Code-search for Optimal TSC Binary Sequences with Low ... - CiteSeerX

Code-search for Optimal TSC Binary Sequences with Low Cross-Correlation Spectrum Panayiotis D. Papadimitriou∗,? and Costas N. Georghiades∗ ∗ ?

Electrical Engineering Department, Texas A&M University, College Station, TX 77843 NOKIA INC., Nokia Mobile Phones, 6000 Connection Dr., MS 2:200, Irving, TX 75039 e-mail: [email protected], [email protected]

Abstract— It has been shown that signature (multi-) sets that meet Welch’s lower bound on the total squared correlation (TSC), maximize the sum capacity of the synchronous code division multiple access (CDMA) channel with equal average input energy constraints. However, a signature set that is TSC-optimum (i.e. achieves the lower bound on TSC), doesn’t necessarily have a low cross-correlation spectrum. Hence, for various system loads there may be users that experience high probability of error due to high cross-correlation of their assigned signature to others in the set. In this paper we present a search method to obtain new optimum TSC binary signature sets with low cross-correlation spectrum. We also show, based on our search method, that if we relax the TSC optimality constraint, codes with better crosscorrelation spectrum may be found. In addition, a catastrophic binary code (i.e. a code having complementary codewords) can be TSC-optimal. Therefore, practically speaking, it is important that the TSC-optimality be sought along with the minimization of the cross-correlation spectrum. Finally our codes are tested on fully-loaded and overloaded synchronous CDMA systems, and simulation results show that they are a promising alternative to the Hadamard-Walsh codes, since they offer higher capacity.

I. I NTRODUCTION In [1] it was shown that the sum capacity of the synchronous code division multiple access (CDMA) channel with equal average input energy constraints is maximized with signature (multi-) sets meeting Welch’s lower bound on the total square correlation (TSC). Specifically for binary signature sets, it has been shown [2] that the sum capacity is maximized for optimum TSC signature sets (sets achieving the lower bound on TSC) of certain cardinality and length [3] (see also [1], [4]), while the rest of the optimum TSC binary signature sets exhibit negligible sum capacity loss as compared to optimum TSC real/complex sequence sets (of the same cardinality and length). On the other hand, optimum TSC sequence sets don’t imply sets with optimum cross-correlation spectrum [5]. If an optimum TSC set [2] has a few pairs with high crosscorrelation, this may result in some users experiencing high probability of error since, for example, the latter (assuming for simplicity equal user powers and BPSK modulation) can be upper-bounded by a function of the sum of the pairwise absolute and/or squared cross-correlations of the sequences of the users with respect to the sequence of the user of interest [6, §3.4]. This situation will be more important in low to moderately loaded systems. Therefore it is critical that the

TSC-optimality be sought along with the minimization of the cross-correlation spectrum. In [5] the authors presented search methods to obtain block codes with low (in some cases optimal) cross-correlation spectrum (“ρ-spectrum”). In another paper [2], Karystinos and Pados derived new TSC-bounds and developed simple designs of binary optimal TSC sequences by appropriately utilizing Hadamard matrices (codes). Although their sequences are TSC-optimal, they are not guaranteed to have low ρ-spectrum. In this paper we merge the concepts of the aforementioned papers to construct optimal-TSC signature sets with low crosscorrelations. In addition we show, based on our search method, that TSC-optimality may prevent the code from having low cross-correlation spectrum. In Section II we review and generalize the search method of [5], in Section III we modify our search method to include TSC optimality and show that the under-loaded code sets have optimal ρ-spectrum. In Section IV we evaluate some of our codes when they are used as spreading sequences in an overloaded synchronous CDMA system. Section V concludes. II. S EQUENCES WITH L OW-C ROSS - CORRELATIONS We define a (K, n) block code CK,n in general as a matrix of size (K ×n) having K binary codewords (sequences) of length n. In a K-user CDMA system, for example, with processing gain n, each user is assigned a single codeword of CK,n . Let each codeword d of length n be a (column) vector with elements from {−1, 1}. We define the pairwise crosscorrelation between two codewords di and dj as 1 ρ = dTi dj , (1) n and the ρ-spectrum Sρ , as all possible 2-tuples (|ρ|, N|ρ| ), where N|ρ| is the number of codeword pairs (excluding selfpairs) with absolute cross-correlation |ρ|, whose maximum value is denoted ¡ ¢ ρmax . Note that the sum of the multiplicities N|ρ| equals K 2 . Definition: Optimal ρ-spectrum [5]. Let the ρ-spectrum Sρ of a (K, n) block code be Sρ = {(0, N0 ), (1/n, N1/n ), . . . , (ρmax , Nρmax )}. Then the optimal ρ-spectrum Sρ∗ over all possible spectra Sρ of (K, n) block codes is the one for which one of the following is true:

1) ρ∗max < ρmax , or 2) ρ∗max = ρmax , and there exists some λ : λ/n = ∗ 0, 1/n, 2/n, . . . , ρmax , for which Nλ/n < Nλ/n , ∗ ∗ N(λ+1)/n = N(λ+1)/n , . . . , Nρmax = Nρmax .1 Codes achieving Sρ∗ are called optimal codes. In the context of a limited search, we also call optimal codes the codes that yield the best Sρ of that search. ∗ The steps of the nested search [5] to find the optimal CK,n code starting from a known CK,ξ code, ξ ≥ 1 are summarized below: 1) Set j = ξ + 1. 2) Search for the K-bit (column) c vector, such that c ∈ / ∗ CK,j−1 2 , and CK,j = [CK,j−1 , c] is optimal. Let CK,j be the optimal code of the search. ∗ 3) Set CK,j = CK,j . 4) Set j = j + 1. 5) If j > n Stop, else go to Step 2. For the search in Step 2 we let the c vector take only columns (their first K bits) of an N -Hadamard code, K ≤ N and N is close to K [5]. We can start the search [5]: (a) with CK,1 being the first column of the N -Hadamard code (ξ = 1), and (b) if N = 2k for some k, we can start with the code CK,k 3 (ξ = k). In this paper we start the search with CK,1 . In order to be consistent with [5], we assume the first element of the Hadamard code is −1 (if not, just flip all the Hadamard code’s elements). III. TSC- OPTIMAL S EQUENCES W ITH L OW ρ-spectrum Codes obtained with the nested search of Section II (hereafter called “Search A”) although have low cross-correlations, they are not guaranteed to be TSC-optimal for all load cases. Therefore we will incorporate in our search the design of [2] to guarantee the TSC-optimality for applicable code sizes [2]. For a K ×n binary code CK,n , TSC is defined as (assuming real sequences) K K 1 XX T TSC = 2 (d dj )2 n i=1 j=1 i

ρX max

N|ρ| ρ2 + K.

CK,n = [CN,n ; A],

(4)

with A a µ × n matrix, µ = K − N [2], A = vT , v ∈ {±1}n , if K = N + 1,

(5)

and if K = N + 2, ½ A=

n

(vT , vT ; vT , −vT ), v ∈ {±1} 2 , n even n−1 (vT , α1 , vT ; vT , α2 , −vT ), v ∈ {±1} 2 , n odd

(6) where α1 , α2 ∈ {±1}. Thus, our search for optimum-TSC codes with low cross-correlations proceeds as follows: Initial∗ ize CK,n to the zero matrix. For each possible A, start the nested search with CK,ξ = [CN,ξ ; Aµ,ξ ] code, where CN,ξ a known code, and Aµ,ξ the left-most sub-matrix of A of size µ × ξ, ξ ≥ 1. The steps are summarized below: 1) Set j = ξ + 1. 2) Search for the N -bit vector c such that c ∈ / CN,j−1 , and CK,j = [CN,j−1 , c; Aµ,j ] is optimal (if j = n, the ∗ optimality shall be checked also over CK,n .) ∗ 3) Let CK,j be the optimal code of Step 2. ∗ ∗ ∗ 4) Set CK,j = CK,j and CN,j = CN,j , where CN,j is the ∗ upper-most sub-matrix of CK,j of size N × j. 5) Set j = j + 1. 6) If j > n, re-start the nested search with the next possible A; if all possible A’s have been searched, Stop, else go to Step 2.

(2)

For the search in Step 2 we again let the vector c be only columns of an N -Hadamard code [2]. In this paper we start the search [5] with CN,1 the first column of the N -Hadamard code (ξ = 1), i.e. CK,1 = [CN,1 ; Aµ,1 ].

