Spreading Code Construction for CDMA - Semantic Scholar
1
Spreading Code Construction for CDMA Zhenning Shi, Christian Schlegel
Abstract |In this letter, we use the well known WalshHardmard codes to construct larger sets of linearly dependent codes such that their matrix of correlations is band-diagnal. Optimal detection for system employing these spreading codes can be eciently accomplished by a trellis decoder. Analysis and simulation show that with a processing gain of N , up to 2N users can be accommodated without much performance degradation. This proposed approach is quite general and larger code set can be constructed analogously. Keywords | Keywords: synchronous CDMA, crosscorrelation, Walsh-Hardmard codes, trellis decoding.
I. Introduction
T is known that optimal detection of a multiple ac-
Icess system, in general, requires a number of op-
decoding complexity.
Assumption:
II. Background
Multiple access system with K active users. Spreading sequences with N chips. Symbol-synchronous system.
The received signal is r = Ad + n; (1) where A is a N K real matrix whose columns are the spreading sequences of the K users, d is a length-K data vector, and n is a sampled white Gaussian noise vector. Passing r through a bank of matched lters, we obtain y = AT r = Rd + z; (2) where R = frij g = AT A is the K K correlation matrix between the spreading sequences, and z = AT n is colored noise. In AWGN, the optimal detector minimizes the Euclidean distance between the received signal and transmission candidates, i.e., d^ = arg min jjr , Adjj2; d2f,1;+1gK (3) which is equivalent to minimizing the quadratic: d^ = arg min(dT Rd , 2dT y ): d2f,1;+1gK (4)
erations that increases exponentially with the number of users. In some special cases, however, optimal detection can be much simpler. In [1], Schlegel and Grant propose a polynomial complexity algorithm for a system where all cross-correlation values are identical. In [2], guidelines for building a set of spreading sequences with a tree-structured relationship of the cross-correlations are presented. Wavelet packets and the work of Ross and Taylor [3] ts nicely into this structure. Spreading sequences built according to this tree structure consist of a linearly dependent set, i.e., the number of spreading sequences are larger than the dimensions spanned, this is refered to as the oversaturated case. An optimal tree joint detection algorithm with a low-order-polynomial complexity is proposed in [2] based on the hirarchical form of the spreading sequence set. In [4] it is shown that if the crosscorrelation between each pair of spreading sequences is nonpositive, then the optimum detection problem can III. Trellis Structure be transformed into the minimum cut problem, which can be solved with polynomial complexity in the num- The quadratic form in (4) can be calculated recurber of users, K . However, an upper bound on the sively. De ne cardinality of such a signal set in N -dimensional space n X n n X X shows that there are less than b3N=2c such sequences J (n) = di dj rij , 2 di yi (5) in the set. i=1 j =1 i=1 In this project, the spreading sequences are constructed such that the correlation matrix is band- as the partial metric. Updating of this partial metric diagonal, and optimal detection can be performed us- is performed according to n ing a trellis structure. For N -dimensional space, the X number of spreading sequences constructed can be up J (n + 1) = J (n) + 2 dn+1,idn+1 rn+1,i;n+1 i=1 to 2N , but the procedure is quite general and more sequences can be constructed at the cost of increased , 2dn+1yn+1 : (6)
2
If the correlation matrix R is band-diagonal, i.e., rij = 0; for ji , j j > L then (6) becomes L X
J (n + 1) = J (n) + 2
,
i=1 2dn+1yn+1 ;
dn+1,idn+1 ri;n+1
(7) which can be calculated by a trellis decoder with 2L states without loss of optimality. The overall complexity is O(2L+1 K ).
