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On Adaptive Minimum Probability of Error Linear Filter Receivers for DS-CDMA Channels Ioannis N. Psaromiligkos, Student Member, IEEE, Stella N. Batalama, Member, IEEE, and Dimitris A. Pados, Member, IEEE

Abstract—Receiver architectures in the form of a linear filter front-end followed by a hard-limiting decision maker are considered for DS-CDMA communication systems. Based on stochastic approximation concepts a recursive algorithm is developed for the adaptive optimization of the linear filter front-end in the minimum BER sense. The recursive form is decision driven and distribution free. For additive white Gaussian noise (AWGN) channels, theoretical analysis of the BER surface of linear filter receivers identifies the subset of the linear filter space where the optimal receiver lies and offers a formal proof of guaranteed global optimization with probability one for the two-user case. To the extent that the output of a linear DS-CDMA filter can be approximated by a Gaussian random variable, a minimum-meansquare-error optimized linear filter approximates the minimum BER solution. Numerical and simulation results indicate that for realistic AWGN DS-CDMA systems with reasonably low signature cross-correlations the linear minimum BER filter and the MMSE filter exhibit approximately the same performance. The linear minimum BER receiver is superior, however, when either the signature cross-correlation is high or the background noise is non-Gaussian. Index Terms—Adaptive equalizers, adaptive filters, code division multiaccess, interference suppression, minimization methods, spread-spectrum communication, stochastic approximation.

I. INTRODUCTION

I

N RESPONSE to an ever-increasing demand for better utilization of the available resources in multicellular mobile radio and personal communication environments, directsequence code-division-multiple-access (DS-CDMA), a specific form of spread spectrum transmission, has recently received considerable interest. In a DS-CDMA communications system the target operation of the receiver is the detection of the transmitted information bit of one (mobile-end) or more (base station) users. At either the base station or the mobileend the receiver operates on the superimposed signals of the individual users and aims at the detection of pertinent information bits according to a given design optimality criterion. Due to the computational complexity requirements (exponential Paper approved by R. Kohno, the Editor for Spread Spectrum Theory and Applications for the IEEE Communications Society. Manuscript received March 29, 1998; revised November 5, 1998. This work was supported by the National Science Foundation under Grant NCR-9725695 and by the Air Force Office of Scientific Research under Grant F49620-99-1-0035. This paper was presented in part at the 1996 Conference on Information Sciences and Systems, Princeton University, Princeton, NJ, March 1996, and at the 1997 Conference on Information Sciences and Systems, Johns Hopkins University, Baltimore, MD, March 1997. The authors are with the Department of Electrical Engineering, State University of New York at Buffalo, Buffalo, NY 14260-2050 USA. Publisher Item Identifier S 0090-6778(99)05227-7.

in the number of users) that the optimal multiuser detector exhibits and the required knowledge of the received user energies [1], [2], proposals for suboptimal reduced complexity receivers are well justified. Suboptimal receiver design criteria as well as constraints on the architecture are usually affected by considerations of mathematical convenience. Minimum mean square error criteria and linear structures fall into this perspective. A representative example of extreme suboptimality is the use of the handy conventional matched filter (MF) receiver in multiuser environments. Although this receiver is the optimum scheme for single-user binary antipodal transmissions in additive white Gaussian noise (AWGN), in a realistic nonorthogonal multiuser environment the MF receiver exhibits unacceptable performance degradation, particularly in the presence of one or more high power interferers (“the near-far problem” [1], [2]). A list of other more successful suboptimal proposals includes the decorrelating receiver [3], [4] which is the zero-forcing solution for multiuser interference rejection in noiseless channels, receivers that are optimized with respect to asymptotic multiuser efficiency and near-far resistance [4], multistage architectures [5], [6], and decision feedback detectors [7]. Supervised backpropagation [8], [9], and unsupervised neural network receivers [10], minimummean-square-error (MMSE) linear receivers [11]–[14], and minimum-variance-distortionless-response (MVDR) linear receivers [15]–[17] are additional examples. Independently of the design optimality criteria for the above receivers, the ultimate performance measure of interest is the probability of error in detecting the transmitted information bit of each user (also known as bit-error-rate or BER). This is exactly the motivation for this work. In this paper we consider the problem of detecting the information bits of a single user in the presence of unknown multiuser interference and additive channel noise. The structure of choice consists of an adaptive -tap linear filter denotes the front-end followed by a sign detector. Here system processing gain (or equivalently the signature length). The weights of the -tap filter are adapted on line in a way that minimizes directly the induced probability of error. Although an analytic closed-form filter solution for the adopted minimum-probability-of-error criterion is not attainable in general, this mathematical intractability is bypassed using results from the well matured theory of stochastic approximation that allow recursive probability of error minimization. Relative to other previously proposed interference suppression approaches, the major advantage of minimum

