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Adaptive Binary Signature Design for Code-Division Multiplexing Lili Wei, Student Member, IEEE, Stella N. Batalama, Member, IEEE, Dimitris A. Pados, Member, IEEE, and Bruce W. Suter, Senior Member, IEEE
Abstract—When data symbols modulate a signature waveform to move across a channel in the presence of disturbance, the signature that maximizes the signal-to-interference-plus-noise ratio (SINR) at the output of the maximum-SINR filter is the smallest-eigenvalue eigenvector of the disturbance autocovariance matrix. In digital communication systems the signature alphabet is finite and digital signature optimization is NP-hard. In this paper, we present a formal search procedure of cost, upon eigenvector decomposition, log-linear in the signature code length that returns the maximum-SINR binary signature vector near arcs of least SINR decrease from the real maximum SINR solution in the Euclidean vector space. The quality of the proposed adaptive binary designs is measured against the theoretical upper bound of the complex/real eigenvector maximizer. Index Terms—Binary sequences, code-division multiple-access (CDMA), code-division multiplexing, signal-to-interference-plusnoise ratio (SINR), signal waveform design, signature sets, spread-spectrum communications, ultra-wideband (UWB) communications.
I. I NTRODUCTION
I
N recent years, there has been renewed interest in optimized signature sets for the growing number of code-division multiplexing applications such as plain or multiple-input multiple-output (MIMO) codedivision multiple-access (CDMA), multiuser orthogonalfrequency-division-multiplexing (OFDM), multiuser ultrawideband (UWB) systems, etc. In the theoretical context of complex/real-valued signature sets, the early work of Welch [1] on total-squared-correlation (TSC) bounds was followed up by direct minimum-TSC design proposals [2]-[5] and iterative distributed optimization algorithms [6]-[8]. Channel and system model generalizations were considered and handled in [9]-[11]. Signature optimization under the user capacity metric [12] or the signal-to-interference-plus-noise ratio (SINR) at the output of the RAKE filter [13] was also studied. It is to be emphasized that all works described above deal with real (or complex) valued signatures, hence their findings constitute only pertinent performance upper bounds for digital communication systems with digital signatures. Recently, new Manuscript received February 11, 2007; revised July 9, 2007; accepted November 4, 2007. The associate editor coordinating the review of this letter and approving it for publication was Y. J. Zhang. This work was supported by the U.S. Air Force Office of Scientific Research under Grant FA9550-04-1-0256. This paper was presented in part at the 2006 IEEE Global Telecommunications Conference (GLOBECOM), San Francisco, CA. L. Wei, S. N. Batalama, and D. A. Pados are with the Department of Electrical Engineering, State University of New York at Buffalo, Buffalo, NY 14260 USA (e-mail: {liliwei, batalama, pados}@eng.buffalo.edu). B. W. Suter resides in Rome, NY USA (e-mail:
[email protected]). Digital Object Identifier 10.1109/TWC.2008.070174.
bounds on the TSC of binary signature sets were found [14] that led to minimum-TSC optimal binary signature set designs for almost all signature lengths and set sizes [14]-[16]. The sum capacity, total asymptotic efficiency, and maximum squared correlation of the minimum-TSC binary sets were evaluated in [17]. The sum capacity of other non-minimumTSC binary sets was calculated in [18] and the user capacity of minimum and non-minimum-TSC binary sets was identified and compared in [19]. A procedure to find minimum-TSC binary signature sets with low cross-correlation spectrum was presented in [20]. In this work, we consider the NP-hard problem of finding adaptively the binary signature that maximizes the SINR at the output of the maximum-SINR filter. It is immediately understood that the complex/real minimum-eigenvalue eigenvector of the disturbance autocovariance matrix constitutes an upper bound benchmark. On the other hand, the binary solutions that have appeared in the literature are: (i) Direct binary quantization of the minimum-eigenvalue eigenvector [21], [22] which is the rank-1-optimal design/approximation as shown in [23]; (ii) the rank-2-optimal proposal of [23] that vastly improves upon the rank-1 design at additional complexity cost of O(LlogL) where L is the signature length under consideration; (iii) the low-cost low-performance conditional Cholesky optimization algorithm in [24] that is eigenvectorfree; and (iv) the code allocation from an orthogonal set described in [25]. The novel scheme that we describe in this present paper performs a binary search near the continuousvalued arcs of least SINR decrease from the real maximum SINR solution and creates a signature candidate list of size O(L). To the best of the authors’ knowledge, the first and only, so far, use of a least-decrease-driven