Constructions of Binary Cross Z-Complementary Pairs With Large CZC Ratio

09/10/2021 ∙ by Hui Zhang, et al. ∙ Southwest Jiaotong University 0

Cross Z-complementary pairs (CZCPs) are a special kind of Z-complementary pairs (ZCPs) having zero autocorrelation sums around the in-phase position and end-shift position, also having zero cross-correlation sums around the end-shift position. It can be utilized as a key component in designing optimal training sequences for broadband spatial modulation (SM) systems over frequency selective channels. In this paper, we focus on designing new CZCPs with large cross Z-complementary ratio (CZC_ratio) by exploring two promising approaches. The first one of CZCPs via properly cascading sequences from a Golay complementary pair (GCP). The proposed construction leads to (28L,13L)-CZCPs, (28L,13L+L/2)-CZCPs and (30L,13L-1)-CZCPs, where L is the length of a binary GCP. Besides, we emphasize that, our proposed CZCPs have the largest CZC_ratio=27/28, compared with known CZCPs but no-perfect CZCPs in the literature. Specially, we proposed optimal binary CZCPs with (28,13)-CZCP and (56,27)-CZCP. The second one of CZCPs based on Boolean functions (BFs), and the construction of CZCPs have the largest CZC_ratio=13/14, compared with known CZCPs but no-perfect CZCPs in the literature.

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1 Introduction

In 1951, Golay 1951-Golay

first introduced a sequence pair whose aperiodic autocorrelation sums (AACSs) are zero at each out-of-phase time shift. Such a pair of sequences is called a Golay complementary sequence pair (GCP), and has been found many applications which include channel estimation

2001-Spasojevic , radar2018-Kumari , and peak power control in orthogonal frequency division multiplexing (OFDM) 1999-GDJ ,2014-Wang , etc. However, one drawback of GCPs is its limited availability. For example, binary GCPs are known to exist for lengths of the form where , and are nonnegative integers 2003-7-Borwein . Due to its scarity, Fan et al. relaxed the autocorrelation constraint of GCPs, and introduced another type of sequences called Z-complementary pair (ZCP) 2007-8-Fan . ZCP consists of a pair of sequences whose aperiodic autocorrelation sums are zero within a zone around the zero shift position, called zero correlation zone (ZCZ). Compared with GCPs, ZCPs can have more flexible lengths. Actually, ZCPs are shown to available for all lengths 2011-1-Li .

In Liu-2020 , Liu et al. investigated the training sequence design for the spatial modulation (SM) system and hence the cross Z-complementary pair (CZCP) was proposed. In contrast to ZCPs, CZCPs deals with cross-correlation sums in addition to calculating autocorrelation sums. They pointed out that CZCPs can be utilized as a key component in designing optimal training sequences for broadband spatial modulation (SM) systems over frequency selective channels. SM is a special class of MIMO techniques which trades multiplexing gain with complexity and performance. Unlike conventional MIMO, an SM system is equipped with multiple transmit antenna (TA) elements but only a single radio-frequency (RF) chain (see 2008-Mesleh ,Yang-2005 and Yang-2016 for more details). CZCPs can be employed as training sequences in an SM system to mitigate the inter-symbol interference (ISI) and the inter-channel interference (ICI) resulting from multi-path propagation Liu-2020 . In Liu-2020 , Liu et al. proposed binary using special types of binary GCPs, where is a Golay number. Besides, the authors also utilizes generalized Boolean functions (GBFs) to construct . It is easy to see that the and achieve the maximum ZCZ width, i.e., by Liu et al. discussed only perfect CZCPs. However, the length of perfect CZCPs is very limited. To check if the complementary sequences of other lengths also display the cross Z-complementary property achieving the maximum ZACZ and ZCCZ width, Fan et al.Fan-2020 proposed binary CZCPs of lengths , and having the ZACZ and ZCCZ width of , and , respectively. In Adhikary-2020 , Adhikary et al. proposed binary and -phase CZCPs of lengths , , and having the ZACZ and ZCCZ width of , and using insertion function, respectively. Besides, The authors proposed the construction of binary and using Barker sequences. These CZCPs are also extended to and , where is the length of a GCP. In Huang-2020 , Huang et al. proposed binary using Boolean functions (BFs), where , . When , , the construction of with . In yang-2021 , Yang et al. proposed binary CZCPs of lengths of the form , where even-length binary ZCPs (EB-ZCPs) of length exists. They also utilised binary CZCPs to construct the quaternary CZCPs.

