JOINT DATA DETECTION AND CHANNEL ESTIMATION, PREFERABLY OF DS/CDMA SIGNALS

Abstract: In this page we present two efficient iterative structures of tractable complexity for joint mutliuser detection and multichannel estimation (JDE) of DS/CDMA signals. The schemes results form an application of the EM and the SAGE algorithm respectively. The EM-JDE receiver updates the data bit sequences in parallel while the SAGE-JDE receiver reestimates them successively. The channel parameters are updated in parallel in both schemes. The EM algorithm provides a set of free parameters, called weight coefficients, which can be selected to optimize its performance. Two optimality criteria are defined and analytical expressions for the corresponding optimized weight coefficients are given. Monte Carlo simulations of a synchronous scenario show that the proposed JDE receivers are near-far resistant and robust against errors in the estimation of the channel parameters. Moreover, very short training sequences are required for the JDE schemes to converge with. Simulations results further demonstrate that the SAGE-JDE receiver exhibits a better performance when the user's bit sequences are updated in the order of increasing signal strength, i.e. the bit sequence of the user with the weakest signal strength is updated first at each stage.

Description: JOINT DATA DETECTION AND CHANNEL ESTIMATION, PREFERABLY OF DS/CDMA SIGNALS The iiwemion relates to methods incorporating, to systems utilising and to tises of the E3VJ algorithm and/or ihe SAGE nJgorilhm for data detection and channel estimation. Among prior art ihe following publications are referred to: WO 00/1 1799, EP 0 964 389, WO 97/05707. WO 00/25438. EP 0 866 568, WO 99/40698, EP 0 654 925, WO 99/01945. BACKGROUND OF THE INVENTION Multiuser detection is known to drastically increase the bandwidth efficiency of CDMA systems compared to conventional detection using RAKE receivers [I]. Verdu derived the optimum multiuser receiver in his seminal paper [2]. However, the complexity of this algorithm grows exponentially with the number of users and the number of multipaths, which prevents any implementation. Thus, suboptimum feasible techniques for multiuser detection have been proposed which still approach the performance of the optimum receiver. Worth mentioning among them are linear multiuser detection [3] and iterative cancellation of the multiple access interference (MAI) from the received signal before making data decisions. The latter method falls into two categories: successive and parallel interference cancellation. An overview of suboptimum multiuser detection techniques can be found in [4] and [5]. The expectation-maximization (EM) algorithm [6] [7] is an iterative method which enables to approximate the maximum-likelihood estimate when a direct calculation of this estimate is computationally prohibitive. One disadvantage of this algorithm is its slow convergence rate. A variety of techniques have been proposed for accelerating the convergence of the EM algo- rithm such as Louis' Turbo EM performing a Newton-Raphson step [8] or the space-alternating generalized expectation-maximization (SAGE) algorithm [9]. Georghiades and Han propose in [10] an EM-based receiver which performs joint data detection and estimation of the time-variant flat Rayleigh fading channel. This scheme necessitates the transmission of only a few preamble bits. Nelson and Poor in [11] extend the EM and SAGE algorithms to a so-called Space-Alternating Missing-Parameter EM (MPEM) algorithm that alternatively updates individual parameters or treat them as probabilistic missing data. All EM, SAGE, and MPEM are applied to derive various multiuser detectors using hard/soft decision feedback for the additive white Gaussian noise (AWGN) channel. Monte Carlo simulations show the near-far resistance of these schemes. The SAGE-based multiuser detector presented in [12] by Raphaeli for frequency-selective channels is identical to that in [11] except the softness of bit decisions can be controlled. Feder and Weinstein showed in [13] that the computational complexity in EM can be reduced by iteratively decomposing the observed signal and estimating the parameters of each signal component. Following this approach, Borran and Nasiri-Kenari develop in [14] a compuationally feasible multiuser detector for the AWGN channel. Finally in [15], Louis' Turbo EM algorithm is used for adaptive channel estimation and sequence detection. Other issues influencing the performance of multiuser detection are the amount of estimated MAI canceled out prior to the detection of each users' bit sequence and the order in which sequences are detected. The degrading effect of MAI can be significantly reduced in parallel interference cancellation, when the amount of removed MAI is increased with the number of stages [16]. Furthermore, the performance of decorrelating decision-feedback multiuser detector (DDF) [17] as well as of the successive interference cancellation (SIC) scheme proposed in [18] can be improved by ranking the users in the order of their signal strength. Experimental investigations into the EM multiuser detector from Borran and Nasiri-Kenari [14] indicate that fidelity of the bit-desicions improves when amount of interference cancellation is proportional to the interference-to-signal ratio. In this paper we apply the EM and the SAGE algorithms to the problem of joint multiuser data detection and channel estimation (JDE) of synchronous DS/CDMA signals in flat Rayleigh fading. In this way we obtain iterative methods of tractable complexity which smartly combine the two processes of data detection and channel estimation. The synchronous assumption and the flat channel model provide a simple means for studying JDE in the uplink of a DS/CDMA system. The JDE receiver architectures presented in this paper can then easily be extended to the asynchronous case and to frequency-selective channels by small modifications. These considerations are currently under investigation. The paper is organized as follows. The signal model of the considered DS/CDMA system is outlined in Section II, followed by a concise description of the EM and SAGE algorithms in Section III. In Section IV the former algorithm is applied to derive the first, so-called EM- JDE receiver. This scheme encompasses weight coefficients which can be selected to optimize its performance. Two optimality criteria are defined and the corresponding optimum weight coefficients are derived. The SAGE-JDE receiver described in Section V results from an application of the SAGE algorithm. Implementation issues such as initialization, complexity and updating strategies are discussed in Section VI. Finally, in Section VII the performance of the JDE algorithms is assessed and compared to that of the SAGE [11] and DDF [17] schemes both supplemented with minimum mean-square-error (MMSE) channel estimation by means of Monte Carlo simulations in the Gaussian and Rayleigh channels. DESCRIPTION OF THE INVENTION We consider a synchronous CDMA system with K active users sharing the same propagation channel. The signal transmitted by each user experiences in the channel flat Rayleigh fading which is assumed to be constant over the observed frame of L bits. The complex baseband representation of the received signal therefore reads K i τ(t) = ∑ ∑ ak * *(* - {i - i)r.) + n(t). (i) fe=l .=1 In the above expression, k t) is the signature waveform of user k and b k [l] € {— l, +l} the bit transmitted by this user during the I th signalling interval. The complex, circularly symmetric Gaussian random variable _. & with zero mean and variance represents the channel attenuation of the signal of the & ,h user. Without loss of generality α^ encompasses the root power of the transmitted signal of User k. Furthermore, n(t) represents complex-valued white Gaussian noise with spectral height N 0 . The signature waveforms pι(t), . . . ,pκ{t) have support [0, T s ), where T s denotes the inverse of the bit rate, and are normalized to have unit energy. We assume that the random elements a = (aι, . . . , aχ) , b = (&_[]], . . . . &κ [l], - • • . b_ [L] , bχ[L]) , and n(. ) are independent as are the components of a and b. The symbol ( • ) designates transposition. Let z[_] = . . . , zχ[.]) τ denote the .K -dimensional complex vector whose components are the output samples of K filters matched to the signature waveforms at the end of the ? th signalling interval. It follows from (1) that z[.] = RB[.]a + n[.], l = l, . .. , L. (2) Here, the K x K matrix R is of the form with p k — f 0 s P j {t)P k (t)dt being the crosscorrelations between the signature waveforms of user k and j. The K x K diagonal matrix B[ ] is given by B[.] == diag(6j [I], . . . , bχ[l]). Because of the above assumptions, the autocovariance matrix of a reads Finally, [l] is a K-dimensional zero-mean Gaussian random vector with covariance matrix NoR- Let Pι(t), ■ ■ ■ ,pκ{t) be selected in such a way that the correlation matrix R is positive definite. Then R can be Cholesky factorized according to R = C T C, where C is a unique lower triangular invertible matrix. Multiplying z[l] by the whitening matrix (C 7 ) "1 yields y[.