Porrentruy, Switzerland. He received the diploma in electrical engineering and mathematics in 1978 and. 1990 respectively, and the doctoral degree in elec-.
1
EM-based joint data detection and channel estimation of DS/CDMA signals Alexander Kocian, Student Member, IEEE, and Bernard H. Fleury Senior Member, IEEE
Abstract— In this paper we present two efficient iterative receiver structures of tractable complexity for joint multiuser detection and multichannel estimation (JDE) of DS/CDMA signals. The schemes result from 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 have excellent multiuser efficiency and are robust against errors in the estimation of the channel parameters. Moreover, very short training sequences are required for the JDE schemes to converge with. Simulation results further demonstrate that the SAGEJDE receiver exhibits a better performance when the users’ 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. Index Terms— Iterative joint multiuser detection and multichannel estimation, EM algorithm, SAGE algorithm, iterative interference cancellation
I. I NTRODUCTION
M
ULTIUSER detection is known to drastically increase the bandwidth efficiency of CDMA systems compared to conventional detection using RAKE receivers [1]. 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 multiple access interference (MAI) in the received signal before making data decision. The latter method can be divided 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 maximumlikelihood estimate when a direct calculation of this estimate is computationally prohibitive. One disadvantage of this This paper was presented in part at the 1st Karlsruhe Workshop on Software Radios, Karlsruhe, Germany, March 2000 and at the 11th IEEE International Symposium on Personal, Indoor and Mobile Radio Communication (PIMRC 2000), London, UK, Sep. 2000. The authors are with the Department of Communication Technology, Aalborg University, DK-9220 Aalborg, Denmark (e-mail: ak,bfl @kom.auc.dk).
algorithm is its slow convergence rate. A variety of techniques have been proposed for accelerating the convergence of the EM algorithm such as Louis’ Turbo EM performing a Newton-Raphson step [8] or the space-alternating generalized expectation-maximization (SAGE) algorithm [9]. Nelson and Poor in [10] extend the EM and SAGE algorithms to a so-called Space-Alternating Missing-Parameter EM (MPEM) algorithm that at each iteration updates a certain subset of parameters while considering the others as probabilistic missing data. Georghiades and Han propose in [11] an EM-based receiver which performs joint data detection and estimation for the time-variant flat Rayleigh fading channel. This scheme necessitates the transmission of only a few preamble bits. In [10], the EM, SAGE, and MPEM algorithms are applied to derive various multiuser detectors using hard or 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 [10] except the soft-decision function for the bits can be controlled by a parameter called temperature. Feder and Weinstein apply in [13] the EM algorithm to the problem of estimating parameters of superimposed signals. Following this approach, Borran and Nasiri-Kenari develop in [14] a computationally feasible multiuser detector for the AWGN channel. Finally, Louis’ Turbo EM algorithm is used in [15] for adaptive channel estimation and sequence detection. Other issues influencing the performance of multiuser detection are the amount of estimated MAI canceled out prior the detection of each users’ bit sequence and the order in which these 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 the decorrelating decision-feedback multiuser detector (DDF) [17] as well as of the successive interference canceler proposed in [18] can be improved by ranking the users in the order of their signal strength. Experimental investigations into the EM multiuser detector presented by Borran and Nasiri-Kenari [14] indicate that accuracy of the bit-decisions improves when amount of interference cancellation is inversely proportional to the signal-to-interference 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 the flat Rayleigh fading channel. 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 fading channel model provide a simple framework for studying JDE in the uplink of a DS/CDMA system. The JDE algorithms presented in this paper can then easily be extended to the asynchronous case and to frequency-selective channels. 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 in the Gaussian and Rayleigh channels and compared to that of the minimum mean-square-error (MMSE) [3], the SAGE [10], and the DDF [17] multiuser detectors, all supplemented with MMSE channel estimation, by means of Monte Carlo simulations. II. S YSTEM D ESCRIPTION We consider a synchronous CDMA system with active users sharing the same propagation channel. In the channel the signal transmitted by each user experiences flat Rayleigh fading which is assumed to be constant over the observed frame of bits. The complex baseband representation of the received signal therefore reads
