WO2004057786A1

WO2004057786A1 - A coding method to create complementary codes with zero correlation window

Info

Publication number: WO2004057786A1
Application number: PCT/CN2002/000904
Authority: WO
Inventors: Shaojun Xu; Daoben Li
Original assignee: Linkair Communications,Inc.
Priority date: 2002-12-20
Filing date: 2002-12-20
Publication date: 2004-07-08
Also published as: AU2002351654A1; CN1695334A

Abstract

The present invention is to provide a coding method to create complementary codes with zero correlation windows. The said coding method includes: Selecting basic complementary codes; Selecting any orthogonal matrix; Selecting any index array; And then expand the said complementary codes to code set using the said orthogonal matrix and the said index array. The said coding method can create a class of complementary codes with the 'Zero Correlation Window' in their auto-correlation functions and cross-correlation functions. The said complementary ZCW code sets can be used in Quasi-Synchronized CDMA system to remove MAI and ISI. The said coding method leads to many ZCW code sets, which have a much wider range of available lengths and window widths of zero correlations. The application of the said codes to quasi-synchronized CDMA system swill remove co-channel interference and influence of multi-path as much as possible.

Description

A Coding Method to Create Complementary Codes with Zero Correlation Window

Field of the invention

The presented invention is related to CDMA system. More specifically, the presented invention is a coding method to create complementary codes with zero correlation window. Background of the invention

The growing popularity of personal communication services coupled with the scarcity of radio bandwidth resources has resulted in the ever-increasing demand for higher spectral efficiency in wireless communications. Traditional multiple access control (MAC) schemes such as FDMA, TDMA can't satisfy such demand any more because of its low spectral efficiency. More and more people think that CDMA will become the main MAC scheme in the next generation wireless communication because of its high spectral efficiency. The difference between CDMA and other MAC schemes is: CDMA capacity has soft-capacity, which is interference limited, i.e. any technique to reduce interference can directly increase the CDMA system capacity. The system capacity of other MAC schemes is a hard-capacity and is decided before design.

Since CDMA system capacity is limited by the system interference level, it is crucial to reduce the system interference in CDMA system. There are many techniques to reduce interference in CDMA system such as Multiple User Detection (MUD), Adaptive Antenna Array, Power Control and so on. In fact the interference between two users stems from imperfect correlation between two spread spectrum codes specific to two users. So it is necessary to find a code set with good Auto- Correlation Function (ACF) and Cross-Correlation Function (CCF) for CDMA system.

To avoid the interference in CDMA system, we hope to find a code set with ideal ACF and ideal CCF. Unfortunately ACF and CCF of a code set are bounded to the Welch bound, which states that ACF and CCF cannot be decreased simultaneously. So it is impossible to find a code set with ideal ACF and ideal CCF. But in many applications, it is not necessary to construct such code set with ideal ACF and ideal CCF in all time shifts. It is enough to ensure the ideal ACF and ideal CCF within maximum time spread. For example, if the maximum time spread is Δ , it is enough that ACF and CCF are ideal within [- Δ , Δ]. In 1997, Prof. Li Daoben found an approach to construct a code set with zero correlation window (ZCW). And his work has been granted a patent (application No. PCT/CNOO/00028). Given the code length, the family size of LS code set is much greater than before, which has great significance in CDMA applications.

LS code provides a binary generation tree to create the complementary codes with one-sided zero correlation windows N from two uncorrelated complementary codes with code length N . The number of access codes of new expanded code set at level M is 2^M+1 , and the code length at level M is 2^M N . Such code set is denoted by 2^M+l,2^M N,N), where 2^M+λ is family size of new expanded code set,

2^M N is the code length and N is one-sided width of zero correlation window.

We denote a complementary code set with zero correlation window by

(κ,L,W ), where K is family size of code set, L is the code length of C component or S component of complementary codes and W is one-sided width of zero correlation window. According to the Welch bound, the following inequality holds:

K ≤ (2L + 2W-2)lW . For LS code the relation between the code length, zero correlation width and family is: K = 2^M* L = 2^MN, W = N , so K = 2L/W . LS code has been close to the theoretical bound when L » W . But family size of LS code is limited to 2 ⁺¹ ,M = 0,1,2, After that, many scholars began to research the spreading codes with zero correlation windows. In 2000, Xinmin Deng and Pingzhi Fan proposed a class of spreading sequence sets with zero correlation windows derived from uncorrelated complementary sets. But the Deng's codes are far from the theoretical bound. Subsequently J.S. Cha provided a construction method to create a class of ternary spreading sequences with zero correlation windows. The spreading codes proposed by J.S. Cha can't approach the bound. In 2001 , Shinya Matsufuji proposed an orthogonal matrix construction method. A class of complementary sequences with zero correlation windows can be obtained from an orthogonal matrix Η._MxM and a pair of uncorrelated complementary codes with code length N . But M and N must be relatively prime to each other in the expansion process, and this greatly limits the range of available code lengths and ZCW width.