} ^ (C T ) 1 z[l] = CB[l} a + n[l) 1 l = l, . . . , L. (4) The complex Gaussian vector n[.] is white, i.e. has covariance matrix Nol/c with lχ denoting the K x K identity matrix. Both the matched-filter-output (2) and the whitened-matched-filter- output (4) are equivalent as they provide the same information about the transmitted bits [4]. However, the latter description is more appropriate for the subsequent analysis because of the property of the noise vector n[l] to be white. For the problem at hand, we need to extend the model in (4) to frames of L bits. In vector notation, the whitened matched filter outputs are given by y = CBa + n, where y = (y[l] T , . . . , y[ ] T ) T , a = (a τ , . . . , a τ ) , and n = ( n[l] , . . . , n[L] j . With the assumptions made above the co- variance matrix of a yields III. THE EM AND SAGE ALGORITHMS A. The EM Algorithm Let B denote a possibly vector-valued parameter to be estimated from some possibly vector- valued observation y with probability density The EM algorithm provides an iterative scheme to approach the maximum likelihood estimate B — arg max (y\B) in case where a direct computation of B is prohibitive. The derivation of the EM algorithm relies on the concept of a hypothetical, so-called complete unobservable data X which, if it could be observed, would ease the calculation of B. The observed random variable y, which is referred to as the incomplete data within the EM framework, is related to X by a mapping X >-» y(X). Since X is not observable, at the i h iteration the EM algorithm computes in a first step, called expectation step (E-step), the estimate of the log-likelihood function The conditional expectation above is given the observation y and assuming that B equals B . In a second step called maximization step (M-step) the parameter vector is updated according to βt' +1] = arg max Q (B|BH) . (7) If {£>M} i s a sequence of estimates generated by the EM algorithm starting from an initial value £> [ °1, then {A (y |#M) } i s non-decreasing (Monotonicity Property). Provided the function Q (B\B') fulfills mild regularity conditions [6] the sequence {Λ (_y|βM) } converges to a fix point [6], [19]. The ability of the EM algorithm to find a global maximum depends on the initialization B^. The convergence rate of the EM algorithm is inversely related to the conditional Fisher information matrix of X given y. This rate is notoriously slow when the dimension of the complete data is large. B. The SA GE A Igorithm The SAGE algorithm proposed by Fessler et al. [9] is a twofold generalization of the EM algorithm: Firstly, rather than updating all parameters simultaneously at iteration i, only a subset 23s of 23 indexed by S = S[i] is updated while keeping the parameters in the complement set 23g — B\Bs fixed; and secondly, the concept of the complete data X is extended to that of the so-called hidden data A$ to which the incomplete data y is related by means of a possibly non- deterministic mapping >-> ■ y(Xs) exhibiting some particular property. A hidden data space would be a complete data space for Bs in the EM framework, if 23g were known [9]. The particular property of the mapping X >-» y Xs) guarantees that the SAGE algorithm exhibits the Monotonicity Property as well. The convergence rate of the SAGE algorithm is usually higher than that of the EM algorithm, because the conditional Fisher information matrix of Xs given y for each set of parameters B is often smaller than that of the complete data X for the entire space B. At the i th iteration the E-step of the SAGE algorithm reads Qs (Bs≠l) B } . ( 8) In the M-step, only Bs is updated, i.e. Bs l ' +1] = argmaxQs (B S \B^) , β s l' +1 I = β g W. (9) IV. THE EM-JDE RECEIVER A. Receiver Design Following a common approach for applying the EM algorithm to the problem of estimation of superimposed signals [13] [20], we decompose the signal at the output of the whitening filter into the sum The summands are defined as x fe [/] = s fc [q + n fc [Z], fe = l . . . A-, (10) where sj.[.] = C f c6*.[.]α / _ with C / . denoting the k th column of C. Clearly, Sj.[.] is the contribution to y[l] of the signal of User k only. The noise vector nj_[.] is white with spectral height β k No- To ensure consistency with the original model (4) the non-negative coefficients β-_, . . . , βκ have to satisfy the constraint K ∑ft = i- (ii) fc=l In other words, the weight coefficients determine the relative portion of total noise τ_ k assigned tO X f c. As the problem at hand is the estimation of the transmitted bits, the complex amplitudes α 1; . . . , ax are unknown nuisance parameters. The EM framework provides a solution for dealing with nuisance parameters. These nuisance parameters are included as a part of the complete data. Together with (10), a natural choice for the complete data is X = {(xι, αι ), . . . . {xχ, aχ)}. The parameter vector to be estimated is B = b and the incomplete data is y — y . Since the components of X are independent, the log-likelihood function of b for the observation X reads K with Λ(x fe> fc |bifc) = Λ(x fc |α fc , b fc ) + Λ(α fc |b f c). Because of the model assumptions, the second summand is independent of b and can therefore be discarded. Moreover, neglecting terms independent of a and b we obtain from (10) h Λ(x fc , α,|b fc ) = ∑ R{C»δ fc [*Kx fc [Z]}. (13) .=ι Here, K{-} and (-) H denote, respectively, the real part and the complex conjugate transpose of the argument and b /. = {b k [l}, . .. , b k [L]) Υ . Because of the special form of A(X\B) in (12), the function Q B\B^) calculated in the E-step of the EM algorithm (see (6)) can be decomposed according to Q (B|βW) = Q (b|bW) = ∑ Q k (b fc |bW) (14) where Q k (b f e|bW) = E {Λ(x fc |α fc ,b fc )|y,bW} . (15) Substituting (13) into (15) yields where (α x fc [f))^ = E {α x fc [f||y,bM}. The latter term can be easily calculated by iterating conditional expectations according to (a* k x k [l} = E {a* k E {x fc [Z]|y,bW, a}} . (17) The conditional distribution of x* . given y, a and assuming that b = b^ is Gaussian with mean (see e.g. [21]) £ {x fe [i]|y,bH, a} = C k b k [l a k + β y[l] - C l aλ . (18) Inserting (18) into (17) yields, (a * k x k [l] = β k (a ] y[l} - ∑ CχΛΨH*A ) (19) where | • | denotes the magnitude of the argument , αι M E {α fc |y, bW} = E , (20) and (a,al E + ( Σ » W ) jfe < 21 > with ∑ β W = β{(aW-a)(aW-a) H | y> bW}. The conditional distribution of the channel coefficient vector a given y and assuming that b = bM is also Gaussian with mean E {a|y , bW } = ∑aaBW ( ^Ε__^ + N 0 (c τ c) ~ C^y (22) and covariance matrix ∑aa W = ∑sa - ∑ssB w (BM∑SSBM + N 0 (c τ c) ~ ^ BM∑sa- (23) Because of (14), the M-step (7) is performed by maximizing the terms Q k ( j.|bM) individually, ϊ+1 argmaxg fc (b fc |bW). Moreover when no coding is used, it follows from (16) that each component of bX +1 ' can be obtained by maximizing the corresponding summand in the right-hand expression in (16), i.e. b k [lf +1] = sgn{3?