!"$#&%('*)+,
(1)
In the above expression, . is the signature waveform of user / and .!021 3546'+87 the bit transmitted by this user during the th signalling interval. The complex, circularly symmetric Gaussian random variable with zero mean and variance 9; : represents the attenuation of the signal of th the / user. Without loss of generality encompasses the root power of the transmitted signal of user / . Furthermore, )? with denoting the DW identity matrix. T
Both the matched-filter-output (2) and the whitened-matchedfilter-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 . to be white. III. T HE EM
AND
SAGE A LGORITHMS
A. The EM Algorithm Let denote a possibly vector-valued parameter to be estimated from some possibly vector-valued observation with probability density . The EM algorithm provides an iterative scheme to approach the maximum likelihood estimate r8Ot rH in case where a direct computation
of is prohibitive. The derivation of the EM algorithm relies on the concept of a hypothetical, so-called complete unobservable data which, if it could be observed, would ease the estimation of . The observed random variable which is referred to as the incomplete data within the EM framework, is related to by a mapping X . Since is not observable, at the th iteration the EM algorithm computes in a first step called expectation step (El step), the estimate
I point of ¢ [6], [19]. The ability of the EM algorithm to > find a global maximum depends on the initialization . The
Property). Provided the function
convergence rate of the EM algorithm is inversely related to the conditional Fisher information matrix of given . This rate is notoriously slow when the dimension of the complete data is large. B. The SAGE Algorithm 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 , only a subset of indexed by is updated while keeping the parameters in the complement set C 8 fixed; and secondly, the concept of the complete data is extended to that of the so-called hidden data to which the incomplete data is related by means of a possibly non-deterministic mapping exhibiting some particular property. A hidden data space would be a complete data space for in the EM framework, if were known [9]. The particular property of the mapping 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 given for each set of parameters is likely smaller than that of the complete data given for the entire space . th iteration, the E-step of the SAGE algorithm reads At the
4 4 ¤ -
YC o¡&¢
In the M-step, only
(7)
i.e. is updated,
§ 5r jt G r8 § -
of user / only. The Gaussian noise vector . is white with spectral height = > . To ensure consistency with the original model (4) the non-negative coefficients 4?-?-@-?4 have to satisfy the constraint
l5-
(10)
Notice that the weight coefficient determines the relative portion of total noise assigned to . As the problem at hand is the estimation of the transmitted ? 4 @ ? @ 4 are unknown nuibits, the complex amplitudes provides a solution to sance parameters. The EM framework dealing with nuisance parameters, namely they are included as a part of the complete data. Together with (9), a natural choice for the complete data is 1 4 4?-?-@-4@ 4 7 . The parameter vector to be estimated is E and the incomplete data is . Since the components of are independent, the loglikelihood function of E for the observation reads
¢ ¢ E ¢ 4 E
6
with
(11)
¢ 4 E ¢ 4 E (' ¢ E
Because of the model assumptions, the second summand is independent of E and can therefore be discarded. Moreover, neglecting terms independent of E we obtain from (9)
~ T . { ¢ 4 E z (12) 1 J7 Here, and $JK denote, respectively, the real part and the complex conjugate of the argument and E C ? 4@-?-?-@4 T . Because of the special form of ¢ G &=w>
(22)
~ T . In deriving (21), we made use of the relation L Because of (13), the M-step (6) is performed by maximizing the terms E E individually, i.e. E § r8Ot rH
E E -
Moreover when no coding is used, it follows from (15) that each component of E § can be separately obtained by maximizing the corresponding summand in the right-hand expression in (15), i.e.
§ t ¡ ¡ ~ T .
