Constructions of Two-Dimensional Golay Complementary Array Pairs - - PowerPoint PPT Presentation

Constructions of Two-Dimensional Golay Complementary Array Pairs Based on Generalized Boolean Functions

Constructions of Two-Dimensional Golay Complementary Array Pairs Based on Generalized Boolean Functions

Cheng-Yu Pai1 and Chao-Yu Chen2

Department of Engineering Science National Cheng Kung University Tainan 701, Taiwan, R.O.C. Email: {n980815051, super2}@mail.ncku.edu.tw

June 21-26, 2020

Cheng-Yu Pai (ES, NCKU) ISIT 2020 June 21-26, 2020 1 / 22

Constructions of Two-Dimensional Golay Complementary Array Pairs Based on Generalized Boolean Functions

Outline

Introduction Definitions

Two-dimensional (2-D) Aperiodic Correlation Functions Golay Complementary Array Pair (GCAP) Golay Complementary Array Mate Generalized Boolean Functions

Proposed Constructions

A Construction of 2-D GCAPs from Boolean Functions A Construction of Golay Complementary Array Mate

Conclusions

Cheng-Yu Pai (ES, NCKU) ISIT 2020 June 21-26, 2020 2 / 22

Constructions of Two-Dimensional Golay Complementary Array Pairs Based on Generalized Boolean Functions

Introduction

The one-dimensional (1-D) Golay complementary pair (GCP) has the well-known autocorrelation property.

The sum of autocorrelations of constituent sequences is zero except for the zero shift.

In [1]∗, a connection between 1-D GCPs and generalized Boolean functions was proposed.

Such GCPs are called the Golay-Davis-Jedweb (GDJ) pairs.

The 2-D GCAP is an extension of 1-D GCP .

The sum of 2-D autocorrelations is zero except for the 2-D zero shift. There are no explicit constructions from Boolean functions.

∗ [1] J. A. Davis and J. Jedwab, “Peak-to-mean power control in OFDM, Golay

complementary sequences, and Reed-Muller codes,” IEEE Trans. Inf. Theory, vol. 45,

• no. 7, pp. 2397–2417, Nov. 1999

Cheng-Yu Pai (ES, NCKU) ISIT 2020 June 21-26, 2020 3 / 22

Constructions of Two-Dimensional Golay Complementary Array Pairs Based on Generalized Boolean Functions

Introduction

Most constructions of 2-D GCAPs need the aid of 1-D sequences

• r existing arrays.

Kronecker product of 1-D GCPs and/or 2-D GCAPs [2]†. Concatenating 1-D GCPs or interleaving 2-D GCAPs [3]‡. Projection of three-dimensional (3-D) GCAPs [4,5]§ ¶. Based on 2-D perfect arrays [6].

† [2] M. Dymond, “Barker arrays: existence, generalization and alternatives,” Ph.D.

thesis, University of London, 1992

‡ [3] S. Matsufuji, R. Shigemitsu, Y. Tanada, and N. Kuroyanagi, “Construction of

complementary arrays,” in Proc. Joint IST Workshop on Mobile Future and Symp. on Trends in Commun., Bratislava, Slovakia, Oct. 2004, pp. 78–81

§ [4] J. Jedwab and M. G. Parker, “Golay complementary array pairs,” Designs,

Codes and Cryptography, vol. 44, no. 1-3, p. 209–216, Sep. 2007

¶ [5] F. Fiedler, J. Jedwab, and M. G. Parker, “A multi-dimensional approach for the

construction and enumeration of Golay complementary sequences,” J. Combinatorial Theory (Series A), vol. 115, no. 5, pp. 753–776, Jul. 2008

[6] F. Zeng and Z. Zhang, “Two dimensional periodic complementary array sets,”

in Proc. IEEE Int. Conf. on Wireless Commun., Netw. and Mobile Comput., Chengdu, China, Sep. 2010, pp. 1–4

Cheng-Yu Pai (ES, NCKU) ISIT 2020 June 21-26, 2020 4 / 22

Constructions of Two-Dimensional Golay Complementary Array Pairs Based on Generalized Boolean Functions

Introduction

Main Contribution

Contributions:

A mapping from generalized Boolean functions to 2-D arrays. A direct construction of 2-D GCAPs based on generalized Boolean functions. A construction of 2-D Golay complementary array mates from Boolean functions.