{c»(< cfcM) W }}- (24) Substituting (1 ) into (24), we obtain sgnj &#(|α fc i 2 ) H + ftC? (a y[l) - ∑ ] ] ( 5) where sgn{-} denotes the signum function. We assume subsequently that the length L of the observed frame is large enough so that the second summand in (21) is negligible compared to the first one (see Appendix). Discarding this term and slightly rewriting, (25) reduces to h[tf** + βka ] ( C n y[l] -∑pMfW H- (26) According to (24), the bits of User k are reestimated at the i th iteration as follows. A regenerated version (ap_ k ' of αx<_ (17) is fed into a single-user (SU) receiver which projects this estimate onto Cj; and finally provides this projection to the input of a conventional coherent detector. The term jJ≠k m ( 6) is an estimate of the MAI at the output of the whitening filter. A block diagram of the proposed EM-JDE receiver is given in Figure 1. We have decomposed a K-user optimization problem into K independent procedures leading to a computationally feasible scheme. B. Optimizing the weight coefficients β k When the EM algorithm is employed for estimation of the parameters of a linear model, its convergence rate is maximized when the weight coefficient are chosen to be equal, i.e. β k — l/K, k = 1, . . . , K [22]. Due to the discrete value of the parameter vector b in our model, this approach to obtaining the weight coefficients is not applicable anymore. Instead, we can select βι, . . . ,βx such that the bit-error-rate of User k is minimized as i tends to infinity, that means β k = arg min (lim P h _ k ≠ b k [l] \ ) . Since an analytical evaluation of this expression is difficult to perform, we rely on suboptimal methods. B.l Optimization via complete data A feasible way to choose the weight coefficients is to minimize the energy of the error in estimating the signal components of User k in front of its SU-receiver while neglecting background noise, i.e. n = 0 [4]: β, = arg min E Sk.lf - s k _l) (27) where || • || designates the norm of the argument. The vector s k [l] can be extracted from the expectation in (19) by s k [lf ] = E {xj . [Z]|y,bH, a W} | n=0 leading to s k [l] b] = C k b k [lfa k + β k . Substituting s k [l] and s k [l_ into (27) and removing terms independent of β k , we obtain β = arg min j - 2β k 5 jE j (b k [l]a k - 6 fc [Z] W α fc «) * ∑ p kj (b,[l]a, - rosruiatmg the cross-terms E {b 3 [l]a 3 * '] Wα / W } = E { h λΨ a f ] Vltf' 1 * 1 1 } o. j ≠ y, we make a negligible error in obtaining the weight coefficients. Hence, The expectations in (28) can be computed as where P uy ' b 3 ~ P b 3 . This bit-error-probability is independent of I because the single- path fading vector a is constant within the entire framelength L. Differentiating with respect to β h , setting the result equal to zero, and solving with respect to βι, . . . , βx yields where T 3 is given by the right-hand expression in (29). Using the relations E + α,M|| 2 } = E σ , the expression V 3 can be rewritten as V, b 3 [l]} (l - P b ] ) + (E - β ,M|| 2 I b 3 [l ≠ b 3 [l}} r AσfjP X (31) It follows immediately from (30) that the optimized coefficients βι, .. . , βx satisfy the constraint (11). Already for moderate signal-to-noise-ratios, the two conditioned variances of the channel estimates in (31) are independent of a certain bit constellation and might therefore be replaced by the actual available ( ∑ss^ ) (23). Inserting this approximation in (30) yields \ k We assume subsequently that L is large enough so that ( ∑s a ) becomes small compared to σ k P b l τ ' and can therefore be discarded (see Appendix). Using these approximations, (32) simplifies further to Observe that β] is a function of the iteration i unlike proposed in the original work [13]. Nevertheless, the EM algorithm still exhibits the Monotonicity Property because the optimized coefficients /3- j , . . . ,β κ meet the constraint (1 1) (see [13]). It should also be mentioned that β k is independent of I since the channel coefficients remain constant over the frame. Furthermore, suppose that EM-JDE provides bit estimates performing close to the single-user (SU) bound P .i t ^ ~ Q( v 2|α. ; M| 2 /No) with Q(x) denoting the error function Q(x) = * e ~χ2 l 2 dx [23]. B.2 Optimization via projected complete data A shortcoming of the attained β k in (30) is that the users are updated independently of the crosscorrelations ρ kj between the signature waveforms. Consequently, a better choice is first to project the cancellation error on Cfc and then to minimize the energy of the projection. This is equivalent to minimize the estimation error at the output of the demodulator of User k. Proceeding in this way yields β, = are min E { C C»» j s fc [Z] w - S f c[.] k β>. Following the steps in Section IV-B.l , we obtain after some straightforward algebraic manipulations a general formula β = o < βk < 1 (34) ∑JLi Ϊ where V k is given in (31). Note that the range of/?/, is K l ≤ i.e. the coefficients βι, . . . , βx violate the initial constrained (11). The implication of this selection of the weight coefficients is twofold: Firstly, the conceptual inteipretation of noise splitting among the users on which the derivation of the EM-JDE receiver relies does not hold anymore. Secondly, experimental investigations indicate that the receiver does not fulfill the Monotonicity Property anymore. However, this fact is not necessarily disadvantegeous for applications to multiuser detection, since achieving a maxima of the likelihood function does not necessarily mean that one particular user, i.e. the weakest one, determining the overall performance is detected most accurately. Similar to (32), for a moderate framelength L (34) reduces to and if L is large enough, the optimized weight coefficients can be approximated by B.3 A Two-User Example The following numerical example will help to gain more insight into the behavior of the optimized weight coefficients β k (33) and β k (36) respectively. We consider the case of K = 2 users transmitting in an AWGN channel. The channel coefficients are assumed to be known to the receiver. To be more specific, the estimates . . . , α # M are replaced by the true values αi, . . . , a in the E- and M-steps of the JDE schemes. Let us first investigate the behavior of β k in three scenarios as depicted in Figure 2a. The received signal-to-noise ratio b ι — ' aι ) of User 1 is fixed at 0 dB, 4 dB and 8 dB respectively, while the signal of User 2 is received with an energy per bit varying from —20 dB to +10 dB with respect to that of User 1. In the limiting case of very high MAI, i.e. 7 2 — > ■ co ', βι — » 1 so that this optimized version of the EM-JDE scheme attempts to update only the weakest user. Hence, the update procedure (26) is based only upon the strong interference of User 2 whose tentative bit decisions are very reliable. In the equal power case the weight coefficients (33) read βi — βi — 1/2. This result is consistent with that obtained in [22]. In the limiting case of very low interference, i.e. . 2 -» 0 2 , β \ — > ■ 1 again. Unlike the case of strong interference, both the bit decisions of User 1 and of User 2 are discarded. Indeed, α 2 tends to 0 and therefore (26) with " "Note that the error function <Q (•) converges to zero faster than 7b 2 tends to infinity so that 7b,2-fb 2 → 0 as 7b _ — > c . 2 As 7 2 → 0, .Pb 2 → 0.5 so that 7b 2^,2 → 0. Q j W = a 3 converges to b_ [i\ t!'+11 - sgn{5t{C^ M}}, i.e. the bits of User 1 are updated by employing the conventional SU-receiver. The behavior of ?*.. (36) in the same scenario as before is depicted in Figure 2b. If we hold the crosscorrelation p_ 2 fixed, i.e. 0.4, then in the limiting cases of 7b, 2 — oo and 7b, 2 -A> 0 the optimized coefficients behave asymptotically in the same way as previously discussed, i.e. βι — > • 1. However in the equal power case, β = /3 2 = 1/(1 + 0.86 rather than 0.5. Focusing on the influence of the crosscorrelations on the optimized coefficients, we assert that β k converges to β k as pn -> 1- At the other hand of the scale, both β\ and β 2 tend to 1 as Pi 2 → 0- In other words, both users are fully updated by using a conventional SU-receiver. Notice, that in this case the constraint (1 1) is violated, because β + β 2 = 2. THE SAGE-JDE RECEIVER A JDE architecture can be derived within the SAGE framework as follows: the parameter vector to be estimated and the incomplete data are B = b and y = y respectively. At each iteration, the bit sequence of only one user is updated. More specifically, Bs ~ h k with k = k[i] = i mod K + 1. Hence, 23§ = b where h is the vector obtained by cancelling the components of b / ,. in b. It can easily be shown, that As = {y, a} is a hidden data for each b f c. Notice that we follow the same approach as for the EM-JDE scheme by incorporating the nuisance parameter vector a into the hidden data. The log-likelihood function of b for the observation Xs can be written as A(X S \B) - Λ(y, a|b) - Λ(y|a, b) + Λ(a|b). Any terms independent of b do not affect the M-step. Dropping these terms, from (4) the log- likelihood function reads L Λ(y|a,b) = ∑ 2κ{(CB[.]a) H y[.]