¤ ¤ 4 (23) 1 J7 denotes the signum function. Substituting (18)
where t into (23), we obtain
. § t ' ~ T
: (24) ~ c c c c
We assume subsequently that the length of the observed frame is large enough so that the second summand in (20)
channel estimation and partial interference cancellation. Then, the cleaned signal is fed into a single-user (SU) receiver consisting of a conventional coherent detector. The term c ] c c . c in (25) is an estimate of the MAI of user / at the output of the matched filter. A block diagram of the proposed EM-JDE receiver is given in Figure 1. We -user optimization problem into have decomposed a independent procedures leading to a computationally feasible scheme. Note that the whitening matrix N~ T has only be used to ease the derivation of the proposed EM-JDE scheme. The bits in (25) can be reestimated without knowledge of ~ . B. Optimizing the Weight Coefficients When the EM algorithm is employed to estimate the parameters of a linear model, its convergence rate is maximized when the weight coefficients are chosen to be equal, i.e. V , /
h4?-?-@-4 [22]. Due to the discrete nature of the parameter vector E in our model, the above setting is not necessarily optimum. Consequently, the vector containing the weight coeffiT cients of all the users QC 4?-@-?-?4 should be selected such that either the joint bit-error-probability or the average bit-error-probability is minimized as tends to infinity. Since this optimization problem is difficult to carry out, we shall resort to suboptimal methods. 1) Optimization via Complete Data: A feasible way to choose the weight coefficients is to minimize the linear mean squared error between the true signal vector of user / in (9) and its estimate . C ¡ . 4 E ¤ :
Cxr5jt G p
!
q . . :
-
(26)
The symbol " J " designates the vector 2-norm of the argument. Taking the expectation with respect to B of (17) yields
Z
` q . o~ 5 . ' \ ~Ic c . c b - (27) c
For the sake of simplicity, we assume # [4] and *%$ . As shown in the Appendix, the estimate B is consistent. Thus in the asymptotic case, the channel can be assumed to
be known to the receiver. Proceeding in this way, (27) becomes
~ . ' ~ c c c c c Substituting and . into (26) and removing terms independent of , we obtain
r8OtG p
:
-
The expectations in (28) can be computed as
c : c . c . : c ¡ c . c ¤& 9 c : c 9 :
where c C c . c . . This bit-error-probability is independent of because the single-path fading vector B is constant within the entire framelength . Differentiating with respect to , setting the result equal to zero, and solving with respect to 4?-@-?-@4 yields
9; : 4 c 9 c : c
A
5-
(29)
Observe that is a function of the iteration unlike the setting proposed in [13]. Nevertheless, the EM algorithm still exhibits the Monotonicity Property because the optimized coefficients 4?-@-?-@4 meet the constraint (10) (see [13]). It should also be mentioned that is independent of since the channel coefficients remain constant over the frame. suppose that the performance of the EM-JDE device S Finally, is close to the SU-bound in a known channel : =w>@ with £ denoting the error function
& d !" : i# [23]. Then by substituting for in the expression for $ , we obtain an approximation (*) the bit-error-probability % in EM-JDE, i.e. % ' & for
:
-
Following the same steps as in Section IV-B.1, we obtain after some straightforward algebraic manipulations -
Since 1 c c $7 A , reduces to k k , the above equation r5jt I p ' : : : (28) ' : S c : c c : c (
~ T . .
o C r5jt G p
] c c c . c . W c ' : ~+c c c . c c
2) Optimization via Projected Complete Data: A shortcoming of the attained in (29) is that these coefficients do ] c between the not take into account the crosscorrelations signature waveforms. Consequently, a better choice is first to project the cancellation error on ~ and then to minimize the mean-squared result. Proceeding in this way yields
: = >,+ , when the channel is unknown to the
receiver. We will see that the setting (29) can be used to optimize the performance of the EM-JDE scheme, even if is small.
9 : ; c ] - : c 9 c :
Note that the range of is -
c
6 -
4
A.