search for finite-alphabet solutions in a communications theory context is by Spasojevic and Georghiades [26] to convert general maximum-likelihood estimates to near maximum-likelihood decisions. The rest of this paper is organized as follows. Section II presents the signal model and pertinent notation. The algorithm for single-user binary signature design in the presence of multipath fading, multiple-access interference (MAI), and additive white Gaussion noise (AWGN) as well as its extension to multiuser binary signature design and assignment are presented in Section III. Section IV is devoted to performance evaluation of the proposed scheme. A few concluding remarks are drawn in Section V. II. S YSTEM M ODEL In this manuscript, we develop and study a new adaptive binary signature optimization algorithm in the general context of
c 2008 IEEE 1536-1276/08$25.00
WEI et al.: ADAPTIVE BINARY SIGNATURE DESIGN FOR CODE-DIVISION MULTIPLEXING
a multiuser CDMA-type environment where K signals/users transmit asynchronously, simultaneously in frequency and time. The kth user transmitted signal, k = 1, 2, . . . , K, is ∞ uk (t) = bk (m) Ek sk (t − mT ) ej(2πfc t+φk ) (1) m=0
where bk (m) ∈ {±1}, m = 0, 1, 2, . . ., is the mth data bit, Ek represents transmitted energy T per bit period T , and sk (t) is the unit-energy normalized ( 0 s2k (t)dt = 1) user signature waveform of the form L−1 sk (l)ψ(t − lTc ) (2) sk (t) = l=0
where, in turn, sk (l) ∈ {±1}, l = 0, 1, . . . , L − 1, are the L signature bits to be designed/optimized, and ψ(t) is the chip waveform of duration Tc = T /L assumed to be given and fixed (for example ideal square pulse, raised cosine, or otherwise); fc and φk in (1) are the common carrier frequency and phase offset pertinent to user k, respectively. The combined received signal waveform due to all K asynchronous transmissions over individual multipath fading channels of impulse response hk (t), k = 1, · · · , K, is r(t) =
K
hk (t)uk (t − τk ) + n(t)
(3)
where τk represents time delay relative to user 1 with τ1 = 0 and n(t) is a white Gaussian noise process. Assuming synchronization with the signal of the user of interest k, k = 1, 2, . . . , K, upon carrier demodulation, chip matchedfiltering and sampling at the chip rate over a presumed multipath extended data bit period of L + N − 1 chips where N is the number of resolvable multipaths, we obtain the data vector r(m) ∈ CL+N −1 of the general form r(m) = Ek bk (m)Hk sk + zk + ik + n, m = 0, 1, . . . , channel matrix ⎤ ... 0 ... 0 ⎥ ⎥ .. ⎥ . ⎥ ⎥ 0 ⎥ ⎥ hk,1 ⎥ ⎥ .. ⎥ . ⎦ . . . hk,N
= where
H ˜ −1 Ek sH k Hk Rk Hk sk
H ˜k = R E (zk + ik + n) (zk + ik + n)
(7)
(8)
is the autocorrelation matrix of the combined channel disturbance. For mathematical convenience we disregard the ˜ k by Rk = ISI component in (8) and approximate R H E (zk + n) (zk + n) , k = 1, 2, · · · , K, which is now independent of sk . Our objective is to find the binary signature sk that optimizes (maximizes) SIN RMMSE,k , i.e. (b)
=
sk,opt
−1 arg max sT HH k Rk Hk s. s∈{±1}L
(9)
The superscript (b) indicates that sk,opt is binary; {·}T is the transpose operator. For notational simplicity we define the L× L matrix −1 (10) Qk = HH k Rk Hk . Then, the SINR-optimum binary sequence is given by (b)
sk,opt = arg max sT Qk s. s∈{±1}L
(11)
In case of a plain AWGN channel (no multipath), the optimum sequence is still given by (11) after replacing Qk (4) by R−1 k . With this in mind, all of our subsequent theoretical developments that assume multipath fading transmissions over AWGN channel can be rederived in a straightforward manner for the plain (no multipath) AWGN channel case, as well. III. B INARY S IGNATURE A SSIGNMENT
(5)
with entries hk,n , n = 1, . . . , N , considered as complex Gaussian random variables to model fading phenomena, zk ∈ CL+N −1 represents comprehensively multiple-accessinterference (MAI) to user k by the other K − 1 users, i.e. √ K Ei bi (m)Hi si , ik ∈ CL+N −1 denotes zk = i=1 i=k multipath induced inter-symbol-interference (ISI) to user k by its own signal, and n is a zero-mean additive Gaussian noise vector with autocorrelation matrix σ 2 IL+N −1 . Information bit detection of user k is achieved via linear minimum-mean-square-error (MMSE) filtering (or, equivalently, max-SINR filtering) as follows ˆbk = sgn Re wH r (6) MMSE,k
where wMMSE,k = cR−1 Hk sk ∈ CL+N −1 , R = E{r rH }, c > 0, {·}H is the Hermitian operator, Re{·} denotes the real part of a complex number, and E{·} represents statistical expectation. The output SINR of the filter wMMSE,k is given by √ 2 H Ek bk Hk sk E wMMSE,k SIN RMMSE,k (sk ) = 2 H E wMMSE,k (zk + ik + n)
(b)
k=1
where Hk ∈ C(L+N −1)×L is the user k ⎡ 0 hk,1 ⎢ hk,2 h k,1 ⎢ ⎢ .. .. ⎢ . . ⎢ h h Hk (L+N −1)×L = ⎢ k,N k,N −1 ⎢ ⎢ 0 h k,N ⎢ ⎢ .. .. ⎣ . . 0 0
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The optimization problem in (11) is equivalent1 to (b)
sk,opt
=
arg max sT Qkr s s∈{±1}L
(12)
where Qkr denotes the real part of the complex, in general, hermitian matrix Qk
Qkr = Re{Qk }.