Summary of the invention The objective of the present invention is to provide a coding method to create complementary codes with zero correlation windows. The said coding method can create a class of complementary codes with the "Zero Correlation Window" in their auto-correlation functions and cross-correlation functions. The said complementary ZCW code sets can be used in Quasi-Synchronized CDMA system to remove MAI and ISI. The said coding method leads to many ZCW code sets, which have a much wider range of available lengths and window widths of zero correlations. The application of the said codes to quasi-synchronized CDMA system swill remove co- channel interference and influence of multi-path as much as possible. LS codes can be also obtained as the special case of the said method.

A coding method to create complementary codes with zero correlation windows, includes:

Selecting two basic uncorrelated complementary codes;

Selecting any orthogonal matrix; Selecting any index array;

And then expanding the said complementary codes to code set using the said orthogonal matrix and the said index array.

Wherein said selecting basic complementary codes includes: Selecting two basic uncorrelated complementary codes with code length N. Wherein said selecting any orthogonal matrix includes: Selecting an MxM orthogonal matrix H _x .

Wherein said selecting any index array includes: Selecting an index array

Wherein said selecting two basic uncorrelated complementary codes with code length N includes: Selecting a basic complementary code (c^,S^j with code length N , where the aperiodic auto-correlation functions of C component and S component sum to zero except at the origin; selecting another complementary code

(cC ,sC ) which is uncorrelated with the code

i.e. their aperiodic cross- correlation functions of C component and S component between two codes sum to zero at all corresponding time shifts; it can be proved that code

and

code :(ι) _<-«^* are uncorrelated with each other; where ' ^~ ' denotes reversing operation;

'*' denotes the complex conjugate operation.

Wherein said selecting an MxM orthogonal matrix H_MxM includes: Selecting

any orthogonal matrix,

^{wnose rows nave tnΘ same} energy.

Wherein said selecting an index array I_2x includes:

where columns of I 2x ' ≤ n ≤ M , are permutations of

Wherein said expansion of the said complementary codes to code set using the said orthogonal matrix H_MxM and the said index array l_2xM includes: Construct two codes:

Two code sets can be obtained according to the following rule:

A = x^(l) « HM_λ xM

B = x⁽²⁾ » H AtxAf

Rows of A and B are expanded codes with length MN , and there are 2M such codes denoted by y^,y^,---,y^ respectively; It can be verified that the

expanded code set form a (2M,MN,N) ZCW code set.

Wherein said selecting basic generalized complementary codes includes: Let P codes be S₁,S₂, ---,S_{P >} all with length N , their auto-correlations sum to zero at all shifts except the origin, then the code (S₁,S₂,---,S^) is called generalized complementary code with P complementary components. The method of the present invention comprising the step of: Let P codes be S₁,S₂,---,S_P, all with length N, their auto-correlations sum to zero at all shifts except the origin, then the code (s₁,S₂,---,S_P) is called generalized complementary code with P complementary components;

There exist at most P uncorrelated generalized complementary codes with

P complementary components, which is denoted by

= l,2,---,P Given p uncorrelated generalized complementary codes:

{sj'ls«,.--,sW},t = l,2,.--,P with code length N , an MxM orthogonal matrix H _x and an index array :

where columns of ϊ_PxM, i® ^■■■

≤n≤M, are permutations of

[l 2 ••• p]^τ , a generalized complementary code set of length MN can be constructed as follows: Construct p codes:

_X X(D ^a <P > _s ^B4P° ' ... > _s ^Λ >0PP ^'

code sets can be obtained according to the following rule:

A(I) __X(I) ._H _A(2) __χ(2)_#H

Λ -x •π _x Rows of A^i≤t^'≤ are expanded codes with length NxM , and there are

PM such codes. The expanded code set forms a (PM,NχM,N) generalized complementary ZCW code set with P complementary components.

Wherein said complementary codes with zero correlation windows includes: binary codes, ternary codes, poly-phase codes and any other ZCW codes in complex field.

Wherein said index array includes:

For an index vector I 2x the four hamming

distances are defined:

where It is obvious that the following equations hold:

D_n+D_n=M, D_2λ+D₂₂=M D_n+D_2l=M, D_l2+D₂₂=M

LS code is the special case that

_M Ai ⁼ 2 ⁼ Ai ⁼ A2 ⁼ ι > M = 2ⁿ, n = 1,2,3,

Another special case can be obtained to make D_n =0 or D_u =0, then A, B are two uncorrelated code sets, i.e. the cross-correlation functions between codes from A and codes from B take zeros at all corresponding shifts;

R_ff(τ) = 0, l≤i≤M, M + l≤j≤2M, -N^+l≤τ≤N^-V,

The equivalent transformation can be applied to generate complementary ZCW code sets.