(30)
-
4
(31)
i.e. the coefficients 4@-?-?-?4 might violate the initial con/ straint (10) for . The implication of this selection of the weight coefficients is twofold: First, the conceptual interpretation of noise splitting among the users on which the derivation of EM-JDE relies does not hold anymore. Secondly, empirical evidence indicates that the scheme does not fulfill the Monotonicity Property anymore. As we will see later, the users’ bit sequences are nevertheless detected more accurately. 3) A Two-User Example: The following numerical example will help gain more insight into the behavior of the optimized users weight coefficients. We consider the case of transmitting in an AWGN channel. The channel coefficients are assumed to be known to the receiver. To be more specific, the estimates 4?-?-@- -4 are replaced by the true values M-steps of EM-JDE. The weight 4@-?-?-?4 in the E- and coefficients
and are then obtained by replacing 9( : with
: in (29) and (30) respectively. Let us first investigate the behavior of in three scenarios as depicted in Figure 2a. The received signal-to-noise ratio 0 is fixed at A dB, dB and 1 dB Ce : = of user respectively, while the signal of user is received with an energy per bit varying from A dB to '+ A dB with respect to that of user . In the limiting case of very high MAI, i.e. 0
1 $ , , the update procedure (25) for user : is based only upon the strong interference of user whose tentative bit decisions are very reliable. Hence, this optimized version of the EM-JDE scheme effectively updates only the weakest user. In the equal power case, the weight coefficients (29) read F . This result is consistent with that : obtained in [22]. In the limiting case of very low MAI, i.e. 0
A 2, again. Unlike in the case of strong MAI, : tends to A and therefore the bit decisions of user are : discarded. Thus, (25) converges to
?H s§ t 1 1 M H 7h754
1 Note that the error function 24365 7 converges to zero faster than 8:9$; < tends to infinity so that 8!9,; =?9 ; is the MMSE estimate of B based on the observation > L 4@J?J@J?4 L . The initial estimate E is the MMSE estimate of E computed from the observation of L while assuming B B > [3]. Experimental evidence shows that a better performance is reached, if the first bits are initialized to equal the MMSE estimate rather than their true value. We refer > > to this method for obtaining B and E as MMSE separate detection and estimation (MMSE-SDE). B. Complexity The joint estimation of the channel coefficients (21) dominates the computational complexity of both EM-JDE and SAGE-JDE. The computation of (21) requires Yz5 { operations per bit. Thus, the time complexity per iteration and bit of the EM-JDE and SAGE-JDE receivers is bounded by Y z S: { { and z respectively. When the average signal-to-noise ratio of the / th user 0 Cl9; : =w> increases to infinity, (21) approaches the decorrelating channel estimate
. z P L { -
(37)
Note that P L is the output signal of the decorrelator. The time complexity per iteration and bit for computing (37) is bounded according to 1 7 and z S: { for the EM-JDE and SAGE-JDE receiver respectively. C. Investigated JDE Architectures
1) EM-JDE: We choose the following two settings of for the proposed EM-JDE scheme: EM-JDE : the weight parameters are set equal to (29).
-
-
EM-JDE , EM-JDE 4 $ : the weight coefficients are set equal to (30). The label $ indicates that the channel estimates are computed by means of (37) instead of (21). 2) SAGE-JDE: So far, the derivation of the SAGE-JDE scheme in Section V is regardless of any sorting of the users. Different strategies exist of how to rank the users. The following two have been implemented: SAGE-JDE , SAGE-JDE 4 $ : The users are sorted > according to their estimated strength : , so that the user with the weakest received signal is ranked first. The label $ means that the channel estimates are computed via (37). SAGE-JDE , SAGE-JDE 4 $ : 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 receiver to cancel out weak MAI. VII. P ERFORMANCE A NALYSIS The feedback and the non-linear structure of the EMJDE and SAGE-JDE schemes make 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 1 users and equal crosscorrelations ]F as follows: ] c 4 k 4 / h4?-@-?-4 1 4 k / . Without loss of generality we focus on user 1. A comparison with other suitable receiver architectures is also made. The SAGE multiuser detector in [10] is identical to our SAGE-JDE receiver, except the channel is assumed to be known and the users are not ranked prior detection. To ensure a fair comparison the SAGE multiuser detector [10] is initialized by using the MMSE-SDE receiver described in Section VIA. We refer to this scheme as SAGE separate detection and estimation (SAGE-SDE) receiver. The DDF multiuser detector [17] supplemented with a separated MMSE channel estimator is considered for comparison, too. We refer to this scheme as DDF separate detection and estimation (DDF-SDE) receiver. The averaged bit-error-rate of user ( ) is depicted in Figure 4 for crosscorrelations ] A - 4 preamble bits and frame length A . Note that averaging is performed over the realizations of the channel coefficients. All the users are received with the same average signal-to-noise ratio 0 , h4 ? @ ? 4 / . Two stages are carried out in both EMJDE and SAGE-JDE receivers. One stage corresponds to the number of iterations required to update every user’s bit sequence once, i.e. one iteration in the EM-JDE scheme and iterations in the SAGE-JDE scheme. The number of stages performed by the SAGE-SDE receiver was limited to two, as well. The performance loss against the SU-bound is the so-called multiuser efficiency in Rayleigh fading [4], or multiuser efficiency (MUE) for short. Evaluating this measure - A at 0 dB, we obtain 35- A dB and A - dB for the EM-JDE and the SAGE-JDE receiver respectively. In comparison, the MMSE-SDE, SAGE-SDE, and DDF-SDE re ceivers achieve a MUE of - dB, - dB, and - dB respectively.