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PSAROMILIGKOS et al.: ON ADAPTIVE MINIMUM PROBABILITY OF ERROR LINEAR FILTER RECEIVERS

probability of error schemes is the incorporation of the performance evaluation measure of choice directly into the learning process. The same criterion was recently adopted in [18]. The algorithm therein is tied to the Gaussian characteristics of the statistics of the additive channel noise, assumes knowledge of the channel noise power level, and requires -point evaluation of the Gaussian probability density function for every filter update. In contrast, the algorithm developed and analyzed in this work is a distribution independent recursive expression that is directly applicable even to cases where the channel noise distribution deviates from the standard Gaussian (for example impulsive channels) and is unknown. In terms of computational complexity, the system parameters are driven directly by the binary receiver decisions. We also demonstrate that, to the extent that the output of a linear DS-CDMA filter can be approximated by a Gaussian random variable, a minimum-mean-square-error optimized linear filter approximates closely the minimum BER solution. This observation is collaborative with the findings in [19] as explained in the sequel. Indeed, our numerical and simulation results indicate that for realistic AWGN DS-CDMA systems with reasonably low signature cross-correlations, the linear minimum BER filter and the MMSE filter exhibit approximately the same performance. We conclude that in studies with low crosscorrelation, -sequence signature assignments and AWGN channels as in [18], the performance of minimum BER and MMSE (LMS) optimized linear filters should appear nearly indistinguishable. The ideal linear minimum BER receiver is, however, superior when either the signature cross-correlation is high or the background noise is non-Gaussian. In the latter case the algorithm in [18] is not applicable. The merits of linear MBER schemes are evident in nonGaussian environments. However, the theoretical analysis of the BER surface of linear filter receivers presented in this work focuses on the AWGN case and offers further insight into the BER optimization of DS-CDMA systems. The subset of the linear filter space where the optimal receiver lies is identified and a formal proof of guaranteed global optimization with probability one is developed for the 2-user case. The rest of this paper is organized as follows. In Section II, we introduce our notation and the signal model. Receiver structures and minimum BER optimization recursions are developed in Section III. Convergence studies and mode analysis of the BER filter surface lead to algorithmic improvements in Section IV. Section V presents numerical and simulation comparisons of the MBER schemes with the conventional MF receiver, the decorrelating receiver, and adaptive MMSE (LMS optimized) linear filters. A few conclusions are drawn in Section VI. II. SIGNAL MODEL We consider a binary DS-CDMA system where users transmit synchronously over a single AWGN channel. Theoretically, we may restrict ourselves to the synchronous case only, since it is well known that every asynchronous system can be modeled as a synchronous one with higher effective virtual users in our case) [12], multiuser population (

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[20]. Then, we may focus on a single symbol period (one-shot detection). The continuous-time contribution of the th user to the baseband received signal is given by

(1) where is the information bit or symbol period. In the above is the received energy, is expression is the signature the th transmitted information bit, and (or spreading waveform), all pertinent to the th user. The signature of the th user is of the form

(2)

is the th coordinate of the spreading where is the chip waveform, is the chip period, sequence, is the system processing gain. Without loss of and generality the signatures are assumed to be normalized to unit energy over the information bit period . is the superposition of the The received signal CDMA signals , , corrupted by additive noise. That is,

(3)

is usually modeled as white Gaussian. where Assuming that at the receiver the signal is chip-matched , we obtain samples filtered and sampled at the chip rate over the period . The discrete-time vector form of (3) can be written as follows: (4)

where bold variables denote vectors of length . In (4), is the signature vector of the th user, and , , are assumed to be linearly independent. We also assume, without loss of generality, that the signatures of , , have positive the interfering users cross-correlation with the signature of the user of interest . Finally, the random vector is WG with autocorrelation , where is the channel noise matrix variance and is the identity matrix. The problem we consider in this work is the detection of the transmitted information bit of the user of interest (user 0) under is the only known signature. In other the assumption that vector space, we wish to detect words, operating in the a binary antipodal signal in the presence of unknown spreadspectrum multiple access interference (MAI) and AWGN. In the following section we present and analyze a minimum BER receiver design strategy together with its theoretical foundation.