Cheng-Yu Pai (ES, NCKU) ISIT 2020 June 21-26, 2020 5 / 22

Constructions of Two-Dimensional Golay Complementary Array Pairs Based on Generalized Boolean Functions

Definition

A q-ary array c of size L1 × L2 can be expressed as c = (cg,i) 0 ≤ g < L1, 0 ≤ i < L2. The corresponding q-PSK modulated array C is C = (Cg,i) = (ξcg,i) 0 ≤ g < L1, 0 ≤ i < L2, where

ξ = e2π√−1/q; cg,i ∈ {0, 1, · · · , q − 1} = Zq; q is an even number.

Cheng-Yu Pai (ES, NCKU) ISIT 2020 June 21-26, 2020 6 / 22

Constructions of Two-Dimensional Golay Complementary Array Pairs Based on Generalized Boolean Functions

2-D Aperiodic Correlation Function

Definition (2-D aperiodic cross-correlation function) We define the 2-D aperiodic cross-correlation function of arrays C and D at shift (u1, u2) as ρ(C, D; u1, u2) =                           

L1−1−u1

• g=0

L2−1−u2

• i=0

ξdg+u1,i+u2−cg,i, 0 ≤ u1 < L1, 0 ≤ u2 < L2;

L1−1−u1

• g=0

L2−1+u2

• i=0

ξdg+u1,i−cg,i−u2, 0 < u1 < L1, −L2 < u2 < 0;

L1−1+u1

• g=0

L2−1+u2

• i=0

ξdg,i−cg−u1,i−u2, −L1 < u1 ≤ 0, −L2 < u2 ≤ 0;

L1−1+u1

• g=0

L2−1−u2

• i=0

ξdg,i+u2−cg−u1,i, −L1 < u1 < 0, 0 < u2 < L2.

Cheng-Yu Pai (ES, NCKU) ISIT 2020 June 21-26, 2020 7 / 22

Constructions of Two-Dimensional Golay Complementary Array Pairs Based on Generalized Boolean Functions

Golay Complementary Array Pair

When C = D, ρ(C, C; u1, u2) is called the 2-D aperiodic autocorrelation function of C and denoted by ρ(C; u1, u2). Definition (Golay Complementary Array Pair) A pair of arrays C and D of size L1 × L2 is called a GCAP , if ρ(C; u1, u2) + ρ(D; u1, u2) =

• 2L1L2,

(u1, u2) = (0, 0) 0, (u1, u2) = (0, 0). C = (ξcg,i), D = (ξdg,i) where c = (cg,i) and d = (dg,i) over Zq for 0 ≤ g < L1, 0 ≤ i < L2. We call (c, d) a q-ary GCAP .

Cheng-Yu Pai (ES, NCKU) ISIT 2020 June 21-26, 2020 8 / 22

Constructions of Two-Dimensional Golay Complementary Array Pairs Based on Generalized Boolean Functions

Golay Complementary Array Mate

Definition (Golay Complementary Array Mate) For two GCAPs (A, B) and (C, D), they are called the Golay complementary array mate of each other if ρ(A, C; u1, u2) + ρ(B, D; u1, u2) = 0 for all (u1, u2).

Cheng-Yu Pai (ES, NCKU) ISIT 2020 June 21-26, 2020 9 / 22

Constructions of Two-Dimensional Golay Complementary Array Pairs Based on Generalized Boolean Functions

Generalized Boolean Functions

A 2-D generalized Boolean function f : (y1, y2, . . . , yn, x1, x2, . . . , xm) ∈ Zn+m

2 mapping

− → f(y1, y2, . . . , yn, x1, x2, . . . , xm) ∈ Zq where xi, yg ∈ {0, 1} for i = 1, 2, . . . , m and g = 1, 2, . . . , n. The monomial of degree r is a product of r variables.

For example, x1x2y1y2 is a monomial of degree 4.

