WO2004057785A1

WO2004057785A1 - A coding method to create mismatched complementary codeswith zero correlation windows

Info

Publication number: WO2004057785A1
Application number: PCT/CN2002/000902
Authority: WO
Inventors: Shaojun Xu; Daoben Li
Original assignee: Linkair Communications Inc
Current assignee: Linkair Communications Inc
Priority date: 2002-12-20
Filing date: 2002-12-20
Publication date: 2004-07-08
Anticipated expiration: 2005-06-20
Also published as: AU2002357418A1; CN1628430A

Abstract

The present invention is to provide a coding method to create mismatched complementary codes with the zero correlation window. The said coding method includes: selecting basic mismatched complementary codes; selecting any orthogonal matrix; selecting any index array; and then expand the said mismatched complementary codes to code set using the said orthogonal matrix and the said index array. The said coding method can create a class of mismatched 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 mismatched codes constructed by the method are close to the theoretical bound.

Description

A Coding Method to Create Mismatched Complementary

Codeswith Zero Correlation Windows

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

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 it is decided before design. Now that CDMA system capacity lies on 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 both ideal ACF and ideal CCF.

But in many applications, it is not necessary to construct such code set with both 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 number. PCT/CN00/00028). Given the code length, the family size of LS code set is much greater than before, and it has great worth 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+l, and the code length at level M is 2^M N . Such code set is denoted by 2^M+X,2^M N,N), where 2^M+Ϊ 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.

In principle matched filtering operation is the optimum operation. In CDMA system, if one code (MA code) is employed to spread, then the same code should be employed to dispread, such codes are called the matched codes. When the dispread code is different from the spread one, it is called a mismatched one. For a matched complementary ZCW code set {c^w,S^w}, 1 < i < K denoted by (κ,L,W), where K is the family size of code set, L is the length of C component or S component and W is width of one-sided zero correlation windows, the relation

2L + 2(W - 1) between K,L and W is: K ≤ * '- according to the theoretical bound. In

the previous work, some code sets approaching the mathematical bound had been proposed such as LS codes proposed by Prof. Li Daoben that satisfy. For LS codes, the relation between code length, family size and zero correlation window width is

2L K = — which is very close to the lower bound on zero correlation windows. But the

W family size of LS codes is limited to 2" where n is any positive integer. Νote:(«) denotes the code employed to spread, and (•)' denotes the code employed to dispread. It will be no longer emphasized at the following sectors.

For a mismatched complementary code with zero correlation windows, the code set employed to spread is

called local code set, 1 < i ≤ K and one employed to dispread is

called mismatched code set that is same as or different from the local one (c^,S^|. For a mismatched complementary ZCW code set, the relation between family size K , length of C component or S component L and width of one side zero correlation windows W is also bounded by

w Summary of the invention

The objective of the present invention is to provide a coding method to create mismatched complementary codes with the zero correlation window. The said coding method can create a class of mismatched 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 mismatched codes constructed by the method are close to the theoretical bound.

A coding method to create mismatched complementary codes with zero correlation windows, includes: Selecting basic mismatched complementary codes;

Selecting any orthogonal matrix; Selecting any index array;

And then expand the said mismatched complementary codes to code set using the said orthogonal matrix and the said index array. wherein said selecting basic mismatched complementary codes includes :

Selecting two basic uncorrelated mismatched complementary codes with code length N. wherein said selecting any orthogonal matrix includes : Selecting an MxM orthogonal matrix U_MxM . wherein said selecting any index array includes : Selecting an index array

The expanded code set forms a mismatched complementary ZCW code set

with (K, L, W) = (2M,MN, N); SO K = — which approaches the theoretical lower

bound. wherein said selecting two basic uncorrelated mismatched complementary codes with code length N includes: _c )₅s( Ϊ two basic uncorrelated mismatched complementary codes

_C(2)_{> S}(2)

with length N .

wherein said selecting two basic uncorrelated mismatched complementary codes with code length Ν comprising the step of:

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

Selecting another mismatched complementary code (c^,S^), (cΗs'^j which is uncorrelated with the first code, i.e. their aperiodic cross-correlation functions of C component and S component between two mismatched codes sum to zero at all corresponding time shifts; To guarantee the orthogonality of the expanded code set, the peaks of the two basic codes should be equal, i.e. R₁(0) = R₂(0);

Given a mismatched complementary code (c^,S^), (c'^S'^j, it is easy to

verify that the mismatched code

[s«r,- 4) is uncorrelated

with the first one; where ' ^~ ' denotes reversing operation;

'*' denotes the complex conjugate operation. wherein said selecting an M M orthogonal matrix H_MxM includes:

Selecting any orthogonal matrix, H _X ={A_I } whose rows have the same energy. wherein said selecting an index array I_2xM includes:

where columns of I 2xM ' ,1 < n ≤ M , are permutations of

wherein said expanding the said mismatched complementary codes to code set using the said orthogonal matrix R_MxM and the said index array l_2xM includes: Construct two mismatched complementary codes:

_x.W ₌ f_c.,<"_>c ",..._>c^«ll^,,s"!'⁾,s"S'⁾ _>..,s"£^l

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

MxM

A' = x'⁽²⁾ « H MxM

B = Y^²^ • ^"HΓ B' = x'⁽²⁾ « H MxM

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

(y^(l)> y'^(l))(y⁽²⁾,y'⁽²⁾) ---,(y^{(2 )},y'^{(2 )}) respectively; It is to verify that the expanded code set form a (2M,MN,N) mismatched complementary ZCW code set;

The equivalent transformation can be applied to generate more mismatched complementary ZCW code sets. wherein said selecting basic mismatched complementary codes includes : Let P mismatched codes be (s₁ ,Sι),(S₂,S₂),---,(s_P,S'p), each with length N , their mismatched auto-correlations sum to zero at all shifts except the origin, then the mismatched code (S,,S₂,---,S_P), (S'₁,S₂,---,S'_P) is called generalized mismatched complementary code with P complementary components. The method of the present invention comprising the step of: Let P mismatched codes be (s₁,S'₁),(S₂,S'₂),---,(S ,S_/J), ail with length N , their mismatched auto-correlations sum to zero at all shifts except the origin, then the mismatched code (s₁,S₂,-",Sp)_> (SJ,S₂,---,Sp) is called generalized mismatched complementary code with P complementary components;

There exist at most P uncorrelated generalized mismatched complementary codes with P complementary components, which is denoted by

Given p uncorrelated generalized mismatched complementary codes

{si^sξ -^S^^S''^S^^' -^S^ji^ ,---^ with code length N, an MxM

orthogonal matrix H_MxM and an index array I_PxM where

columns of I_PxM, i® ■•■

≤n≤M, are permutations of [l 2 ••• Pf, a generalized mismatched complementary code set of length MN can be constructed as follows: Construct p codes:

ft) ( ,.⁽¹⁾ ,.⁽¹⁾ . J e°.1® J c^a'24° »^β ^2 ' ... '° C 2 s ,s ^•<-' -s P J.

^'

X — O] ,⁽_.,

j>

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

Ά. — x • a._} M , Λ — x • a_MxM , ^A (²) - v(²) . H A'® - x' • H

Α (^») _ - _x X(^/») _# • _H ⁿ xA^ ' ^Λ( '( ) - - x'W • • H "M '

Rows of A^,A'^,l<z≤P are expanded mismatched codes with length NxM , and there are PM such codes;

The expanded mismatched code set forms a (PM,NχM,N) generalized mismatched complementary ZCW code set with P complementary components, wherein said mismatched complementary codes with zero correlation windows includes:binary codes, ternary codes, poly-phase codes and any other ZCW codes in complex field.

A general construction method is present to create a class of mismatched complementary codes with zero correlation windows or generalized mismatched complementary codes with zero correlation windows. The invention provides the following benefits:

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

The said coding method can provide mismatched complementary ZCW codes with arbitrary positive ZCW width. Mismatched complementary codes with different ZCW width are determined by the actual channel condition.

The said coding method can provide mismatched complementary ZCW codes with arbitrary positive family size.