} - ||CB[/]a|| 2 . (37) 1=1 With a slight abuse of notation Q k (•) = Qs (•) the E-step (8) reads in this case Q f c (b f clbH) = E {Λ(y|a, bfc, b f cH) |y,bH} . (38) Substituting (37) into (38) and dropping terms independent of b k yields Q k (b ; bW) = (39) with aiX and (α^) as given in (20) and (21) respectively. In an uncoded CDMA system the M-step (9) is performed by maximizing each summand of the right-hand expression in (39) individually: ] ■ (40) If the observed frame length L is large enough, so that the second term in (21) can be neglected (see Appendix), the right hand expression in (40) can be approximated leading to According to (41), the tentative decisions of the bits are used to calculate an estimate of the MAI which is increasingly reliable with iteration i. The signals at the output of the whitening filter are cleaned up with this estimated MAI, weighted by α ^, projected onto C k , and finally fed into a conventional coherent detector. A block diagram of the SAGE-JDE algorithm is depicted in Figure 3. Note that the users' bit sequences are updated successively while the channel coefficients are reestimated in parallel after completion of each sequence up-dating. IMPLEMENTATION ISSUES A. Initialization The initialization of the EM-JDE and SAGE-JDE schemes are performed as follows: The first p bits transmitted by each user are known to the receiver. The initial channel estimate a'°l is the MMSE estimate of a based on the observation y[l], • ■ • ,y[p]. The initial estimate b^ is the MMSE estimate of b computed from the observation of y while assuming a = a'°l [3]. Experimental evidence shows that a better performance is reached, if the first p bits are initialized to equal the MMSE estimate rather than their true value. We refer to this method for obtaining a^ and b !o] as MMSE separate detection and estimation (MMSE-SDE). B. Complexity The joint estimation of the channel coefficients (22) dominates the computational complexity of both EM-JDE and SAGE-JDE algorithms. The computation of (22) requires 0{JK 3 L 2 ) operations per bit. Thus, the time complexity per iteration and bit of the EM-JDE and SAGE- JDE scheme is bounded by 0(K 2 L 2 ) and ϋ(K 3 L 2 ) respectively. When the average signal- to-noise ratio of the k th user 7 b . .. — j " increases to infinity, (22) approaches the decorrelating channel estimate a k = ) ∑ [l]{C-'y[l}} k . (42) L 1=1 Note that C _1 y[.] is the output signal of the decorrelator. A time complexity per iteration and bit of this estimate is bounded according to (K) and ϋ(K 2 ) for the EM-JDE and SAGE- JDE receiver respectively. C. Investigated JDE architectures C.l EM-JDE We choose the following two settings of/?/,, for the proposed EM-JDE receiver: • EM-JDE (β k ): the parameters β k are optimized according to (33). . EM-JDE (β k ), EM-JDE (/? . , oo): the weight coefficients β k are optimized according to (36). The label oo indicates that the channel estimates are computed by means of (42) instead of (22). C.2 SAGE-JDE So far, the derivation of the SAGE-JDE algorithm in Section V is regardless of any sorting of the K users. Different strategies exist of how to rank the users. The following two have been implemented: . SAGE-JDE (t), SAGE-JDE (|, oo): The users are ranked according to their estimated strength l α k I 2 ' so tnat me user w i m tne weakest received signal is ranked first. The label oo means that the channel estimates are computed via (42) instead of (22). . SAGE-JDE (I), SAGE-JDE (4-, oo): The users are ranked in the order of decreasing strength. As shown below the selected sorting procedure drastically affects the ability of the SAGE- JDE algorithm to cancel out weak MAI. PERFORMANCE ANALYSI S The feedback structure of the EM-JDE and SAGE-JDE schemes makes a theoretical analysis of their performance cumbersome. Hence, Monte Carlo simulations were carried out to assess the bit-error-performance of the individual users. The system parameters are chosen as follows: K — 8 users and equal crosscorrelations p = p jk , j, k = 1, . . . , 8, j k. Without loss of generality we focus on User 1. A comparison with other suitable receiver architectures is made. The SAGE multiuser detector in [11] is identical to our SAGE-JDE receiver, except the channel is assumed to be known and the users are updated in the order 1, . . . , K. For the comparison to be fair, the SAGE multiuser detector from [11] is combined with a receiver performing MMSE-SDE as described in Section VI-A. We refer to this scheme as SAGE-separate detection and estimation (SAGE-SDE) receiver. The DDF multiuser detector [17] supported with a separated MMSE multi-channel estimator has been considered another candidate for comparison. We refer to this scheme as DDF-separate detection and estimation (DDF-SDE) receiver. The averaged bit-error-rate P b l of User 1 is depicted in Figure 4 for equal crosscorrelations p — 0.4, p = 4 preamble bits and frame length L = 40. Note that the average is computed over the realizations of the channel coefficients. All the users are received with the same average signal-to-noise ratio _ k , k = 1, . . . , K. Two stages are carried out in both EM-JDE (β k ) and SAGE-JDE (t). One stage corresponds to the number of iterations required to update every user's bit once, i.e. one iteration and K iterations in the EM-JDE scheme and in the SAGE- JDE scheme respectively. In the competing SAGE-SDE receiver the number of stages was limited to two, too. The performance loss against the SU-bound is the so-called multiuser efficiency in Rayleigh fading 7 1 [4], or multiuser efficiency for short. Evaluating this measure at b: ι = 30 dB, we obtain 1.0 dB and 0.9 dB for EM-JDE (β k ) and SAGE-JDE (|) respectively. The multiuser efficiency of the MMSE-SDE scheme is 2.6 dB. The SAGE-SDE and DDF-SDE receivers achieve a multiuser efficiency of 2.4 dB and 2.3 dB respectively. The multiuser efficiency for User 1 is reported in Figure 5 as a function of the number of stages. It can be seen that depending on the value of p convergence is already achieved after 1 stage for p — 0.2 (low crosscorrelation between the signature waveforms) and 3 stages for p — 0.8 (high crosscorrelation) respectively. The multiuser efficiency of SAGE-JDE (T) is 0.7 dB for p = 0.2 and 3.3 dB for p — 0.8 corresponding to an improvement against MMSE-SDE of 0.8 dB and 3.8 dB respectively. In turn, EM-JDE (/? /. ) shows a multiuser efficiency of about 0.7 dB for p = 0.2 and 4.2 dB for p = 0.8. This gap in efficiency is caused twofold: Firstly, the channel coefficients are updated only once every stage in the EM-JDE scheme rather than K times in the SAGE-JDE algorithm. Secondly, it is well known that successive interference cancellation such as SAGE-JDE outperforms parallel interference cancellation like EM-JDE when the received signals have distinctly different strengths [5]. The multiuser efficiency of SAGE-SDE (DDF- SDE) results in 1.2 dB (1.4 dB) for p = 0.2 and 7.3 dB (8.3 dB) for p = 0.8. The influence of L on the multiuser efficiency is depicted in Figure 6 for the SAGE-JDE scheme with p = 0.4. Two stages are carried out. For τ ι → oo the performance of the MMSE detector approaches that of the decorrelating detector, whose performance is known to be limited by noise enhancement depending on the inverse of R [4]. Since p is fixed, for a certain p the multiuser efficiency curve of the MMSE-SDE scheme is independent of L. On the other hand for a fixed L, the multiuser efficiency rises with increasing p because the MMSE of a becomes more reliable. Also seen from Figure 6, for a certain p the SAGE-JDE algorithm becomes first progressively more efficient but then stagnates as a function of L. The first effect is explained partially by the fact that with each iteration the channel estimates aid the estimated bits to become more reliable and vice versa. Provided the bits of the entire frames are known, the variance of the estimate of a is known to be inverse proportional to L. The remaining gap is mainly caused by a residual error in suppression of the MAI as discussed next. Analogous conclusions can be drawn for the behavior of the EM-JDE scheme. The simulation results of the SAGE-SDE and DDF-SDE receivers are not included in the figure. The multiuser efficiency of those schemes is about 0.5 dB better than that of the MMSE-SDE for all values of p independent of L. Note that the feedback loop incorporated in SAGE-SDE and DDF-SDE influences their bit-error-performance [11] [17]. It is interesting to investigate the near-far resistance of JDE. For this purpose, we assume subsequently an AWGN channel, p = 0.4, p = 4, and L = 40. First, the channel coefficients are assumed to be known to the receiver. Figure 7 illustrates the simulated bit-error-rate of User 1 for the EM-JDE and SAGE-JDE schemes within the following scenario: the received signal-to-noise ratio b .