(13)
If we relax, for a moment, our constraint that the signature (sequence) alphabet is binary and assume, instead, that s can be real-valued (s ∈ RL ) with the same norm (sT s = L), then the corresponding optimization problem becomes (r)
sk,opt = arg
max
s∈RL ,sT s=L
sT Qkr s
(14)
1 Since s ∈ {±1}L ⊂ RL and sT Q s is a real scalar, sT Q s = k k sT Qkr s.
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where the superscript (r) indicates that sk,opt is real-valued. The optimization √ in (14) is carried out over a hypersphere in RL of radius L centered at the origin (sT s = L). Let f (s) denote the cost function in (14),
f (s) = sT Qkr s,
(15)
and let {qk,1 , qk,2 , · · · , qk,L } be the L eigenvectors of Qkr with corresponding eigenvalues λk,1 ≥ λk,2 ≥ · · · ≥ λk,L . The real-valued sequence s that maximizes f (s) is well known and equal to the eigenvector that corresponds to the maximum eigenvalue of the matrix Qkr , i.e. (r)
sk,opt = arg
max
s∈RL ,sT s=L
f (s) = qk,1 .
(16)
Consider now arcs on the searching hypersphere (sT s = L) qk,1 −ρv , vT qk,1 = 0, vT v = 1, ρ ∈ R, that pass of the form √ 2 1+ρ
through the point qk,1 . The orthogonal to qk,1 direction v that defines the arc of least decrease in SINR (sT Qkr s, s ∈ RL ) (r) from the optimum real signature point sk,opt = qk,1 is
set of all distinct such points after reindexing in ascending order2 (i.e. ρ1 < ρ2 < · · · < ρMi and ρm = ρl for some l ∈ {1, 2, · · · , L}, m = 1, 2, · · · , Mi ). Then, the partition of R (−∞, ρ1 ], (ρ1 , ρ2 ], · · · , (ρMi , ∞) exhibits the following properties. (i) For all ρ in each of the intervals (−∞, ρ1 ], (ρ1 , ρ2 ] · · · , (ρMi , ∞), there exists exactly one binary sequence of the form sgn(qk,1 − ρqk,i ). (ii) Two binary sequences of the form sgn(qk,1 − ρqk,i ) that correspond to adjacent intervals differ in at least one coordinate. Proof (i) Fix i ∈ {2, 3, · · · , L}. Since ρ ∈ (ρn , ρn+1 ], ρ > ρn > ρn−1 > · · · > ρ1 . But ρ1 = ρc1 , ρ2 = ρc2 , · · · , ρn = ρcn for some c1 , c2 , · · · , cn ∈ {1, · · · , L}. The latter implies that the c1 th, c2 th, · · · , and cn th coordinates of the sequences sgn(qk,1 − ρqk,i ) are fixed at −sgn(qk,i [c1 ]), −sgn(qk,i [c2 ]), · · · , −sgn(qk,i [cn ]), correspondingly, for all ρ ∈ (ρn , ρn+1 ], i.e. sgn(qk,1 [c1 ] − ρqk,i [c1 ]) sgn(qk,1 [c2 ] − ρqk,i [c2 ])
qk,1 − ρv min {f (qk,1 )−f ( )}, ρ ∈ R. vT qk,1 =0,vT v=1 1 + ρ2 (17) Generalizing to L − 1 mutually orthogonal directions, we obtain qk,1 − ρv )}, qk,i = arg min {f (qk,1 ) − f ( v∈Ωi 1 + ρ2 i = 2, · · · , L, ρ ∈ R, (18) qk,2 = arg
sgn(qk,1 [cn ] − ρqk,i [cn ])
Lemma 1 Let {qk,1 , qk,2 , · · · , qk,L } be the L eigenvectors of Qkr = Re{Qk }, Qk given by (10), with corresponding eigenvalues λk,1 ≥ λk,2 ≥ · · · ≥ λk,L . Then, for any given i ∈ {2, 3, · · · , L}, the binary sequence that is closest in q −ρqk,i √ 2 , ρ ∈ R, is of the form the l2 sense to the arc k,1 sgn(qk,1 − ρqk,i ).
