OPTIMAL PERFORMANCE OF AN OVERSATURATED OCDMA/OCDMA SYSTEM BASED ON SCRAMBLED ORTHOGONAL BASES Frederik Vanhaverbeke and Marc Moeneclaey TELIN/DIGCOM, Ghent University, Sint-Pietersnieuwstraat 41, B-9000 Gent, Belgium,
[email protected] Abstract - In this paper we consider a multiple access system with oversaturation of the channel by means of a combined OCDMA/OCDMA scheme with scrambled orthogonal bases. We investigate the performance and the associated maximum channel capacity (channel load) for uncoded BPSK modulation with binary valued chip sequences and optimum multiuser detection for spreading factor N = 4, 8, 16 and 32. It is found that, for scrambling sequences that change randomly from symbol-interval to symbol-interval, the allowable channel overload is zero for N = 4, 8, 16, and is only about 3% for N = 32. Major improvements on the allowed channel capacity are achieved by selecting a ‘good’ scrambling sequence and keeping it the same for all symbol-intervals. In this way the allowed channel overload can be increased to 25 % for N = 8, to 31.25 % for N = 16 and to at least 31.25 % for N = 32. Keywords – channel oversaturation, optimal detection, CDMA. I. INTRODUCTION It is well-known that in a multiple access system based on code-division multiple access (CDMA) with spreading factor N, N is the maximum number of orthogonal users. Although almost perfect synchronization of the different users is required to achieve this limit, synchronization can always be achieved in the downlink. Even in the uplink, some systems like for example multicarrier-CDMA can maintain orthogonality by application of an appropriate cyclic prefix [1] and single-tap equalization. In order to cope with a number of users that exceeds the limit N in a perfectly synchronized CDMA system, new allocation schemes for the oversaturated channel were presented in [2] and [3]. Both schemes proposed to use orthogonal sequences for the first N users and to add some other sequences for the additional users. These methods, however, did not focus on maximization of the total capacity of the system. In addition to this, they allowed the chips to take on any real or complex value, whereas most of the practical CDMA systems focus on chip values from a very restricted (e.g. binary) alphabet. The present authors introduced another way of obtaining oversaturation in CDMA, labeled OCDMA/ OCDMA (O/O), by making a set of K > N binary vectors out of two orthogonal sets of vectors that are randomized to each other [4]. In this way, every user suffers only from interference
0-7803-7589-0/02/$17.00 ©2002 IEEE
caused by users belonging to the other set. In order to achieve an easy scalability, the first N users are assigned all orthogonal vectors from the first set. Every additional user is assigned a vector from the second orthogonal set. Multiuser detection [5] is required in order to achieve an acceptable performance of the O/O system, and optimal multiuser detection serves as a benchmark for the evaluation of suboptimal multiuser detectors. Unfortunately, in general it yields a complexity that increases exponentially with the total number of users K. The crosscorrelation matrix of the O/O system does not fit the requirements to perform optimum detection with polynomial complexity [6], [7], so we are forced to run an