In this paper, we proposed two approaches to construct CZCPs. The first one of CZCPs via properly cascading sequences from a Golay complementary pair (GCP). The proposed construction leads to , and , where is the length of a binary GCP. Specially, we propose two optimal binary CZCPs with and . The second one of CZCPs based on BFs, and the construction of CZCPs have the largest CZC ratio, compared with known CZCPs but not perfect CZCPs in the literature.

This paper is organized as follows. In Section 2, we give some basic notation and definitions which will be needed in this paper. Section 3.1 and 3.2 are the core of the paper in which we present our two approaches for designing CZCPs. The first one (Section 3.1) uses the concatenation of Golay sequences of length while the second one (Section 3.2) employees Boolean functions. We also make a comparison of our results of CZCPs with known CZCPs in the literature. Section 4 concludes this paper.

2 Preliminaries

In this section, we give some basic notation and definitions which will be used throughout the paper:

2.1 Notations

  • denotes a binary sequence of length .

  • denotes the conjugation of .

  • denotes the reverse of .

  • denotes the negation of .

  • denotes the concatenation of and .

  • denotes the all-zero vector of length

    .

  • denotes the binary representations of .

  • is the binary complement of .

  • and are denoted by and , respectively.

2.2 Z-Complementary Pair

Definition 1

The aperiodic cross-correlation function (ACCF) of two complex-valued sequences and of length at shift is defined as

Note that . If , is called the aperiodic autocorrelation function (AACF) of , and refer to .

Definition 2

A pair of sequences and of length is called a -complementary sequence pair (ZCP) with ZCZ width , denoted by -ZCP, if they satisfy the following:

,   for ,

where . If , then is called a Golay sequence pair (GCP).

2.3 Cross Z-Complementary Pair

In contrast to ZCPs, CZCPs deals with cross-correlation sums in addition to calculating autocorrelation sums.

Definition 3

(Liu-2020 ) Let be a pair of sequences of identical length . For positive integers and with , define two intervals as and . Then is called an if it possesses symmetric zero (out-of-phase) AACF sums for time-shifts over and zero ACCF sums for time-shifts over . In short, it needs to satisfy the following two conditions:

(1)

From two conditions given in (1), a CZCP has two zero autocorrelation zones (ZACZs) and one zero cross-correlation zone (ZCCZ).

Definition 4

(Liu-2020 ) Let be a binary . is called perfect if ( even). In this case, a perfect reduces to a sequence pair, called strengthened GCP.

Definition 5

(Adhikary-2020 ) For an , the cross Z-complementary ratio is defined as

where denotes the maximum value of for a given length . Note that since . When which implies that is achieved, such is called an optimal .

Note that for strengthened GCPs, , otherwise it is since of (1).

2.4 Boolean Function

A Boolean function is a function consisting of variables , where for . For a Boolean function with variables, we define the associated sequence and let where is the binary representation vector of the integer .

Obviously, the length of a sequence constructed by the Boolean function with variable is . But, sometimes, we need sequence lengths that are not limited to be powers of . Therefore, we define the truncated sequence , where is the result by removing the last elements from , that is, . Note that , where .

3 A Construction of CZCPs From GCPs

The ultimate objective of this section is to design CZCPs. We shall follow two directions as presented in the next subsections 3.1 and 3.2.

3.1 Our first approach for constructing CZCPs

In this section, we will propose two constructions of binary CZCPs from binary GCPs. Before then, we introduce some lemmas which will be used in our construction.