1+ρ
Proposition 1 Let {qk,1 , qk,2 , · · · , qk,L } be the L eigen vectors of Qkr = Re{Qk }, Qk given by (10), with corresponding eigenvalues λk,1 ≥ λk,2 ≥ · · · ≥ λk,L . For q [n] for each given i ∈ {2, 3, · · · , L}, we define ρn = qk,1 k,i [n] all n ∈ {1, · · · , L} such that qk,i [n] = 0 where qk,1 [n], qk,i [n], n = 1, · · · , L, is the n-th element of vector qk,1 and qk,i , respectively. Let {ρ1 , ρ2 , · · · , ρMi }, Mi ≤ L, be the
−sgn(qk,i [c1 ]) −sgn(qk,i [c2 ]) −sgn(qk,i [cn ]). (20)
At the same time, ρ ≤ ρn+1 < ρn+2 < · · · < ρMi and ρn+1 = ρcn+1 , ρn+2 = ρcn+2 , · · · , ρMi = ρcM for some i cn+1 , cn+2 , · · · , cMi ∈ {1, 2, · · · , L} \ {c1 , c2 , · · · , cn }. Hence, the cn+1 th, cn+2 th, · · · , cMi th coordinates of the sequences sgn(qk,1 − ρqk,i ) are fixed at sgn(qk,i [cn+1 ]), sgn(qk,i [cn+2 ]), · · · , sgn(qk,i [cMi ]), correspondingly, for all ρ ∈ (ρn , ρn+1 ], i.e.
where Ωi = {v : vT qk,j = 0, j = 1, 2, · · · , i − 1, vT v = 1}. The slowest descent arcs qk,1 − ρqk,i , i = 2, · · · , L, ρ ∈ R, (19) 1 + ρ2 trace the searching hypersphere, extend from −qk,i to qk,i , (r) and pass through the optimum real signature sk,opt . On each arc, the function sT Qkr s, s ∈ RL , takes values in [λk,i L, λk,1 L], i = 2, · · · , L. Our objective is to identify the binary sequences that are closest in the l2 sense to the above least decrease arcs. The following Lemma, Proposition, and Corollary provide the theoretical foundation of our proposed SINR-optimized binary signature search algorithm. The proof of the Lemma is straightforward and omitted.
= = ··· =
sgn(qk,1 [cn+1 ] − ρqk,i [cn+1 ]) sgn(qk,1 [cn+2 ] − ρqk,i [cn+2 ]) sgn(qk,1 [cMi ] − ρqk,i [cMi ])
= = ··· =
sgn(qk,i [cn+1 ]) sgn(qk,i [cn+2 ]) sgn(qk,i [cMi ]).(21)
If Mi = L, (20) and (21) define only one sequence of the form sgn(qk,1 − ρqk,i ) for ρ ∈ (ρn , ρn+1 ]. If Mi < L, then there exists 0-valued coordinate(s) of qk,i and/or ρn = ρm for some m, n ∈ {1, 2, · · · , L}. In the former case, sgn(qk,1 [n] − ρqk,i [n]) = sgn(qk,1 [n]) for all n ∈ {1, · · · , L} such that qk,i [n] = 0; in the latter case, sgn(qk,1 [m] − ρqk,i [m]) = sgn(qk,1 [n] − ρqk,i [n]). Thus, when Mi < L, again there exists only one binary sequence of the form sgn(qk,1 −ρqk,i ) for ρ ∈ (ρn , ρn+1 ]. We can prove in a similar manner that the same holds true when ρ ∈ (−∞, ρ1 ] or ρ ∈ (ρMi , ∞). (ii) Part (ii) follows directly from the proof of Part (i). The following corollary is a direct consequence of Proposition 1. Corollary 1 For any given i ∈ {2, 3, · · · , L}, the set of all binary sequences of the form sgn(qk,1 − ρqk,i ), ρ ∈ R, has cardinality at most L + 1. We are now ready to present our proposed binary signature search algorithm. 2 If
qk,i [n] = 0 for all n = 1, · · · , L, then Mi = L. Otherwise, Mi < L.