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this single-letter distortion measure

is as follows:

(9) and Since the vectors , respectively, we have

Fig. 1. Adaptive linear minimum BER receiver.

come from hypothesis

III. ADAPTIVE MINIMUM BER RECEIVERS

and (10)

The problem of detecting the transmitted bit of user 0 can be cast as a binary hypothesis testing problem. Given the received vector in (4) we wish to determine which one of the following hypotheses is true: Information bit of user 0 is

, i.e.,

Equation (10) allows us to reduce the problem of finding the that minimizes the probability of error to value of that minimizes the expected that of finding the value of . value of Let in addition

Information bit of user 0 is

, i.e.,

(11)

We consider a linear filter receiver structure of the following simple form (Fig. 1): sgn

, with its th element

(5)

where is the filter tap weight vector. In this context, is a decision in favor of hypothesis and is a . The probability of error at the output decision in favor of of the receiver is given by

(6) and are the where and , respectively. In prior probabilities of hypothesis . To emphasize the general, we assume that dependency of the probability of error induced by the linear filter receiver on the weight vector , we will denote the error . Our goal is to adapt the weight vector probability by such that the induced probability of error becomes minimum. We will refer to the resulting receiver as the linear minimum bit-error-rate (LMBER) detector. Let sgn

be a vector of the same length as , , defined by

(7)

be the output of the receiver in Fig. 1. We define the singleas follows letter distortion measure (8) and are assumed to be received data vectors where and , respectively. This is the modification for from the antipodal 1 case of the distortion measure defined in measures the distortion at the output of [24]. Intuitively, and the receiver. The terms are both zero when the receiver makes the correct decision and strictly positive otherwise. The two terms are weighted by the prior probabilities of the two hypotheses, so failure of the receiver to detect the most likely hypothesis will result in a proportionally large penalty. As a result, the expected value of

(12) for some and denotes the where th coordinate unit vector. The sequences and , , are assumed to be pilot (training) sequences of and , respectively. We received data from hypothesis and , , note that in this setup consist of independent identically distributed vectors under and , respectively. The variance of all comhypothesis is finite for all . The regression function ponents of is twice continuously differentiable with bounded secin (3) is white Gaussian ond derivative at least as long as [24]. Finally, the above selection of the gain sequence guarantees that . Then, for a monotonically decreasing such that , sequence of positive numbers , and , the recursion (13) converges with probability one (w.p. 1) to the value of , that minimizes provided that for every such that exists a positive number implies and

, say there

(14)

The foundation of the recursive algorithm in (13) is provided by [21] which is an extension to multivariate regressions of the well-known Kiefer–Wolfowitz stochastic approximation method [22] for finding the extrema of a regression function. to Theoretically [cf. (14)], optimization w.p. 1 requires , , have a unique minimum. However, even if does not have a unique minimum, the algorithm maintains convergence w.p. 1 if we are able to identify a subset of where is unimodal and we restrict in that subset. These issues are addressed in Section IV.

PSAROMILIGKOS et al.: ON ADAPTIVE MINIMUM PROBABILITY OF ERROR LINEAR FILTER RECEIVERS

At every stage of recursion (13) each component of the is estimated simultagradient of the regression function neously by (12). Common selection for the sequence in (12) in (13) is , , and and the learning gain , , respectively. These selections, however, , , are arbitrary and any sequences that satisfy , , are acceptable. The following difference approximations

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The following observations for the tap weight vector hold for any number of active users: , since sgn sgn 1) [12]. such that we have 2) For any vector . . 3) For any vector , in the set of Observation 1 implies that we may restrict . This is assumed for the rest unit norm vectors denoted by of this section unless otherwise specified. Expression (17) shows that the probability of error is a function of the cross-correlations between and the signature ( ). Let vectors

(15) with

(18)

or denote the vector of the cross-correlations between the normaland the signature vectors. Then, the probability ized filter of error as a function of is given by (16) can also be used in place of (12) [23]. We note that (12) and (15) give two-sided difference approximations for each , while (16) is a one-sided component of the gradient of approximation. Also, the three approximations use a different number of samples per step. More specifically, (12) utilizes better the available samples since it uses only 2 samples per samples. step, while (15) uses 2 samples, and (16) uses Recycling the available samples may result to an even better but caution approximation of the optimum weight vector should be taken to avoid overfitting [25]. Summarizing the results presented in this section, the receiver architecture under consideration is the conventional linear tap-weight filter followed by a sign detector. The taps are optimized adaptively in a supervised minimum BER sense, by a decision-driven recursion (13) that is independent of the channel noise model. In the next section, we investigate the BER surface of linear receivers for the AWGN channel case. IV. THEORETICAL ANALYSIS