ι of this user is fixed at 8 dB, while all the other users are received with an energy per bit varying from —10 dB to +10 dB with respect to User 1. The stronger the MAI is, the better the EM-JDE, SAGE-JDE, SAGE-SDE, and DDF-SDE schemes perform, because the more reliable the MAI estimates are. Hence for high MAI, all the schemes perform close to the SU-bound. Only the EM-JDE scheme with β = 1/Λ " can not take advantage of the accurate estimates of strong MAI, because the weakest user determining the overall performance is just partially updated. If the MAI is strong, the performance of the MMSE-SDE detector is known to be limited due to noise enhancement [4]. For weak MAI, all algorithms exhibit a performance similar to that of the MMSE-SDE scheme, since subtracting weak MAI from the whitened matched filter signal in (25) and (40) doubles the interference with some probability. Moreover, SAGE-JDE (|) is more near-far resistant than both SAGE-JDE (4) and SAGE-SDE, because the weakest user is detected first and thus its strong MAI is subtracted with highest reliability. The bit-error-rate of User 1 achieved with the EM-JDE and SAGE-JDE schemes when the channel is unknown is depicted in Figure 8. The bit-error performance of both schemes is close to that achieved when the channel is known (see Figure 7) unlike the SAGE-SDE and DDF-SDE schemes whose performance drastically impairs when the channel is unknown. This leads to the conclusion that the proposed JDE algorithms are robust against channel estimation errors unlike the SAGE-SDE and DDF-SDE receivers. For SAGE-JDE (4.), the user of interest achieves a better performance for low MAI than in a known channel environment. No really convincing explanation of this surprising behavior at first glance could be found yet. Both in EM-JDE (β k ) and in SAGE-JDE (p), for the selected value of L the computational intensive channel estimate (22) can be replaced by that in (42) without significant loss in performance. Figure 9 shows that the appealing properties of JDE is still preserved, if the signal energy of the user of interest is only 7b.ι = 4 dB. In particular, SAGE-JDE (t) is able to cope with low MAI much more efficiently than the MMSE-SDE scheme can do. So far in AWGN environment, we have seen that on the one hand EM-JDE (β k ) performs similar to EM-JDE (βk) and on the other hand that the up-sorting procedure in the SAGE- JDE scheme leads to a better performance when the MAI is weak. In this sense, mainly strong users benefit from the up-sorting procedure in the SAGE-JDE receiver. In flat Rayleigh fading the averaged bit-error-rate Pb of User 1 is obtained by averaging .i over the realizations of («ι ax). As a consequence, P b _ is dominated by the values of P i when | ι| is significantly smaller than \aχ\. Due to this fact, the multiuser efficiency [4] inform us about the behavior of the weakest user in the system. This multiuser efficiency of User 1 is depicted in Figure 10 for the EM-JDE scheme as well as for the SAGE-JDE receiver as a function of the normalized crosscorrelations between the signature waveforms. The channel coefficients are assumed to be known to the receiver. As p increases, the multiuser efficiency of the EM-JDE algorithms progressively decreases. In particular, EM-JDE (βk) outperforms EM-JDE (β k ) over the whole range of p. Again, EM-JDE with β k — l/K follows roughly the multiuser efficiency curve of the MMSE-SDE scheme. Furthermore, this figure reveals that in the SAGE-JDE receiver even the weaker users benefit from the ranking procedures as p increases beyond 0.6. Generally spoken, the SAGE-JDE scheme performs better than the EM-JDE receiver. A possible explanation for this behavior is that successive interference cancellation (e.g. SAGE-JDE) outperforms parallel interference cancellation (e.g. EM-JDE) when the received signals have distinctly different strengths [5]. If the channel is unknown, both the EM-JDE and SAGE-JDE schemes outperform the SAGE- SDE and DDF-SDE algorithms as illustrated in Figure 11. The degradation of η of about 15% occurring for p « 0 is caused by the error in the estimation of a. Updating the channel coefficients via (42) causes a further performance degradation of only a few percent. CONCLUSIONS We developed two efficient iterative receiver structures for joint multiuser detection and multichannel estimation (JDE). One (EM-JDE) is based on the EM algorithm, the other (SAGE-JDE) results from an application of the SAGE algorithm. Monte-Carlo simulations were carried out to assess the performance of both receiver architectures in terms of bit-error-rate and asymptotic multiuser efficiency in Rayleigh fading. It turned out that a few preamble bits are enough for the JDE schemes to converge. A comparison with other suitable receiver architectures was made. Both the SAGE multiuser detector and the decorrelating decision-feedback multiuser detector were supplemented with a separate minimum-mean-squared error (MMSE) multi-channel estimator. We refer to these schemes as SAGE-SDE and DDF-SDE respectively. Moreover, a receiver performing MMSE separate multiuser detection and channel estimation (MMSE-SDE) was simulated for comparison pur- pose. In a known channel, all MMSE-SDE, SAGE-SDE, DDF-SDE, EM-JDE, and SAGE- JDE schemes perform similarly. In an unknown channel however, both the EM-JDE and the SAGE-JDE scheme remain still near-far resistant, while the bit-error performance of the MMSE- SDE , SAGE-SDE, and DDF-SDE receivers dramatically impairs. These results demonstrate the robustness of schemes which smartly combine data detection and channel estimation in multiuser systems unlike architectures where both processes are implemented separately. The EM-JDE scheme performs best when the constrained regarding the sum of the weight coefficients (see (35)) is relaxed unlike proposed in the original work of the EM algorithm. In rum, the SAGE-JDE scheme exhibits a better performance when the users' bit sequences are cyclically updated in the order of increasing signal strength, i.e. the bit sequence of the user with the weakest signal strength is updated first in each stage. Notice, that this strategy is in the reverse order of the the ranking commonly used in sequential multi-user detection where the users are decoded in the order of decreasing received power. APPENDIX In examining (21 ), the question arises as to whether the error covariance matrix of the channel estimates ∑aa approaches 0 as the observation length L tends to infinity. We start from (23). This equation can also be expressed in the form -1 _ .H - l KL - BWc τ (cB w ∑a a B [2l C τ + No^ " CBW∑a with l denoting the 7 -dimensional identity matrix. Observe that the conjecture lim ∑aa W = 0 (A 1) i→ c is true if and only if lim . BBW W CC TT ((CCBBWW∑∑ 5S55 BBW W CC TT + + λMf 0 .l KL ) ∞W∑a - l KL . (A 2) L→c After a few straightforward matrix manipulations, (A 2) yields lim fcBW∑ 5S BWc τ + - I L . (A 3) Suppose that are the eigenvalues and the corresponding orthonor- mal eigenverctors of CBM∑ as B W C. Then we can write CBW∑ggBWC = £) Ajy fc vJ and (A 3) becomes Solving the charcteπstic equation ( CBM∑ aa BWC ) v = λv, we obtain [24] Λ . X 0 + X 1 L + . . . + χ κ L κ K k < K 0 K + l < k < KL The number of eigenvalues larger than zero and the degree of the polynomial in (A 5) corresponds to the dimension of ∑ aa (3) and the number of equal blocks per row/column in ∑ a5 (5) respectively. The exact value of the coefficients χ k , k = 0, . . . , K depends on a certain bit constellation besides ∑ aa and C. In the special case R = I, i.e. C = I, the polynomial in (A 5) is linear in L with χι = σ 2 Inserting (A 5) into (A 4) and invoking the rule of L'Hospital, we S et ∑ f eli V A. V A. T — ΪKL corresponding to the definition of the eigenvectors [24]. Thus, (A 1) follows REFERENCES [I] S Verdu and S Shamai, "Spectral efficiency of CDMA with random spreading," IEEE Trans Information Theory, vol 45, Mar 1999 [2] S Verdu, "Minimum probability of error for asynchronous gaussian multiple-access channels," IEEE Trans Information Theory, vol 32, Jan 1986 [3] A Kle , G K Kaleh, and P W Baier, "Zero forcing and minimum mean-square-error equalization of multiuser detection in code-division multiple-access channels," IEEE Trans Vehicular Technology, vol 45, May 1996 [4] S Verdu, Multiuser Detection, Cambridge University Press, 1998 [5] A Duel-Hallen, J Holtzman, and Z Zvonar, "Multiuser detection for CDMA systems," IEEE Pers Communications, vol 2, Apr 1995 [6] A Dempster, N Laird, and D Rubm, "Maximum likelihood from incomplete data via the EM algorithm," J Royal Statist Soc , Ser S, vol 39, Jan 1977 [7] T K Moon, "The expectation-maximization algorithm," IEEE Signal Processing Magazine, Nov 1996 [8] T A Louis, "Finding the observed information matrix when using the EM algorithm," J Royal Statist Soc , Ser B, vol 44, 1982 [9] J A Fessler and A O Hero, "Space-alternating generalized expectation-maximization algoπthm," IEEE Trans Signal Processing, vol 42, Oct 1994 [10] C N Georghiades and J C Han, "Sequence estimation m the presence of random parameters via the EM algoπthm " IEEE Trans Communications, vol 45, Mar 1997 [I I] L B Nelson and H V Poor, "Iterative multiuser receivers for CDMA channels An EM-based approach," IEEE Trans Communications, vol 44, Dec 1996 [12] D Raphaeh, "Suboptimal maximum-likelihood multiuser detection of synchronous CDMA on frequency selecme multi- path channels," IEEE Trans Communications, vol 48, May 2000 [13] M Feder and E Weinstein, "Parameter estimation of superimposed signals using the EM algoπthm," lEEE Tians Acoustics, Speech and Signal Processing, vol 36, Apr 1988 [14] M J Borran and M Nasiπ-Kenaπ, "An efficient detection technique for synchronous CDMA communication systems based on the Expectation Maximization algoπthm," IEEE Trans Vehicular Technology, vol 49, Sep 2000 [15] H Zamiπ-Jafaπan and S Pasupathy, "Adaptive MLSDE using the EM algorithm," IEEE Trans Communications, vol 47 Aug 1999 [16] D Divsalar amd M K Simon and D Raphaeh, "Improved parallel interference cancellation for CDMA,' IEEE Trans Communications, vol 46, Feb ! 