exhaustive search that turns out to be exponential in the number (K-N) of excess users. In this paper, we focus on BPSK modulation without coding and with binary chip sequences. Throughout this paper, we assume that the channel is nondispersive AWGN, that the different user signals are in perfect time synchronism and have the same carrier phase. Because of the intractability of the exhaustive search for high values (K-N), we restrict our attention to spreading factors N ≤ 32. In section II, the O/O system with scrambled orthogonal bases is explained along with its characteristics. In section III, we discuss the optimum detection and minimum Euclidean distance of this system. In section IV, the simulation results are presented, and finally, in section V, we draw the conclusions. II. SCRAMBLED OCDMA/OCDMA Consider a CDMA system with a spreading factor N, and assume that we want to accommodate a number of users K= N + M. We restrict our attention to the detection after chip-matched filtering at the receiver, so that the sequences of the users can be represented by vectors. The first N users (set1-users) are assigned orthonormal vectors s11, s21, …, sN1 from basis S1 = [s11 s21 … sN1]. The additional M set2-users are assigned the first M vectors from a second orthonormal basis S2 = [s12 s22 … sN2]. All sequences are assumed to belong to {1/√N, -1/√N}N. All orthonormal sequences can be grouped together into the signature matrix S = [S1 | S2 ]. The two orthonormal bases of the O/O system can be constructed by assigning all Walsh-Hadamard vectors WHi(N) (i = 1,…,N) of length N [8] to the set1-users, and the same vectors, overlaid by a scrambling vector Λ = {Λ1,…,ΛN}∈ {+1,-1}N, to the set2-users. This is actually
PIMRC 2002
inspired by the principle of downlink separation between different cells in wideband-CDMA [8], where downlink sequences of different cells are composed of the same orthogonal sequences, but overlaid by different scrambling sequences. Note that this construction puts a restriction on S2, since in this case S1 and S2 are related by diagonal matrix Ψ =diag{Λ1,…,ΛN}:
S 2 = Ψ.S1 (1) A consequence of choosing scrambled orthogonal bases for our system, is that submatrix Γ N ,0 ( N ) = S1T .S 2 of the crosscorrelation matrix ST.S (T denotes transposition), has a special structure, which is considered in more detail in the appendix. Fig. 1 shows Γ 8,0 (8) , which exhibits a ‘mirror’ structure down to submatrices of order 2. In general, Γ N ,0 ( N ) consists of only N different components ρj that
(
j
)
each occur N times, with ρ j = Λ.WH / N .
III. OPTIMUM DETECTION AND MINIMUM EUCLIDEAN DISTANCE
for the O/O sequence set as compared to more general sequence sets. The asymptotic BER after optimum detection can be approximated by [9]: BER =
columns of Γ N ,0 ( N ) ; and n = [n1…nN] denotes independent
n =1
æd ö ( i , n ) Q ç min ÷ 2 σ è ø
ò
1 2π
é ρ1 êρ ê 2 êρ3 ê ρ4 Γ 8,0 (8) = ê êρ ê 5 êρ6 êρ ê 7 êë ρ 8
ρ2 ρ1 ρ4
ρ3 ρ6 ρ5 ρ8 ρ7
∞
x
rewrite expression (6) as æd ö BER = L(Λ, A, B ).Qç min ÷ è 2σ ø
(6)
exp
( )du . −u 2 2
We can
(7)
ρ3 ρ4 ρ1 ρ2 ρ7 ρ8 ρ5 ρ6
ρ4 ρ3 ρ2 ρ1 ρ8 ρ7 ρ6 ρ5
ρ5 ρ6 ρ7 ρ8 ρ1 ρ2 ρ3 ρ4
ρ6 ρ5 ρ8 ρ7 ρ2 ρ1 ρ4 ρ3
ρ7 ρ8 ρ5
ρ8 ù ρ 7 úú ρ6 ú ú ρ5 ú ρ4 ú ú ρ3 ú ρ2 ú ú ρ1 úû
ρ6 ρ3 ρ4 ρ1 ρ2
Fig. 1 : Build-up of submatrix Γ8,0(8) In the following, L(Λ, A, B ) separately.