WEI et al.: ADAPTIVE BINARY SIGNATURE DESIGN FOR CODE-DIVISION MULTIPLEXING
Binary Signature Design Algorithm For i = 2, · · · , P (P ≤ L) do: Step 1 Partition the real-line domain of ρ into intervals defined by the points (cf. Proposition 1) ρn = qk,1 [n]/qk,i [n],
n = 1, · · · , L,
(22)
where qk,i [n] = 0, n = 1, · · · , L, is the nth element of qk,i . Then, let {ρ1 , ρ2 , · · · , ρMi }, Mi ≤ L, be the set of all distinct such points after reindexing ρn in ascending order, ρ1 < ρ2 < · · · < ρMi where ρ1 = ρc1 , ρ2 = ρc2 , · · · , ρMi = ρcM for some c1 , c2 , · · · , cMi ∈ {1, · · · , L}. i Step 2 Find the unique binary sequence sgn(qk,1 − ρqk,i ) for ρ in each of the intervals (−∞, ρ1 ], (ρ1 , ρ2 ], · · · , (ρMi , ∞). There are Mi + 1 binary sequences in total (cf. Proposition 1) i denoted by s0k , s1k , · · · , sM that can be computed recursively k as follows: s0k = sgn[qk,1 ],
sl+1 k
=
slk
−
2s0k [cJ +l ]ecJ+l ,
l = 0, 1, · · · , Mi − J,
− 2s0k [cJ +l ]ecJ+l , l = −1, −2, · · · , 1 − J, slk = sl+1 k
(23) (24) (25)
where J ∈ {1, 2, · · · , Mi } is such that ρJ−1 < 0 < ρJ , s0k [cJ+l ] is the cJ+l th element of s0k and ecJ+l denotes the unit-length vector in RL whose coordinates are all zero except the cJ+l th which is one. In addition, according to Proposition 1, for each vector i s0k , s1k , · · · , sM k , in case of existing 0-valued coordinate(s) of qk,i , set sgn(qk,1 [n] − ρqk,i [n]) = sgn(qk,1 [n]) for all n ∈ {1, · · · , L} such that qk,i [n] = 0; in case of ρn = ρm for some m, n ∈ {1, 2, · · · , L}, set sgn(qk,1 [m] − ρqk,i [m]) = sgn(qk,1 [n] − ρqk,i [n]). Step 3 Evaluate SIN R(slk ) for each binary signature slk returned by Step 2, l = 0, · · · , Mi , and choose the binary signature that exhibits maximum SINR. We note that the above procedure identifies and composes Mi + 1 (Mi ≤ L) binary sequences for each one of a total of P − 1 slowest descent arcs (i = 2, 3, · · · , P ≤ L) Pand selects the one that gives maximum SINR among all i=2 Mi + 1 signatures3 . Our experimental studies indicate that P = 2 or 3 (i.e. one or two slowest descent arcs) is sufficient to closely approximate the performance level reached when all possible slowest descent arcs are considered (P = L). So far, we considered the problem of designing the signature of one user of interest while considering the other users as interferers. Our binary signature design algorithm can be extended to cover multiuser binary signature assignment in a sequential user-after-user manner in what we call a multiuser adaptation cycle. Several multiuser adaptation cycles are carried out until numerical convergence of the binary signature set is observed. Our proposed multiuser assignment algorithm is outlined below. 3 The binary sequence s0 = sgn[q k,1 ] obtained by direct quantization of k the real maximizer (the rank-one-optimal binary solution [23]) is present and common in all slowest descent arcs.