The MUE of user is reported in Figure 5 as a function of the number of stages. It can be seen that the convergence rate depends on the value of ] . Convergence is already achieved stage for ] A - (low crosscorrelation between after the signature waveforms) and stages for ] A - 1 (high crosscorrelation). The MUE of the SAGE-JDE receiver is A - dB for ] A - and q- dB for ] A - 1 corresponding to an improvement against the MMSE-SDE receiver of A - 1 dB and - 1 dB respectively. In turn, the EM-JDE scheme shows a MUE of about A - dB for ]U A - and - dB
] A for 1 . The cause of this gap in efficiency compared to the SAGE-JDE receiver is twofold: First, the channel coefficients are updated only once every stage in the EMJDE scheme rather than times as performed in the SAGEJDE receiver. 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 MUE of the SAGE-SDE (DDF-SDE) receiver equals to 3h- dB ( 35- dB) for ] A - and - dB ( 1 - dB) for ] A - 1 . The influence of on the MUE of the SAGE-JDE scheme is depicted in Figure 6 while considering ] A - . Two stages are carried out. As 0 and increase, the performance of the MMSE data detector (channel estimator) approaches that of the decorrelating data detector (channel estimator) whose performance is known to be limited by noise enhancement depending on the inverse of P [4]. Since ] is fixed, for a certain , the MUE curve of the MMSE-SDE scheme is roughly independent of . In contrast for a fixed , the MUE rises with increasing because the MMSE estimate of B becomes more reliable. Also seen from Figure 6, for a certain the SAGE-JDE receiver becomes first progressively more efficient but then stagnates as is increased. The first effect is explained partially by the fact that at each iteration the channel estimates aid the estimated bits to become more reliable and vice versa. The remaining gap is mainly caused by the residual error in suppression of the MAI. Analogous conclusions can be drawn for the behavior of the EM-JDE scheme. The simulation results for the SAGE-SDE and DDF-SDE receivers are not included in the figure. The MUE of those schemes is about A - dB better than that of the MMSE-SDE scheme for all values of regardless of the value of . Note that multiuser detectors incorporating bit-decision feedback such as the SAGE-SDE and DDF-SDE schemes do, perform better than those without [17] like the MMSE-SDE scheme. The MUE of user is depicted in Figure 7 for the EMJDE and SAGE-JDE schemes as a function of the normalized crosscorrelations between the signature waveforms. The channel coefficients are assumed to be known to the receiver. As ] increases, the MUE of the EM-JDE receivers progressively decreases. In particular, the EM-JDE scheme outperforms
the EM-JDE scheme over the whole range of ] . Since the users are not optimally updated, the MUE curve of the EM-JDE receiver with } follows roughly that of the MMSE-SDE scheme. Furthermore, this figure reveals that in average the ranking procedure affects the performance of the SAGE-JDE scheme as ] increases beyond A - . Generally spoken, the SAGE-JDE scheme performs better than the EM-
JDE receiver. A possible explanation for this behavior is again that successive interference cancellation performs superior to parallel interference cancellation when the received signals have distinctly different strengths. If the channel is unknown, both the EM-JDE and SAGEJDE schemes outperform the SAGE-SDE and DDF-SDE receivers as illustrated in Figure 8. The degradation of of about 15% occurring for ] & A is caused by the error in the estimation of B . Updating the channel coefficients via (37) causes a further performance degradation of only a few percent. We now investigate the performance of JDE as a function of the signal-to-noise ratio of the interferers in AWGN channel for ] A - , , and A . First, the channel coefficients are assumed to be known to the receiver. Figure 9 reports the bit-error-rate using the EM-JDE and SAGEJDE schemes in the following scenario: the received signalto-noise ratio 0 of user is fixed at 1 dB, while all the other users are received with an energy per bit varying from 3 A dB to '+ A dB with respect to that of user . 