We begin with a pair of uncorrelated complementary codes, and then expand it to code set with longer length using an orthogonal matrix and an index array. The complementary codes proposed by present invention have the following properties:

The proposed codes include binary codes, ternary codes, poly-phase codes and any other ZCW codes in complex field.

The new construction method can provide complementary ZCW codes with arbitrary positive ZCW width. Complementary codes with different ZCW width are determined by the actual channel condition.

The new construction method can provide complementary ZCW codes with arbitrary positive family size.

The new construction method can be extended to create generalized complementary codes with zero correlation windows.

The three necessary elements to create a complementary code set is a pair of uncorrelated complementary codes, an orthogonal matrix and an index array. The zero correlation window width of expanded code set is equal to length of each component of the original complementary codes. The side-lobed distribution of expanded code set lies on the orthogonal matrix and the index array.

Equivalent transformations don't change the ZCW properties of proposed complementary ZCW codes.

All codes constructed by the proposed construction method are close to the theoretical bound.

The benefit of the present invention is to provide a coding method to create complementary codes with zero correlation windows. The said coding method can create a class of complementary codes with the "Zero Correlation Window" in their auto-correlation functions and cross-correlation functions. The said complementary ZCW code sets can be used in Quasi-Synchronized CDMA system to remove MAI and ISI. The said coding method leads to many ZCW code sets, which have a much wider range of available lengths and window widths of zero correlations. They will be applied to quasi-synchronized CDMA systems, where co-channel interference and influence of multi-path can be removed as much as possible.

Preferred embodiments of the invention

The present invention is to provide a coding method to create complementary codes with zero correlation windows. The new complementary ZCW code sets can be used in Quasi-Synchronized CDMA system to remove MAI and ISI. The complementary ZCW code sets are generated from two basic uncorrelated complementary codes with code length N , an M xM orthogonal matrix B._MxM and an index array I_2xM . The complementary code set is denoted by (κ,L,w) = (2M,MN,N), where M and N are arbitrary positive integer. The proposed complementary code sets also satisfy K - 2LIW . However the family size of code sets is not limited to 2 ⁺¹ and code length is not only 2^N.

It is shown that new construction method leads to many ZCW code sets which have a much wider range of available lengths and window widths of zero correlations. They can be successfully applied to quasi-synchronized CDMA systems, where co-channel interference and influence of multi-path can be removed as much as possible. LS codes can be also obtained according to the special case of the new construction method.

Complementary codes with Zero Correlation Windows:

The objective of the present invention is to provide a new coding method to create a class of complementary codes with the "Zero Correlation Window" in their auto-correlation functions and cross-correlation functions. Due to the creation of the "zero correlation window", the fatal near-far effect in traditional CDMA radio communications is removed. All codes constructed by the method are close to the theoretical bound. Using the method, the complementary ZCW code set (2M,MN,N) can be obtained, where 2M is the family size of the code set, MN is the code length of C component or S component and N is the code length of the basic complementary codes.

To achieve the above objective, the coding method of complementary codes with "Zero Correlation Window" includes the following steps:

(1) Selecting a basic complementary code

with code length N , where the aperiodic auto-correlation functions of C component and S component sum to zero except at the origin.

(2) Selecting another complementary code (c^,S^j which is uncorrelated with the code (c^,S^), i.e. the aperiodic cross-correlation functions of C component and S component between two codes sum to zero at all corresponding time shifts. r-

It can be proved that code (c^.S^j and code uncorrelated v J with each other. where ' ^~ ' denotes reversing operation.

'*' denotes the complex conjugate operation.

(3) Selecting an orthogonal matrix,

whose rows have the same energy.

(4) Selecting an index array

^l2xM

are permutations of

(5) According to the above four steps, three necessary elements to construct an expanded ZCW code set are obtained, two basic uncorrelated complementary codes (c^,s^) and (c^,S^), an orthogonal matrix H _xΛ and an index array

^L2xM . Then the expanded code set can be obtained according to the following rules:

(5.a) Construct two codes: ') /[>) « = (C /, „fi '« ,C „& , <A^lM> O 'l ⁾ . ,^•(

.,'(2) „ (2) r(2) = f . ..C^.S^.S^.-.S*'^'

(5.b) Two code sets can be obtained according to the following rule: A = x • T1_M M ~

B = ⁽²⁾ « H MxM

(5.c). Rows of A and B are expanded codes with length MN , and there are 2M such codes denoted by y^,y®,-",y^ respectively. It can be verified that the expanded code set form a (2M,MN,N) ZCW code set.