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Iterative Algorithm for Multiuser Signature Assignment (i) Select an arbitrary initialization signature set [s1 , · · · , sK ]. (ii) For k = 1, 2, · · · , K do: (a) Calculate Qk ; (b) design and assign sk by invoking the Binary Signature Design Algorithm. (iii) Repeat Step (ii) until numerical convergence of the binary signature set [s1 , · · · , sK ] is observed. IV. S IMULATION S TUDIES In this section, we evaluate and compare the performance of the proposed binary signature optimization algorithm against the following theoretical benchmarks: (i) The complex maximum-eigenvalue eigenvector of Qk , denoted in the figures as “complex max-EV,” which is the theoretical maximumSINR signature solution over the complex field and (ii) the real maximum-eigenvalue eigenvector of Qkr = Re{Qk }, denoted as “real max-EV,” which is the theoretical maximumSINR signature solution over the real field. Comparisons with the adaptive binary design algorithms of [24] and [25] are also carried out along with conventional static designs (WalshHadamard, Gold). We consider a code-division multiplexing signal model as described in Section II, (equations (1)-(8)). We assume that each user signal experiences N = 3 independent paths and the corresponding channel coefficients are zero-mean complex Gaussian random variables of equal power. Then, following the notation of Section II, the total average received SNR for user k is N 2 2 E E E |h (n)| E h k k k k n=1 = . (26) SN Rk = σ2 σ2 In all studies below, we initialize the signature set arbitrarily and execute one signature set update, which we call a multiuser adaptation cycle, by running sequentially user-after-user a chosen signature design algorithm. In Fig. 1, we plot the pre-detection SINR of a representative user per adaptation cycle averaged over 1,000 random channel realizations. We assume K = 8 active users in the system with signature length L = 16 and SNR values set as follows: SN R1−3 = 8dB, SN R4−6 = 9dB, and SN R7−8 = 10dB. All algorithms are seen to converge effectively in no more than five cycles. A fixed Walsh-Hadamard signature assignment is also included in the study to challenge, potentially, the notion of signature adaptivity. The Cholesky based procedure [24] and code allocation solution in [25] exhibit quite similar performance. The proposed algorithm is seen to perform statistically close to the real-field eigenvector maximizer. The static (non-adaptive) Walsh-Hadamard signature assignment exhibits rather poor performance, as expected. In Fig. 2, we repeat the same study as in Fig. 1 for a system with processing gain L = 31 and K = 16 users with SNR values SN R1−4 = 8dB, SN R5−8 = 9dB, SN R9−12 = 10dB and SN R13−16 = 11dB. The code allocation algorithm of [25] is not applicable due to the non-existence of an orthogonal binary code set for L = 31 (or, to that respect, for lengths that are not multiples of four). A (non-adaptive) Gold
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Fig. 3. Average pre-detection SINR for a representative user versus multiuser adaptation cycle (K=31, L=31).
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Fig. 4. Average pre-detection SINR for a representative user versus sample support (K=16, L=31).
signature assignment is included in the comparisons which fails disappointingly to withstand the asynchronous multipath environment. In Fig. 3, we increase the user load of Fig. 2 to 100% (K = L = 31), with SN R1−8 = 8dB, SN R9−16 = 9dB, SN R17−24 = 10dB and SN R25−31 = 11dB, to test its effect on the convergence of the multiuser adaptation cycles. The study demonstrates that, still, five or so adaptation cycles are about enough. In Fig. 4 we plot the average pre-detection SINR of a representative user versus the size of the data record used to sampleaverage estimate4 the disturbance signal autocorrelation matrix for the same system as in Fig 2. The figure shows the average pre-detection SINR after the third multiuser adaptation cycle (averaged over 1,000 random channel realizations per
adaptation cycle). Finally, to illustrate the performance of the signature design scheme under single-user adaptation, in Fig. 5, we plot the biterror-rate (BER) of a representative user versus SNR when the size of the data record used for the estimation of the received signal autocorrelation matrix is set equal to 3(L + N − 1) and all interferers have fixed non-adapted Gold signature assignments. In the figure we also include, as a reference, the performance curve of the proposed scheme under the assumption of perfectly known received signal autocorrelation matrix. The number of users as well as the signature length and SNR values are set as in Fig. 2. The conclusions are similar to those in Figs. 1-4 obtained under the SINR metric and multiuser adaptation.
4 In
field applications, the autocorrelation matrix of the disturbance to user k, Rk = E (zk + n) (zk + n)H , is not known in general and ˆ k (T ) = may be sample-average estimated through T signal observations, R H 1 T (z (t) + n(t)) (z (t) + n(t)) . k k t=1 T
V. C ONCLUSIONS In this paper we considered the problem of adaptive signature optimization for code-division communications in AWGN and multipath fading environments. In contrast to previous
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Fig. 5. BER versus SNR under single-user adaptation (K=16, L=31, data record size = 3(L+N-1)).
