The Necessary Condition of Families of Odd-periodic Perfect Complementary Sequence Pairs

Jin Hui-long1,2 Liang Guo-dong1 Liu Zhi-Hua3 Xu Cheng-qian2

(1. The Faculty of Electronic， Hebei Normal University， Shijiazhuang 050031, China)

(2. The College of Information Science and Engineering， YanShan University，Qinhuangdao 066004, China)

(3.. The College of Information of Technical， Hebei Normal University， Shijiazhuang 050031, China)

JHL981981@yahoo.com.cn

Abstract The theory of families of odd-periodic perfect

complementary sequence pairs(OPCSPF) is given. OPCSPF can be derived from the odd-perfect almost binary sequence and sequence pairs. Some characteristics of OPCSPF are discussed. The Fourier spectrum characteristics of OPCSPF are studied, and the necessary conditions for existence of OPCSPF are given. The method for con- structing an OPCSPF ii given and proved. Key words: coding theory; odd-periodic; the family of sequence pairs 0. Introduction Signals with good correlation properties have been widely used in modern communications, radar, sonar, navigation, location, guidance, space moni- toring and control, signal processing, source coding and other fields[1]. Periodic binary complementary sequence sets is signals of this kind, which have such properties: the sum of aperiodic autocorre- lation of sequences is zero everywhere except 0 shift[2], in practice odd-periodic correlation function is of same importance with periodic correlation function, on certain conditions, periodic binary complementary sequences sets and odd-periodic binary sequences sets can transform into each other[3-4]. Ever since 1999, when the theory of sequences pairs was raised[5], many scholars have researched on it , and it has developed from Supported by National Natural Sciences Foundation (60872061) and Natural Sciences Foundation of Hebei Province (F2008000855)，（2007472）

sequences pairs to almost sequences pairs, families of sequences pairs and other fields, and it has achieved certain fruits[6-7] in characteristics of the spectrum, the necessary condition for existence, methods for construction and so on. In this paper, we synthesize the two research methods above, advance a new kind of binary sequences pairs sets-families of odd-periodic perfect complementary sequences pairs(OPCSPF), study three properties of OPCSPF, use Fourier Transform to analyze the spectrums of OPCSPF, and achieve the necessary condition for the existence of OPCSPF. These conditions can greatly narrow down the scope of OPCSPF when we search it, and improve the efficiency to search OPCSPF.

1. Basic concept In order to facilitate the discussion of the issue,

let sequences a and 'a be 2 sequences of length

N , be labeled as ))1(,),1(),0(( −= Naaaa ='a

),0(( 'a ))1(,),1( '' −Naa .

Definition 1[3] : For sequence )(na , each

element of which is 1+ or 1− . We use the symbol of periodic cyclic alternate reverse sequence to write out the odd-periodic sequence )(ˆ na of odd sequence

)(na : [ ]ˆ( ) ( 1) ([ ] )n NNa n a n= − , [ ]x is the biggest integer

that is smaller than x , [ ] ( )modNn n N= . Definition 2: Binary sequences pairs ),(( na

))(' na [5] of length N is called odd-periodic perfect

binary sequences pairs, if its odd-periodic autocorre-

2009 International Conference on Computational Intelligence and Security

978-0-7695-3931-7/09 $26.00 © 2009 IEEE

DOI 10.1109/CIS.2009.227

303

lation function (OPCF) '( , )( )

a aψ τ satisfies

'

1'

( , )0

0ˆ( ) ( ) ( )

0 0

N

a aj

Ea j a j

τψ τ τ

τ

−

=

=⎧= + = ⎨ ≠⎩∑

Definition 3: Let }10))(),({( ' −≤≤ Pinana ii be a

family of binary sequences pairs of period N, it is called a family of odd-periodic perfect com- plementtary sequences pairs, labeled as

),( 'ii

NP aaOPCSPF , if and only if it satisfies the

expression⎩⎨⎧

≠=

=∑−

=000

)(1

0),( '

ττ

τψEP

iaa ii

.

