WO2003081797A1

WO2003081797A1 - A coding method to create general spread spectrum sequence with zero correlation window

Info

Publication number: WO2003081797A1
Application number: PCT/CN2002/000193
Authority: WO
Inventors: Shaojun Xu; Yan Gao; Daoben Li
Original assignee: Linkair Communications, Inc.
Priority date: 2002-03-22
Filing date: 2002-03-22
Publication date: 2003-10-02
Also published as: CN1613193A; AU2002250774A1

Abstract

The present invention provide a coding method to create general spread spectrum sequence with zero correlation window, wherein general spread spectrum sequence with zero correlation window include binary codes at least in complex field. The present invention is to solve the problems remains in prior art, including the fatal near-far effects in traditional CDMA radio communications and leads to many new codes which have a much wider range of available lengths and window widths of zero correlations. All codes constructed by the method are close to the theoretical bound.

Description

A Coding Method to Create General Spread Spectrum Sequence With Zero Correlation Window

Field of the invention

The present invention relates to the field of telecommunications, and more particularly to a coding method to create general spread spectrum sequence 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 already can't satisfy such demand because of 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 is a soft- capacity and it lies in the interference level, i.e. any technique to reduce interference can directly increase the CDMA system capacity. However, system capacity of other MAC schemes is a hard-capacity and it is decided before design.

Now that CDMA system capacity lies in the system interference level, how to reduce the system interference is crucial to increase the CDMA system capacity. 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 a user receiving from other users, and 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 group 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 group with ideal

ACF and ideal CCF. Unfortunately ACF and CCF of any code group are bounded by Welch bound. According to Welch bound, ACF and CCF cannot be decreased simultaneously. So it is impossible to find a code group with ideal ACF and ideal CCF.

But in many applications, it is not necessary to construct such code group with ideal ACF and ideal CCF in all time shifts. And it is enough for a synchronized CDMA system to ensure the ideal ACF and ideal CCF within maximum time spread of the channel. For example, if the maximum time spread of the channel 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 spread spectrum multiple access code with zero correlation window (ZCW). And his work has been granted a patent (patent application number is PCT/CN00/00028 ). Given the code length, the size of LS code set is much greater than before, and it has great worth in CDMA applications.

LS code provides a generation tree to create the complementary orthogonal codes with zero correlation windows of width 2N-1 from a pair of uncorrelated complementary orthogonal codes with code length N. The family size of access codes of the expanded code group at level M is 2^(M+I), and the code length at level M is 2^MN. Such code group is denoted by (2 ^(M+1), 2^MN, N), wherein 2 ^(M+1) is number of access codes of new expanded code group, 2^MN is the code length and N is one side window width of zero correlation. A code group with zero correlation window is denoted by (K, L, W), where K is family size of access codes of code group, L is the code length of C part or S part of complementary codes and W is one side window width of zero correlation. According to the modified Welch bound, the following inequality holds: K^(2L+2W-2)/W. For LS code K=2 ^(M+I), L=2^MN, W=N, so K=2L W. LS code has approached to the theoretical bound when L > > W But family size of access codes of LS code is limited to 2 ^(M+1).

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 mutually orthogonal 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 H_M„_M expansion method to create a class of complementary orthogonal ZCW codes from a pair of uncorrelated complementary orthogonal codes with code length N. But M and N must be prime each other in his 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 general spread spectrum sequence with zero correlation window. The present invention is a new coding method to create a class of general spread spectrum multiple access codes that have the "Zero Correlation Window" in their auto-correlation functions and cross-correlation functions. The new construction method leads to many new codes which have a much wider range of available lengths and window widths of zero correlations. Due to the creation of the "zero correlation window", the fatal near-far effects in traditional CDMA radio communications will be overcome. All codes constructed by the method approach to the theoretical bound. The present invention provide a coding method to create general spread spectrum sequence with zero correlation window, wherein general spread spectrum sequence with zero correlation window include binary codes in complex field. wherein said general spread spectrum with zero correlation window sequence include ternary codes in complex field. wherein said general spread spectrum sequence with zero correlation window include poly-phase codes in complex field. wherein said general spread spectrum sequence with zero correlation window include any other ZCW codes in complex field. wherein said general spread spectrum sequence with zero correlation window include binary codes , ternary codes , poly-phase codes and any other ZCW codes in complex field. wherein said binary codes include :

In binary field, family size of LS code is only 2", wherein n is a positive integer; because there exist orthogonal matrices H,₂.,₂, we can construct binary ZCW codes with family size 12*2 according to new construction method; so said method will provide wider range of family size of binary ZCW codes. wherein said ternary codes include :