we
will
discuss
dmin
and
A. dmin The minimum Euclidean distance dmin can be found as
where d (ε) = 2
which has a complexity in the order of 2M only. This shows that the optimal detection complexity is significantly lower
bit
distance dmin, and Q( x) =
(a' , b') = arg éêmin y − s(a, b, A, B) ùú (4) ë a ,b û which has a complexity in the order of 2K. For the O/O system however, it is immediately verified, by means of (3), that (4) can be reduced to ìb ' = arg min y − s sgn(y − Γ M ( N ).b), b, A, B N ,0 ï b (5) í M ïîa' = sgn( y − Γ N ,0 ( N ).b' )
)}
i∈ I d min
ån
is the number of neighbors at distance dmin from the constellation point i, nbit(i,n) is the number of bits in which the i-th constellation point differs from its n-th neighbor at
d min =
(
N d min ( i )
pi
the constellation points that have at least one neighbor at minimum (Euclidean) distance dmin, pi is the a priori probability of the i-th constellation point (pi = 2-K), N d min (i )
Gaussian noise samples with variance σ2 = N0/2. In general, the optimum multiuser detector selects the datavectors a’ and b’ according to the maximum likelihood rule :
{
å
In the above expression, the index i enumerates the constellation points s(a,b,A,B), I d min is the set of indices of
Here we consider the optimum multiuser detector performance of the O/O system when transmitting in BPSK over a nondispersive AWGN channel. Assuming the users are synchronized in time, we restrict our observation to one bit interval (one-shot problem). Taking into account that all users have the same carrier phase, chip-matched filtering, sampling at the chip-rate and premultiplying by the orthogonal matrix S1T yields the following real-valued observation vector y: y = s (a, b, A, B ) + n = A.a + B.Γ M (3) N ,0 ( N ).b + n In the above, a = [a1 …aN]T and b = [b1 … bM]T are composed of databits of the set1-users and the set2-users respectively; A = diag{A1,…,AN} and B = diag{B1,…,BM} are diagonal matrices containing the amplitudes of the set1users and set2-users; Γ M N , 0 ( N ) consists of the first M
1 N +M
min
ε∈{−1,0 ,1}K ,ε ≠ 0
N
d (ε )
with
M
åAε s + åB ε s = (ε ... ε ) and ε = (ε 1 1 i i i
i =1
ε1T
(8)
1 1
j
2 1 j j
j =1
1 N
T 2
2 1
(
and ε = ε1T
ε T2
)
T
)
2 ... ε M .
Considering the fact that d (ε ) = d (− ε ) , evaluation of (8) by means of an exhaustive search would require the evaluation of 3K/2 different vectors ε i . This exhaustive
search becomes intractable for N ≥ 32. Note that the crosscorrelation matrix ST.S is singular, so that dmin can not be computed in a simpler way than exhaustive search, like the one proposed in [10]. In the case of perfect and uniform
the number of users for a particular scrambling sequence.
power control (i.e., Ai = Bj = A for i = 1,…,N and j = 1,…,M), d (ε ) reduces to d pc (ε) = 2 A ε + 2.ε1T .Γ M N ,0 ( N ).ε 2
(9)
where ε denotes the number of nonzero components of ε . In the saturated case (M=0), with Ai = 1 for i = 1,…,N, we know that dmin = 2, which implies that dmin ≤ dopt= 2 in the oversaturated case (M>0). There are exactly 2N possible matrices Γ M N,0 ( N ) and the highest dmin will be obtained for these matrices Γ M N,0 ( N ) that maximize (9). Some remarks can be made about dmin: • Suppose we apply power control for set1- and set2users, so that all users of a set have the same amplitude: A1 = … =AN = A and B1 = … = BM =B. Fixing the overall energy in the system to (N+M) yields NA2 + MB2 = N+M. From this expression, it is clear that A < 1 or B < 1 if A ≠ B. Assume A < 1, then d(1,0,…,0) = 2A < 2, so that dmin < 2. For B < 1, we obtain analogously: dmin < 2. This implies that the optimal minimum distance dopt = 2 cannot be achieved for any M > 0 if users of different sets are assigned different energies. That’s why we focus on the case A = B in this paper. • The following holds for a particular Γ M N,0 ( N ) :
1 if ρ * = max ρ k ( N ) > then d min < 2 (10) 1≤ k ≤ N 2 * Indeed, selecting i an j such that (Γ M N ,0 ( N )) i , j = ρ , and taking εj = -εi.sgn(ρ*) ≠ 0, εn = 0 for n ∉(i,j) yields d (ε ) < 2 . An implication of (10) is that scrambling vectors, scaled to norm one, that differ in less than N/4 positions from a Walsh-Hadamard vector (or its inverse), cause the system to have dmin < 2. The total fraction of such scrambling sequences is
θN =
2N 2
N
N / 4 −1
.
å i =0
N! (i )!.( N − i )!