Definition 4[4]: The family of odd-periodic perfect complementary sequences pairs

),( 'ii

NP bbOPCSPF is the “pairs companion” of

another family of odd-periodic perfect complement-

tary sequences pairs ),( 'ii

NP aaOPCSPF , if and only

if:

0)(1

0),( ' =∑

−

=

τψP

iba ii

，对 τ∀

0)(1

0),( ' =∑

−

=

τψP

iba ii

，对 τ∀ .

2. Transformation properties

Let )},{( 'ii aa be a family of odd-periodic

perfect complementary sequences pairs of period N,

and )},{( 'ii bb be it’s the family of its transfor-

mation sequences pairs, 10 −≤≤ Pi , here are 3 properties.

Property 1 (Negative element transform)

)()( nanb ii −= , )()( '' nanb ii −= where 10 −≤≤ Nn ,

10 −≤≤ Pi , then the family of sequences pairs

)},{( 'ii bb is a family of odd-periodic perfect

complementary sequences pairs. Proof.

∑∑∑−

=

−

=

−

=

+=1

0

1

0

'1

0),( )(ˆ)()('

P

i

N

jii

P

ibb jbjb

iiττψ

)(ˆ)1)(()1( '1

0

1

0

τ+−−=∑∑−

=

−

=

jaja i

P

ii

N

j

)(ˆ)( '1

0

1

0

τ+=∑∑−

=

−

=

jaja i

P

i

N

ji

)(1

0),( ' τψ∑

−

=

=P

iaa ii

For )},{( 'ii aa is a family of odd-periodic

perfect complementary sequences pairs, )},{( 'ii bb is

a family of odd-periodic perfect complementary sequences pairs.

Property 2 (Linear Phase Transformation) If

)()1()( nanb in

i −= ， )()1()( '' nanb in

i −= , where ≤≤ n0

1−N , 10 −≤≤ Pi , then sequences pairs )},{( 'ii bb

is a family of odd-periodic perfect complementary sequences pairs.

Proof.

∑∑∑−

=

−

=

−

=

+=1

0

'1

0

1

0),( )(ˆ)('

P

ii

N

ji

P

ibb jbjb

iiτψ

)]([)1()1)(()1( '])([1

0

)(1

0Ni

NjP

i

ji

N

j

j jaja τττ +−−−= +−

=

+−

=∑∑

304

)]([)1)(()1( '1

0

])([1

0Ni

P

i

Nji

N

j

jaja τττ +−−=∑∑−

=

+−

=

)(ˆ)()1( '1

0

1

0

ττ +−=∑∑−

=

−

=

jaja i

P

ii

N

j

∑∑−

=

−

=

+−=1

0

'1

0

)(ˆ)()1(P

ii

N

ji jaja ττ

)()1(1

0),( ' τψτ∑

−

=

−=P

iaa ii

10 −≤≤ Nτ

For )},{( 'ii aa is a family of odd-periodic

perfect complementary sequences pairs and from the definition of families of odd-periodic perfect complementary sequences pairs, we know that

)},{( 'ii bb is a family of odd-periodic perfect

complementary sequences pairs. Property 3 (Reverse transform) if =)(nbi

)1( nNai −− ， )1()( '' nNanb ii −−= ,where ≤≤ n0

,1−N 10 −≤≤ Pi , then sequences pairs )},{( 'ii bb

is a family of odd-periodic perfect complementary sequences pairs.

Proof.

∑ ∑∑−

=

−

=

−

=

+=1

0

1

0

1

0

'),( )(ˆ)()('

P

i

P

i

N

jiibb jbjb

iiττψ

∑∑−

=

−

=

+−−−−=1

0

1

0

' )1(ˆ)1(P

i

N

jii jNajNa τ

)0(ˆ)0()1(ˆ)1( '' ττ +++++ iiii aaaa

+++++=∑−

=

)1(ˆ)1()0(ˆ)0(( '1

0

' ττ ii

P

iii aaaa

))1(ˆ)1()2(ˆ)2( '' ττ +−−++−− NaNaNaNa iiii

∑∑−

=

−

=

+=1

0

1

0

' )(ˆ)(P

i

N

jii jaja τ

∑−

=

=1

0),( )('

P

iaa ii

τψ

For )},{( 'ii aa is a family of odd-periodic

perfect complementary sequences pairs, we know

that )},{( 'ii bb is family of odd-periodic perfect

complementary sequences pairs.