(19)

and exhibits the following properties: is strictly decreasing with respect to i) is even with respect to , ii)

. , i.e.

iii)

is strictly increasing (decreasing) with respect to , , the absolute value of the cross-correlation ( ). when , , , , , iv) For any

If

then

Under the AWGN channel assumption and independent and equiprobable information bits, the probability of error at the is output of a linear receiver

(17) where bits.

is the th bit of the th combination of interfering

In Fig. 2, we show a typical plot of for the 2were to take any value in [0, 1] [ 1, user case. If , then would have a unique global minimum at 1] and uncountably many global maxima at points for any . However, according to the following proposition this is not the case. The proof is included in the Appendix. that contains all vecProposition 1: The subset of tors defined by (18) is the surface and the interior of a and only the surface if . hyperellipsoid if This hyperellipsoid will be denoted by , while its interior and , respectively. and its surface will be denoted by

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Fig. 2. The probability of error Pe () as a function of the cross-correlation parameters for a 2-user system. Fig. 3.

We also define the following subsets of

The sets E0 , E+ , El , Er+ , and Er0 .

: (20) (21) (22)

and (23) and, more specifA brief discussion on the function ically, the location of its minima is considered useful at this point. Property iv) implies that the unique minimum of lies in . Motivated by this observation we restrict in . by We define the set

with

(24) and the set

by (25)

and for the 2-user case. In Fig. 3 we show the sets The definition of these two sets implies that for any there exists with . Therefore, the lies in . Moreover, recalling the global minimum of , may take any value proof of Proposition 1, when . In that case, if , Properties i) and iii) state in is strictly decreasing with respect to and strictly that , . This implies increasing with respect to cannot have a minimum in or in . Thus, we that have proved the following proposition: Proposition 2: For any , the global minimum of lies in the set defined by (25). If then all minima lie in . of In other words, if then is unimodal in if and only if it is unimodal in . Still, no conclusion can from the above be drawn about the number of minima in discussion. This problem is addressed in the following theorem

Fig. 4. Local maximum in Er+ .

for the special case of a 2-user system. The proof can be found in the Appendix. then has a unique Theorem 1: If . minimum in The staple result of this section now follows. The proof is given in the Appendix. Theorem 2: has no minima a) The probability of error function , where is the set of normalized vectors in of the subspace spanned by the user signatures. has a minimum at if and only if b) has a minimum at , where . Based on the above theorem and the fact that the global lies in we conclude that the global minimum of lies in the following set of vectors minimum of (26) The implications of Proposition 2 and Theorems 1 and 2 are quite important. First, in the case of a 2-user system if we to have unit norm and positive cross-correlation constrain is unimodal in with , then due to Theorem 2 (27) Thus, at each iteration step of the recursive algorithm for the determination of the LMBER filter tap weight vector we

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(a)

(b)

(c)

Fig. 5. (a) Separation coefficient (sepco) as a function of the signature cross-correlation. (b) Bit error rate as a function of the signature cross-correlation. (c) LMBER and MMSE filter cross-correlation as a function of the signature cross-correlation.

restrict

to the class

using the following adjustments: and

if

(28)

Then, the algorithm converges w.p. 1 to the optimum value user system the global minimum [23]. In the case of a and these adjustments remain valid. still lies in The following section is devoted to some numerical studies and simulation comparisons. V. NUMERICAL RESULTS

AND

SIMULATIONS

The natural benchmark for the LMBER receiver is the LMS optimized (adaptive MMSE) linear filter [27]. To the extent that the output of a linear DS-CDMA filter can be approximated by a Gaussian random variable, a minimummean-square-error optimized linear filter approximates the minimum BER solution. According to [19], under the Gaussian approximation (GA) the probability of error at the output of is a linear filter (29)