In binary field, there only possibly exist complementary codes for N=2^a10^b26^c, so the width of zero correlation windows of binary codes only takes 1, 2, 4, 8, ^•••; but there exist ternary complementary codes with zero correlation windows for any positive N, so ternary complementary codes with any ZCW length can be constructed; furthermore, there exist binary orthogonal matrices H_{M *M} only when M is equal to

2,4,8, 12, ^{• • •} , while there exist ternary matrices H_M * _M for any positive M; said method leads to many new codes which have a much wider range of available lengths and window widths of zero correlations, thus we can select proper ZCW length and family size of complementary codes according to the channel condition and system demand. wherein said poly-phase codes include :

In binary field, there only possibly exist complementary codes for N=2^a10^b26^c,so the width of zero correlation windows of binary codes only takes 1, 2, 4, 8, ^{•• ■}; but there exist poly-phase complementary codes with zero correlation windows for any positive N, so poly-phase complementary codes with any ZCW length can be constructed; furthermore, there exist binary orthogonal matrices H_MAM only when M is equal to 2,4,8,12, ^{■ • •}, while there exist poly-phase matrices H_M*M for any positive M; said method leads to many new codes which have a much wider range of available lengths and window widths of zero correlations, thus we can select proper ZCW length and family size of complementary codes according to the channel condition and system demand, wherein said the method includes the following steps:

Selecting a complementary orthogonal code (Cl, SI) with code length N, where the aperiodic auto-correlation functions of code C and code S oppose each other but also complement each other except at the origin; selecting another complementary orthogonal code (C2, S2) which is uncorrelated with the code (Cl, SI), i.e. their aperiodic cross-correlation functions of code C and code S between two codes sum to zero at all corresponding time shifts;

It can be proved that code (Cl, Sl) and code SI ,-Cl^* j are uncorrelated with each other, where ' ^~ ' denotes reversing operation, '*' denotes the complex conjugate operation; selecting any orthogonal matrix, H_MxM which has M rows and M columns, the new codes can be constructed:

H_MxM ® Cl, H_MxM ® Sl

H_MxM ® C2, H MxM <g> S2 where ' <8> ' denotes the Kronecker product; the auto-correlation functions or cross-correlation functions of the expanded code group will form a zero correlation window around the origin with the size of window - N. wherein said the method includes the following steps:

( Cl Sl \ If < _\2 J *^{s a} P^{a r} °^ ^uncoπ"elated complementary orthogonal codes with code length

N, and the ternary orthogonal matrix H 3x3 + - 0 is adopted, wherein "+" denotes +1, "0" 0 0 + denotes 0 and "-" denotes -1 ; the expanded code group is:

when N=5, a pair of uncorrelated ternary complementary orthogonal codes is:

according to the said expansion method, the expanded code group: Cl Cl 0 SI SI 0 (+ + + -0 + + +-000000, +-++0+-++000000^ Cl - Cl 0 SI - SI 0 +++-0 +000000, +-+ +0-+--000000

0 0 Cl 0 0 SI 0000000000+++-0, 0000000000+-++0 C2 C2 0 S2 S2 0 ++-+0++-+000000, + 0+ 000000 C2 - C2 0 S2 - S2 0 ++-+0--+-000000, + 0-+++000000

0 0 C20 0 S2 0000000000 + + -+0, 0000000000 + 0 the width of zero correlation window will be N=5; family size of the expanded code group is 6; the equivalent transformation can be applied to generate basically ternary complementary ZCW code group. wherein said the method includes the following steps:

The uncorrelated complementary orthogonal quadri-phase code pair is:

Cl Sl^l f 010, 002^1 , . . . . . , J C2 S2 ⁼ 200 212 ' ^{erein ger e es e}

The expansion matrix is: '0 0

H ^[ 2.x.2. = 02

The expanded code group is: ^f Cl Cl SI SI ^010010, 002002 Cl - Cl SI - SI 010230, 002220

C2 C2 S2 S2 200200,212212 C2 - C2 S2 -S2 ^200022, 212030

It can be verified that the zero correlation window width of expanded code group is 3, and family size of expanded code group is 4;

If the orthogonal matrix H_MxM is defined:

a ZCW code group (2M, 3M, 3) is obtained;

The proposed ZCW codes are expanded by an orthogonal matrix Η_M._M from a pair of uncorrelated complementary orthogonal codes with code length N; the new expanded code group is the Kronecker product of the original uncorrelated complementary orthogonal code pair and orthogonal expansion matrix, The expanded code group can be denoted by (K, L, W)= (2M, MN, N), wherein M, N are any positive integer;

The proposed code groups satisfy: K=2L/W, but the family size of access codes of code group is not limited to K=2 ^(M+1) any more, and code length are also not only 2^MN; said method leads to many new codes which have a much wider range of available lengths and window widths of zero correlations; said method can construct complementary codes with any even family size K, zero correlation window width W and code length L which satisfy K=2L/W; said method is also available for generalized complementary codes, such complementary code triads and complementary quads etc.