(3)

In case of large n, i.e. for unmanageable search complexity, one can set (for example in the case K = N + 1), v = [u; u; . . . ; u], where u is an n/ζ-bit vector (for some integer ζ, and n even), to reduce the search complexity, sacrificing possibly in the ρ-spectrum (for n odd, set v accordingly.)

and in terms of our ρ-spectrum, (2) can be written as TSC = 2

Let CK,n be of the form

|ρ|=0

We consider, similar to [2], the following cases: A. Overloaded case: K ≥ n Assume there exists an N -Hadamard matrix, such that [2], N = 4b K+1 4 c, and N ≥ n. Then, [2] K ∈ {N − 1, N, N + 1, N + 2}. For the first two possible values of K, Search A yields optimal TSC sequences with low ρ-spectrum. For K = N + 1 or K = N + 2 the following search applies: 1 For a code with n odd, λ is necessarily odd (from (1)) and for n even, λ is even. However, for simplicity, we do not make this distinction. 2 Meaning c is not a column of C K,j−1 . 3C K,k contains the first K codewords of systematic code CN,k (see [5]).

The following examples demonstrate the search method of this Section. Example 1: We searched for a code of K = 34 codewords of length n = 18 (N = 32). The result was a new TSC∗ optimal code C34,18 (TSC=66), with ρmax = 6/18 = 0.3333 and ρ-spectrum {Nρmax , Nρmax −1/n , Nρmax −2/n , . . . , 0} = {20, 0, 216, 0, 252, 0, 73}, as compared to [2] which has TSC=66, but ρmax = 12/18 = 0.6666. The codes are shown

in the {0, 1} format. 

∗ C34,18

              =              

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 1 1

0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0

0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 1 1

0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

0 1 1 0 1 0 0 1 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 1 0 0 1 0 1 1 0 0 0

0 1 0 1 0 1 0 1 1 0 1 0 1 0 1 0 0 1 0 1 0 1 0 1 1 0 1 0 1 0 1 0 0 0

0 0 1 1 1 1 0 0 1 1 0 0 0 0 1 1 0 0 1 1 1 1 0 0 1 1 0 0 0 0 1 1 0 0

0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 0 1

0 1 0 1 1 0 1 0 1 0 1 0 0 1 0 1 0 1 0 1 1 0 1 0 1 0 1 0 0 1 0 1 1 0

0 0 1 1 0 0 1 1 1 1 0 0 1 1 0 0 0 0 1 1 0 0 1 1 1 1 0 0 1 1 0 0 0 1

0 1 1 0 0 1 1 0 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 0 1 1 0 0 1 1 0 1 0

0 1 0 1 0 1 0 1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 0 1 0 1 0 1 0 1 0 1

0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 1 0

0 0 1 1 0 0 1 1 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 0 0 1 1 0 0 1 1 0 1

0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 0 1

0 1 0 1 1 0 1 0 0 1 0 1 1 0 1 0 1 0 1 0 0 1 0 1 1 0 1 0 0 1 0 1 0 1

with a full search over v, has TSC= 12.43 and ρmax = 5/7. ∗ On the other hand, Search A yield C9,7 (N = 12), which has TSC= 13.41, but lower ρmax = 3/7.  0000000   0000000 

                             

? C9,7

∗ C33,31

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0

0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0

0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0

0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0

1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 0

1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 0

0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0

0 1 1 0 0 1 1 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 0 0 1 1 0 0 1 1 1

1 1 0 0 1 1 0 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 0 1 1 0 0 1 1 0 0

1 1 0 1 0 0 1 0 1 1 0 1 0 0 1 0 1 1 0 1 0 0 1 0 1 1 0 1 0 0 1 0

0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1

0 1 1 0 0 1 1 1 1 0 0 1 1 0 0 0 0 1 1 0 0 1 1 1 1 0 0 1 1 0 0 1

1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 1

0 1 1 1 1 0 0 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 0 0 1 1 1 1 0 0 1

0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 1

0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1

1 0 1 0 1 0 1 1 0 1 0 1 0 1 0 0 1 0 1 0 1 0 1 1 0 1 0 1 0 1 0 1

1 1 0 1 0 0 1 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 1 0 0 1 0 1 1 0 1

0 1 1 1 1 0 0 1 1 0 0 0 0 1 1 0 0 1 1 1 1 0 0 1 1 0 0 0 0 1 1 1

1 0 1 1 0 1 0 0 1 0 1 1 0 1 0 1 0 1 0 0 1 0 1 1 0 1 0 0 1 0 1 0

1 0 1 1 0 1 0 1 0 1 0 0 1 0 1 0 1 0 1 1 0 1 0 1 0 1 0 0 1 0 1 0

1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 0

0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0

1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 0

1 0 1 1 0 1 0 1 0 1 0 0 1 0 1 1 0 1 0 0 1 0 1 0 1 0 1 1 0 1 0 0

affordable search complexity.