The new construction method leads to many ZCW code sets which have a much wider range of available lengths and window widths of zero correlations. Two special cases are emphasized here:

For an index vector I 2x , the four hamming

distances are defined before introducing the two special cases:

where . It is obvious that the following equations hold:

D_u+D_l2=M, D_2l+D₂₂=M

D_n+D_2l=M, D_U+D₂₂=M

Special Case 1:

LS code is the special case that

M D_n=D_l2=D_2l=D₂₂=—, M = 2ⁿ, n = l,2,3,-.

Special Case II:

Another special case can be obtained to make D_n=0 or >₁₂=0,then A, B are two uncorrelated code sets, i.e. the cross-correlation functions between codes from A and codes from B take zeros at all corresponding shifts. i?_y(r) = 0, l≤i≤M, M + l≤j≤2M, -MN + l≤τ≤MN-l

The equivalent transformation can be applied to generate complementary

ZCW code sets.

Generalized Complementary Codes with Zero Correlation Windows:

The above method can be extended to create a class of generalized complementary codes with zero correlation windows.

Let P codes be S₁,S₂,---,S , all with length N, their auto-correlations sum to zero at all shifts except the origin, then the code

is called generalized complementary code with P complementary components.

There exist at most P uncorrelated generalized complementary codes with

P complementary components, which is denoted by

= l,2,---,P

Given P uncorrelated generalized complementary codes

with code length N, an MxM orthogonal matrix

H _x and an index array I_PxM where columns of I PxM '

Y ι > ■••

≤M, are permutations of [l 2 ••• P]^T , a generalized complementary code set of length MN can be constructed as follows: (a). Construct p codes:

(0

Bf. S ^&<ι²° > . > _S°1 > ^°2 » S°'2² ^•,»2 s^<> S'²° ^■■,Op

_x x(2)_ — _sϊ,²⁾ ,_sθj²⁾ ,.^•.^•. ,_sθj²⁾ ,_sa₂ ²⁾ j_sd4₂ ²⁾ _>... »_s ^ai2²⁾... & sp'.⁽²⁾ > ^&p^{2) ■}> ... > s^J'pfe* w sf (p)

2 'Λ

⁾ _s°i1 ^p) s ^,32^p) > _S°42^P) s ^•,S'*f ip , Op . _si^p)

(b). P code sets can be obtained according to the following rule:

Λfx (²) = χ(²) . H MxM

Λ - x • a_MxM

(c). Rows of A^,l ≤ i ≤ P are expanded codes with length Nx , and there are PM such codes. The expanded code set forms a {PM,NχM,N) generalized complementary ZCW code set with P complementary components.

The present invention will be described in detail with reference to the preferred embodiments:

First an example of binary complementary code with code length Ν=2 is given:

C^(l) = (+ +), S^w = (+ -), wherein '+' means '1' and '-' means '-1'. It is true that the aperiodic auto-correlation of code (c^,S^j is equal to zero except at the origin, i.e. C component and S component are complementary each other. Now define:

The aperiodic auto-correlation function of code C^ is: Rf {τ)= ∑ ^Tc^[- _τj ,

;=0 where τ is the time shift.

The aperiodic auto-correlation function of code S^ is: R[{τ) = ∑ ^ ₊tJ »

/=o where τ is the time shift.

And the aperiodic auto-correlation function of code (c^,S^) is:

Table 1 is for the aperiodic auto-correlation values of code (c^,S^j.

Table 1 : Auto-Correlation of code (c^(l),S^(l))

Given a complementary code, another complementary code, which is uncorrelated with it can be obtained according to the uncorrelated complementary code construction method mentioned above.

We define:

The cross-correlation functions between code

are:

;=o =0

Table 2 is for the auto-correlation values of code

Table 3 is for the cross-correlation values between code (c^,S^j and code

Table 2: Auto-Correlation of Code (c⁽²⁾,S⁽²⁾).

Table 3: Cross-Correlation between code (c^(l),S^(l)) and code (c⁽²⁾,S⁽²⁾).

This is one of the basic forms for the uncorrelated complementary codes with each code length 2. Other forms can be derived from re-ordering of C^ and ²

S^ and S^ , swapping C and S, rotation, order reverse, and alternative negation etc. It should be noted that only the operation of code C with code C and code S with code S should take place when making the operation of correlation or matching filtering. Code C and code S will not encounter on operation.