998 [17] A Duel-Hallen, "Decorrelating decision-feedback multiuser detector for synchronous code-division multiple-access channel," IEEE Trans Communications, vol 41, Feb 1993 [18] P Patel and J Holtzman, "Analysis of a simple successive interference cancellation scheme m a DS/CDMA system,' IEEE Journal on Selected Areas in Communications, vol 12, June 1994 [19] C F J Wu, "On the convergence properties of the EM algoπthm," Ann Slat , vol 11, 1983 [20] B H Fleury, M Tschudin, R Heddergott, D Dahlhaus, and K I Pedersen, "Channel parameter estimation in mobile radio environments using the SAGE algoπthm," IEEE Journal on Selected Areas in Communications, vol 17, Mar 1999 [21] H V Poor, An introduction to signal detection and estimation, Springer- Verlag, 1994 [22] J A Fessler and A O Hero, "Complete-spaces and generalized EM algorithm," in Proc IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP 93), Minneapolis, Minnesota, April 1993 [23] J G Proakis, Digital Communications, McGraw-Hill, Inc , 1995 [24] G H Golub and C F van Loan, Matrix Computations, The John Hopkins University Press, 3 edition, 1996 [25] A Kocian and B H Fleury, "Iterative joint symbol detection and channel estimation for DS/CDMA via the SAGE algorithm " in Proc 11th IEEE International Symposium on Personal Indoor and Mobile Radio Commumcation (PIMRC 2000) London, UK, Sep 2000 [26] L Wei and L K Rasmussen, "A near ideal noise whitening filter for an asynchronous time-varying CDMA system ' IEEE Trans Communications, vol 44, pp 1355-1361, Oct 1996 [27] P Fan and M Darnell Sequence Design for Communications Applications, John Wiley and Sons, 1996 The signal model of a DS/CDMA system is outlined in Section 2 In Section 4 the latter scheme is applied to derive the SAGE-JDE scheme. Implementation issues such as initialization and complexity are discussed in Section 5. Finally, the performance of the SAGE-JDE scheme is assessed and compared to that of successive interference cancellation (SIC) combined with minimum mean-square-error (MMSE) channel estimation by means of Monte Carlo simulations in Section 6. 2. Description of a possible First System We consider a synchronous CDMA system with K active users sharing the same communication channel. The signal transmitted by each user experiences in the channel flat Rayleigh fading which is assumed to be constant over the observed frame of L bits. The complex baseband representation of the received signal therefore reads K L r W = ∑∑ * 6 *M 'kit - (I - l)T s ) +n(t). (1) fc = l 1 = 1 In the above expression, s k (t) is the signature waveform of user k and b k [l] € {— 1, +1} the bit transmitted by this user during the Z th signalling interval. The complex circularly symmetric Gaussian random variable a k with zero mean and variance represents the channel attenuation of the signal of user k. Without loss of generality a k encompasses the root power of the transmitted signal of user k. The signature waveforms s_ (£), . . . , s _ ■ (£) have support [0, T s ), where T s denotes the inverse of the data rate, and are normalized to have unit energy. Furthermore, n(t) represents complex-valued white Gaussian noise with spectral height No- We assume that the random elements a £= {a . . . , a κ , b £ (b[l , .. . , b[ ] T ) T = ( [\], . . . , b K [l}, . . . , b_ [L}, . . . , b K { }) T , and n(t) are independent as are the components of a. The time-discrete signal at the output of a bank of matched filters, matched to the signature waveforms, can be expressed as z - RBa + ή, (2) where z is a A'L-dimensional complex vector and B is the KL x KL diagonal matrix B = diag(b). The KL x where R = PKϊ ■ 1 with p jk = J 0 ' s 3 (t)s k (t)dt being the crosscorrelations between the signature waveforms. Furthermore, a = (a τ , . . , a τ ) . With the above assumptions, the covaπ- ance matπx of a reads ∑aa — with *? ∑aa — 0 > κ while, ή is a .KX-dim circularly symmetπc zero-mean Gaussian vector with covanance matrix Λ/OB-- Let sι(t), - , sχ(t) be selected in such a way that the correlation matπx R is positive definite Then R can be Cholesky factoπzed according to R = C T C, where C is an unique lower tπangular lnvertible matπx. Here, (•) designates transposition Equivalently, R = C T C with B. The SAGE Algorithm C = The SAGE algoπthm proposed by Fessler et al [7] generalizes the EM algoπthm Rather than updating all parameters simultaneously at iteration ., only a subset Bs of B indexed by S = S[ι] is updated while keeping the param¬ Multiplying z by the whitening matπx (C τ ) 1 yields eters in the complement βg = B\Bs fixed The convergence rate of the SAGE algoπthm is higher than that of - (<?)- z = CBS + n (3) the EM algorithm. At the ι th iteration the E-step reads The zero-mean circularly symmetric Gaussian vector n is white, i e has covanance matπx Nόl, where I denotes the s (BslBt' l) ± E {\(x\Bs, l ] ) \y, &' l} . (4) KL x KL identity matπx. Both the matched filter outputs (2) and the whitened In the M-step, only 23s is updated, i.e. matched filter outputs (3) are equivalent as they provide ' +11 = argmax Q s (β s |β [ ' l) , the same information about the transmitted bits [ J However, the latter descπption is more appropπate for the sub,[ι +1] B\ sequent analysis because of the property of the noise vec= B> 1 (5) tor n to be white. Provided X is a complete data for each selected subset Bs under the assumption that the parameters in the complement βg are known, the SAGE algonthm exhibits the monotonicity property. 4. Application to Joint Symbol Detection and Channel Estimation In this section we apply the SAGE algoπthm to JDE of DS/CDMA signals. The parameter vector to be estimated data is selected to be X = {y, a} in order to remove the unknown nuisance parameter vector a The log-likehhood function of b for the observation X reads A(X\B) = Λ(y, a|b) = Λ(y|a, b) + Λ(a|b) According to the model assumptions, the second summand is independent of b and can therefore be discarded Neglecting terms independent of a and b, we obtain from (3) L Λ(y|a, b) = ∑ 23l{y[f] H CB[I]a} - ||CB[/]a|| 2 (6) (=1 Here, !R( ), ( ) , and || || denote the real part, the complex conjugate transpose, and the norm of the argument respectively At iteration z, only the bits of user k = k[ι] = . mod K+l are updated Hence, B s = b k = (bι_[i], , b k [L}) and B s — bj is the vector obtained by cancelling the components of b f c in b With the slightly abuse of notation Qk ( ) — Qs ( ) the E-step (4) in this case becomes Q k (bfelb t 'l) = £ {Λ(y|a,bfc,b ] )|y, b t ' l } (7) where sgn( ) denotes the signum function Substituting (6) into (7) and dropping terms independent of b f c yields L of the observed Q k (b f cl In this expressi According to (12), the signal., at the output of the whiten¬ (α α M + (.εSi) jt (9) ing filter are cleaned up with increasingly reliable tenta¬ The conditional distπbution of a given y and assuming tive decisions of the mterfeπng users as . grows, projected that b = b^' 1 is Gaussian with mean (see e g [12]) onto C f c, weighted by a k , and finally fed into a single- user (SU) detector The block diagram of the SAGE-JDE E {a|y, !> [ ' ] } = scheme is depicted in Figure \2.. Note, that the users' bit sequences are updated successively while the channel co¬ ∑aaBt' ^Bt' l∑aaBt' l + N 0 ~1 ) " 1 C- 1 y (10) efficients are reestimated in parallel on completion of each sequence up-dating and covanance matπx 5. Implementation Issues A. Initialization ∑ aa - ∑ aa B [ ' ) (Bt' l∑ δa Bt' ] + M 0 B.