The said coding method leads to a large class of mismatched complementary codes with zero correlation windows by selecting various basic mismatched complementary codes, orthogonal matrices and index arrays.

If the code set employed to dispread is same as the local one, the mismatched code set leads to matched code set. Thus the new construction method can also provide matched complementary codes with zero correlation windows. The said coding method can be extended to create generalized mismatched complementary codes with zero correlation windows.

The three necessary elements to create a mismatched complementary code set are a pair of uncorrelated mismatched 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 mismatched 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 mismatched complementary ZCW codes. All mismatched codes constructed by the proposed construction method are close to the theoretical bound.

Preferred embodiments of the invention

The objective of the present invention is to provide a coding method to create a class of mismatched 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 mismatched codes constructed by the method are close to the theoretical bound. Using the method, the mismatched 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 mismatched complementary codes.

To achieve the above objective, the coding method of mismatched complementary codes with "Zero Correlation Window" includes the following steps: (1) Selecting a basic mismatched complementary code

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

(2) Selecting another mismatched complementary code

_wnj_cn j_S uncorrelated with the first code, i.e. their aperiodic cross- correlation functions of C component and S component between two mismatched codes sum to zero at all corresponding time shifts. To guarantee the orthogonality of the expanded code set, the peaks of the two basic codes should be equal, i.e. R₁(0) = R₂(0).

Given a mismatched complementary code (c^,S^j, (c'^S'^j, it is easy to

verify that the mismatched code [s^,(l)]^* ,-[c'^(l)]^* I , | [s^(l)]^*,-[c^(l) M is uncorrelated

with the first one. where ' ^~ ' denotes reversing operation.

'*' denotes the complex conjugate operation.

(3) Selecting any 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 mismatched complementary ZCW code set are obtained, two basic uncorrelated mismatched complementary codes

and

(c⁽²⁾,S⁽²⁾), (c⁽²⁾,S'⁽²⁾), an orthogonal matrix ΕL_MxM and an index array I_2xM.Then the expanded mismatched complementary code set can be obtained according to the following rules:

(5.a) Construct two mismatched complementary codes:

(0 _,^•(>)

<« =(< 5 -' : -,s

x'⁽¹⁾=( ^,,,c*^,,...,c^,s'*^,,s^^,,...,s'<

x (2) -r C'¹ , & o ²> o ²> , (.)

•,C S'^> ,s ;S^l l

'⁽²⁾= ²⁾ &² ,c^,s^₅s^,..,s^^r

(5.b) Two mismatched complementary code sets can be obtained according to the following rule:

A = x^(l)«H M_Λ xM ^'

A' = x'⁽²⁾*H MxM

B = x⁽²⁾ • H MxM

B' = x'⁽²⁾.H MxM

(5.c). Rows of A, A' and B, B' are expanded mismatched complementary codes with length MN , and there are 2M such codes denoted by

(y^(l)> y'^(l))(y⁽²⁾,y'⁽²⁾)---,(y^{(2 )},y'^(w)) respectively. It is to verify that the expanded code set form a (2M,MN,N) mismatched complementary ZCW code set. The equivalent transformation can be applied to generate more mismatched complementary ZCW code sets.

Generalized Mismatched Complementary Codes with Zero Correlation Windows: The above method can be extended to create a class of generalized mismatched complementary codes with zero correlation windows.

Let P mismatched codes be (S_{l 5}s;),(s₂,S₂),-..,(Sp,Sp), all with length N , their mismatched auto-correlations sum to zero at all shifts except the origin, then the mismatched code (s₁,S₂,---,Sp), (s;,S₂,--,S_p) is called generalized mismatched complementary code with P complementary components.