1 0 1 0 1 0 1 0

0 1 1 0 0 1 1 0

0 0 0 1 1 1 1 0

1 1 0 0 1 1 0 0

1 0 1 1 0 1 0 0

0 1 1 1 1 0 0 1

    ∗ = , C9,7  

0 0 0 0 0 0 0 0

1 1 0 1 1 1 0 0

0 1 1 0 1 1 1 0

0 1 0 1 1 0 1 1

0 0 1 0 1 1 0 1

1 1 0 0 0 1 0 1

1 0 1 1 0 1 1 1

   

Example 4. Catastrophic TSC-optimal codes: Let the C6,4 TSC-optimal code (TSC=10)

1 1 0 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 1 0 0 1 0

1 0 1 1 0 1 0 0 1 0 1 1 0 1 0 0 1 0 1 1 0 1 0 0 1 0 1 1 0 1 0 0

0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 0

1 0 1 0 1 0 1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 0 1 0 1 0 1 0 1 1

                           

Example 3: In this example, we show that there are cases where a TSC-optimal code found with the search of this section, has higher ρmax than a non TSC-optimal code found ? with Search A; i.e. a TSC-optimal code C9,7 (N = 8) found 4 For

0 0 0 0 0 0 0 0

So we see that in this context, the TSC-optimality may prevent the code from having a low ρmax . In addition a catastrophic code (i.e. a code having ρmax = 1, or, equivalently, having complementary codewords) can be TSC-optimal, as we explain in the following example and also observed in [4].

Example 2: In this example, we consider the ρmax of some sequences from the asynchronous CDMA case [7], just for benchmarking, since our codes are designed for the synchronous case, i.e. minimum cross-correlation spectrum at zero lag. The Gold code derived from the primitive polynomials 458 , and 758 (polynomial degree, ξ = 5) [8, p. 117] consists of 33 sequences of length 31 and has ρmax = 9/31, with multiplicity Nρmax = 6 (as well as being TSC-optimal [2] with TSC=36.097). We used our aforementioned search by setting v = [u; u[1:10] ], where u is a 21-bit4 vector, and u[1:10] the first 10 elements of u. The result is a TSC-optimal code ∗ with ρmax = 9/31 (same as the Gold code), but with C33,31 lower multiplicity Nρmax = 4.  0000000000000000000000000000000                =             

  = 

C6,4 = [H; zT , zT ; zT , −zT ]

(7)

where H a 4 × 4 Hadamard matrix, and z ∈ {±1}2 , [2]. It can be easily shown that all the possible 4 codes of (7) are ∗ code found with Search catastrophic. On the other hand, a C6,4 A (N = 8) has ρmax = 0.5, and TSC=10.  0000  ∗ = C6,4

0 0 0 0 0

1 0 1 0 1

0 1 1 0 0

1 0 1 1 0



Therefore it is important that the TSC-optimality is sought along with the minimization of the ρ-spectrum, which may yield in the relaxation of the TSC-optimality if the resulting code is catastrophic or has large ρ-spectrum. Lastly we would like to mention that in the case of K = N + 2, and n odd, there is one more degree of freedom in the search, that of α1 , α2 . For example, for the case of C34,19 code, α1 = −α2 yields a code with better ρ-spectrum, than if α1 = α2 ; both cases though result in TSC optimal codes. B. Underloaded case: K ≤ n In [9] it was shown that the TSC-optimal codes (K ≤ n) of [2], [10] have also minimum ρmax . Here we will generalize their results to show that these codes have in addition optimal ρ-spectrum. Assume there exists an N -Hadamard matrix, such that [9], N = 4b n+2 4 c, and N ≥ K. Then, n ∈ {N − 2, N − 1, N, N + 1}. Note that the results of this subsection are also valid for the cases where N 0 ≥ K ≥ n (N 0 the size of a Hadamard matrix), for the corresponding values of n (e.g. n ∈ {N 0 − 2, N 0 − 1}.) 1) n = N : It is obvious that if we take an orthogonal code CN,N and remove any number of rows, then the resulting code, say CK,N , K ≤ N , has ¡still¢ ρmax = 0, hence has optimal ρspectrum, i.e. Sρ = {(0, K 2 )}.