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

All codes presented in said method approach to the theoretical bound. The aim of present invention is to solve the problems remains in prior art, including the fatal near-far effects in traditional CDMA radio communications. The present invention leads to many new codes which have a much wider range of available lengths and window widths of zero correlations. All codes constructed by the method approach to the theoretical bound.

Detail of the invention A new construction method is proposed to create a class of general spreading ZCW codes including binary codes, ternary codes and poly-phase codes etc. The proposed ZCW codes are expanded by an orthogonal matrix H_M._M from a pair of uncorrelated complementary orthogonal codes with code length N. The expanded code group is the Kronecker product of the original uncorrelated complementary orthogonal code pair and orthogonal expansion matrix. The expanded code group can be denoted by (K, L, W)= (2M, MN, N), wherein M, N are any positive integer. The proposed code groups satisfy: K=2L W. But the family size of access codes of code group is not limited to K=2 ^(M+1) any more, and code length are also not only 2^MN. It is shown that new construction method leads to many new codes that have a much wider range of available lengths and window widths of zero correlations. The present invention will be detailedly described with reference to the preferred embodiments and the tables.

First we give an example of ternary complementary code with code length N=5: Cl = (+ +0 + -), SI = (+ -0 + +), wherein '+' denotes ' 1 ' and '-' denotes '-1 '. It is true that the aperiodic auto-correlation of code (Cl,Sl) is equal to zero except at the origin, i.e. code C and code S are complementary each other. Now define:

The aperiodic auto-correlation function of code C is: R_cl(r) = Σ c,( )c^*.(ι + r),

wherein τ is the time shift. N-\-τ

The aperiodic auto-correlation function of code S is: R^, (τ) = ∑Σ -s, (t)s^*ι (/ + τ) ,

wherein τ is the time shift.

And the aperiodic auto-correlation function of code 1 is: R, (r) = R_cl (r)+ R_sl (r) Table 1 is for the aperiodic auto-correlation values of code 1.

Table 1: Auto-Correlation of Code 1

According to the said uncorrelated complementary orthogonal code construction method from a known code, we can obtain another uncorrelated complementary orthogonal code.

(C2, 52) = (Sϊ⁷ ,-CV )= (+ +0 - +,+ - 0 — )

We define:

The cross-correlation between code 1 and code 2 are:

R (τ)= ∑ c.(/)c^* ₂(t + r), R »= if -s,( Λ(ι + r)

Table 2 is for the auto-correlation values of code 2.

Table 3 is for the cross-correlation values between code 1 and code 2.

Table 2: Auto-Correlation of Code 2.

Table 3: Cross-Correlation between Code 1 and Code 2.

There is only one basic form for the uncorrelated complementary orthogonal codes with family size of access code 2 and each code length 5. Other forms can be derived from reordering of Cl and C2, SI and S2, 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 any orthogonal matrix, the expanded code group with longer length can be derived from the original uncorrelated complementary orthogonal codes. Suppose the

orthogonal is H_2x2 = then we can achieve new code group with family size 6 and

code length 15.

Cl l 0 SI SI 0 ^f+ +0 + - + +0 + -00000, + -0 + + + -0 + +00000^Λ

Cl - Cl 0 SI -SI 0 + +0 + 0 - +00000, + -0 + + - +0 - -00000

0 0 Cl 0 0 SI 0000000000 + +0 + -, 0000000000 + -0 + +

C2 C2 0 S2 S2 0 + +0 - + + +0 - +00000, + -0 - - + -0 — 00000

C2 - C2 0 S2 - S2 0 + +0 - + - -0 + -00000, + -0 +0 + +00000

0 0 C2 0 0 S2 0000000000 + +0-+, 0000000000 - +0 + +

Table 4 is for the auto-correlation and cross-correlation functions of the new expanded code group. Table 4: Auto-Correlation and Cross-Correlation of the New Expanded Code

The width of zero correlation window of new expanded code group is equal to 5. And the family size of new code group is 6. In binary field, it is impossible to construct complementary codes with family size 6 and ZCW width 5.