60
50
40
30
20
IV. Construction of spreading sequences
Walsh-Hardmard codes are a set of orthogonal codes which are built recursively from the basic building block +1 H2 = +1 +1 ,1 Fig. 1. Code Set Correlation Matrix according to = 2jjhj jj 2n H2n : H2n+1 = H p H2n ,H2n = 2 N; (9) However, for length N , the maximum number of such where a ; b 2 f,1; +1g. i i orthogonal spreading sequences is N . We introduce adThe signal minimum distance of the new signal set ditional spreading sequences by interpolating the orig- is inal Walsh-Hardmard codes. The cost of this are: The spreading sequences become correlated, which K causes a system performance loss. 2N = min jj X(ai , bi)w jj Dmin users can not be detected independently, and this i i=1 increases complexity. The proposed spreading sequence set is formed as = 2jjw2j , w2j +2jj follows j +4 j +3 X X hk jj = 2jj0:5 hk , 0:5 w2i,1 = hi ; k=j +1 k=j 10
10
w2i = 1=2
i+3 X
i = 1; 2; : : :; N , 3;
hk ;
k=i hN ,3+i ;
w2N ,6+i = i = 1; 2; 3; (8) where hi is the ith vector in HN . The new spreading sequence set has size 2N , 3. The band-diagonal correlation matrix of the new spreading code set is shown in Figure (1). These spreading sequences are correlated over a nite span with band-diagonal correlation matrix in (2). Their p signal minimum distance is shortened by a factor of 2 w.r.t orthogonal sequences, whose signal minimum distance is N Dmin = min jj
K X i=1
(ai , bi)hi jj
20
30
40
50
60
p
= jjhj , hj +4 jj = 2N:
(10) This is equivalent to a 3 dB energy loss. Since the memory of the correlation matrix L = 6 the decoding complexity is O(27K ). Using this method, we can further extend the spreading code set to include more sequences. However, this leads to a higher correlation between sequences and requires more states in the trellis decoder. With more sequences generated, it proved dif cult to construct a set which could be analytically shown to possess the largest signal minimum distance, thus larger sets containing 3N or even more sequences are obtained by trial and error. The following is an attempt to form a (3N-5)-sequence set based on the
3
Walsh-Hardmard code set. Compared to the original orthogonal code set, it combines an a ordable energy loss with reasonable complexity incured by its correlation memory. The set is
0
10
−1
10
v 3i,2 = w2i,1 = hi
p
v3i = 1= 3
i+2 X
k=i hN ,3+i
k=i
hk
−2
10
hk
BER
v 3i,1 = w2i = 1=2
i+3 X
−3
10
i = 1; 2; : : :; N , 3
i = 1; 2; 3 v 3N ,8+i = Its signal minimum distance is
(11)
K 3N = minjj X(ai , bi)v jj Dmin ki i=1 = 2jjv3j , v 3j ,1jj j +3 j +2 X p X = 2jj1= 3 hk , 1=2 hk jj k=j k=j q p = 2 [(1= 3 , 1=2)2 + 1=4]N
p
10
−5
10
VI. Conclusion
We have presented a method to increase data rates of synchronous CDMA systems using interpolated Hard-
0
2
4
6
8
10
12
Fig. 2. Simulation Results
mard spreading codes. The performance of such a system is comparable to that of using higher order modulations for which it may be an alternative.
[2] Parameters: Sequences: Interpolated Hardmard-Walsh w1 ; w2 ; : : :. { w2i,1 = hiPis a Hardmard-Walsh code, [3] i = 1; 2; : : :; N { w2i = 1=2 ik+3 =i hk Code Length: N = 32. [4] Decoder: 64-state (L = 6) trellis decoder. Energy Loss (minimum signal distance): 3 dB.
In Figure (2), the ratio K=N is up to 1.91, and the corresponding performance loss is about 4 dB, while for smaller number of users the loss is 3 dB as predicted by the signal minimum distance. In high SNR ( BER 10,3 ), all these simulated curves lie between the perfromance curves of BPSK and 4-PAM. In fact, the performance of 4-PAM and K = 61 users is virtually identical. Note that both systems o er the same data rate.
−2
SNR (dB)
(12) = 2 0:2679N: This is eqivalent to a 5:7 dB energy loss. Since L = 9, thus 29 states are required at each stage of the trellis [1] decoder, and the decoding complexity is O(210K ). V. Simulations
single user BPSK 7user 15user 25user 33user 47user 61user
−4
References C. Schlegel and A. Grant, \Polynomial Complexity Optimal Detection of Certain Multiple Access Systems", submitted to IEEE Transactions on Information Theory, May, 1999. R. E. Learned, A. S. Willsky and D. M.Boroson, \Low complexity optimal joint detection for oversaturated multiple access communications". IEEE Transactions on Signal Processing, vol. 45, no. 1, Jan., 1997. J. A. F. Ross and D. P. Taylor, \Vector assignment scheme for M + N users in N -dimensional global additive channel", Electron. Lett.. vol. 28, Aug., 1992. C. Sankaran and A. Ephremides, \Solving a class of optimum multiuser detection problems with polynomial complexity", IEEE Transactions on Information Theory, vol. 44, no. 5, Sept., 1998.