(11)
Numerical evaluation of (11) yields θ8=56%, θ16 = 34% and θ32 = 6.7 %, which indicates that the portion of scrambling sequences yielding max ρ k ( N ) > 1 / 2 , 1≤ k ≤ N
•
decreases with increasing spreading factor. For a particular Γ N ,0 ( N ) , the following property holds for the O/O systems with K1 and K2 users (dmin(Ki) is the minimum Euclidean distance with Ki users): K1 < K2 Þ dmin(K1) ≥ dmin(K2)
(12)
This is obvious because if we achieve dmin(K1) in the O/O system with K1 users for ε = ε min ( K1 ) , selecting
(
)
ε' = ε Tmin ( K1 ) 0 for the system with K2 users yields
d (ε') = d min ( K1 ) . Property (12) implies that we cannot increase the minimum Euclidean distance by increasing
B.
L(Λ, A, B ) From (6), L(Λ, A, B ) is given by
L (Λ , A , B ) =
2−K K
N d min ( i )
å ån
i∈ I d min
bit
(i, n )
(13)
n =1
To each ε that satisfies d (ε ) = d min correspond 2 constellation points s(a,b,A,B) that have a neighbor at Euclidean distance dmin; such a constellation point and its neighbor differ by ε bits. Hence, (13) becomes K− ε
L(Λ, A, B ) =
å
d (ε ) = d min
ε K
2
−ε
(14)
Due to the mirror structure of Γ 8,0 ( N ) in figure 1, with perfect and uniform power control (A1 = … =AN = B1 = … = BM = A) and as M increases, many different vectors ε will produce the same value d (ε ) , so that L(Λ, A, B ) can be expected to increase significantly with increasing overload. IV. MAXIMUM CAPACITY We define the capacity of an O/O system as the total number of users Kmax the system can accommodate so that the SNR-degradation, with respect to a single-user system, at an average BER = 10-5, over a run of 107 transmitted bits, is less than 0.4dB for every number of users K ≤ Kmax. For N = 8, 16, we evaluate the asymptotic performance of scrambling sequences that fulfil this capacity criterion. In this section, we focus exclusively on a system with perfect and uniform power control. First, we consider the O/O system with the scrambling sequences Λ being generated randomly from one bit interval to the next. The capacity of this system is a lower bound for the capacity that can be achieved for the best particular scrambling sequence over the entire set of possible scrambling sequences. Simulations of this system show very poor results: the maximum load Kmax/N does not exceed 100 % for N = 8, 16 and is only 33/32 for N = 32. Beside this, considering the BERperformance for a particular channel load factor K/N, we find that the BER decreases with increasing spreading factor. The total number of scrambling sequences is given by 2N, but as the scrambling sequences Λ and - Λ yield the same BER, actually 2N-1 sequences need to be considered. For N = 8, 16 and 32, this amounts to 128, 32,768 and 2.109, respectively. It is clear from these numbers that we can assess the performance of every individual scrambling sequence for N = 8 and 16 only. For N = 32, we restrict ourselves to a subset of 20,000 different scrambling sequences (which is still only 0.001 % of the total), chosen completely at random from the entire set of size 2.109.
load. For N = 8, the maximum load is given by 125 % while it has increased to 131.25 % for N = 16.
Percentage scrambling sequences
100 90 80 N=8
70
N=16
60
N=32
50 40 30 20 10 0 100
105
110
115
120
125
130
135
140
Load factor K/N (%)
Fig. 2 : Percentage of good scrambling sequences as a function of load factor. In figure 2, for spreading factors N = 8, 16 and 32, the percentage of scrambling sequences having a capacity of at least K/N are shown. We discuss the results for the different spreading factors separately. For convenience, we denote L(Λ, A, A ) by L. A. N = 8. By inspection of (11), we see that at least 56 % of the scrambling sequences yield dmin < 2. Only about 40% of the sequences achieve a capacity of K = 9 and all of them achieve dmin = 2 and L = 2.2. About 4% of the sequences also have dmin = 2 and L = 2.2, but do not fulfil the capacity criterion. For K = 10, 25% of the scrambling sequences have dmin = 2 and L = 3.2, while for all other sequences dmin = 0. Only 3% of the sequences are selected by the capacity criterion and all of them yield dmin = 2. For K = 11, all scrambling sequences give rise to dmin = 0, and none of them fulfils the capacity criterion. From these results, we conclude that the capacity criterion selects scrambling sequences with good asymptotic performance, although some of the sequences with