+ + + —

Given an orthogonal matrix H_4x4 = + + — +

+ — + + , several expanded code sets

+ — — — will be discussed based on different index matrices:

( 1 ) Let the index array be I_2x4 = 1 2 1 2 2 1 2 1 , then the expanded code set

can be obtained according to the construction method mentioned above:

_x« = (c« ²> cW ² sW s<²> s« s<²>)

(2) _C(l)

_ _Y(D >H ²> -cW

4x4 _C(D _ _C(2) _C(l) c« - ²> -cW

_C(2) _C(l) _C(2) V

B ,(2) ®H_4x4 = c⁽²⁾ C^(l) -C⁽²⁾ _c(2) _ _c(l) _c(2)

_C ⁽²⁾ -cW -C

Rows of A and B are expanded codes with length 2 x4 = 8 , and there are 8 such codes denoted by y^,y^,---,y respectively. The expanded code set forms a (8,8,2) ZCW code set. Table 4: ACF and CCF of Code y W

Table 4 shows only the ACF and CCF of code y®, and similar correlation properties for other codes from the expanded code set. The expanded code set forms a (8,8,2) ZCW code set. It is shown that this example is special case I i.e. LS code.

(2) Let the index array be I_2x4 = 1 1 1 1

2222 , then the expanded code set

can be obtained according to the construction method mentioned above:

_x(i) ₌ (_c(i) _c(i) _c(i) _c(i)_{5 S}(i) _s(i) _s(i) _s(D)

x® = (c ²⁾ C ²\ S S S⁽²> S<²>)

f_c(ι) _c(ι) _c(ι) __c(ι)_{) s}(ι) _s(ι) _S(D __s(ι

-_v(ι) ®H_4x4 = _cd⁾ _c ⁽ι⁾ __c« cW'sW S^(l) -S^(l) S« ctø -C^(l) C^(l) C^(l), s⁽¹⁾ -s⁽¹⁾ s^(l) s⁽¹⁾ c« -ctø -cW -όi s⁽¹⁾ -s« -sW _sW

Rows of A and B are expanded codes with length 2x 4 = 8 , and there are 8 such codes denoted by

respectively. The expanded code set form a

(8,8,2) ZCW code set.

Table 5 shows only the ACF and CCF of code y^, and similar correlation properties for other codes from the expanded code set. The expanded code set also forms a (8, 8, 2) ZCW code set. This is an example of special case II. It is shown in table 5 that two code subsets A and B are uncorrelated code sets. The ZCW width of code set in (2) is equal to that in (1 ), but the side-lobe distributions are different between two code sets. It provides us a way to design a code set with desired side-lobe distribution outside the "zero correlation window".

(3) Let the index array be I_2x4 = 1 1 1 2 2 2 2 1 , then the expanded code set

can be obtained according to the construction method mentioned above: ,(2) ₌ (_C(2) _C(2) _c(2) _c(l)_{j S}(2) _S(2) _s(2) gfo)

f_C« _C(l) _C(D __C(2)₅ g(l) _S(D _S(l) _g(2 _C(l) _C(D __C( _C(2)_s' _S(D _S(l) __S(1) _S(2)

A = x^(l) ® H_4x4 =

_CW _C⁽¹⁾ C^(l) d²⁾ S^(l) -S^(l) S^(l) _S(2)

_CW -cW -cW - ²>, s⁽¹> -s« -s«

f_c(2) _c(2) _c(2) __c(l)_{5 s}(2) g(2) _S(2) _g(l)>

B _ = x(2) <8>H_4x4 = _c ⁽²⁾ _c -c cW ' s^ s<²> -s<²⁾ s⁽¹> ⁽²⁾ _c⁽²⁾ C⁽²⁾ C^(l), s⁽²⁾ -s⁽²⁾ s⁽²⁾ s⁽¹⁾ ⁽²⁾ -C⁽²⁾ -C⁽²⁾ -&\ s⁽²⁾ -s⁽²⁾ -s⁽²⁾ -s⁽¹⁾

respectively. The expanded code set form a

(8,8,2) ZCW code set.

Table 6 shows only the ACF and CCF of code y^ , and similar correlation properties for other codes from the expanded code set. The expanded code set also forms a (8, 8, 2) ZCW code set. The maximum side-lobe of this code set is 8, but the maximum side-lobe of the two code sets mentioned above is equal to 12. Changing the index array can lead to different code sets with same ZCW width but different side-lobe distribution.

Changing the orthogonal matrix and maintaining the index array, we can also obtain different code sets with same ZCW width. Given I,_v 2

2 , = 121 x4 2121 the

orthogonal matrix is changed to , then the expanded code set

can be obtained according to the new construction method: _χ(l)₌(_C(l) _C(2) _C(l) _C(2)_{5 S}(l) _S(2) _S(l) _s(₂)) _χ(2)₌(_C(2) _C(l) _C(₂) _C(l)_{j S}(2) _s(l) _S(₂) _s(l))

_C ) _C(2) _C(l) _C(2)_{j s}w _s<²> s« s⁽²n

_C(l) __C(2) _c(l) __c(2) _S( __S(2) _S(l) _g(2)