Cheng-Yu Pai (ES, NCKU) ISIT 2020 June 21-26, 2020 10 / 22

Constructions of Two-Dimensional Golay Complementary Array Pairs Based on Generalized Boolean Functions

Generalized Boolean Functions

Let the associated array of size 2n × 2m f =      f0,0 f0,1 · · · f0,2m−1 f1,0 f1,1 · · · f1,2m−1 . . . . . . ... . . . f2n−1,0 f2n−1,1 · · · f2n−1,2m−1      where

fg,i = f((g1, g2, · · · , gn), (i1, i2, · · · , im)); (g1, g2, · · · , gn) is the binary representation of the integer g = n

h=1 gh2h−1;

(i1, i2, · · · , im) is the binary representation of the integer i = m

j=1 ij2j−1.

Cheng-Yu Pai (ES, NCKU) ISIT 2020 June 21-26, 2020 11 / 22

Constructions of Two-Dimensional Golay Complementary Array Pairs Based on Generalized Boolean Functions

Generalized Boolean Functions

Example 1 Let q = 4, n = 2, and m = 3. The generalized Boolean function is f = 3x1 + 2x2x3 + 2y1 + y2. The associated array is f =     3 3 3 2 1 2 1 2 1 2 1 3 1 1 1 3 2 3 2 3 2 3 1    

22×23

.

Cheng-Yu Pai (ES, NCKU) ISIT 2020 June 21-26, 2020 12 / 22

Constructions of Two-Dimensional Golay Complementary Array Pairs Based on Generalized Boolean Functions

A Construction of 2-D GCAPs

Theorem 1 For a q-ary array c, let π1 be a permutation of {1, 2, · · · , m} and π2 be a permutation of {1, 2, · · · , n}. Let the generalized Boolean function f =q 2  

m−1

• i=1

xπ1(i)xπ1(i+1) +

n−1

• g=1

yπ2(g)yπ2(g+1)   + q 2xπ1(m)yπ2(1) +

m

• i=1

pixi +

n

• g=1

λgyg + p0 (1) where pi, λg ∈ Zq.

Cheng-Yu Pai (ES, NCKU) ISIT 2020 June 21-26, 2020 13 / 22

Constructions of Two-Dimensional Golay Complementary Array Pairs Based on Generalized Boolean Functions

A Construction of 2-D GCAPs

Theorem 1 (Cont’d) The array pair (c, d) =

• f, f + q

2xπ1(1)

• (2)

is a q-ary GCAP of size 2n × 2m.

Cheng-Yu Pai (ES, NCKU) ISIT 2020 June 21-26, 2020 14 / 22

Constructions of Two-Dimensional Golay Complementary Array Pairs Based on Generalized Boolean Functions

Example

Example 2 Taking q = 2, m = 3, and n = 2, we let π1 = (2, 1, 3) and π2 = (1, 2). The Boolean function is f = x2x1 + x1x3 + y1y2 + x3y1 by setting pi = 0 for i = 0, 1, 2, 3 and λg = 0 for g = 1, 2. (c, d) = (f, f + x2) is a GCAP of size 4 × 8 given by c =     1 1 1 1 1 1 1 1 1 1 1 1     , d =     1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1     .

Cheng-Yu Pai (ES, NCKU) ISIT 2020 June 21-26, 2020 15 / 22

Constructions of Two-Dimensional Golay Complementary Array Pairs Based on Generalized Boolean Functions

Example

Example 2 (Cont’d) C = (−1)cg,i and D = (−1)dg,i, 0 ≤ g < L1, 0 ≤ i < L2.

(ρ(C; u1, u2))u1=−3∼3,u2=0∼7 =          −1 −2 3 −1 −2 −1 −2 4 −2 2 4 2 3 −2 −1 −1 −2 −1 32 1 2 −3 1 2 1 −2 4 −2 2 4 2 −3 2 1 1 2 1          , (ρ(D; u1, u2))u1=−3∼3,u2=0∼7 =          1 2 −3 1 2 1 2 −4 2 −2 −4 −2 −3 2 1 1 2 1 32 −1 −2 3 −1 −2 −1 2 −4 2 −2 −4 −2 3 −2 −1 −1 −2 −1          .

Cheng-Yu Pai (ES, NCKU) ISIT 2020 June 21-26, 2020 16 / 22

Constructions of Two-Dimensional Golay Complementary Array Pairs Based on Generalized Boolean Functions

A Construction of Golay Complementary Array Mate

Theorem 2 An array pair (c′, d′) is a Golay complementary array mate of the array (c, d) given in (2) if (c′, d′) =

• f + q

2yπ2(n), f + q 2xπ1(1) + q 2yπ2(n)

• where f is the associated array to the Boolean function f in (1).