good asymptotic performance are rejected. B. N = 16 Although at least 34% of the scrambling sequences have dmin < 2, for low overloads (K=17,18) some of them still meet the capacity criterion. Over 90% of the scrambling sequences achieve the capacity K = 17. For K = 18, 19, 20 and 21, dmin and L were calculated for a subset of the scrambling sequences that achieve the respective capacities. For K = 18, about 93 % of these sequences yield dmin = 2 with average value for L equal to 1.24. About 7 % of the sequences have dmin = 1.73. For K = 19, 20, 21, all sequences achieve dmin = 2, and the average L is 1.42, 1.6 and 2.17 respectively. For K = 22, evaluation of dmin over a subset of the scrambling sequences that achieve capacity K = 21, shows that all have dmin = 0. From N = 8 to N = 16, there is a clear increase in percentage of scrambling sequences achieving a particular
C. N = 32 For N = 32, figure 2 gives only an indication of what can be expected, since we are forced to consider only a very small part (0.001 %) of the total number of scrambling sequences. Over the 20,000 randomly chosen sequences, the maximum load is 131.25 %. However, the results obtained with the scrambling sequences that change randomly from one bit interval to the other, showed the superiority in terms of BER of N=32 over N=16 for every load factor, which might indicate that the set of 20,000 sequences we considered is not representative for the entire set of scrambling sequences. Probably, better scrambling sequences can be found within the entire set. According to (5), the complexity to evaluate the capacity criterion is of the order of 2M.107/K, while the evaluation of dmin by means of (8) is of the order of 3N+M/2. For (N,K) = (16,19), the complexity of evaluating dmin is more than 100 times that of the capacity criterion. For N = 32, evaluation of dmin is even intractable, while evaluation of the capacity criterion remains feasible. Hence, the capacity criterion selects scrambling sequences with good asymptotic properties with a much lower complexity than (8). V. CONCLUSION AND REMARKS In this paper, we considered the performance of the O/O system with scrambled orthogonal bases and uncoded BPSK modulation, optimal multiuser detection and perfect power control. We found that the achievable channel overload with random scrambling sequences was very small for spreading factors N = 8, 16 and 32. The BER-performance for a particular channel load was shown to improve for increasing spreading factor. A much better performance is obtained by selecting a particular spreading sequence that remains the same for all successive symbol-intervals. In this case, the achievable channel overload for N = 8 is 125 % and rises to 131.25 % for N = 16. For N = 32, the achievable overload is at least 131.25 %. For high overloads, all scrambling sequences that achieve that capacity yield a minimum Euclidean distance dmin = 2 between the constellation points. The capacity criterion, where good scrambling sequences are found by evaluation of the BER curve, is a fast and efficient way to detect scrambling sequences for which the O/O system has good asymptotic BER performance. APPENDIX We proof that Γ N ,0 ( N ) = S1T .S 2 for N = 2k, is given by the recursive expression : é Γ N / 2,0 (N ) Γ N / 2, N / 2 (N )ù Γ N ,0 (N ) = ê ú ëΓ N / 2, N / 2 ( N ) Γ N / 2, 0 ( N ) û
(I)
é ρ ( N ) ρ 2+i (N )ù Γ 2,i (N ) = ê 1+i ú (i = 0,2) ë ρ 2+i (N ) ρ1+i (N )û
with
(
(I-2)
)
and ρ j (N ) = Λ.WH j ( N ) / N : the scalar product of the unit-norm scrambling sequence with the j-th WalshHadamard vector of length N. If (I) holds, (Γ N ,0 ( N ) )i , j is given by
(Γ (N ))
N
å
Λ.Ψ i , j (N )
Λ s .Ψ ( N ) = N s =1 N Hence, if we want to prove (I), it suffices to show that the matrix QN,0(N) of vectors (QN,0(N))i,j = Ψ i, j (N ) , obeys to N ,0
i, j
=
1
i, j s
é Q N / 2, 0 ( N ) Q N / 2, N / 2 ( N ) ù Q N ,0 ( N ) = ê ú ëQ N / 2, N / 2 ( N ) Q N / 2,0 ( N ) û
(II)
é WH 1+i ( N ) WH 2 + i ( N )ù Q 2,i (N ) = ê (II-2) ú 2+i 1+ i ëê WH ( N ) WH ( N ) ûú and WHk(N): kth N-dimensional Walsh-Hadamard vector. It is straightforward to check that (II-2) holds for i = 0 and N = 2. Starting from this, we proof expression (II) by induction, supposing that (II) is valid for N. The normalized Walsh-Hadamard matrix for 2N is: WN ù 1 é WN W2 N = êW ú 2 ë N − WN û with
where WN denotes the Walsh-Hadamard matrix of order N.
