ ,j − c g ′ ,i = 0 , (80)for ≤ π ˆ α ( ˆ β − ≤ n and X c ∈ G ξ c h,j − c g,i + ξ c h,j ′ − c g,i ′ = 0 . (81)for n < π ˆ α ( ˆ β − ≤ n + m .According to the above two cases, the equality in (76) holds. Example 5: Let us take q = 2 , n = 2 , and m = 3 for example. According to Theorem 17, welet I = { , , } , I = { , } , π (1) = 4 , π (2) = 2 , π (3) = 5 , π (1) = 1 , π (2) = 3 , and theBoolean function f = z z + z z + z z , which can be represented by f = x y + y x + y x according to (17). The set G = { f , f + z , f + z , f + z + z } forms a q -ary (4 , , -GCASwhere f = , f + z = , f + z = , f + z + z = . According to the mapping in (17), we have z = x , z = x , z = y , z = y , and z = y . Therefore, we have F = ξ f = ( ξ f g,i ) , F = ξ f + z = ( ξ f g,i + i ) , F = ξ f + z =( ξ f g,i + g ) , and F = ξ f + z + z = ( ξ f g,i + i + g ) , where ≤ g < , ≤ i < and g = P s =1 g s s − , i = P l =1 i l l − . Their aperiodic autocorrelations are given, respectively, in (82) February 9, 2021 DRAFT0 ( ρ ( F ; u , u )) u = − ∼ ,u = − ∼ = − − − − − − − − − − − − − − 10 4 0 0 0 − − − − − − , (82) ( ρ ( F ; u , u )) u = − ∼ ,u = − ∼ = − − − − − − − − 11 0 11 0 − − − − − − − − − 11 0 3 0 − − − − − − − , (83)to (85). We can observe that the sums of their autocorrelations are all zero for ( u , u ) = (0 , . ( ρ ( F ; u , u )) u = − ∼ ,u = − ∼ = − − − − 10 4 0 0 0 − − − − − − 10 8 0 0 0 8 0 32 0 8 0 0 0 8 0 − − − − − − − − − − , (84) February 9, 2021 DRAFT1 ( ρ ( F ; u , u )) u = − ∼ ,u = − ∼ = − − − − − − − − − 11 0 3 0 − − − − − − − 11 0 11 0 − − − − − − − − . (85)Compared to [14], Theorem 17 constructs aperiodic GCASs and does not require any perfectarray as a kernel. 2-D GCASs can be generated directly from generalized Boolean functions.IV. C ONCLUSION In this paper, novel constructions of 2-D GCAPs, 2-D GCASs, and 2-D Golay complementaryarray mates based on 2-D generalized Boolean functions have been proposed. First, we give thebasic construction of 2-D GCAPs of size n × m from Boolean functions in Theorem 9. Then,we propose extended constructions of 2-D GCAPs and 2-D Golay complementary array mates,respectively, in Theorem 12 and Theorem 16, which can include the results in [40, Th.6] and [40,Th.7] as special cases. By adopting the proper mapping given in (17), Theorem 12 presents anelegant and explicit expression for 2-D GCAPs in terms of 2-D Boolean functions. In addition, theupper bounds on row sequence PAPR and column sequence PAPR of the proposed 2-D GCAPsare derived in Corollary 13 and Corollary 14. 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