Spreading Code Construction for CDMA Zhenning Shi, Christian Schlegel
Abstract |In this letter, we use the well known WalshHardmard codes to construct larger sets of linearly dependent codes such that their matrix of correlations is band-diagnal. Optimal detection for system employing these spreading codes can be eciently accomplished by a trellis decoder. Analysis and simulation show that with a processing gain of N , up to 2N users can be accommodated without much performance degradation. This proposed approach is quite general and larger code set can be constructed analogously. Keywords | Keywords: synchronous CDMA, crosscorrelation, Walsh-Hardmard codes, trellis decoding.
I. Introduction
T is known that optimal detection of a multiple ac-
Icess system, in general, requires a number of op-
decoding complexity.
Assumption:
II. Background
Multiple access system with K active users. Spreading sequences with N chips. Symbol-synchronous system.
The received signal is r = Ad + n; (1) where A is a N K real matrix whose columns are the spreading sequences of the K users, d is a length-K data vector, and n is a sampled white Gaussian noise vector. Passing r through a bank of matched lters, we obtain y = AT r = Rd + z; (2) where R = frij g = AT A is the K K correlation matrix between the spreading sequences, and z = AT n is colored noise. In AWGN, the optimal detector minimizes the Euclidean distance between the received signal and transmission candidates, i.e., d^ = arg min jjr , Adjj2; d2f,1;+1gK (3) which is equivalent to minimizing the quadratic: d^ = arg min(dT Rd , 2dT y ): d2f,1;+1gK (4)
erations that increases exponentially with the number of users. In some special cases, however, optimal detection can be much simpler. In [1], Schlegel and Grant propose a polynomial complexity algorithm for a system where all cross-correlation values are identical. In [2], guidelines for building a set of spreading sequences with a tree-structured relationship of the cross-correlations are presented. Wavelet packets and the work of Ross and Taylor [3] ts nicely into this structure. Spreading sequences built according to this tree structure consist of a linearly dependent set, i.e., the number of spreading sequences are larger than the dimensions spanned, this is refered to as the oversaturated case. An optimal tree joint detection algorithm with a low-order-polynomial complexity is proposed in [2] based on the hirarchical form of the spreading sequence set. In [4] it is shown that if the crosscorrelation between each pair of spreading sequences is nonpositive, then the optimum detection problem can III. Trellis Structure be transformed into the minimum cut problem, which can be solved with polynomial complexity in the num- The quadratic form in (4) can be calculated recurber of users, K . However, an upper bound on the sively. De ne cardinality of such a signal set in N -dimensional space n X n n X X shows that there are less than b3N=2c such sequences J (n) = di dj rij , 2 di yi (5) in the set. i=1 j =1 i=1 In this project, the spreading sequences are constructed such that the correlation matrix is band- as the partial metric. Updating of this partial metric diagonal, and optimal detection can be performed us- is performed according to n ing a trellis structure. For N -dimensional space, the X number of spreading sequences constructed can be up J (n + 1) = J (n) + 2 dn+1,idn+1 rn+1,i;n+1 i=1 to 2N , but the procedure is quite general and more sequences can be constructed at the cost of increased , 2dn+1yn+1 : (6)
2
If the correlation matrix R is band-diagonal, i.e., rij = 0; for ji , j j > L then (6) becomes L X
J (n + 1) = J (n) + 2
,
i=1 2dn+1yn+1 ;
dn+1,idn+1 ri;n+1
(7) which can be calculated by a trellis decoder with 2L states without loss of optimality. The overall complexity is O(2L+1 K ).