3. The necessary condition Theorem 1 PN , the product of sequence length

N of the family of odd-periodic perfect complementary sequences pairs )},{( '

ii aa and the number of sequences pairs P, is even.

Proof From definition 3, when 0≠τ , we have

0)(ˆ)()(1

0

1

0

'1

0),( ' =+=∑∑∑

−

=

−

=

−

=

P

i

N

jii

P

iaa jaja

iiττψ

There are PN terms in the product-sum form, and each term is 1± , so PN must be even if product-sum form is 0.

Theorem 2 Let '{( ( ), ( )),0 1,i ia n a n i P≤ ≤ − 0 1n N≤ ≤ − be a family of odd-periodic perfect complementary sequences pairs of period N.

iaN is the number of )(nai which are "1"+ , )(ˆ ' τ+nai is the sequence that )(' nai odd-periodic shift τ ,

'ˆiaN is the number of )(ˆ ' τ+nai which are "1"+ ,

iθ is the number when )(nai and )(ˆ ' τ+nai are both "1"+ , Zi ∈θ and 0≥iθ , then

∑ ∑−

=

−

=

+=+1

0

1

0ˆ 22

)()( '

P

i

P

iiaa

PNNNii

θ

Proof From definition 3, when 0≠τ ,

305

0))1(ˆ)1()0(ˆ)0((1

0

'' =+−−+++∑−

=

P

iiiii NaNaaa ττ

For iθ is the number when )(nai and )(ˆ ' τ+nai

are both "1"+ , we rearrange the terms of

sequences )(nai and )(ˆ ' τ+nai , keeping the corre-

spondence between )(nai and )(ˆ ' τ+nai ,

−−−

++−−+++−−−−++

−

++

0ˆ0

'0

000

'0

0

)(ˆ

)(

θθτ

θθ

a

a

Nna

Nna

−−−

++−−+++−−−−++

−

++

1ˆ1

'1

111

'1

1

)(ˆ

)(

θθτ

θθ

a

a

Nna

Nna

−−−

++−−+++−−−−++

−

++

−−−

−−−

−

−

1ˆ1

'1

111

'1

1

)(ˆ

)(

PaPP

PaPP

P

P

Nna

Nna

θθτ

θθ

Clearly, for 0)()(1

0

1

0

' =+∑∑−

=

−

=

P

i

N

jii jaja τ , when the

value of i range from 0 to 1−P , the number

when the values of )(nai and )(ˆ ' τ+nai are not

same is 2)(PN ,

∑ ∑−

=

−

=

=−+1

0

1

0ˆ 2)(2)( '

P

i

P

iiaa PNNN

iiθ ，

∑ ∑−

=

−

=

+=+1

0

1

0ˆ 22)()( '

P

i

P

iiaa PNNN

iiθ .

Theorem 3 Let }10),,{( ' −≤≤ Piaa ii be a

NPOPCSPF , )(kF

ia and 'ˆ ( )ia

F k are the Fourier

Transformations of ia and 'ˆia , then

'

1 1

ˆ0 0

1( ) ( ) ( )i i

P N

a ai j

F j F jN− −

= =

−∑∑ =1

0

2P

ii

PN d−

=

− ∑

id — the Hamming distance between the former

sequence and the odd-periodic sequence of the latter sequence in every sequences pair when

sequence 'ia doesn’t shift, where 1,,1,0 −= Pi .

Notice id bellow is same as above.