work on this subject that focused on real/complex-valued signature sets, we considered binary signature alphabets and thus pursued digital signature optimization. We proposed and studied a suboptimal algorithm that evaluates arcs of least SINR decrease from the real maximum SINR solution, finds O(L) binary sequences -where L is the signature length- that are closest to these arcs in the l2 sense, and -after direct SINR evaluation- returns the binary sequence that maximizes the SINR at the output of the maximum SINR filter. R EFERENCES [1] L. R. Welch, “Lower bounds on the maximum cross correlation of signals,” IEEE Trans. Inform. Theory, vol. IT-20, pp. 397–399, May 1974. [2] J. L. Massey and T. Mittelholzer, “Welch’s bound and sequence sets for code division multiple access systems,” Sequences II: Methods in Communication Security and Computer Science, vol. 47, pp. 63–78, Springer-Verlag, New York, 1993. [3] 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. [4] P. Viswanath, V. Anantharam, and D. N. C. Tse, “Optimal sequences, power control, and user capacity of synchronous CDMA systems with linear MMSE multiuser receivers,” IEEE Trans. Inform. Theory, vol. 45, pp. 1968–1983, Sept. 1999. [5] P. Cotae, “An algorithm for obtaining Welch bound equality sequences ¨ Int. J. Electron. Commun., vol. 55, pp. for S-CDMA channels,” AEU. 95–99, Mar. 2001. [6] S. Ulukus and R. D. Yates, “Iterative construction of optimum signature sequence sets in synchronous CDMA systems,” IEEE Trans. Inform. Theory, vol. 47, pp. 1989–1998, July 2001. [7] C. Rose, S. Ulukus, and R. D. Yates, “Wireless systems and interference avoidance,” IEEE Trans. Wireless Commun., vol. 1, pp. 415–428, July 2002. [8] P. Anigstein and V. Anantharam, “Ensuring convergence of the MMSE iteration for interference avoidance to the global optimum,” IEEE Trans. Inform. Theory, vol. 49, pp. 873–885, Apr. 2003. [9] H. Boche and S. Stanczak, “Iterative algorithm for finding optimal resource allocation in symbol asynchronous CDMA channels with different SINR requirements,” in Proc. 36th Asilomar Conf. on Signals, Systems and Computers, Pacific Grove, CA, Nov. 2002, vol. 2, pp. 1909–1913. [10] J. Luo, S. Ulukus, and A. Ephremides, “Optimal sequences and sum capacity of symbol asynchronous CDMA systems,” IEEE Trans. Inform. Theory, vol. 51, pp. 2760–2769, Aug. 2005. [11] O. Popescu and C. Rose, “Sum capacity and TSC bounds in collaborative multibase wireless systems,” IEEE Trans. Inform. Theory, vol. 50, pp. 2433–2438, Oct. 2004.
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[12] T. Guess, “User-capacity-maximization in synchronous CDMA subject to RMS-bandlimited signature waveforms,” IEEE Trans. Commun., vol. 52, pp. 457–466, Mar. 2004. [13] G. S. Rajappan and M. L. Honig, “Signature sequence adaptation for DS-CDMA with multipath,” IEEE J. Select. Areas Commun., vol. 20, pp. 384–395, Feb. 2002. [14] G. N. Karystinos and D. A. Pados, “New bounds on the total squared correlation and optimum design of DS-CDMA binary signature sets,” IEEE Trans. Commun., vol. 51, pp. 48–51, Jan. 2003. [15] C. Ding, M. Golin, and T. Klφve, “Meeting the Welch and KarystinosPados bounds on DS-CDMA binary signature sets,” Designs, Codes and Cryptography, vol. 30, pp. 73–84, Aug. 2003. [16] V. P. Ipatov, “On the Karystinos-Pados bounds and optimal binary DSCDMA signature ensembles,” IEEE Commun. Lett., vol. 8, pp. 81–83, Feb. 2004. [17] G. N. Karystinos and D. A. Pados, “The maximum squared correlation, total asymptotic efficiency, and sum capacity of minimum total-squaredcorrelation binary signature sets,” IEEE Trans. Inform. Theory, vol. 51, pp. 348–355, Jan. 2005. [18] F. Vanhaverbeke and M. Moeneclaey, “Sum capacity of equal-power users in overloaded channels,” IEEE Trans. Inform. Theory, vol. 53, pp. 228–233, Feb. 2005. [19] F. Vanhaverbeke and M. Moeneclaey, “Binary signature sets for increased user capacity on the downlink of CDMA Systems,” IEEE Trans. Wireless Commun., vol. 5, pp. 1795–1804, July 2006. [20] P. D. Papadimitriou and C. N. Georghiades, “Code-search for optimal TSC binary sequences with low crosscorrelation spectrum,” in Proc. IEEE MILCOM, Boston, MA, Oct. 2003, vol. 2, pp. 1071–1076. [21] T. F. Wong and T. M. Lok, “Transmitter adaptation in multicode DSCDMA systems,” IEEE J. Select. Areas Commun., vol. 19, pp. 69–82, Jan. 2001. [22] C. W. Sung and H. Y. Kwan, “Heuristic algorithms for binary sequence assignment in DS-CDMA systems,” in Proc. IEEE Intern. Symp. Personal, Indoor and Mobile Radio Commun., Sept. 2002, vol. 5, pp. 2327– 2331. [23] G. N. Karystinos and D. A. Pados, “Rank-2-optimal adaptive design of binary spreading codes,” IEEE Trans. Inform. Theory, vol. 53, pp. 3075–3080, Sept. 2007. [24] G. N. Karystinos and D. A. Pados, “Adaptive assignment of binary user spreading codes in DS-CDMA systems,” in Proc. SPIE, Orlando, FL, Apr. 2001, vol. 4395, pp. 137–144. [25] H. Y. Kwan and T. M. Lok, “Binary-code-allocation scheme in DSCDMA systems,” IEEE Trans. Veh. Technol., vol. 56, pp. 134–145, Jan. 2007. [26] P. Spasojevic and C. N. Georghiades, “The slowest descent method and its application to sequence estimation,” IEEE Trans. Commun., vol. 49, pp. 1592–1604, Sept. 2001.