{S« S«,.-.,S«},{S^,S^,.--,S^}- = 1,2,.--,P

Given p uncorrelated generalized mismatched complementary codes

{sf⁾,S₂ ⁾,---,S_p ⁾},{s ^(/),S₂ ^(/),.--,S^^')}- = l,2,-..,P with code length N , an MxM

orthogonal matrix H _x and an index array l_PxM where

columns of l_PxM , [ i ^■•^■

≤ n ≤M , are permutations of [l 2 ••• P]^τ , a generalized mismatched complementary code set of length MN can be constructed as follows:

(a). Construct p codes:

, r(Wι) (A* 4" o » „< '>) « . s[² ' 'i⁽.' S4'l"'⁾ o':»

= [S'{ , , s^s®l _ _s ^i5<2>⁽¹⁾ S ■ 2 »°2 • S ^M

..⁾

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

Α A (2) _ — _X X(²) . H n^xw , A A'(²) - - x X'⁽²> • H "MxM

- ₅> Α ΛP) _ — X _x(/^>) . • _H "MxM ' ^ AP) — - _X X'{P) . " H "MxM

(c). Rows of A^,A'^,l < t < P are expanded mismatched codes with length N M , and there are PM such codes. The expanded mismatched code set forms a (PM,N M,N) generalized mismatched complementary ZCW code set with P complementary components. 0 The present invention will be in detail described with reference to the preferred embodiments.

First an example of binary mismatched complementary code with code length N=5 is given:

C« = (+ - + - +), S« = (- + — -), C« = ( ) , S'W = (- + — -). 5 where '+' means '1' and '-' means '-1'. It is true that the aperiodic auto-correlation of mismatched code (c^,S^), (c'^,S'^) 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 mismatched code C^ , C'^ is: 0 R (r) = ^N~∑ ^τ cjfi c$ \ , where τ is the time shift.

.=o

The aperiodic auto-correlation function of mismatched code S^, S'^ is:

Rϊ(τ) = ^N~∑ ^τs [s' J , where τ is the time shift.

.=o

And the aperiodic auto-correlation function of mismatched complementary code (C«,S«), (c'W S'«) is: R₁(r) = R₁ ^c(τ)+ R (r) 5 Table 1 is for the aperiodic auto-correlation values of mismatched complementary code (c^(l),S^(l)), (c'^(l),S'^(l)).

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

(C«,s'«)

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

(c⁽²⁾,s⁽²⁾)=f[s'⁽¹⁾r₅-[c'⁽¹⁾]^* = (---₊-, ₊₊₊₊₊)

We define:

The cross-correlation functions between mismatched complementary code (c^(l),S^(l)), (c^(l),S'^(l)) and mismatched complementary code (c^,(2),S'⁽²⁾) are:

.=0 ι=0

Table 2 is for the auto-correlation values of mismatched complementary code

(c<²>, sW), (c'⁽²⁾,s' ). Table 3 is for the cross-correlation values between mismatched complementary code

and mismatched complementary code

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

Table 3: Cross-Correlation between mismatched complementary code (c^'.S^j,

(C'⁽¹⁾,S'⁽¹⁾) and mismatched complementary code (c⁽²⁾,S⁽²⁾), (c^,(2),S'⁽²⁾).

This is one of the basic forms for the uncorrelated mismatched complementary codes with each code length 5. Other forms can be derived from re-ordering of C^ and C^, 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 = + + - +

+ — + + and an index array

+ — — —

1112

12x4 - 2221 then the expanded mismatched complementary code set can be

obtained according to the construction method mentioned above: _χ(l)₌(_C(l) _C(l) _C(l) _C(2)_{5 S}(l) _S(l) _S(l) _S(2))

_X'⁽0 ₌ (_C ⁽0 c'W C^(l) C'⁽²⁾, S'W S'^(l) S'W S'⁽²>) _χ(2) ₌(_C(2) _C(2) _C(2) _c(l)_{5 s}(2) _S(2) _s(2) ₀(.)) _X'(2)₌(_C'(2) _C(2) _C(2) _C« _S'(2) _S»(2) _S,(2) c'-))

^{( x)} c® c⁽¹⁾ -c<²⁾, s^{1) s⁽¹⁾ s⁽¹⁾ -s⁽²⁾"ι

A - =A x ) C&⁾ C^(l) -C^(l) C⁽²⁾,' s⁽¹⁾ s⁽¹⁾ -s⁽¹⁾ s⁽²⁾

"4x4 - _CW -c^(l) C^(l) C⁽²⁾' s⁽¹⁾ -s⁽¹⁾ s⁽¹⁾ s⁽²⁾ c*« -c« - -c , s« -s« -s« -s⁽²⁾

