Hybrid higher-order statistics learning in multiuser detection

Depósito de Investigación de la Universidad de Sevilla https://idus.us.es/ This is an Accepted Manuscript of an article published by Cambridge University Press Applied Catalysis A: General 897 (2010), available at: https://doi.org/10.1016/j.apcata.2010.05.027 Copyright 2010. En idUS Licencia Creative Commons CC BY-NC-ND 1 Hybrid Higher Order Statistics Learning in Multiuser Detection Antonio J. Caama˜ no Member, IEEE,, Rafael Boloix-Tortosa, Javier Ramos Member, IEEE,, Juan J. Murillo-Fuentes Member, IEEE, Abstract—In this work we explore the significance of second and higher order statistics learning in communications systems. The final goal in spread spectrum communication systems is to receive a signal of interest completely free from interference caused by other concurrent signals. To achieve this end we exploit the structure of the interference, designing Second Order Statistics (SOS) detectors, such as the Minimum Square Error (MMSE) (known to be optimum linear detectors in noisy environments) in conjuction with Higher Order Statistics (HOS) techniques, such as the Blind Source Separation (BSS) and the Independent Component Analysis (ICA) (known to exhibit such remarkable properties in learning such as scale invariance and superefficiency). Such Hybrid Higher Order Statistics (HyHOS) approach enables us to alleviate BSS/ICA algorithms of one of their main problems, that is, their sensitiviness to high levels of noise as we benefit from their speed of learning and independence of the initial settings of the problem. We successfully applied the results of this approach to the design of multiuser detectors in Code Division Multiple Access (CDMA) channels. Index Terms—Array Signal Processing, blind source separation, higher order statistics, independent component analysis, unsupervised learning, CDMA. I. INTRODUCTION INTERFERENCE limitation due to the simultaneous access of multiple users systems has been the stimulus to the development of a powerful family of Signal Processing techniques, namely Multiuser Detection (MUD). This techniques have been extensively applied to Direct Sequence Code Division Multiple Access (DS-CDMA) systems. Thus, most of last generation digital communication systems such as GPS, wireless 802.11b, UMTS, etc, may take advantage of any improvement on this topic. In the blind case the algorithms should cope not only with noise and the near-far problem but with no training sequence available. In this sense, the minimum mean square error (MMSE) criteria [1] provides a good blind linear solution to the problem. Due to its computational complexity, some other alternative algorithms have been proposed [2], [3], [4]. On the other hand, blind source separation (BSS) and independent component analysis (ICA) have lately been main fields of research due to their great number of applications A.J. Caama˜ no and Javier Ramos are with the DTSC, EPSTelecomunicaciones, Universidad Carlos III de Madrid. Butarque, 15, 28911 Legan´ es, Madrid, Spain. Tl: +34 916248736. Fax: +34 916248749. E-mail: [email protected] J.J. Murillo-Fuentes and R. Boloix-Tolosa are with ATSC. Escuela Superior de Ingenieros. Universidad de Sevilla. Paseo de los Descubrimientos sn. 41092 Sevilla. Spain. Tl: +34 954487333. Fax: +34 954487372. E-mail: [email protected]. in different areas. ICA and BSS have been usually applied (see [5] and references therein) to communications signals (mainly in array processing), biomedical signals such as ECG or EEG, monitoring, as an alternative to principal component analysis in image or financial data processing, in monitoring, etc... Besides, they have been lately applied to other problems such as spread-spectrum based communications [6], [7], digital watermarking of images [8], audio spectrum basis functions computation [9], image classification, encoding or compression [10],... Several methods have been proposed as solutions to the BSS/ICA problem. We bring out here two main approaches. On the one hand, we have techniques based on the cancellation of estimation equations by using the steepest descent algorithm [11], [12], [13], [7]. These methods usually have the maximum likelihood (ML) approach as starting point and are not usually robust as the probability density functions (p.d.f.) are given a priori through the separation [14], also score [12], functions. Here robustness stands for the methods to be available for mixtures of any statistical distributions. On the other hand, we have contrast functions. Contrast are cost functions whose minimization yields the solution to the BSS/ICA. These methods usually face the BSS/ICA problem starting from lower to higher order moments analysis. In this sense the mixtures are first reduced to zero mean signals. Then a whitening process is carried out to achieve decorrelation. Finally, higher order moments are used to compute a unitary transformation to diagonalize the associated higher order tensor of the whitened outputs. If at most one marginal fourth-order cumulant is null, this unitary matrix may be computed by diagonalizing the fourth order tensor. This tensor may be diagonalized by either canceling the whole set of cumulants out of the diagonal (minimization of the mutual information, MI), or by the