The existence of zero elements of ternary complementary ZCW codes results in different processing gains between different members in the code group. If we extend ternary complementary orthogonal codes to poly-phase complementary orthogonal codes, Complementary orthogonal ZCW code without zero elements will be obtained. The following example is given: The original pair of uncorrelated complementary orthogonal quadri-phase codes is:

Cl Slλ f 010, 002^1 , . . , . . _t j-2π r C2_? S_?2j ^=" l 2n0n0, 2 _{? 1}1 _?2j ' ^whereιn integer l denotes

The expansion matrix is:

The expanded code group is:

' l l SI SI ^Λ f 010010, 002002^ l - Cl SI - SI 010230, 002220 C2 C2 S2 S2 200200, 212212 C2 - C2 S2 - S2 200022, 212030

It can be verified that the zero correlation window width of expanded code group is 3, and family size of expanded code group is 4.

If the orthogonal matrix H_MxM is defined:

A ZCW code group (2M, 3M, 3) is obtained.

Table 5 is for examples of ternary uncorrelated complementary orthogonal codes for any length.

Table 6 is for examples of ternary orthogonal matrices for any length.

Table 7 is for examples of poly-phase uncorrelated complementary orthogonal codes for any length.

Table5: Ternary Uncorrelated Complementary Tableό: Ternary Orthogonal

Orthogonal Codes Expansion Matrices

Table 7: Poly-Phase Uncorrelated Table 8: Poly-Phase Orthogonal Expansion Complementary Orthogonal Codes Matrices.

Wherein integer i denote the complex number j±-2* e ^M .

Up to now, we can construct complementary codes with any even family size K, zero correlation window width W and code length L which satisfy K=2L/W. The construction method is also available for generalized complementary codes, such complementary code triads and complementary quads etc. For example there are four uncorrelated complementary orthogonal code quads with code length 4: Code 1: ++++, +-+-, -++, -++- Code 2: -+-+, — , +--+, ++-- Code 3 : --++, -++-, ++++, +-+-

Code 4: +--+, ++--, -+-+, —

Their auto-correlations of each part sum to zero except at the origin and their cross- correlations of each part sum to zero at all the corresponding shifts.

Expanding the original uncorrelated complementary orthogonal code quads by an orthogonal matrix, we can obtain a generalized complementary ZCW code (4M, 4M, 4). According to Frank's research, for quadri-phase complementary codes, the possible lengths up to 100 are the following.

Binary complementary codes: 1,2,4,8,10,16,20,26,32,40,52,64,80,100. Other quadriphase complementary codes: 3,5,6,12,13,18,24,30,36,48,50,60,72,78,96 (lengths 7,9,11,15,17 do not exist).

Quadriphase triads: Except 50, all above plus: 7,9,11,14, 15, 17, 19, 21, 22, 25, 27, 33, 37, 39, 41, 42, 45,49,51,53,54,57,58,61,63,65,66,73,75,81,90,97,99. Quadriphase quads: All lengths except 71,89. π More than four sequences: All lengths.

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

The reference include:

[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] N.M. Sidelnikov, On mutual correlation of sequences, Soviet math. Dokl., vol.12, pp. 197-201, 1971. [5] P.Z. Fan, Ν. Suehiro, Ν. 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 general spread spectrum sequence with zero correlation window, wherein general spread spectrum sequence with zero correlation window include binary codes at least in complex field.

2. A method of claim 1, wherein said general spread spectrum with zero correlation window sequence include ternary codes in complex field.

3. A method of claim 1, wherein said general spread spectrum sequence with zero correlation window include poly-phase codes in complex field.

4. A method of claim 1, wherein said general spread spectrum sequence with zero correlation window include any other ZCW codes in complex field.

5. A method of claim 1, wherein said general spread spectrum sequence with zero correlation window include binary codes , ternary codes , poly-phase codes and any other ZCW codes in complex field.

6. A method of claim 1, wherein said binary codes include : In binary field, family size of LS code is only T, wherein n is a positive integer; because there exist orthogonal matrices H₁₂.₁₂, we can construct binary ZCW codes with family size 12*2; so said method will provide wider range of family size of binary ZCW codes.