A = = x ^0) H4χ4 - _C(l) _C(2) __c(l) __C(2)^' _SW _S&⁾ -S« -S<²> cW _C⁽²⁾ -C^(l) C⁽²⁾ s« -s(²) -s« s<²> _j

_c(2) _c(l) _c(2) _c(₁)_{) s}(2) _s(l) _s(2) _s(l) -

_C(2) __C(l) _C(2) __C(1)₅ g(2) __s(l) _s(2) _g(l)

op C^(l) -C⁽²⁾ -C^(l)[ s⁽²⁾ s^w -s⁽²⁾ -s⁽¹⁾

_C ⁽²⁾ _c« -C⁽²> ^x s⁽²> -sW -s^ s⁽¹⁾

Rows of A and B are expanded codes with length 2x4 = 8, and there are 8 such codes denoted by y^,y^,---,y respectively. The expanded code set form a (8,8,2) ZCW code set.

Table 7: ACF and CCF of Code y^(l)

Table 7 shows only the ACF and CCF of code y^, and similar correlation properties for other codes from the expanded code set. The expanded code set also forms a (8, 8, 2) ZCW code set. From the above examples, it can be concluded that a complementary code set with ZCW=(2M,MN,N) can be generated from two uncorrelated complementary codes, an orthogonal matrix and an index array. The side-lobe distributions are different for different orthogonal matrices and index arrays.

Up to now, a construction method has been present to create a class of complementary codes with zero correlation windows. Given a complementary code set created by the proposed construction method, some other complementary code sets can be obtained by the transform of the previous one. These transforms are listed below exclusively:

• Swapping the C components and S components. • Swapping the C^(l) and C⁽²⁾; S^(l) and S⁽²⁾ .

• Reverse the code order

• Negation of each code

• Negation the every other bits in code C and S. For example for codes (++-+, ⁺— ). (⁺⁺⁺-. ⁺-⁺⁺); negation the even chips, then (+---, ++-+), (+-++, +++-) or negation the odd chips, then (-+++, --+-), (-+--, — +)

• Rotation in complex plane; for example, say the code (++-+, +---), (+++-, +- ++); the rotation a degree is

(JΨC Jtøci ^+a) _ i.'Pei ⁺²∞) A<Pq ⁺³«) _ [ _ si ^+Q _ ₀A<P_n +^2a) ₀Aφ_n +3«) N

f ^φ"2 .

where φ_Cι ,φ_C2 ,φ dX\ύ φ_Sι &re the initial angles. It is easy to verify that the correlation functions of these resultant codes have the same property as the original two codes. However, the distribution of side lobes maybe changed outside of "zero correlation window".

• Any other equivalent transforms of the code sets created by the method mentioned above.

Generalized Complementary Codes with Zero Correlation Windows

The above method can be extended to create a class of generalized complementary codes with zero correlation windows. An example is given below to illustrate it.

(a). Let P = A uncorrelated generalized complementary codes with P = 4 complementary components be:

(sf⁾ sW s s«)= (_{+ +}, +- +-, ++) (s{²⁾ s₂ ²⁾ s⁽²⁾ s⁽²⁾)= (_{+ +}, +-, -+, --) (sf³> s<³> s?> sj>)= (+-, ++, ++, +-) (s{⁴⁾ s₂ ⁴⁾ sW S₄ ⁴>)= (+ - + +, —, -+)

+ + + +

(b). Let an orthogonal matrix be H + — + —

4x4 + 4- - —

+ — — +

(c). Let an index array be I

(d). According to the construction method mentioned above, an expanded generalized complementary code set can be obtained:

The expanded generalized complementary code set is obtained:

+ + + + + + + -: + - + - + - + +, + - + + , + + + + + + + -- + + -+, + -- + + , + -- + - + + +, + + -- -- + - + + + + +: 4- -. + -. - + - - + - + - + - + +, + + + + + + + - + + -- + -- + - + + +, + -- + + , + + -- + + - + + - + - + + + +_: + + + + + - + -, + + + +, + -- + + + -- + -- + + + - + + -- + --+, + + + + + - + -, + - + - + + + + + - + - + + + + - + -+, + + + + - + -- + -- + + + -- + + + + + -, + + + + - + -+, + - + - + + + + + -+- + - + - + + + +, - + - + -- + +, - + + - + + -- + --+: + -- + + + --, - + + , -- + + - + - + + + + + - + -+, + " + , -+-+++-- + -- + + + -- - + + -. + -- + -- + +, - + + - + + + +, + + + - + - + - + - + - + + + + + + + + + -, -- + + + + + -, - + + - + -+ + + -- + + , + + -- + + -+, + + -+, - + - + + + _. + - - + --, + + + + +, -- + + +, - + + - - + --

+ -- + - + + +. + + + -, + __? - + - + - + + + Rows of the above code set are expanded generalized complementary codes with length 2x4 = 8 , and there are 16 such codes denoted by

respectively. The expanded code set form a (16,8,2) generalized complementary ZCW code set with 4 complementary components.