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 can not cope with strong MAI because in (25), the balance between own signal and observation minus MAI is not chosen optimally. The performance of the MMSE-SDE detector (estimator) 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 (36) doubles the interference with some probability. Moreover, the SAGE-JDE scheme performs better than both SAGE-JDE and SAGE-SDE receivers do because the weakest user is detected first and thus its strong MAI is subtracted with highest reliability. In this particular scenario, the EM-JDE receiver performs similar to the EM-JDE receiver. The bit-error-rate of user achieved with the EM-JDE and SAGE-JDE schemes when the channel is unknown is depicted in Figure 10. The performance of both receivers is close to that achieved when the channel is known (see Figure 9) 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 SDE schemes. For SAGE-JDE , user 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. For the selected value of , the computationally - intensive channel estimate (21) can be replaced by that in (37) without significant loss in performance in both EM-JDE and SAGE-JDE schemes. Figure 11 shows that the appealing properties of the JDE schemes are still preserved even if the signal energy of the user of interest is low, i.e. 0 dB. In particular, SAGE-JDE is able to cope with low MAI more efficiently than the MMSESDE scheme can do.
VIII. C ONCLUSIONS We developed two efficient iterative receiver structures for joint multiuser detection and multichannel estimation (JDE). One (EM-JDE) is based on the EM algorithm while the other (SAGE-JDE) results from an application of the SAGE algorithm. Monte-Carlo simulations were carried out to assess the performance of both receivers in terms of bit-error-rate and multiuser efficiency in the fading Rayleigh and Gaussian channels. It turned out that a few preamble bits are enough for the JDE schemes to converge. A comparison with other previously published multiuser detectors was also made. The minimummean-squared error (MMSE), the SAGE [10], and the decorrelating decision-feedback (DDF) [17] multiuser detectors were supplemented with a separate MMSE channel estimator. We refer to these schemes as MMSE-SDE, SAGE-SDE, and DDF-SDE receivers respectively. The abbreviation SDE stands for separate detection and estimation. In known channel, all investigated schemes perform similarly. In unknown channel however, the JDE schemes keep on having excellent multiuser efficency, while the performance of the SDE receivers dramatically impairs. These results demonstrate the robustness of schemes which smartly combine data detection and channel estimation in multi-user systems unlike architectures where both processes are implemented separately. The EM-JDE scheme performs best when the constraint (31) on the range of the weight coefficients is relaxed. This constraint ensures that this scheme is derived within the theoretical framework. In turn, 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 ranking commonly used in sequential multi-user detection where the users are decoded in the order of decreasing received power.
A PPENDIX In this appendix, we show that the channel vector estimate
B in (21) is consistent. First, we prove that the channel vector estimate is asymptotically unbiased. More specifically,
¡£B B 4 E ¤ B 4
¥ ps
(A 1)
almost surely. We start with the expectation of B in (21) given B and assuming E E :
¡£B B 4 E ¤ W
u hv v R PSR . = > '
R . ¡£L BE ¤ -
Inserting (4) in the expectation, solving, and expanding with , we get u v5v ' R . P R ¡ B B 4 E ¤ & = R . P R B W
Invoking strong law of large numbers, the term the R . P R converges almost surely to ¡ R . P R ¤ = , as $ . Thus, (A 1) follows. u vhthat v Finally, we need to show the error covariance matrix of the channel estimates in (22) approaches # as the observation length tends to infinity, i.e.