maximization of the diagonal entries (minimization of the marginal entropy, ME [15]) as the SICA method used in this paper. Some blind source separation (BSS) algorithms have been proposed as fully blind MUD’s, in the sense that spreading codes are unknown [16], [17], [7], [18]. However this blind techniques are not usually noise robust. We propose in this paper to introduce a similar technique to BSS such as independent component analysis (ICA) [19], [5] into the structure of the MMSE MUD to exploit its proven noise robustness. We will compare the performance of a block ICA algorithm such as SICA [19] to an adaptive method to compute the matrix such as M-EASI algorithm [16]. The paper is organized as follows. Section II summarizes 2 the matrix model, main assumptions and definitions for a DSCDMA communication system. The MMSE detector will be also addressed. Then, BSS/ICA techniques are introduced in Section III, where we will focus on the SICA algorithm. Section IV is devoted to relate both of these approaches BSS/ICA and the MMSE to propose a new HOS based solution, the so called BH MUD detector. These are referred to throughout this paper as HyHOS I and II respectively. Section V includes experiments to compare MMSE with the Hybrid High Order Statistics detectors developed in this work and to show noise resistance en superefficient learning. Section VI is devoted to conclusions. II. COMMUNICATION SYSTEM MODEL:THE MMSE Consider a CDMA transmitter [1].The transmitted baseband signal, assuming that the propagation channel of each user is memoryless, can be represented as x(t) = L X l=1 n X i=1 χisici(l−τi)g(t−lTc−τi),0≤t≤Ts(1) where Lis the length of the spreading code, nis the number of active users, χnis a scaling factor specified by the power control loop, snis the symbol transmitted by user n,g(t) represents the transmitter pulse shaping filter, Tcis the chip period, and τnis the delay of each user. The spreading code of user nis denoted by cn. Note that individual chips in cn are denoted by the elements in [cn(1),...,cn(L)]Twith each cn(·)∈ ±1. Equation 1 usually is expressed as: x(t) = N X n=1 χnsncn(t−τn)(2) where cn(t)is the normalized signature waveform assigned to the n-th user. If the transmitted pulse-shape filters are Nyquist, we assume the synchronous case and the receiver uses a chiprate sampled matched filter followed by a serial to parallel converter, the matrix form of the system equation is as follows: xt=Atst+nt(3) where xtis a L×1vector, Atis a L×nmatrix representing the channel response and ntis a noise vector. The memoryless easiest solution is a synchronized matched filter (MF). However, if the difference between powers of users is high enough and user’s codes are not orthogonal, we have the near-far problem and other well-known SOS based detectors such as the MMSE have much better a performance. A. Linear Multiuser Detection A linear multiuser detector Cgives an estimation of the original transmitted signals as y(t) = Cx(t) = CAs(t) + Cn(t)(4) The centralized detector minimizing the mean-square-error (MSE) for each user k, MSEk=E|yk(t)−sk(t)|2, is the CMMSE CMMSE =R−1 vv H∗=H∗HW ∗ vWvH∗(5) where v(t) = H∗x(t)and Wvis the decorrelating matrix for vector v. Here, as in the following, ∗denotes transposeconjugate. In [17] the authors define a whitening–rotation detector (WR) as the CW R matrix that minimizes the MSE sum of MSEk,k= 1, ..., n. A posible structure for this receiver was given as follows CW R =JQW (6) where Wis a m×mwhitener, Qam×mrotation matrix, and J = [I0] is n×m. In the context of the BSS, the problem consist of computing the matrix WBthat minimizes statistical dependence at the output. Other previous BSS approaches to fully blind MUD’s use a SVD decomposition. However, this involves a higher computational complexity. The natural or relative gradient (NG) has been also used as a steepest descent algorithm for adaptive approach in both BSS and MUD [20]. We face in this paper the development of a HOS based detector. Let’s first introduce the HOS-BSS/ICA tools we are going to use. III. BSS AND SICA A. Main assuptions and model Blind separation of sources (BSS) involve the task of obtaining a non-observable set of signals, the so-called sources, from another set of observable signals regarded as mixtures. Here, the adjective ”blind” stands for the fact that neither the original sources nor the mixture itself are known. Usually, and in the context of this paper, the assumption of spatial statistical independence is the key to achieve separation. In this sense, the related [21] and more general technique of independent component analysis (ICA) computes the projection of a set of components (mixtures in BSS) that minimizes the statistical dependence of the ouputs [15]. In its simplest form, the BSS reduces to the following matrix form. The m=nmixtures sampled at time t,xt, are instantaneous linear combinations of the nsources st, i.e., x(t) = As(t)(7) Usually, and in the context of this paper, statistical independence of the inputs is assumed. Thus, if xtis a stationary ergodic random sequence and the mixing matrix Ais nonsingular, it is possible to estimate a separation matrix Bto obtain the sources as y=Bx =BAs. yt=Bxt=BAst=Cstt= 1,2,... (8) In Blind Source Separation (BSS) we compute matrix Bso that