2) n = N ± 1: If now, from the aforementioned CK,N code we remove one column, or add any one arbitrary column [5], [9], it is easily shown from (1), that the resulting code CK,N −1 , or CK,N +1 will have ρmax = 1/(N −1) or 1/(N +1) respectively (due to the missing or additional column), ¡ ¢ and so they have optimal ρ-spectrum, i.e. Sρ = {(ρmax , K 2 )}. 3) n = N − 2: Since by definition N is a multiple of 4, n = N − 2 ≡ 2 (mod 4). We know that an n × n Hadamard (orthogonal) matrix exists only if n = 0 (mod 4), except for the trivial cases of n=1, and n = 2, for which C2,2 has optimal ρ-spectrum, i.e. Sρ = {(0, 1)}. So when n ≡ 2 (mod 4), and n > 2, ρmax 6= 0. In such a CK,N −2 code5 (i.e. a code resulting by removing two columns of CK,N of §III-B.1), the absence of the two columns will contribute a maximum of 2/(N − 2) to the ρmax (= 0) of the CK,N . Therefore a code CK,N −2 has ρmax = 2/(N − 2), (see also [9]). In [10], there is a construction of a TSC-optimal CK,N −2 code. In our words, the CK,N code, which has resulted by removing N − K rows from an N Hadamard matrix, can be written, through possible row exchanges (row exchanges don’t alter the ρ-spectrum of the code), in the form [B, CK,N −2 ], where B = [v1 , v1 ; v2 , −v2 ], v1 a bK/2c-bit all 1’s vector, and v2 a dK/2e-bit all 1’s vector. Then [10], CK,N −2 is a TSC-optimal code with 4 4 bK/2c(bK/2c − 1) + 2 dK/2e(dK/2e − 1) n2 n (8) Next, we ’ll try to assess if the aforementioned CK,N −2 has optimal ρ-spectrum. Let the C 0 K,N −2 code TSC = K +

C 0 K,N −2 = C˜K,N \ {c1 , c2 }

(9)

where C˜K,N equals CK,N with probably some rows exchanged, and c1 , c2 are any two columns of C˜K,N . Let D = [c1 , c2 ], a K × 2 matrix having α rows (not necessarily consecutive) of the form [±1, ±1], and β = K − α rows of the form [±1, ∓1]. Then it is easy to show that C 0 K,N −2 (n = N − 2) has ρspectrum ¡ ¢ ¡2 1 ¢ 1 Sρ = { 0, αβ , , α(α − 1) + β(β − 1) }. (10) n 2 2 Now, we need to find the values of α and β that yield optimal ρ-spectrum (10). One way is to find α and β that maximize the multiplicity N0 = αβ (since the sum of multiplicities is fixed, this implies that N2/n will be minimized), such that α + β = K. It can be shown that N0 is maximized for ½ K/2, for K even, α= (11) bK/2c or dK/2e, for K odd. Hence, for these values of α (β = K −α), the C 0 K,N −2 code (as well CK,N −2 , [10]) has optimal ρ-spectrum in addition to be TSC-optimal [9], [10] (the TSC is given by (8) ).

TABLE I TSC OPTIMAL CK,P (K, P ) (256, 168) (256, 192) (256, 224) (256, 240) (256, 248) (256, 256)

can be drawn for the case of n = N + 2, N =

ρmax bound 0.0453 0.0362 0.0237 0.0162 0.0112 0.0000

TSC 390.1 341.3 292.6 273.1 264.3 256.0

IV. P RACTICAL C ONSIDERATIONS In this Section we evaluate the performance of our overloaded sequences in a synchronous CDMA system. For the sake of simplicity, we assume all users have unit power. Consider the following discrete-time QPSK synchronous CDMA system model, with K users and processing gain P , r=