60
50
40
30
20
IV. Construction of spreading sequences
Walsh-Hardmard codes are a set of orthogonal codes which are built recursively from the basic building block +1 H2 = +1 +1 ,1 Fig. 1. Code Set Correlation Matrix according to = 2jjhj jj 2n H2n : H2n+1 = H p H2n ,H2n = 2 N; (9) However, for length N , the maximum number of such where a ; b 2 f,1; +1g. i i orthogonal spreading sequences is N . We introduce adThe signal minimum distance of the new signal set ditional spreading sequences by interpolating the orig- is inal Walsh-Hardmard codes. The cost of this are: The spreading sequences become correlated, which K causes a system performance loss. 2N = min jj X(ai , bi)w jj Dmin users can not be detected independently, and this i i=1 increases complexity. The proposed spreading sequence set is formed as = 2jjw2j , w2j +2jj follows j +4 j +3 X X hk jj = 2jj0:5 hk , 0:5 w2i,1 = hi ; k=j +1 k=j 10
10
w2i = 1=2
i+3 X
i = 1; 2; : : :; N , 3;
hk ;
k=i hN ,3+i ;
w2N ,6+i = i = 1; 2; 3; (8) where hi is the ith vector in HN . The new spreading sequence set has size 2N , 3. The band-diagonal correlation matrix of the new spreading code set is shown in Figure (1). These spreading sequences are correlated over a nite span with band-diagonal correlation matrix in (2). Their p signal minimum distance is shortened by a factor of 2 w.r.t orthogonal sequences, whose signal minimum distance is N Dmin = min jj
K X i=1
(ai , bi)hi jj
20
30
40
50
60
p
= jjhj , hj +4 jj = 2N:
(10) This is equivalent to a 3 dB energy loss. Since the memory of the correlation matrix L = 6 the decoding complexity is O(27K ). Using this method, we can further extend the spreading code set to include more sequences. However, this leads to a higher correlation between sequences and requires more states in the trellis decoder. With more sequences generated, it proved dif cult to construct a set which could be analytically shown to possess the largest signal minimum distance, thus larger sets containing 3N or even more sequences are obtained by trial and error. The following is an attempt to form a (3N-5)-sequence set based on the
3
Walsh-Hardmard code set. Compared to the original orthogonal code set, it combines an a ordable energy loss with reasonable complexity incured by its correlation memory. The set is
0
10
−1
10
v 3i,2 = w2i,1 = hi
p
v3i = 1= 3
i+2 X
k=i hN ,3+i
k=i
hk
−2
10
hk
BER
v 3i,1 = w2i = 1=2
i+3 X
−3
10
i = 1; 2; : : :; N , 3
i = 1; 2; 3 v 3N ,8+i = Its signal minimum distance is
(11)
K 3N = minjj X(ai , bi)v jj Dmin ki i=1 = 2jjv3j , v 3j ,1jj j +3 j +2 X p X = 2jj1= 3 hk , 1=2 hk jj k=j k=j q p = 2 [(1= 3 , 1=2)2 + 1=4]N
p
10
−5
10
VI. Conclusion
We have presented a method to increase data rates of synchronous CDMA systems using interpolated Hard-
0
2
4
6
8
10
12
Fig. 2. Simulation Results
mard spreading codes. The performance of such a system is comparable to that of using higher order modulations for which it may be an alternative.
[2] Parameters: Sequences: Interpolated Hardmard-Walsh w1 ; w2 ; : : :. { w2i,1 = hiPis a Hardmard-Walsh code, [3] i = 1; 2; : : :; N { w2i = 1=2 ik+3 =i hk Code Length: N = 32. [4] Decoder: 64-state (L = 6) trellis decoder. Energy Loss (minimum signal distance): 3 dB.
In Figure (2), the ratio K=N is up to 1.91, and the corresponding performance loss is about 4 dB, while for smaller number of users the loss is 3 dB as predicted by the signal minimum distance. In high SNR ( BER 10,3 ), all these simulated curves lie between the perfromance curves of BPSK and 4-PAM. In fact, the performance of 4-PAM and K = 61 users is virtually identical. Note that both systems o er the same data rate.
−2
SNR (dB)
(12) = 2 0:2679N: This is eqivalent to a 5:7 dB energy loss. Since L = 9, thus 29 states are required at each stage of the trellis [1] decoder, and the decoding complexity is O(210K ). V. Simulations
single user BPSK 7user 15user 25user 33user 47user 61user
−4
References C. Schlegel and A. Grant, \Polynomial Complexity Optimal Detection of Certain Multiple Access Systems", submitted to IEEE Transactions on Information Theory, May, 1999. R. E. Learned, A. S. Willsky and D. M.Boroson, \Low complexity optimal joint detection for oversaturated multiple access communications". IEEE Transactions on Signal Processing, vol. 45, no. 1, Jan., 1997. J. A. F. Ross and D. P. Taylor, \Vector assignment scheme for M + N users in N -dimensional global additive channel", Electron. Lett.. vol. 28, Aug., 1992. C. Sankaran and A. Ephremides, \Solving a class of optimum multiuser detection problems with polynomial complexity", IEEE Transactions on Information Theory, vol. 44, no. 5, Sept., 1998.