Lili Wei (S’05) received B.S. and M.S. degrees in electrical engineering (both with excellence) from Shanghai Jiao Tong University, China in 1997 and 2000, respectively, and the M.S. degree in electrical engineering from the State University of New York at Buffalo in 2005. From 2000 to 2001, she worked as a research and development engineer in Wuhan Research Institute of Posts and Telecommunications, Wuhan, China. After that, she joined the Chinese Academy of Telecommunication Technology, Beijing, China, and held a research and development engineer position there until August 2003. From September 2003, she has been a Ph.D. student in the Communications and Signals Laboratory of the Department of Electrical Engineering, State University of New York at Buffalo, Buffalo, NY. Ms. Wei is student member of the IEEE Communications Society.
Stella N. Batalama (S’91, M’94) received the Diploma degree in computer engineering and science (5-year program) from the University of Patras, Greece in 1989 and the Ph.D. degree in electrical engineering from the University of Virginia, Charlottesville, VA, in 1994. From 1989 to 1990 she was with the Computer Technology Institute, Patras, Greece. In 1995 she joined the Department of Electrical Engineering, State University of New York at Buffalo, Buffalo, NY, where she is presently a Professor. During the summers of 1997-2002 she was Visiting Faculty in the U.S. Air Force Research Laboratory (AFRL), Rome, NY. From Aug. 2003 to July 2004 she served as the Acting Director of the AFRL Center for Integrated Transmission and Exploitation (CITE), Rome NY.
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Her research interests include small-sample-support adaptive filtering and receiver design, adaptive multiuser detection, robust spread-spectrum communications, supervised and unsupervised optimization, distributed detection, sensor networks, covert communications and steganography. Dr. Batalama was an associate editor for the IEEE Communications Letters (2000-2005) and has been serving as an associate editor for the IEEE Transactions on Communications since 2002.
Dimitris A. Pados (M’95) was born in Athens, Greece, on October 22, 1966. He received the Diploma degree in computer science and engineering (fiveyear program) from the University of Patras, Patras, Greece, in 1989, and the Ph.D. degree in electrical engineering from the University of Virginia, Charlottesville, VA, in 1994. From 1994 to 1997, he held an Assistant Professor position in the Department of Electrical and Computer Engineering and the Center for Telecommunications Studies, University of Louisiana, Lafayette. Since August 1997, he has been with the Department of Electrical Engineering, State University of New York at Buffalo, where he is presently a Professor. His research interests are in the general areas of communication theory and adaptive signal processing with an emphasis on wireless multiple access communications, spread-spectrum theory and applications, coding and sequences. Dr. Pados is a member of the IEEE Communications, Information Theory, Signal Processing, and Computational Intelligence Societies. He served as an Associate Editor for the IEEE Signal Processing Letters from 2001 to 2004 and the IEEE Transactions on Neural Networks from 2001 to 2005. He received a 2001 IEEE International Conference on Telecommunications best paper award and the 2003 IEEE Transactions on Neural Networks Outstanding Paper Award for articles that he coauthored with his students.
Bruce W. Suter (S’69, M’72, SM’92) received the B.S. and M.S. degrees in electrical engineering in 1972 and the Ph.D. degree in computer science in 1988, all from the University of South Florida, Tampa, FL. Since 1998, he has been with the Information Directorate of the Air Force Research Laboratory, Rome, NY, where he was the Founding Director of the Center for Integrated Transmission and Exploitation. Dr. Suter has authored over a hundred publications. He is also the author of the book Multirate and Wavelet Signal Processing (Academic Press, 1998). His research interests include multiscale signal and image processing, cross layer optimization, networking of unmanned aerial systems, and wireless communications. His professional background includes industrial experience with Honeywell Inc., St. Petersburg, FL and with Litton Industries, Woodland Hills, CA, and academic experience at the University of Alabama, Birmingham, AL and the Air Force Institute of Technology, Wright-Patterson AFB, OH. Dr. Suter’s honors include Fellow of the Air Force Research Laboratory, the Arthur S. Flemming Award: Science Category, and the General Ronald W. Yates Award for Excellence in Technology Transfer. He served as an associate editor of the IEEE Transactions on Signal Processing. Dr. Suter is a member of Tau Beta Pi and Eta Kappa Nu.