K X

bk sk + n

(12)

k=1

where r = [r0 , r1 , . . . , rP −1 ]T is the received chip vector, bk is the k th -user’s QPSK symbol, sk = [s0,k , s1,k , . . . , sP −1,k ]T the k th -user’s signature, and n = [n0 , n1 , . . . , nP −1 ]T is the noise vector, with nj i.i.d., zero-mean, circularly symmetric complex Gaussian random variables having variance N0 . √ The signature sequences equal sk = dk / P , where dk (∈ {±1}P ) is the k th codeword of the CK,P code. The TSC optimal codes used6 are summarized in Table I. In the same Table we have listed also Levenshtein’s bound on ρmax of binary codes [11], which for the corresponding codes’ parameters coincide with the Welch bound on ρmax [12] which is valid also for complex codes. Hence although our codes have ρmax close to the bound, we believe the bound is quite loose for the corresponding codes of Table I, except of course for the trivial case of the orthogonal code C256,256 . The performance of the matched-filter (MF) receiver for various overload factors f , f ≡ K−P P 100%, is given in Figure 1 for K = 256, as the average-over-the-users bit error rate vs. the signal to noise ratio per bit. As expected, the performance degrades severely with high overload factors. Therefore an advanced receiver is required. In Figure 2 we plot the performance of a 5-stage7 partial Parallel Interference Canceller (PIC) [13], [14]. We can say, based on Figure 2, that the performance of overloaded synchronous CDMA with K = 256 users and partial PIC is acceptable for even 50% overload. Since in practice the AWGN channel appears in cellular systems very rarely, e.g. when you are very close to the base station, we evaluate the performance of the codes of Table I over wireless channels also, like the Pedestrian A (PedA), and Vehicular A (VehA) channels [15], for which we assume block fading. We assume also a chip rate of 1.2288Mcps, which 6 All

5 Similar conclusions 4b n+1 c, [2]. 4

ρmax 0.0833 0.0625 0.0446 0.0333 0.0323 0.0000

CODES .

the codes used in the paper are available by the authors. comprises of a conventional first stage and four interference cancellation stages. 7 It

0

0

10

10

P=168, f=52.4% P=192, f=33.3% P=224, f=14.3% P=256, f=0% P=256, f=0% (Rake)

−1

10

−2

−1

10

BER

BER

10

−3

10

−4

−2

10

10

P=168, f=52.4% P=192, f=33.3% P=224, f=14.3% P=240, f=6.7% P=248, f=3.2% P=256, f=0%

−5

10

−6

10

−3

0

2

4

6

8

10 SNR (dB)

12

14

16

18

10

20

0

2.5

5

7.5

10

b

Fig. 1.

12.5 SNR (dB)

15

17.5

Matched filter receiver; K = 256 over AWGN.

Fig. 3.

22.5

25

5-stage partial PIC; K = 256 over PedA channel (block fading).

0

0

10

10

P=168, f=52.4% P=192, f=33.3% P=224, f=14.3% P=256, f=0% P=256, f=0% (Rake)

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5-stage partial PIC; K = 256 over AWGN.

makes PedA and VehA channels, 2-path and 5-path channels respectively, and perfect channel estimation. In addition, we incorporate to our simulation model quadrature spreading (prior to transmission of the sum chip signal) similar to that of an existing 3G CDMA wireless standard [16, §9.3.1.3.4]. The simulation results for various loads and for the PedA channel, are shown in Figure 3, where we see that in 33% and 50% overload we are away about 1.9dB and 5.2dB (at 5% BER) respectively from the orthogonal codes (P = 256), while dramatically increasing system capacity. On the other hand for the difficult VehA channel (since it has more paths, the increased multipath-multiuser interference becomes severe for the overloaded sequences), it seems from Figure 4, that up to 33% overloaded system is acceptable with the current receiver. In the Figures 3-4 we have also included the performance

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5-stage partial PIC; K = 256 over VehA channel (block fading).

of a Rake receiver. Therefore comparing to the Rake performance, we can claim that we can achieve same performance while enhancing system capacity by 33%, but at the same time, we have to pay in higher receiver complexity. We believe that with receivers which take into account the known ρ-spectrum of the spreading sequences, better performance can be achieved, but this is out of the scope of this paper. V. C ONCLUSIONS We considered code design for the synchronous CDMA system. Search methods were given to construct TSC-optimal codes with low cross-correlation spectrum and it was shown that the underloaded codes (K ≤ n) have optimal ρ-spectrum. The designed codes were based on both the TSC and ρspectrum criteria.