Cheng-Yu Pai (ES, NCKU) ISIT 2020 June 21-26, 2020 17 / 22

Constructions of Two-Dimensional Golay Complementary Array Pairs Based on Generalized Boolean Functions

Example

Example 3 Let us follow the same notations given in Example 2. We can obtain a complementary array mate (c′, d′) = (f + y2, f + x2 + y2) given by c′ =     1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1     , d′ =     1 1 1 1 1 1 1 1 1 1 1 1     .

Cheng-Yu Pai (ES, NCKU) ISIT 2020 June 21-26, 2020 18 / 22

Constructions of Two-Dimensional Golay Complementary Array Pairs Based on Generalized Boolean Functions

Example

Example 3 (Cont’d)

(ρ(C, C′; u1, u2))u1=−3∼3,u2=0∼7 =           −1 −2 3 −1 −2 −1 −2 4 −2 2 4 2 5 2 −7 1 2 1 5 −6 1 −3 −6 −3 2 −4 2 −2 −4 −2 3 −2 −1 −1 −2 −1           , (ρ(D, D′; u1, u2))u1=−3∼3,u2=0∼7 =           1 2 −3 1 2 1 2 −4 2 −2 −4 −2 −5 −2 7 −1 −2 −1 −5 6 −1 3 6 3 −2 4 −2 2 4 2 −3 2 1 1 2 1           .

Cheng-Yu Pai (ES, NCKU) ISIT 2020 June 21-26, 2020 19 / 22

Constructions of Two-Dimensional Golay Complementary Array Pairs Based on Generalized Boolean Functions

Conclusions

Our Results:

A new construction of q-ary GCAPs of size 2n × 2m has been proposed based on 2-D generalized Boolean functions. A construction of Golay complementary array mates is provided. The proposed constructions do not require any existing arrays or special 1-D sequences as kernels.

Future Topics:

Application of the constructed 2-D GCAPs to 2-D synchronization. Extension to the 2-D complete complementary code (CCC).

Cheng-Yu Pai (ES, NCKU) ISIT 2020 June 21-26, 2020 20 / 22

Constructions of Two-Dimensional Golay Complementary Array Pairs Based on Generalized Boolean Functions

Thank you !!

Cheng-Yu Pai1 and Chao-Yu Chen2

Department of Engineering Science National Cheng Kung University Tainan 701, Taiwan, R.O.C. Email:

1n98081505@mail.ncku.edu.tw

Cheng-Yu Pai (ES, NCKU) ISIT 2020 June 21-26, 2020 21 / 22

Constructions of Two-Dimensional Golay Complementary Array Pairs Based on Generalized Boolean Functions

References I

[1] J. A. Davis and J. Jedwab, “Peak-to-mean power control in OFDM, Golay complementary sequences, and Reed-Muller codes,” IEEE Trans. Inf. Theory, vol. 45, no. 7, pp. 2397–2417,

• Nov. 1999.

[2] M. Dymond, “Barker arrays: existence, generalization and alternatives,” Ph.D. thesis, University of London, 1992. [3] S. Matsufuji, R. Shigemitsu, Y. Tanada, and N. Kuroyanagi, “Construction of complementary arrays,” in Proc. Joint IST Workshop on Mobile Future and Symp. on Trends in Commun., Bratislava, Slovakia, Oct. 2004, pp. 78–81. [4] J. Jedwab and M. G. Parker, “Golay complementary array pairs,” Designs, Codes and Cryptography, vol. 44, no. 1-3, p. 209–216, Sep. 2007. [5] F. Fiedler, J. Jedwab, and M. G. Parker, “A multi-dimensional approach for the construction and enumeration of Golay complementary sequences,” J. Combinatorial Theory (Series A),

• vol. 115, no. 5, pp. 753–776, Jul. 2008.

[6] F. Zeng and Z. Zhang, “Two dimensional periodic complementary array sets,” in Proc. IEEE

• Int. Conf. on Wireless Commun., Netw. and Mobile Comput., Chengdu, China, Sep. 2010,
• pp. 1–4.

Cheng-Yu Pai (ES, NCKU) ISIT 2020 June 21-26, 2020 22 / 22