Lemma 1

(Huang-2020 ) Let and be two sequences of length over . If and for all , then the pair satisfies the following

Lemma 2

(1974-Turyn ) Let and be the first and the second binary GCPs of lengths and , respectively. Then is a GCP of length , where

Lemma 3

(Adhikary-2020 ) Let be a GCP of length and be an . Also, let

Then is an .

Now we are ready to present the main result.

Theorem 1

Let be a binary of length . Define the sequence pair with

where

Then, the sequence pair is a binary .

Proof

Case 1: Firstly, we compute the value of as follows:

  • For . When , we can obtain

    When , we can obtain

    Then, we have , .

  • For . When , we can obtain

    When , we have

    Then, we have , .

  • Similarly, for and , .

  • For ,

    Then .

    Based on the above discussion, we can obtain

Case 2: We compute the value of .

Obviously, we can see that and are sequences of length , , , . Then, by Lemma 1, we have

Remark 1

When in Theorem 1, the sequence pair is a binary . Moreover, .

Let and . By Theorem 1, we have

Then, we list the sums of their AACF and ACCF, respectively, as follows:

Hence, is a . Moreover, .

Next, we present one example to illustrate our above theorem.

Example 1

Let and . By Theorem 1, we have

where

Then, we list the sums of their AACF and ACCF, respectively, as follows:

Hence, is a . Moreover, .

Corollary 1

If is a Golay pair and in Theorem 1. Moreover, , , . Then is , and . Specially, when , .

Proof

It is easy to see that and are sequences of length , , , . Then, by Lemma 1, we have

Example 2

Let and . By Theorem 1, we can obtain

where

Then, we list the sums of their AACF and ACCF, respectively, as follows:

Hence, is a . Moreover, .

Theorem 2

Let be a GCP of length and be an . Also, let be an , where not is a Golay number. Moreover, , and . Define the sequence pair with

Then is an and .

Remark 2

If is a GCP of length and be an (strengthened GCP) in Theorem 2, we can obtain

Theorem 3

Let be a binary of length . Define the sequence pair with

where

Then, the sequence pair is a binary .

Proof

The proof process is similar to that of Theorem 1, so we omit it here.

Remark 3

When in Theorem 3, the sequence pair is a binary .

Example 3

Let and . By Theorem 3, we have

where

Then, we list the sums of their AACF and ACCF, respectively, as follows:

Hence, is a . Moreover, .

A summary of known CZCPs till date is given in Table 1. Compared with known CZCPs (but the length not Golay number) in Adhikary-2020 , Fan-2020 and yang-2021 , we proposed the construction of CZCPs have the largest .

Table 1  Summary of CZCPs

ref. Constraint Method
Adhikary-2020 Insertion function
Adhikary-2020 Insertion function
Adhikary-2020 Insertion function
Adhikary-2020 Barker sequence
Adhikary-2020 Kronecker product
Fan-2020 GCP and Kronecker product
Fan-2020 GCP and Kronecker product
Fan-2020 GCP and Kronecker product
Fan-2020 GCP and Kronecker product
yang-2021 ZCP and concatenation
yang-2021 ZCP and concatenation
yang-2021 ZCP and concatenation
yang-2021 ZCP and concatenation
yang-2021 ZCP and concatenation
Remark 1 - GCP and concatenation
Corollary 1 , GCP and concatenation
Theorem 1 GCP and concatenation
Corollary 1 , GCP and concatenation
Theorem 3 GCP and concatenation

3.2 Constructions of CZCPs from BFs

In this section, two constructions of CZCPs based on BFs are given and the CZC ratios of the CZCPs are also discussed. Compared with known CZCPs in the literature, we proposed CZCPs have the largest CZC. Before then, we introduce the following lemma which is useful for our construction.

Lemma 4

(1999-GDJ ) Let be a permutation of the set . Define

where and . Let be the binary representations of the integer , then the two pairs of associated binary sequences

all form binary GCPs of length .

Theorem 4

Let be integer and be a permutation of . Define two binary BFs with variables as:

(3)

where