Table 8 lists the ACF and CCF of code y^(l)

Table 8 shows only the ACF and CCF of code y^, and similar correlation properties for other codes from the expanded code set. The expanded code set also forms a (16, 8, 2) generalized complementary ZCW code set with 4 complementary components.

Up to now, a construction method has been present to create a class of generalized complementary codes with zero correlation windows. Given a generalized complementary code set created by the proposed construction method, some other generalized complementary code sets can be obtained by the transform of the previous one. These transforms are listed below exclusively:

• Changing the positions of P complementary components: S_{l 9}S₂,-",S_P .

• Reverse the code order

• Negation of each code • Negation the every other bits in each complementary component.

• Rotation in complex plane.

• Any other equivalent transforms of the code sets that could be created by the method mentioned above.

Benefit: A general construction method is presented to create a class of complementary codes with zero correlation windows or generalized complementary codes with zero correlation windows. The invention provides the following benefits: i. A general construction method is presented to create a class of complementary codes with zero correlation windows or generalized complementary codes with zero correlation windows. ii. The proposed codes include binary codes, ternary codes, poly-phase codes and any other ZCW codes in complex field. iii. The new construction method can provide complementary ZCW codes with arbitrary positive ZCW width. Complementary codes with different ZCW width are determined by the actual channel condition. iv. The new construction method can provide complementary ZCW codes with arbitrary positive family size. v. The new construction method can be extended to create generalized complementary codes with zero correlation windows. vi. The three necessary elements to create a complementary code set is a pair of uncorrelated complementary codes, an orthogonal matrix and an index array. The zero correlation window width of expanded code set is equal to length of each component of the original complementary codes. The side lobe distribution of expanded code set lies on the orthogonal matrix and the index array. vii. Equivalent transformations don't change the ZCW properties of proposed complementary ZCW codes. viii. All codes constructed by the proposed construction method are close to the theoretical bound.

Although the invention has been described in detail with reference only to a preferred embodiment, those skill in the art will appreciate that various modifications can be made without departing from the invention. Accordingly, the invention is defined only by the following claims, which are intend to embrace all equivalents thereof.

Reference:

[1] D.B. Li, "High spectrum efficient multiple access code", Proc. of Future Telecommunications Forum (FTP'99), Beijing, pp.44-48, 7-8 December 1999.

[2] P.Z. Fan and M. Darnell, "Sequence Design for Communications Applications", John Wiley, RSP, 1996.

[3] L.R. Welch, Lower bounds on the maximum cross correlation of signals, IEEE Trans. Inform. Theory, vol. IT-20, pp. 397-399, 1974.

[4] V.M. Sidelnikov, On mutual correlation of sequences, Soviet math. Dokl., vol.12, pp. 197-201 , 1971. [5] P.Z. Fan, N. Suehiro, N. Kuroyanagi and X.M. Deng, "A class of binary sequences with zero correlation zone," IEE Electron. Lett., vol.35, pp. 777-779, 1999.

[6] X.M. Deng and P.Z. Fan, Spreading sequence sets with zero correlation zone, IEE Electron. Lett., vol. 36, pp. 993-994. [7] R.L. Frank, Polyphase Complementary Codes, IEEE Trans. Inform. Theory, vol. IT-26, pp. 641-647, 1980.

[8] L.S. Cha, Class of ternary spreading sequences with zero correlation duration, IEE Electron. Lett., vol. 37, pp. 636-637.

Claims

1. A coding method to create complementary codes with zero correlation windows, includes:

Selecting basic complementary codes; Selecting any orthogonal matrix;

Selecting any index array;

2. According to the method of claim 1 , wherein said selecting basic complementary codes includes:

Selecting two basic uncorrelated complementary codes with code length N, where N takes any positive integer value.

3. According to the method of claim 1 , wherein said selecting any orthogonal matrix includes: Selecting an M M orthogonal matrix H_{MxM >} where M takes any positive integer value.

4. According to the method of claim 1 , wherein said selecting any index array includes:

Selecting an index array l_2xM .

5. According to the method of claim 1 comprising the step of:

Selecting two basic uncorrelated complementary codes with code length N;

Selecting an MxM orthogonal matrix H_MxM ;

Selecting an index array l_2xM ;

And then expand the said complementary codes to code set using the said orthogonal matrix H_MxM and the said index array I_2xM .