In addition, we evaluated the performance of our overloaded codes for a synchronous CDMA system with QPSK modulation in an AWGN channel, as well as the Pedestrian A and Vehicular A wireless channels, with both a matched-filter and a partial PIC receiver. It was shown that acceptable performance is provided even for 33% − 50% overload factors (depending of course on the transmission channel and the receiver). Better results can be obtained by using more sophisticated receivers that exploit the known cross-correlation spectrum of the spreading sequences. Finally, since multipath transmission seems to limit the overload factor in synchronous CDMA, an alternative application is the deployment of our overloaded codes in a suitably designed multi-carrier CDMA (MC-CDMA) system (e.g. a system with DS-CDMA followed by OFDM modulation), since there the received signal is affected only by flat Rayleigh fading and noise [17]. ACKNOWLEDGMENT The authors would like to thank G.N. Karystinos and D.A. Pados, for providing [3] and pre-prints of [2], [9]. R EFERENCES [1] M. Rupf and J. L. Massey, “Optimum sequence multisets for synchronous code-division multiple-access channels,” IEEE Trans. Inform. Theory, vol. 40, pp. 1261–1266, July 1994. [2] G. N. Karystinos and D. A. Pados, “New bounds on the total-squaredcorrelation and optimum design of DS-CDMA binary signature sets,” IEEE Trans. Com., vol. 51, no. 1, pp. 48–51, Jan. 2003. [3] ——, “Fundamental code division multiplexing properties of minimum total-squared-correlation binary signature sets,” in Conf. on Information Sciences and Systems, 2003.

[4] J. L. Massey and T. Mittelholzer, “Welch’s bound and sequence sets for code-division multiple-access systems,” in Sequences II (Methods in Communication, Security, and Computer Science), R. Capocelli, A. D. Santis, and U. Vaccaro, Eds. Springer-Verlag, 1993, pp. 63–78. [5] P. D. Papadimitriou and C. N. Georghiades, “On binary code design for the non-coherent block fading channel,” in IEEE Global Communications Conference (GLOBECOM), 2003. [6] S. Verd´u, Multiuser Detection. Cambridge University Press, 1998. [7] D. V. Sarwate and M. B. Pursley, “Crosscorrelation properties of pseudorandom and related sequences,” Proc. of the IEEE, vol. 68, no. 5, pp. 593–619, May 1980. [8] R. L. Peterson, R. E. Ziemer, and D. E. Borth, Introduction to Spread Spectrum Communications. Prentice-Hall, 1995. [9] G. N. Karystinos and D. A. Pados, “Binary CDMA signature sets with concurrently minimum total-squared-correlation and maximum-squaredcorrelation,” in IEEE Int. Conf. Com. (ICC), 2003. [10] C. Ding, M. Golin, and T. Kløve, “Meeting the Welch and KarystinosPados bounds on DS-CDMA binary signature sets,” Hong Kong Univ. of Science and Tech., Tech. Rep. HKUST-TCSC-2002-03, 2002. [11] V. I. Levenshtein, “Bounds for self-complementary codes and their applications,” in Eurocode ’92: International Symp. on Coding Theory and Applications, ser. CISM Courses and Lectures, P. Camion, P. Charpin, and S. Harari, Eds., vol. 339. Springer-Verlag, Wien - New York, 1993, pp. 159–171. [12] L. R. Welch, “Lower bounds on the maximum cross correlation of signals,” IEEE Trans. Inform. Theory, vol. 20, pp. 397–399, 1974. [13] M. K. Varanasi and B. Aazhang, “Multistage detection in asynchronous code-division multiple-access communications,” IEEE Trans. Com., vol. 38, pp. 509–519, Apr. 1990. [14] D. Divsalar, M. K. Simon, and D. Raphaeli, “Improved parallel interference cancellation for CDMA,” IEEE Trans. Com., vol. 46, pp. 258–268, Feb. 1998. [15] “Guidelines for evaluation of radio transmission technologies for IMT2000,” Rec. ITU-R M.1225. [16] “cdma2000 high rate packet data air interface specification,” 3GPP2, C.S0024, v2.0, Oct. 2000. [17] S. Kaiser, “OFDM-CDMA versus DS-CDMA: Performance evaluation for fading channels,” in IEEE Int. Conf. Com. (ICC), 1995.