~ l ) 'β^∑ aa " The initialization of the SAGE-JDE algoπthm is per¬ When no coding is used, it follows from (8) that the M- formed as follows The first p bits transmitted by each step (5) is performed by maximizing each summand of user are known to the receiver The initial channel estithe nght hand expression in (8) mate, a'°l is the MMSE estimate of a based on the obb k [l) , +,] = sgn {^ {a^y[l} H C k servation y[l], , y[p] The initial estimate bf°' is the MMSE estimate of b computed from the observation of - ∑. Pk ] [l] ( k ) l, i } } . (1 1 ) y while assuming a s= at°l [ ] Expeπmental evidence j≠k shows that a better performance is reached, if the first p their true value. We refer to this method for obtaining b'°l as MMSE separate detection and estimation (MMSE- SDE). B. Complexity Using this approximation, channel estimates can be obtained with time complexity per iteration and bit of O(K 2 ) . C. Sorting of the Users So far the derivation of the SAGE-JDE algorithm in Section 4 is obtained for an arbitrary sorting of the K users. Different strategies exist of how to rank the users. The following two have been implemented: mated by (13). SAGE-JDE (I): The users are ranked in the order of decreasing strength. Similar to above, SAGE- JDE (4, dec) designates decorrelating channel estimates. As shown below the selected sorting procedure drastically affects the ability of the SAGE-JDE algorithm to cancel out weak interferers. investigated when the channel coefficients are known to the receiver. Figure ($ illustrates the simulated bit error 6. Performance Analysis rate of User 1 for the following scenario: its energy per bit b i = - t-, with |-| denoting the magnitude of the The feedback structure of the SAGE-JDE algorithm argument, is fixed at 8 dB, while all the other users are makes a theoretical analysis of the behavior of this scheme received with the same energy per bit varying from -2 dB cumbersome. Monte Carlo simulations have been carried through +18. dB. The stronger the interferers are, the betout to investigate the near-far resistance of the different ter the SAGE-JDE and SIC algorithms perform, because implemented versions of the SAGE-JDE algorithm. The the estimate of the subtracted MAI is more reliable. The system parameters are chosen as follows: K = 8 users, performance of the MMSE-SDE detector is known to be L p — 4 preamble bits, frame length L = 40, and equal limited due to noise enhancement, if the MAI is strong [ ]. cross-correlation coefficients p jk = p = 0.4, j, k — Note, that only SAGE-JDE (t) is able to combat weak in1, . . . , 8, j φ k. Two stages corresponding to 2K iterterferers, because the MAI of the weaker user can be estiations have been carried out. The performance of a SIC mated more reliably in (12). scheme combined with the MMSE channel estimator used The bit-error-rate of User 1 in the unknown channel is deto initialize the SAGE-JDE algorithm in Section 5.A has picted in Figure Vi, In this scenario the SAGE-JDE algobeen simulated for comparison purpose. Without loss of rithm outperforms SIC in any case. It can be concluded generality we focus on User 1. that the SAGE-JDE algorithm is robust against channel The performance of the SAGE-JDE scheme has been first estimation errors unlike SIC. For the SAGE-JDE (i), the user of interest appears to be more near-far resistant for low MAI than in a known channel environment. No really convincing explanation of this behavior surprising at first glance could be found. Finally, for the selected value of L the complicated channel estimate (10) can be replaced by the simple decorrelating estimate (13) without significant loss in performance. The averaged multiuser efficiency η [ ] in flat Rayleigh fading is depicted in Figure 4 as a function of the normalized cross-correlations in the case of a known channel. Both SAGE-JDE and the SIC algorithms achieve a similar ή for p below 0.5. As p increases beyond this value, η of the SIC scheme drops rapidly to values close to zero, while that of the SAGE algorithm progressively decreases. As expected, SAGE-JDE (t) outperforms SAGE-JDE (J,). If the channel is unknown, the SAGE-JDE algorithms outperform the SIC algorithm as illustrated in Figure 5. The degradation of ή of about 15% which occurs for p R_. 0 is caused by the error in the estimation of a. Updating the channel coefficients via (13) causes a further performance degradation of only a few percent. Conclusions We apply the SAGE algorithm to the problem of joint data detection and channel estimation (JDE) to obtain an iterative receiver architecture of tractable complexity. Different versions of this so-called SAGE-JDE result depending on the order in which the users' bit sequences are cyclically updated. Monte-Carlo simulations show that SAGE-JDE is near-far resistant and robust against channel estimation errors. Moreover, a better performance is achieved when the users are ranked in order of increasing strength, i.e. when the bit sequence of the user with the weakest strength is updated first. In Section 4 the latter scheme is applied to derive the SAGE- JDE scheme Implementation issues such as initialization and complexity are discussed in Section 5 Finally, the performance of the SAGE-JDE scheme is assessed and compared to the decorrelating detector [ ] supported with a separate decorrelating channel estimator (DEC-SDE) by means of Monte Carlo simulations in Section 6 2. Description of a preferred Second System We consider an asynchronous CDMA scenario with K active users The transmitted signal of each user experiences time- vaπant frequency-selective Rayleigh fading We assume that data transmission starts at time 0 and ends at time (L — Ϊ)T S The data symbol duration T s is much smaller than the coherence time of the channel so that the channel can be considered as slowly varying The bandwidth B_ of the signature waveforms is much larger than the coherence bandwidth of the channel Thus the w th time-variant frequency-selective channel can be modeled as a tapped delay line with tap-spacing 1/B_ The number of considered paths in the receiver is fixed to W < \ T__B^\ + 1, where T m denotes the multipath spread and |_-J is the greatest integer not larger than the argument The complex baseband representation of the received signal therefore reads X pi. \t- -; tion Let Z [.] = (^[.}, . . , z [t ... . z_ n >[l}. .x uiY denote the JΪW-dimensional complex vector containing the The signal model of a output samples of K matched filters m bin I = 0, . . , L — 1 DS/CDMA system is outlined in Section 2 for W particular rays corresponding to the sampling instants IT, + τ< 0) + (w - l)B c , 1 < it < K. I < w < W. It follows from (1) that where z[ϊ] The JJTW P ( uJ w [0] = / w(*-n-(»-l)/ft) with x pi-(t- »-(uj'-l)/B c )c.t ∑aa = being the zero-offset crosscorrelation and σl/W Let pι(t),...,pκ(t) be selected in such a way that the cor¬ PΪV' l] = I"' dt-τ*- [w-l) IB _) relation matrix R is positive definite Then, R can be factoπzed according to R = C T C, where C is a unique lower tπangular x Pk , (tA- - τ k , - («/ - I)/ B c ) dt invertible matrix with following structure - 1, 0] / we know that R influence the coevery KW x KW in (4) depends subsequent analysis we note that is spanned by KW col(C< 1} [.,Δ.], .. , by the whitening matrix (C ) _1 l e has covanance identity mamodel in (3) and the in (5) are equivaabout the transmitted more appropriate property of the noise that the length L of the observed Cj ;u '' [I, Al] in C (4), is In the M-step, only Bs is updated, i e ,+11 = aιg max Qs ( β s|B W ) , 4. Application to Joint Symbol Detection and Channel Estimation In this section we apply the SAGE algoπthm to JDE of DS/CDMA signals The parameter vector to be estimated is β = {b} and the incomplete data is y = y The complete data is selected to be X = {y, a} The log-likelihood function of β for the observation X reads Λ(A-|β) = Λ(y a|b) = Λ{y|a b) + Λ(a|b) According to the model assumptions, the second term is independent of b and can therefore be discarded Neglecting terms independent of b, we obtain from ( 5 ) Λ(y|a, b) = Y 23? i y[/] H T C[ΔZJB[. - Al]u[l - Δ.]a I 1=0 I Λ(=0 J - a T t [.] H B[.] ( ∑ R[Δ.]B[. - Al]ιA_l - Al] ^ Λ1=0 R[Δ.] T B[. + Al]v[l + Al a (10) Here, ! ( ) and ( ) denote the real part and the complex conjugate transpose of the argument respectively At iteration %, only the bits of user k = k[ι] = _ mod K 4- 1 are updated Hence, Bs = bt = (fa [1], , fa [L]) and β s = b k is the vector obtained by cancelling the components of bj. in b With a slight abuse of notation Q_ ( ) = Q s ( ) and the E-step (8) in this case becomes Q_ (bι|bW) - £ {Λ(y|a b* , bW) |y, bW } (1 1) At the J* iteration the E-step of the SAGE algorithm reads In this expression Qs (13) (αϊ * oir ,, ) l,J = I ( - ,W oif'' W The conditional distribution of a given y and assuming that b = b''' is Gaussian with mean (see e g [14]) x (B w ι/∑ 5a j H B w + W 0 ~ 1 ) ~1 C- 1 (15) and covariance matπx w = ∑ sa - Σ M „ H BM X (B ' J ι ∑aai/ H B [,] + i -1 ) The path-delay of the to* ray of the λ* user can be identified -Mi ■ (w - 1)1 B c , l < w < W. When no coding is used, the M-step (9) is performed by maximizing (12) and finally fed into a single-user (SU) detector along with the other W — 1 multipath components Note, that the users' bit 6ι[q t,+1J = sgn{ sequences and the Doppler shift of all the paths are updated successively while the channel coefficients are reestimated in parallel on completion of each sequence up-dating The block sj £ y [l + Al] H c[^[Al]e l2 - T> V + 'X* W X ( X [>] diagram of the SAGE-JDE scheme is depicted in Figure ^7. iϋ=l ^ Δl=0 The joint estimation of the channel coefficients ( 15) dominates - - ΔJ'] the computational complexity of the algorithm The computation of (15) requires 0(K i L~ W l ) operations per bit For the average signal-to-noise ratio of the k u eτ k — A k — -c (15) approaches the decorrelated estimate For every single ray w, the signals at the output of the whitening filter are cleaned up with increasingly reliable tentative r,W decisions of the interfering users as . grows, projected onto C ^[AI] for different time-offsets Δ. to cancel mtersymbol The shortcut μ stands for (lW-\- (w — l))K-X- k Using this apinterference, weighted by αϊ ( " ,) ' 1 ' and by e p(— j2πlT s f d h ) proximation, channel estimates can be obtained with time com- plexity per iteration and bit of 0(K 3 LW 3 ) If a is not updated while reestimating the components of id, a negligible error is made, but the computational complexity can be reduced by a factor KW Finally, if only the Ws out of W strongest paths provided by the DEC-SDE scheme are estimated, computational complexity can be reduced by another factor WjWs The label SAGE-SDE(oo) indicates that above complexity reducing steps have been implemented The computational complexity per bit is oϊO(K LW ) 6. Performance Analysis tipath components are unknown to the receiver. Figure illustrates the simulated bit-error-rate of User 1 for the SAGE- JDE scheme within the following scenario: the received signal- to-noise ratio 7t > ,ι of this user is fixed at 4 dB, while all the other users are received with an energy per bit varying from —10 dB to 4-10 dB with respect to User 1. First, the SAGE-JDE scheme attempts to estimate one paths only, i.e. Ws = W = 1. The stronger the MAI is, the better the SAGE-JDE scheme performs, because the more reliable the MAI estimates are. Hence SAGE- JDE performs close to the SU-bound. If the MAI is strong, the performance of the DEC-SDE detector is known to be limited due to noise enhancement [ ]. Hence, for high MAI P^ is closed to SU-bound. For weak MAI, we observe a performance degradation against the SU-bound, because subtracting weak interferers from the whitened matched filter signal in (16) doubles the interference with some probability. If the SAGE- JDE scheme attempts to estimate more multipath components than actually exist, i.e. Ws > W, its performance curve drifts away from the SU-bound. Despite this fact SAGE-JDE still outperforms DEC-SDE for every Ws. Conclusions Monte Carlo simulations of an asynchronous scenario in time- variant frequency selective channels show that the proposed receiver is near-far resistant and robust against errors in the estimation of the channel parameters. This is a consequence of the smart combination of data detection and channel estimation in JDE schemes unlike architectures where both processes are implemented separately. The only requirement for mantaining the excellent near-far resistance of SAGE-JDE is that the number of considered paths does not exceed the number of resolvable ones. It can be shown that very short training sequences are required for the SAGE-JDE scheme to converge.

Claims: CLAIMS 1. A method for multi-user decision and/or multi-channel estimation (JDE) in a \\ ireless communication network comprising a plurality of users, said method comprising the slep or updating a data bit sequence by applying a expectation-maximisation ( EM) algorithm. 2. A method according to claim 1. said method furthermore comprising the step of re-estimating the data bit sequence by applying a space-alternating generalised expectation- maximisation (SAGE) algoπthm. 3. A method for multi-user decision and/or multi-channel estimation (JDE) in a wireless communication network comprising a plurality of users, said method comprising the step of re- estimating the data bit sequence by applying a space-alternating generalised expectation- maximisation (SAGE) algoπthm. 4. A method according to claim 3. said method furthermore comprising the step of updating a data bit sequence by applying a expectation-maximisation (EM) algoπthm. 5. A method according to any of claims 1-4. said method comprising utilising either of the algorithms EM and SAGE, alternatively utilising both of the algorithms, for JDE of synchronous DS/CDMA signals. 6. A method according to any of claims 1-5, where the users are ranked according to estimated strengths of the their signals, respectively, where the signal is computed as the square of a value of a complex circularly symmetric Gaussian random value a^, i.e. |a f , and where the user k (k = 0,1,2 ... up to K users) being received as having the weakest signal is ranked first. 7. A method according to any of claims 1-6, said method comprising utilising a EM-JDE receiver for receiving a signal (y), and said method furthermore comprising decomposing said signal (y) into a sum of parameters (x k ) from K different users, said parameters (xj . ) themselves each consisting of the addition of a transmitted signal (s \. ) and a corresponding noise vector (n from a user k, i.e. y = ∑ (s k + n k ), where k = 1 ,2 ... up to K users. 8. A system for receiving a signal and for decomposing of the signal and/or for estimation of channel, where a receiver comprises a scheme for data detection and channel eslimation. preferably joint detection and estimation (JDE). and where data bit sequences of a signal (y) from a plurality K of users are capable of being updated successively while channel coefficients are capable of being re-estimated in parallel. 9. A system for receiving a signal and for decomposing of the signal and/or for estimation of channel, where a receiver comprises a scheme for data detection and channel estimation, preferably joint detection and estimation (JDE), and where data bit sequences of a signal (y) from a plurality K of users are capable of being updated successively while channel coefficients are capable of being re-estimated simultaneously. 10. Use of an EM algorithm in an EM-JDE receiver for updating data bit sequences of a signal from a user in a wireless communication network of a plurality of users by iteratively optimising a maximum likelihood estimate of the data bit sequences received. 1 1. Use of a SAGE algorithm in a SAGE-JDE receiver for re-estimating data bit sequences of a signal from a user in a wireless communication network of a plurality of users by cyclically updating the data bit sequences in the order of increasing signal strength. 12. Use of both an EM algorithm and a SAGE algorithm in a combined EM-JDE receiver and SAGE-JDE receiver for updating data bit sequences and for re-estimating data bit sequences of a signal from a user in a wireless communication network of a plurality of users by iteratively optimising a maximum likelihood estimate and by cyclically updating the data bit sequences in the order of increasing signal strength. 13. Use of an EM algorithm for iterative symbol detection and channel estimation, preferably multichannel estimation, in a macro-cellular environment. 14. Use of a SAGE algorithm for iterative symbol detection and channel estimation, preferably multichannel estimation, in a macro-cellular environment. 15. Use of both an EM algorithm and a SAGE algorithm for iterative symbol detection and channel estimation in a macro-cellular environment.

Inventors: FLEURY BERNARD HENRI, KOCIAN ALEXANDER, FLEURY, BERNARD, HENRI, KOCIAN, ALEXANDER
Applicants: FLEURY BERNARD HENRI, KOCIAN ALEXANDER, FLEURY, BERNARD, HENRI, KOCIAN, ALEXANDER
Application year: 2001
Main classification: H04B1
IPC codes: H04B1, H04L25