We note that the ratio under the square root is the signalto-interference-plus-noise ratio which is maximized by the MMSE filter. Therefore, it is expected that among the class of linear receivers for which the GA is accurate the MMSE solution exhibits near-optimal performance. This is not true, however, when the GA for the MMSE solution is inaccurate, as for example in the case of high signature cross-correlations

or when non-Gaussian channel noise contribution is present. We investigate these two cases below. For DS-CDMA transmissions over plain AWGN channels, Poor and Verdu [19] conjecture that the maximum deviation of the MMSE filter output from the Gaussian distribution—in the Kullback–Leibler sense—occurs in the case of only one active interferer. Even for this single interferer setup they show that the GA is accurate except in those cases where the signature cross-correlation . It is, thus, implied that if the GA is accurate for the LMBER then the performance of the MMSE and the LMBER receiver will be nearly identical (except, of ). course, when To quantify the accuracy of the Gaussian approximation at the output of the LMBER filter we examine a 2-user DSCDMA system and we define the “separation coefficient” (sepco) as the distance between the two centers (maxima) of the Gaussian mixture at the output of the linear filter (MMSE ( ). This coefficient or LMBER) under hypothesis acts as a measure of accuracy for the approximation of the Gaussian mixture at the output of the linear filter by a single Gaussian with the same mean and variance [19]. For the 2user system under consideration we can show that the sepco , where and coefficient reduces to are the signature and the energy, respectively, of the interferer. We fix SNR at 7 dB and SNR at 12 dB and we plot the sepco coefficient as a function of the signature cross-correlation for both the LMBER and the MMSE filter. The results are shown in Fig. 5(a). We observe that the centers of the Gaussian mixtures are almost equally separated at the output of both receivers except for those cases where the absolute signature cross-correlation approaches one and the separation at the output of the LMBER filter becomes lower than the

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Fig. 6. Bit error rate versus SNR for user 0 (SNR1

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= 7 dB, SNR2 = 5 dB, SNR3 = 5 dB, SNR4 = 3 dB, and SNR5 = 3 dB).

separation at the output of the MMSE filter. We conclude that the MMSE and the LMBER filter shall exhibit approximately the same BER when the Gaussian approximation is accurate. For the same 2-user study as in Fig. 5(a), we plot the BER of the LMBER and the MMSE filter as a function of the signature (where cross-correlation . Fig. 5(b) shows that for according to [19] the distribution at the output of the MMSE is approximately Gaussian), the two receivers exhibit identical BER performance. For more than two users in AWGN and reasonable signature cross-correlation values our studies have shown that the two receivers maintain similar performance, which is in full agreement with the conjecture of [19, p. 865]. The same holds true for multipath slowly fading channels, that is, when the fading coefficients remain constant during the training period. However, as the absolute cross-correlation approaches unity—and the distribution at the MMSE output can no longer be approximated by a Gaussian—the MMSE performance deteriorates faster than the LMBER. Finally, we plot the cross-correlation between the LMBER and the MMSE filter as a function of the signature cross-correlation [Fig. 5(c)]. The plot shows that when the Gaussian approximation holds (low values of ) the two filters are almost identical, while approaching they are significantly different for values of one. With respect to performance evaluation and the relative merits of the LMBER and the MMSE filter in non-Gaussian environments we perform the following case studies. We consider a 6-user DS-CDMA system with synchronous transmissions over phase synchronous nonfading channels. The cross-correlation of the interfering user signatures with the signature of the user of interest is 0.73, 0.2, 0.2, 0.3, and 0.06, respectively. The non-Gaussian environment is modeled by an -contamination mixture of two Gaussians, ,

where and , represent zero mean Gaussian 10 000. pdf’s with variance ratio Fig. 6 plots the BER of the MF, the ideal MMSE, the adaptive MMSE (LMS recursion), the decorrelator, and the LMBER as a function of the SNR of the user of interest over the 0 to 18 dB range. The SNR of the interfering users is fixed at 7, 5, 5, 3, and 3 dB, respectively. A sufficiently large data record is assumed available to ensure convergence of the LMBER and the LMS-implemented MMSE receiver. , for the The learning sequences are for the LMS recursion. LMBER, and Fig. 7 repeats the study in Fig. 6 and plots the BER as a function of the sample support. The SNR of the user of interest is fixed at 14 dB. Fig. 8 shows the BER as a function of the near–far ratio (NFR). The SNR of the user of interest is fixed at 15 dB while the SNR of the interferers is varied according to SNR NFR dB, SNR NFR dB, SNR NFR dB, NFR dB, and SNR NFR dB. SNR Fig. 9 shows the BER as a function of the SNR of all active users (under perfect power control) over the 0 to 18 dB range. We note that the LMBER recursion requires no significant computational overhead compared to the LMS algorithm. Real valued filter implementations require 2.5 multiplications per for the LMS, where is sample for the LMBER and the system processing gain. We conclude with the observation that although minimum BER and MMSE are two different optimization criteria, in practice, for Gaussian DS-CDMA scenarios with reasonably low signature cross-correlations the two resulting linear receiver structures exhibit similar performance. The LMBER is, however, superior when either the signature cross-correlation is high or the background noise is non-Gaussian.