¥s p
u v5v # 4
(A 2)
almost surely. We start from (22). Multiplying both sides with , we obtain
u vhv u v5v ' R PSR =w>& w = >
[14] M. J. Borran and M. Nasiri-Kenari, “An efficient detection technique for synchronous CDMA communication systems based on the Expectation Maximization algorithm,” IEEE Trans. Vehicular Technology, vol. 49, pp. 1663–1668, Sept. 2000. [15] H. Zamiri-Jafarian and S. Pasupathy, “Adaptive MLSDE using the EM algorithm,” IEEE Trans. Communications, vol. 47, pp. 1181–1193, Aug. 1999. [16] D. Divsalar, M. K. Simon, and D. Raphaeli, “Improved parallel interference cancellation for CDMA,” IEEE Trans. Communications, vol. 46, pp. 258–268, Feb. 1998. [17] A. Duel-Hallen, “Decorrelating decision-feedback multiuser detector for synchronous code-division multiple-access channel,” IEEE Trans. Communications, vol. 41, pp. 285–290, Feb. 1993. [18] P. Patel and J. Holtzman, “Analysis of a simple successive interference cancellation scheme in a DS/CDMA system,” IEEE Journal on Selected Areas in Communications, vol. 12, pp. 796–807, June 1994. [19] C. F. J. Wu, “On the convergence properties of the EM algorithm,” Ann. Stat., vol. 11, pp. 95–103, 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 algorithm,” IEEE Journal on Selected Areas in Communications, vol. 17, pp. 434–450, Mar. 1999. [21] H. V. Poor, An introduction to signal detection and estimation, Springer, 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, Apr. 1993, pp. IV/1–IV/4. [23] J. G. Proakis, Digital Communications, McGraw-Hill, Inc., 3. edition, 1995.
u v5v the same argumentation as above, it can be shown Employing u vhv converges almost that surely to =+> as tends to infinity. Consequently, converges almost surely to # with rate . This proves (A 2). R EFERENCES [1] S. Verd´u and S. Shamai, “Spectral efficiency of CDMA with random spreading,” IEEE Trans. Information Theory, vol. 45, pp. 622–640, Mar. 1999. [2] S. Verd´u, “Minimum probability of error for asynchronous Gaussian multiple-access channels,” IEEE Trans. Information Theory, vol. 32, pp. 85–96, Jan. 1986. [3] A. Klein, 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, pp. 276–287, May 1996. [4] S. Verd´u, 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, pp. 46–57, Apr. 1995. [6] A. Dempster, N. Laird, and D. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” J. Royal Statist. Soc., Ser. B, vol. 39, pp. 1–38, Jan. 1977. [7] T. K. Moon, “The expectation-maximization algorithm,” IEEE Signal Processing Magazine, pp. 47–59, Nov. 1996. [8] T. A. Louis, “Finding the observed information matrix when using the EM algorithm,” J. Royal Statist. Soc., Ser. B, vol. 44, pp. 226–233, 1982. [9] J. A. Fessler and A. O. Hero, “Space-alternating generalized expectationmaximization algorithm,” IEEE Trans. Signal Processing, vol. 42, pp. 2664–2677, Oct. 1994. [10] L. B. Nelson and H. V. Poor, “Iterative multiuser receivers for CDMA channels: An EM-based approach,” IEEE Trans. Communications, vol. 44, pp. 1700–1710, Dec. 1996. [11] C. N. Georghiades and J. C. Han, “Sequence estimation in the presence of random parameters via the EM algorithm,” IEEE Trans. Communications, vol. 45, pp. 300–308, Mar. 1997. [12] D. Raphaeli, “Suboptimal maximum-likelihood multiuser detection of synchronous CDMA on frequency-selective multipath channels,” IEEE Trans. Communications, vol. 48, pp. 875–885, May 2000. [13] M. Feder and E. Weinstein, “Parameter estimation of superimposed signals using the EM algorithm,” IEEE Trans. Acoustics, Speech, and Signal Processing, vol. 36, pp. 477–489, Apr. 1988.