+ - + - + + - + - + + - + - + + + + -+, - + -•

+-+-++-+-+-+-+ + -, - + -- — _{+ +}__++++++++

+ - + - + - + - + - + - + - + + -, - + -- —+-+++-+ +++++ +- + - + - + - + -- + - + - + + + -+, - + -- ^■-+-++++-+++ J

(_c(2) _c(2) _c(2) __c(l)^ _s(2) _s(2) g(2) __s(l)>

_C(2) _C(2) __c(2) _C(l) _S(2) _g(2) __S(2) _s(l)

B = __χ(2) <g>H 4x4 _C(2) __C(2) _C(2) _C(l)_{5 S}(2) __s(2) g(2) _s(l)

_C(2) __C(2) __C(2) __C(1) _s(2) __s(2) __s(2) __s(l) f - + + + -- + - + - + + + + + + + + + + + + + + + + - + + +

• + + - + + + - + + - + -+, + + + + + + + + + + +

^■ + - + + + - + + - + - + -+, + + + + + + + + + + - +

^■ + - + + + - + + + + - + - + - + -, + + + + + + - + + + / _c,(2) _c,(2) _C(2) g'(2) _S'(2) _S'(2) _c(2) _c,(2) __c(2) _ = v x'(²) ®H_4x4 = C'W, _S'(2) _S'(2) __S 2)

B '' _C'(2) __C'(2) Q>(2) _C/(1)₅ g.(2) __S'(2) _S'(2)

C'(²) _ '(²) __C'(²) _c'W, S'⁽²⁾ -S'⁽²⁾ -S'⁽²⁾

- + - + + - + -- + - + - + - + + + - + - + -- + - + - + - + - + - + - + - + - + - + - + - + - + -- +

- + _+- ₊ - .- + + -4.- + 4.-4, + +_y

Rows of A, A' and B, B' are expanded codes with length 5x4 = 20, and there are 8 such codes denoted by (y^,y'^)( ^,y'^)---,(y^,y'^) respectively. The expanded code set forms a (8,20,5) mismatched complementary ZCW code set.

Table 4: ACF and CCF of Mismatched complementary Code y ^ , y''¹' for

+ + + - + + - + 12

"4x4 ^— + - + + and I 11

2x4 2221

Table 4 shows only the ACF and CCF of code y^, y'^, and similar correlation properties for other codes from the expanded code set. It is shown that the expanded code set forms a (8,20,5) mismatched complementary ZCW code set from the above example.

Given two basic uncorrelated mismatched complementary codes with code length N , a class of mismatched complementary code sets with ZCW = (2M,MN,N) can be obtained by selecting various orthogonal matrices H _x and index arrays I_2xM . The widths of zero correlation windows are same for all these code sets. But the side lobe distributions of these code sets are different, which lies on the orthogonal matrix H _x and index array I_2xM . The above

+ + + — example is the special case for H_4x4 = + + — +

+ - + + and I_2x4 = 1 1 1 2 2 2 2 1 Some

+ — — — other examples for different orthogonal matrices and index arrays are listed in the following tables:

Table 5 lists the ACF and CCF of mismatched complementary code y^\ y'^

+ + + — for H_4x4 = + + — +

+ - + + and I_2x4 = 1 2 1 2 2 1 2 1

+ — — —

Table 5: ACF and CCF of Mismatched Complementary Code jr¹' , y'^ for

Table 6 lists the ACF and CCF of mismatched complementary code y^ , y'^

+ + + + for H_4x4 = + — + —

+ + — — and I_2x4 = 1 1 1 2

2 2 2 1

+ — — +

Table 6: ACF and CCF of Mismatched Complementary Code y^, y'^ for

+ + + +

+ - + -

"4x4 ^— + + - - and I_2x4 = 1 1 1 2 2 2 2 1

+ — — +

From the above examples, it can be concluded that a mismatched complementary code set with ZCW=(2M,MN,N) can be generated from two basic uncorrelated mismatched complementary codes, an orthogonal matrix and an index array. There are different side-lobe distributions for various orthogonal matrices and index arrays.