If we adopt the notation Ψ ik, j (2 N ) for the vector of length N
that consists of the N first (k=1) or N last (k=2) elements of Ψ i , j (2 N ) and if I = {1,2,…,N} and J = {N+1,N+2,…,2N}, we distinguish two situations : • (i ∈ I, j ∈ I) or (i ∈ J, j ∈ J) : In this case, 1 ì i, j i, j Ψ i , j (N ) i, j ∈ I ïΨ 1 (2 N ) = Ψ 2 (2 N ) = 2 ï í 1 i , j i , j ïΨ (2 N ) = Ψ (2 N ) = Ψ i − N , j − N (N ) i, j ∈ J 2 ïî 1 2
In this way, Ψ i , j (2 N ) = 1/√2.[ Ψ i, j (N ) | Ψ i, j (N ) ]T constitutes one of the first N Walsh-Hadamard vectors of length 2N, since we assumed that (II) was met for N. • ( i ∈ I, j ∈ J ) or ( i ∈ J, j ∈ I ) : In this case, ì i, j 1 i, j Ψ i, j − N (N ) ïΨ 1 (2 N ) = − Ψ 2 (2 N ) = 2 ï í 1 ïΨ i , j (2 N ) = − Ψ i , j (2 N ) = Ψ i − N , j (N ) 2 ï 1 2 î
i∈I j∈J
Combining the two cases, we obtain : é Q N ,0 ( 2 N ) Q N , N (2 N )ù Q 2 N ,0 ( N ) = ê ú ëQ N , N (2 N ) Q N ,0 (2 N ) û which proves (II) and so also (I). REFERENCES [1] S. Hara, R. Prasad, “ Overview of Multicarrier CDMA”, IEEE Communications Magazine, Vol. 35, pp. 126-133, Dec. 1997. [2] R. E. Learned, A. S. Willisky and D. M. Boroson, “Low complexity joint detection for oversaturated multiple access communications,” IEEE Trans. Signal Processing, vol. 45, pp. 113-122, January 1997. [3] 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, August 1992. [4] F. Vanhaverbeke, M. Moeneclaey, H. Sari, “DS/CDMA with Two Sets of Orthogonal Spreading Sequences and Iterative Detection,” IEEE Communications Letters, vol. 4, pp. 289-291, September 2000. [5] S. Verdu, Multiuser detection, Cambridge University Press, New York 1998. [6] C. Sankaran and A. Ephremides, “Solving a class of optimum multiuser detection problems with polynomial complexity,” IEEE Transactions on Information Theory, vol. 44, September 1998. [7] C. Schlegel and A. Grant, ”Polynomial Complexity Optimal Detection of Certain Multiple Access Systems,” Int. Symp. on Information Theory, Honolulu, Nov. 5-8, 2000. [8]
E. H. Dinan and B. Jabbari, “Spreading Codes for Direct Sequence CDMA and Wideband CDMA Cellular Networks,” IEEE Commun. Mag.,pp 48-54, Sept. 1998.
[9] G. C. Clark and J. B. Cain, ″Error-correction coding for digital communications,″ Plenum Press, 1981.
i∈J j∈I
Here we have Ψ i , j (2 N ) = 1/√2.[ Ψ i, j (N ) | - Ψ i, j ( N ) ]T which is one of the last N Walsh-Hadamard vectors of length 2N which has the properties given by (I), valid for N.
[10] C. Schlegel and L. Wei, “A Simple Way to Compute the Minimum Distance in Multiuser CDMA Systems,” IEEE Trans. Com., pp. 532-535, vol. 45, May 1997.