Alexander Kocian was born in Vienna, Austria, on January 22, 1971. He received the Dipl. Ing. degree (with distinction) in electrical engineering from Vienna University of Technology in 1997. He is currently working towards his Ph.D. degree at the Department of Communication Technology, Aalborg University, Denmark. From 1997-1999 he was with the Spread Spectrum Team at the Communication Technology Laboratory at the Swiss Federal Institute of Technology (ETH), Zurich, Switzerland. In 1999, he joined the faculty of Aalborg University. He has been a Visiting Research Scholar at the Wireless Systems Laboratory, Georgia Institute of Technology, Atlanta, USA, in 2001. His research interests include joint data detection and channel estimation in multiple-access communication systems and characterization of multiple-input multiple-output (MIMO) channels.
Bernard H. Fleury was born on March 28, 1954 in Porrentruy, Switzerland. He received the diploma in electrical engineering and mathematics in 1978 and 1990 respectively, and the doctoral degree in electrical engineering in 1990 from the Swiss Federal Institute of Technology Zurich (ETHZ), Switzerland. During 1978-85 and 1988-92 he was a Teaching and Research Assistant at the Communication Technology Laboratory (CTL) and at the Statistics Seminar at ETHZ. In 1992 he joined the CTL again where he headed the Spread Spectrum Team from 1994. Since 1997 he has been with the Center for PersonKommunikation, Aalborg University, Denmark initially as a Guest Professor and from July 2000 as a Professor in Digital Communications. Bernard H. Fleury is head of the division Digital Communications at the Department of Communication Technology. In 1999 he was elected IEEE Senior Member. Since October 2002 he has been in charge of the Ph.D. Study Programme “Wireless Communications” of the International Doctoral School of Technology and Science at Aalborg University. His current fields of interest include stochastic modelling of the radio channel, high-resolution methods for the estimation of the parameters of the radio channel, characterization of multiple-input multiple-output (MIMO) channels, and advanced techniques for joint channel parameter estimation and data detection/decoding in multi-user communication systems.
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JDE scheme based on the EM algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comparison of the optimized weight coefficients a) and b) provided by the EM-JDE scheme for a two-user system with a) arbitrary ] and b) ] A - . The signal-to-noise ratio of user is fixed at A dB, dB and 1 dB : : respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . JDE scheme based on the SAGE algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Performance in flat Rayleigh fading (] A - , , A , stages). . . . . . . . . . . . . . . . . . . . . . . Behavior of the multiuser efficiency versus the number of stages with ] as a parameter ( , A , 0 = 0
= -?-?- = 0 = A dB). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . : Behavior of the multiuser efficiency in flat Rayleigh fading as a function of with as a parameter (]S A - , stages, 0 = 0 = -?-?- = 0 = A dB). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . : Behavior of the multiuser efficiency in flat Rayleigh fading of the EM-JDE and SAGE-JDE receivers. The channel coefficients are known to the receivers ( , A , stages, 0 = 0 = -?-@- = 0 = A dB). . . . . . . . : Behavior of the multiuser efficiency in flat Rayleigh fading of the EM-JDE and SAGE-JDE receivers. The channel coefficients are unknown to the receivers ( , A , stages, 0 = 0 = -?-?- = 0 = A dB). . . . . . . : Behavior of the bit-error-rate as a function of the signal-to-noise ratio of the interferers in AWGN channel. The channel coefficients are known to the receivers (0 1 dB, ] A - , , A , stages). . . . . . . . . . Behavior of the bit-error-rate as a function of the signal-to-noise ratio of the interferers in AWGN channel. The channel coefficients are unknown to the receivers (0 1 dB, ] A - , , A , stages). . . . . . . . . Behavior of the bit-error-rate as a function of the signal-to-noise ratio of the interferers in AWGN channel. The channel coefficients are unknown to the receivers (0 dB, ] A - , , A , stages). . . . . . . . .
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SU - receiver
ag replacements
Joint channel estimation and partial interference cancellation
.. .
SU - receiver
No converged ?
Yes
Fig. 1.
JDE scheme based on the EM algorithm.
PSfrag replacements
1 0 ;2 54
&%
[dB]