Up to now, a construction method has been present to create a class of mismatched complementary codes with zero correlation windows. Given a mismatched complementary code set created by the proposed construction method, some other mismatched 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

( JΨcγ J(Ψcι ^+a) _ ./faq +S^β) y(*>q +3«) Jψs _ K<Psι +") _ tø-i ^+2a. _ Nfø-₁ ^+3α) \

( JVc₂ A<Pc₂ +<*) j(φ_C2 +2a) _ j(φ_C2 +3α) jφ_s_ _ ψ._ +^a) Ji<P._ ⁺² . -< n +^3α) \ .

where φ_Cι ,φ ,φ_Sj ar.ά <p_Sι are 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 Mismatched Complementary Codes with Zero Correlation Windows

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

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

(s« s« s« s«)= (+-₊-₊, — +-, -+ , +++++)

(_s;(i) _s,(ι) _s,(ι) _s,(ι))_{= ( >} _ _ _ ₊_₅ _ _{+ f} _ ₊ _ ₊ _)

(si²⁾ S₂ ²> Sψ

(+- + -+, + + + -+, - + , )

(s' ²⁾ S'₂<²> s )= { , +++-+, -+ , +-+-+)

(s|³>

+-, +-+-+, +++++, - + ---) (s[ $' S' s )=(— +-, , -+-+-, -+— -)

(sf⁴> s₂ ⁴> s<⁴> sW)=(— +- -+-+- + + + ++, + - + ₊ +) (s;^ s₂ ⁽⁴⁾ s ⁾ sf⁾)=(— -+- +++++, -+-+-, +-+++)

(b). Let an orthogonal matrix be H_2x2 = + + + -

(c). Let an index array be I 4x2

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

_A (i) ₌r^+-+-+ — ^+_> — +-+-+-+,-+ — +++++, +++++-+ — ^" + _ + ₅ _ _{+ +} _ ₊ --.₊₅ _ + _ + __ +

A'W = + + + -+, + - + + + + +, - + + - + -, - + - + - + - + + +

_A(2) ₌ ^' ₊ _ ₊ _ ₊ __{+j +}_ ₊ _ _{+ + + +} _-._{+5 + + + + +} _ _{+ 5} _ + ^" + -- + - + - + - + - + + -, + + + + + + - + + +, - + +,+ + + + _A'(2) ₌ + , + + + -+, - + - + -- + , - + + - + - +

₊ _ _{+ + + + +j +} _ _ ₊ _ ₊ _ ₊ _ _{+ + +5} - + + - + -

A^ = + - + - + -+, - + - + + -, + + + + + - + , +-++++++++ +__+_+_ _ + _ + _ + + + _-. _+j + - _{+ + +^} __ +

_A'(3) ₌ ^" + _{5 + + + + + +} _ _ ₊ _ ₊ __ _{+ 5} +_+++-+-+- + - + + + + +, + + + + + + + + -+, - + - + - + - + + +, +- + + + + - + - +

₊ _-|-_ ₊ -|-_ _{+ + +} _4-- + _ + - - + _{+ + + + +j +} _ _{+ + +} ^"

A« = + - + - + + + + -+, + + + - + + - + -+, - + , +

A'W = + -, + + + - + + + + + +, - + + - + -, +- + - + + - + + +

• + + + -+, + + + - + , - + + - + -+, +- + - + - +

Rows of A^,A'^,l</<4 are expanded mismatched codes with length 2x5 = 10, and there are 2x4 = 8 such codes denoted by ^y( y'Wj^y()₅y'(2)|...^_y(8)₅y(8)| -j-_{ne ΘX}p_anc|_ecj mismatched code set forms a

(8,10,5) generalized mismatched complementary ZCW code set with 4 complementary components.

Table 7 lists the ACF and CCF of generalized mismatched complementary code

(y<",y'»)

Table 7: ACF and CCF of Generalized Mismatched Complementary Code (y^.y'^j

Table 7 shows only the ACF and CCF of generalized mismatched complementary code y , y'^, and similar correlation properties for other codes from the expanded code set. The expanded code set forms a (8, 10, 5) generalized mismatched complementary ZCW code set with 4 complementary components.