7. A method of claim 2, wherein said ternary codes include : In binary field, there only possibly exist complementary codes for N=2^a10^b26^c, so the width of zero correlation windows of binary codes only takes 1, 2, 4, 8, ^{• ••}; but there exist ternary complementary codes with zero correlation windows for any positive N, so ternary complementary codes with any ZCW length can be constructed; furthermore, there exist binary orthogonal matrices H_M*_M only when M is equal to 2,4,8,12, ^{• • •}, while there exist ternary matrices H_M*_M for any positive M; said method leads to many new codes which have a much wider range of available lengths and window widths of zero correlations, thus we can select proper ZCW length and family size of complementary codes according to the channel condition and system demand.

8. A method of claim 3, wherein said poly-phase codes include : In binary field, there only possibly exist complementary codes for N=2^a10^b26^c,so the width of zero correlation windows of binary codes only takes 1, 2, 4, 8, ^{• ••}; but there exist poly-phase complementary codes with zero correlation windows for any positive N, so poly-phase complementary codes with any ZCW length can be constructed; furthermore, there exist binary orthogonal matrices H_M*_M only when M is equal to 2,4,8,12, ^{• • •}, while there exist poly-phase matrices H_M*_M for any positive M; said method leads to many new codes which have a much wider range of available lengths and window widths of zero correlations, thus we can select proper ZCW length and family size of complementary codes according to the channel condition and system demand.

9. A method of claim 1, wherein said the method includes the following steps: Selecting a complementary orthogonal code (Cl, SI) with code length N, where the aperiodic auto-correlation functions of code C and code S oppose each other but also complement each other except at the origin; selecting another complementary mismatched code (C2, S2) which is uncorrelated with the code (Cl, SI), i.e. their aperiodic cross-correlation functions of code C and code S between two codes sum to zero at all time shifts;

It can be proved that code (Cl, Si) and code SI ,-Cl^* j are uncorrelated with each other, where ' ^ denotes reversing operation, '*' denotes the complex conjugate operation; selecting any orthogonal matrix, H_MxM which has M rows and M columns, the new codes can be constructed:

(H_MxM ® Cl, H_MxM ® Sl) [H_MXM ® C2, H_MXM ® S2) where ' Θ ' denotes the Kronecker product; the auto-correlation functions or cross-correlation functions of the expanded code group will form a zero correlation window around the origin with the size of window ^ N.

10. A method of claim 2, wherein said the method includes the following steps:

( Cl Sl If ₍-2 _<? 2 ^^{s a} P^a*^r °f ^uncorrelated complementary orthogonal codes with code length

when N=5, a pair of ternary uncorrelated complementary orthogonal codes is: Cl Sl + + + -0, + - + +0 C2 S2 ^~ + + - +0, + 0 according to the said expansion method, the expanded code group:

'Cl Cl 0 SI SI 0 ^ '+ + + -0 + + + -000000, + - + +0 + - + +000000^Λ l - Cl 0 SI - SI 0 + + + -0 +000000, + - + +0 - + - -000000

0 0 Cl 0 0 SI 0000000000+++-0, 0000000000+-+ +0 C2 C2 0 S2 S2 0 + +-+0++ -+000000, + 0+ 000000 C2 - C2 0 S2 - S2 0 + + -+0 —+ -000000, + 0-+ + +000000

0 0 C20 0 S2 0000000000 + + - +0, 0000000000 + 0 the width of zero correlation window will be N=5; family size of the expanded code group is 6; the equivalent transformation can be applied to generate basically ternary complementary ZCW code group.

11. A method of claim 3 or 4, wherein said the method includes the following steps: The uncorrelated complementary orthogonal quadri-phase code pair is:

The expansion matrix is:

The expanded code group is:

Cl Cl SI SI ^ 010010, 002002^" Cl - Cl SI - SI 010230, 002220 C2 C2 S2 S2 200200, 212212 C2 - C2 S2 - S2 200022, 212030

If the orthogonal matrix H_MxM is defined:

H MxM

a ZCW code group (2M, 3M, 3) is obtained;

The proposed ZCW codes are expanded by an orthogonal matrix Η_M._M from a pair of mutually orthogonal complementary codes with code length N; the new expanded code group is the Kronecker product of the uncorrelated complementary orthogonal pair and orthogonal expansion matrix, The expanded code group can be denoted by (K, L, W)= (2M, MN, N), wherein M, N are any positive integer;

12. A method of claim 1 or 2 or 3 or 4 or 5, equivalent transformations don't change the

ZCW properties of proposed complementary ZCW codes.

13. A method of claim 1 or 2 or 3 or 4 or 5, all codes presented in said method are close to the theoretical bound.