PSAROMILIGKOS et al.: ON ADAPTIVE MINIMUM PROBABILITY OF ERROR LINEAR FILTER RECEIVERS

Fig. 7. Bit error rate versus number of samples for user 0 (SNR0

= 14 dB, SNR1 = 7 dB, SNR2 = 5 dB, SNR3 = 5 dB, SNR4 = 3 dB, and SNR5 = 3 dB).

Fig. 8. Bit error rate as a function of the near–far ratio (SNR0 = 15 dB, SNR1 = NFR SNR4 = NFR 2 dB, and SNR5 = NFR 2 dB).

2

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2

VI. CONCLUSIONS We considered the problem of detecting the information bits of a DS-CDMA user of interest in the presence of unknown multiple access interference (MAI). We confined ourselves within the class of linear FIR filters followed by hard-limiting decision making and we aimed at the incorporation of the receiver performance evaluation measure (BER) directly into

2 4 dB, SNR2

= NFR

2 4 dB, SNR3

= NFR

2 3 dB,

the design process. Based on the well matured ideas of stochastic approximation, a recursive algorithm was developed that adapts the filter taps in a way that minimizes the induced BER. The recursion form feeds directly on the decisions made and is distribution independent. Therefore, no algorithmic change is necessary when the additive ambient channel noise deviates from the standard white Gaussian behavior (impulsive

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Fig. 9. Bit error rate versus SNR under perfect power control.

channels). On the other hand, in direct comparison with MMSE criteria, the less accurate the Gaussian approximation at the output of the linear filter is, the more appropriate minimum BER optimization becomes. Such is the case of high signature cross-correlation or non-Gaussian background noise. The BER surface of linear filters does not share the convenient unimodality of MSE linear filter surfaces. Theoretical analysis for AWGN channels identified the subset of the filter space where the global minimum resides and pertinent algorithmic modifications were suggested. A formal proof for guaranteed global optimization w.p. 1 was given for the 2-user case. APPENDIX A. Proof of Proposition 1 , where We rewrite (18) in matrix form matrix with rows the user signatures:

is the

B. Proof of Theorem 1 has a unique minimum in To show that has no local maxima in . to show that Let (see Fig. 3)

it suffices

(31) and (32) and be any two Let also with . Since is the lower side of points in which implies that . an ellipse, and we conclude that no extremum exists So, . in Referring to Fig. 4, let us assume that the point is a local maximum. Then, there exist two points , in with (33)

(30) Let be the space spanned by the linearly independent vec, and its orthogonal complement. tors , there exist unique vectors For any vector and such that . Then, . and the equality holds if and In addition, . only if then and . In that case, the If vectors define the surface of a hyperellipsoid [26]. and then, as before, defines a point If then on the surface of the hyperellipsoid. If, however, and traces the interior of the hyperellipsoid.

be the contour line that passes through these points. Let A contour line is defined as the set of points for which for an arbitrary constant , i.e., : . A contour line defines implicitly a function such that , since when . It can be shown that for any

(34) and (35)

PSAROMILIGKOS et al.: ON ADAPTIVE MINIMUM PROBABILITY OF ERROR LINEAR FILTER RECEIVERS

with (36) and

(37) and and the contour Therefore, and concave. lines are increasing with respect to Since the contour line is concave, the straight line that and lies below . In conpasses through the points is convex. Therefore trast, the line formed by the points in lie below the straight line. Combining the all points of above we conclude that for any point in between and , there exists a point in with and . . Since lies on Hence, we have . This contradicts (33). We conclude that it is not possible for a local maximum to exist and exhibits a unique minimum. in

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Moreover, since , the above expression arbitrarily close to implies that there exists a vector that induces smaller probability of error. As a result is not a minimum of . is invertible, (38) defines a 1-to-1 mapping (b) Since onto the unit sphere of . from the ellipsoid surface from to defined by Therefore, the mapping is 1-to-1 and onto. The extension of that mapping to is a linear functional from to . is a minimum of . Then, Suppose that such that there is implies or implies or implies (41) ,