6. According to the method of claim 5, wherein said selecting two basic uncorrelated complementary codes with code length N includes:

Selecting a basic complementary code (c^,S^) with code length N , where the aperiodic auto-correlation functions of C component and S component sum to zero except at the origin;

Selecting another complementary code (c ,S ) which is uncorrelated with the code

i.e. their aperiodic cross-correlation functions of C component and S component between two codes sum to zero at all corresponding time shifts; * \ Said code (c^(l),S^(l)) and code g(ι) ^cW^* uncorrelated with each other;

V where ' ^~ ' denotes reversing operation;

'*' denotes the complex conjugate operation.

7. According to the method of claim 5, wherein said selecting an M M orthogonal matrix R_MxM includes:

Selecting any orthogonal matrix,

whose rows have the same energy.

8. According to the method of claim 5, wherein said selecting an index array

I_2x includes:

where columns of I 2xM ' < n ≤ M , are permutations of

9. According to the method of claim 5, wherein said expand the said complementary codes to code set using the said orthogonal matrix H_MxM and the said index array I_2x includes:

Construct two codes:

rW = r ^}, .cW , ,-w

;S

Two code sets can be obtained according to the following rule:

A = x^(l) « H ^lM, xM

B = x⁽²⁾ « H MxM

Rows of A and B are expanded codes with length MN , and there are 2M such codes denoted by

respectively; It can be verified that the expanded code set form a (2M,MN,N) ZCW code set, i.e. the expanded code set has the zero correlation window width N.

10. According to the method of claim 5 comprising the step of:

Selecting a basic complementary code

with code length N , where the aperiodic auto-correlation functions of C component and S component sum to zero except at the origin;

Selecting another complementary code

which is uncorrelated with the code

i.e. their aperiodic cross-correlation functions of C component and S component between two codes sum to zero at all corresponding time shifts; Selecting any orthogonal matrix, H_{A x} ={A_{! MχM} whose rows have the same energy; selecting an index array I_2x ,

where columns of I 2xM ' < n < M , are permutations of

Construct two codes:

_[« -(c'P.c«^l.....c* s*^,.s*^,.-.s«)

^ -.(c^.c^.-.c^.s^.s^"....^*),

Two code sets can be obtained according to the following rule:

A = x^(l) « E ^LMxM

B = x • H _x

Rows of A and B are expanded codes with length MN , and there are 2M such codes denoted by y^,y^,---,y^ respectively; It can be verified that the expanded code set form a (2M,MN,N) ZCW code set.

11. According to the method of claim 1 , wherein said selecting basic generalized complementary codes includes:

Let P codes be S₁,S₂, ---,S_{P >} all with length N , their auto-correlations sum to zero at all shifts except the origin, then the code (s_l,S₂,---,S_p) is called generalized complementary code with P complementary components.

12. According to the method of claim 11 , comprising the step of:

Let P codes be S₁,S₂, --.,S_jP , all with length N , their auto-correlations sum to zero at all shifts except the origin, then the code (s^S_j,---^) is called generalized complementary code with P complementary components;

There exist at most P uncorrelated generalized complementary codes with

P complementary components, which is denoted by

= l,2, ---,P Given P uncorrelated generalized complementary codes:

with code length N , an M x M orthogonal matrix H _x and an index array :

where colui Tin s of I_p ,-(2)

≤ M , are permutations of

[l 2 •■• p]^r , a generalized complementary code set of length MN can be constructed as follows: Construct p codes: _xd) » •• . ', o'₂M ... S_P ,Sp ,- - -,Sft j

_χ(2)

(p) . ,« ) W

= S? ^aι 5 ,sf »S₂ P _S4^P) . ,^•(

Op , O p ,

P code sets can be obtained according to the following rule:

^{Λ x Ά}MXM

Rows of A^,l ≤i ≤P are expanded codes with length N M , and there are PM such codes; The expanded code set forms a (PM,NχM,N) generalized complementary ZCW code set with P complementary components.

13. According to the method of claim 1 or 5, wherein said complementary codes with zero correlation windows includes: binary codes, ternary codes, poly-phase codes and any other ZCW codes in complex field.

14. According to the method of claim 10, wherein said index array includes:

For an index vector I_2x amming

distances are defined:

D₂₁ = ∑ δ(l,i®} D₂₂ = ∑ δfai®) m=l m=\

A|| x — y where δ(x,y)=\ ' , It is obvious that the following equations hold:

[0, else

D_u+D_l2=M, D₂ +D₂₂=M

D_n+D_2l=M, D_X2+D₂₂=M

LS code is the special case that

M ι=A₂=Aι=A₂=— , M = 2 n = l,2,3,-]

Another special case can be obtained to make Aι^{=0 or} ₂ ⁼⁰ _>th^en A,

B are two uncorrelated code sets, i.e. the cross-correlation functions between codes from A and codes from B take zeros at all corresponding shifts; Ri_j(τ) = 0, l≤i≤M, M + l≤j≤2M, -N,N₂4-l<r<N₁N₂ -1;