Up to now, a construction method has been present to create a class of generalized mismatched complementary codes with zero correlation windows. Given a generalized mismatched complementary code set created by the proposed construction method, some other generalized mismatched 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: S₁,S₂,-.-,Sp .

• 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.

A general construction method is present to create a class of mismatched complementary codes with zero correlation windows or generalized mismatched complementary codes with zero correlation windows. The invention provides the following benefits: i. The proposed mismatched complementary codes include binary codes, ternary codes, poly-phase codes and any other mismatched complementary ZCW codes in complex field. ii. The new construction method provides mismatched complementary

ZCW codes with arbitrary positive ZCW width. Mismatched complementary codes with different ZCW width are determined by the actual channel condition. iii. The new construction method provides mismatched complementary

ZCW codes with arbitrary positive family size. iv. The new construction method leads to a large class of mismatched complementary codes with zero correlation windows by selecting various basic mismatched complementary codes, orthogonal matrices and index arrays. v. If the code set employed to dispread is same as the local one, the mismatched code set leads to matched code set. Thus the new construction method can also provide matched complementary codes with zero correlation windows. vi. The new construction method can be extended to create generalized mismatched complementary codes with zero correlation windows. vii. The three necessary elements to create a mismatched complementary code set are a pair of uncorrelated mismatched 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 mismatched complementary codes. The side lobe distribution of expanded code set lies on the orthogonal matrix and the index array. viii. Equivalent transformations don't change the ZCW properties of proposed mismatched complementary ZCW codes. ix. All mismatched 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 mismatched complementary codes with zero correlation windows, includes:

Selecting basic mismatched complementary codes; Selecting any orthogonal matrix;

Selecting any index array;

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

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

Selecting two basic uncorrelated mismatched complementary codes with code length N.

3. According to the method of claim 1 , wherein said selecting any orthogonal matrix includes : Selecting an M x M orthogonal matrix H _x .

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 mismatched complementary codes with code length N;

Selecting an MxM orthogonal matrix K_MxM ;

Selecting an index array I_2xM ;

And then expand the said mismatched complementary codes to code set using the said orthogonal matrix H _x and the said index array I_2x ;

The expanded code set forms a mismatched complementary ZCW code set

with (K,L,W) = (2M,MN,N); SO K = — which approaches the theoretical lower

bound.

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

two basic uncorrelated mismatched complementary codes with length N .

C'⁽²⁾,S'⁽²⁾

7. According to the method of claim 5, wherein said selecting two basic uncorrelated mismatched complementary codes with code length Ν comprising the step of:

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

Selecting another mismatched complementary code

which is uncorrelated with the first code, i.e. their aperiodic cross-correlation functions of C component and S component between two mismatched codes sum to zero at all corresponding time shifts; To guarantee the orthogonality of the expanded code set, the peaks of the two basic codes should be equal, i.e. R_l{0) = R₂{0);

Given a mismatched complementary code it is easy to

verify that the mismatched code [s⁽¹)j " is uncorrelated

with the first one; where ' ^~ ' denotes reversing operation;

'*' denotes the complex conjugate operation.

8. According to the method of claim 5, wherein said selecting an M M orthogonal matrix B._MxM includes:

Selecting any orthogonal matrix,

} whose rows have the same energy.

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

I_2xM includes:

where columns of I 2xM (2) ,1 < n ≤ M , are permutations of

10. According to the method of claim 5, wherein said expand the said mismatched complementary codes to code set using the said orthogonal matrix Η._MxM and the said index array l_2xM includes:

Construct two mismatched complementary codes:

''⁽¹⁾ = rc^^,) ₎c^^,) _>.-._>c'^,s'*⁾,s^⁾ _>-₎s'*⁾

,-(2) r(2) = = c ²⁾, , 0^M ,s ²⁾ s'²⁾. ,s^|

A = x^(l)«H MxM

A' = x'⁽²⁾.H MxM B = x⁽²⁾»H M, xM

T r>' = - γ ^X'(²) # ^mH "-MxM

Rows of A, A' and B, B' are expanded mismatched complementary codes with length MN , and there are 2M such codes denoted by (_y( , y ι⁾)ι(y⁽2⁾ _5y'^{( )}^...₎(_J ⁽2 )_jy,(2 ) _respectjve[y; It is to verify that the expanded code set form a (2M,MN,N) mismatched complementary ZCW code set;

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

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

Selecting a basic mismatched complementary code

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

Selecting another mismatched complementary code

which is uncorrelated with the first code, i.e. their aperiodic cross-correlation functions of C component and S component between two mismatched codes sum to zero at all corresponding time shifts. To guarantee the orthogonality of the expanded code set, the peaks of the two basic codes should be equal, i.e. Λ₁(0) = Λ₂(0);

Given a mismatched complementary code (c^,S^j, (c'^,S'^), it is easy to

verify that the mismatched code f [s'⁽¹⁾]^* -[^c'^(l)] ^* ) , is uncorrelated

with the first one; where ' ^~ ' denotes reversing operation;

'*' denotes the complex conjugate operation;

Selecting any orthogonal matrix, H_MXM={/J_I } whose rows have the same energy;

Selecting an index array:

^l2xM

are permutations of

According to the above four steps, three necessary elements to construct an expanded mismatched complementary ZCW code set are obtained, two basic uncorrelated mismatched complementary codes (c^,S^), (c'^,s'^) and

(c⁽²⁾,S^{( )}), (c⁽²⁾,S'⁽²⁾), an orthogonal matrix ΕL_MxM and an index array I_2x ; Then the expanded mismatched complementary code set can be obtained according to the following rules:

Construct two mismatched complementary codes: ι» -(c*^,,c**....,c«,s*^,.s*^,.^....s*).

_['⁽'⁾ =fc"!^,,,c*',...,c"ϊ^,,s'*^,,s' '⁾,...,s"'

^ =(c^,c ,..,c&^,s^,..^ (?)

Two mismatched complementary code sets can be obtained according to the following rule: A = x^(l) . H MxM

A' = x'⁽²⁾ . H MxM

B = χ(²⁾ « H, B' = v ^X'( ^W2) * "MxM >

(y^(l)> y'^(l)Xy⁽²⁾,y'⁽²⁾)---,(y^(2M),y'^{2Λ/)) respectively; It is to verify that the expanded code set form a (2M,MN,N) mismatched complementary ZCW code set;

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

Let P mismatched codes be

all with length N , their mismatched auto-correlations sum to zero at all shifts except the origin, then the mismatched code (s₁,S₂,..-,S_p), (Sj,S₂,-.-,S'p) is called generalized mismatched complementary code with P complementary components.

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

Let P mismatched codes be (s₁,s;),(S₂,S'₂),-",(Sp,S ), all with length N , their mismatched auto-correlations sum to zero at all shifts except the origin, then the mismatched code (s_{l 5}S₂,..-,Sp), (SJ,S₂,---,S'_P) is called generalized mismatched complementary code with P complementary components;

{s« S₂'⁾,..-,S«},

l,2₅.. -,P; Given p uncorrelated generalized mismatched complementary codes

- -,s ⁾]i = l,2,- --,P with code length N , an M xM orthogonal matrix ΕL_MxM and an index array l_PxM where

columns of l_PxM, {$ i •••

, are permutations of [l 2 ••• P , a generalized mismatched complementary code set of length MN can be constructed as follows: Construct p codes:

()_ —r I_S°'l ⁾ -_S°'ι ⁾ >...._S°'lif⁾ -°_s2^f) '°e2^/,) '... >^a _S'2i;⁾ .

_S ^P) .. _S'i^p)

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

A A ⁾_ -_X X(^I).H "-M M ' ^Ά'W- -X'W. •H "M M '

^ 'J.H ^LM,xM ' ^ ^{Λ )}= ~_X ^Λ'W ".H "M_; xM '

Rows of A^,A'^,l<t≤P are expanded mismatched codes with length N M , and there are PM such codes;

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

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