However, since C. Proof of Theorem 2 be an orthonormal basis (a) Let (we recall that the signature vectors of the subspace are assumed to be linearly independent). We let , , be the unique vector of the with the cross-correlations of an arbitrary vector . Let be the basis vectors matrix with rows the vectors . Since the rows of and defined by (30) span the same subspace, there exists an . Post-multiplying both invertible matrix such that we see that for any given sides of the above equality by we have vector (38) with We consider now an arbitrary (the case can be treated similarly). This vector—as any vector—can be decomposed uniquely into , where is in and is orthogonal to . , the vector lies in the interior Since such that the of . This means that there exists a scalar , where , lies in vector of the hyperellipsoid and . the surface implies that all vectors In addition, the convexity of , , defined by the points on the and are in and line segment connecting . Let , where is the th coordinate of . It is easy to verify that the vector (39) is of unit norm and

. Thus, (40)

: implies (42) where of

. Expression (42) implies that the vector such that is a minimum . The converse can be proved similarly. ACKNOWLEDGMENT

The authors wish to thank the anonymous reviewers for their valuable comments and suggestions on the original version of this manuscript. REFERENCES [1] S. Verdu, “Minimum probability of error for asynchronous Gaussian multiple-access channels,” IEEE Trans. Inform. Theory, vol. 32, pp. 85–96, Jan. 1986. [2] , “Optimum multiuser asymptotic efficiency,” IEEE Trans. Commun., vol. 34, pp. 890–897, Sept. 1986. [3] K. S. Schneider, “Optimum detection of code division multiplexed signals,” IEEE Trans. Aerospace Electron. Syst., vol. 15, pp. 181–185, Jan. 1989. [4] R. Lupas and S. Verdu, “Linear multiuser detectors for synchronous code-division multiple-access channels,” IEEE Trans. Inform. Theory, vol. 35, pp. 123–136, Jan. 1989. [5] M. K. Varanasi and B. Aazhang, “Multistage detection in asynchronous code-division multiple-access communications,” IEEE Trans. Commun., vol. 38, pp. 509–519, Apr. 1990. [6] , “Near-optimum detection in synchronous code-division multiple-access systems,” IEEE Trans. Commun., vol. 39, pp. 725–736, May 1991. [7] A. Duel-Hallen, “A family of multiuser decision-feedback detectors for asynchronous code-division multiple access channels,” IEEE Trans. Commun., vol. 43, pp. 421–434, Feb./Mar./Apr. 1995. [8] B. Aazhang, B. P. Paris, and G. Orsak, “Neural networks for multiuser detection in CDMA communication,” IEEE Trans. Commun., vol. 40, pp. 1212–1222, July 1992. [9] U. Mitra and H. V. Poor, “Adaptive receiver algorithms for near–far resistant CDMA,” IEEE Trans. Commun., vol. 43, pp. 1713–1724, Feb./Mar./Apr. 1995.

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[19] [20] [21] [22] [23]

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Ioannis N. Psaromiligkos (S’96), for photograph and biography, see p. 917 of the June 1999 issue of this TRANSACTIONS.

Stella N. Batalama (S’91–M’94), for photograph and biography, see p. 916 of the June 1999 issue of this TRANSACTIONS.

Dimitris A. Pados (M’95) was born in Athens, Greece, on October 22, 1966. He received the Diploma degree in computer engineering and science from the University of Patras, Patras, Greece, in 1989 and the Ph.D. degree in electrical engineering from the University of Virginia, Charlottesville, in 1994. He was an Applications Manager at the Digital Systems and Telecommunications Laboratory, Computer Technology Institute, Patras, Greece, form 1989 to 1990. From 1990 to 1994 he was a Research Assistant in the communications Systems Laboratory, Department of Electrical Engineering, University of Virginia, Charlottesville. From 1994 to 1997 he was an Assistant Professor in the Department of Electrical and Computer Engineering at the Center for Telecommunications Studies, University of Southwestern Louisiana, Lafayette. Since August 1997, he has been an Assistant Professor with the Department of Electrical Engineering, State University of New York at Buffalo. His research interests are in the areas of wireless multiple access communications, detection of spread-spectrum signals, adaptive antenna and radar arrays, and neural networks.