Proof

Let )(nvi )(ˆ ' τ+= nana ii ）（ , from the properties

of Fourier Transformation, we get

'

1( )

ˆ0

1( ) ( ) ( ) ( )i i i

Nk j

v a aj

F k F j F k j WNτ

−− −

=

= −∑

So

'

1

ˆ0

1(0) ( ) ( ) ( )i i i

Nj

v a aj

F F j F j WNτ

−

=

= −∑

For

)()(ˆ)()()0( ),(

1

0

'1

0' τψτiii aa

N

jii

N

jiv jajajvF =+== ∑∑

−

=

−

=

From the formula above , }10),,{( ' −≤≤ Piaa ii is a

NPOPCSPF , we can get

'

1 1

ˆ0 0

1( ) ( ) ( )i i

P Nj

a ai j

F j F j WNτ

− −

= =

− =∑∑ ∑−

=

1

0),( )('

P

iaa ii

τψ

⎪⎩

⎪⎨

⎧

≠

=−= ∑−

=00

021

0τ

τP

iidPN

So '

1 1

ˆ0 0

1( ) ( ) ( )i i

P N

a ai j

F j F jN− −

= =

−∑∑1

0

2P

ii

PN d−

=

= − ∑.

306

4. The construction method Theorem 4 For a set of families of odd-

periodic perfect complementary sequences pairs '( , )N

P i iOPCSPF a a and its pairs companion '( , )N

P i iOPCSPF b b , the family of interlaced

sequences pairs: ),,(),(),{( '''iiiiii bbaacc ⊗= ≤≤ i0

}1−P is a new family of odd-periodic perfect

complementary sequences pairs ),( '2ii

NP ccOPCSPF .

Proof For

)()()2( ),(),(),( ''' τψτψτψiiiiii bbaacc += , 0 1Nτ≤ ≤ − ,

We can get

⎩⎨⎧

≠=

=+=∑ ∑∑−

=

−

=

−

=000

)()()2(1

0

1

0),(),(

1

0),( '''

ττ

τψτψτψEP

i

P

ibbaa

P

icc iiiiii

, For

)()1()12( ),(),(),( ''' τψτψτψiiiiii abbacc ++=+ ,

so

∑∑∑−

=

−

=

−

=

++=+1

0),(

1

0),(

1

0),( )()1()12( '''

P

iab

P

iba

P

icc iiiiii

τψτψτψ

0= . 5. The Conclusion

This paper is based on odd-periodic perfect sequences, brings the concept of “families of pairs” into it, and constructs families of odd- periodic perfect complementary sequences pairs (OPCSPF), these are new binary sequences pairs sets. OPCSPF can get much more and complicated sequence, based on several old methods for studying perfect signals. This paper is based on the study of transformation properties of odd-periodic perfect sequences pairs, necessary condition and spectrum properties, studies the properties and spectrum properties of OPCSPF, offers the

necessary condition of OPCSPF, and it can greatly reduce the workload when we search by computer on this condition. The methods for constructing an OPCSPF are given and proved. The families of periodic perfect complementary sequence pairs can be constructed correspondingly by using the equivalent relationships. References [1] Yang Yixian. Theory and design of perfect Signal [M], Beijing: Posts & Telecom Press, 1996: 15-45． [2] L Bomer and M Antweiler．Periodic complement- tary binary sequences[J].IEEE Trans on Inform Theory, 1990,36(6):1487-1497. [3] H D Luke. Binary odd-periodic complementary sequences [J]. IEEE Trans On Inform Theory, 1997, 43(1):365-367． [4] Wen Hong, Hu Fei, Fu Chusheng, Jin Fan. Odd- periodic complementary sequence sets. Chinese Journal of Radio Science. 2006,26(1):70-73. [5] Zhao Xiaoqun, He Wencai, Wang Zhongwen and so on. Study on Perfect Binary Arrays Pairs[J]. Acta Elec- tronica Sinica, 1999,27(1):34-35. [6] Xu Chengqian, Jin Huilong. Families of Almost Perfect Periodic Complementary Binary Sequences Pairs[J]. Systems Engineering and Electronics. 2003, 25(9): 1086-1089. [7] Jin Huilong, Xu Chengqian. The necessary con- ditions of families of Bent complementary functions pairs. Journal of Electronics & Information Technology [J], 2008,30(6):1397-1399

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