Cis ideally the identity matrix. However, the original scaling and arrangement cannot be estimated from the independence assumption. In this sense, Cis a non mixing matrix if it has one and only one non-zero entry in each column and each row. The related [21] and more general problem of independent component analysis (ICA) [15] consists of obtaining the change of basis Bthat projects components xt into a set ytas statistically independent as possible (sources). As matrix Bcan be decomposed into the product of a 3 whitening Wand a unitary Vmatrix, this sphering stage gives us signals zi(t): yt=Bxt=V W Ast=V ztt= 1,2,... (9) In addition to the source independence hypothesis, there are other minor considerations. First, although the instantaneous mixture model considered here applies to many of the applications referred at the Introduction, it may be extended to the convolutive case [22], [23] when needed. Besides, we may consider here the case were the number of mixtures m is greater or equal to the number of sources n, i.e. m≥n. In the case m>n, a subspace approach such as a singular value decomposition may be used to project the mixture space onto the signal subspace, reducing the effect of weak noises. For noisy mixtures see [24], [25], [26]. We thus assume the noiseless case with m=n. In addition, a necessary and sufficient condition for the waveform-preserving source estimation to be feasible is that no more than one gaussian distributed source be present in the mixture [24], [15], [27]. This model in (7) may be identified with the narrowband msensor linear-array application, the synchronous-CDMA case with spreading factor L=m, or the general instantaneous BSS/ICA model. In synchronous CDMA communications, matrix Amay be decomposed into A=Hχ, where χis a diagonal matrix with the amplitudes of each user and His a matrix whose columns are the spreading codes. Notice that BSS/ICA applied to CDMA would lead to a detector where neither a training sequence nor the spreading codes themselves are needed. However, noise vector in (3) is not included in the BSS/ICA model in (7). Thus, if noise is high enough, the performance of the BSS/ICA model deteriorates. Adaptive solutions to this problem are usually based on maximum likelihood and the natural (or relative) gradient [11], [28], [29]. On the contrary, most of the off-line solutions to ICA [15], [30] minimize one criterion, contrast function, or cancellation of multiple criteria. For instantaneous linear mixtures of two sources, direct methods, which consist of directly estimating the mixing matrix from the mixtures, are also possible. Direct methods suggested in [31], [32], [33] solve polynomial equations based on cumulants up to fourthorder. The solutions of the equations provide the entries of the mixing matrix. However, methods based on the optimization of criteria usually have a better performance. In this sense we consider here the contrast function based on the minimization of the marginal entropies. B. Minimum marginal entrophy based contrast A contrast function φ(·)maps Y, the set of random vectors y(multivariate p.d.f.’s), on R. It has a minimum when the entries of yare statistically independent, i.e, if yhas independent component φ(y)≤φ(Ay)∀Anon-singular. Thus, the minimization of a contrast function yields the solution to the BSS/ICA problem. If the number of mixtures m > 2, these contrasts are complex functions to minimize. The ‘Jacobi optimization’ [15] consist of solving the n-dimensional problem by decomposing it in a set of 2-dimensional optimizations. The rotation matrix Vis decomposed into g=m(m−1)/2Givens rotations. Thus, minimization of the contrasts for each angle θh,h= 1, ..., g, drives the system to the solution. We will first focus on a 2-dimenisonal contrast, the SICA. Under the whiteness constraint, i.e. E[yyT] = I, it yields the following orthogonal (denoted by φ2) contrast [21], φME 2(y)c = n X i=1 H[yi](10) where the criteria to minimize is the marginal entropy H[y]. Here, as in the following, c =means equality up to an additive constant c. Besides, we will denote by µy ijkl =E[yiy∗ jyky∗ l] and µy ij =E[yiy∗ j]the fourth order and second-order moments. If we approximate the possible distributions for sby and Edgeworth expansion, rewrite the result in terms of second-order and fourth-order cumulants, Cy ijkl =µy ijkl −µy ijµy kl −µy ikµy jl − µy ilµy jk and then minimize it for all possible distributions [15], it follows that φME 2(y)≈1 48φME 24 (y) = −1 48 X i (Cy iiii)2(11) where Cy iiii =µy iiii −3µ2 ii is the autocumulant of output yi, symbol ∗denoting complex conjugation. Thus, the minimization of the fourth-order cumulant based ME contrast yields the diagonalization of the associated tensor when at most one marginal fourth-order cumulant is null. in the following, we will face the real case. The extension to complex-valued sources is somehow immediate. By using the complex notation and under the whitening constraint, the independent components yp(t)and yq(t)in the two-dimensional approach yield y∗(t) = r(t)ejρpq (t)=r(t)ej(θpq +βpq (t)) =ejθpq z∗(t)(12) where j=√−1. The whitenned mixture vector z∗(t) = zp(t) + jzq(t)are a simple rotation of the normalized (unit variance) sources ¯s∗(t) = ¯sp(t) + j¯sq(t) = r(t)ejα. Notice that at the solution ρ(t) = θ+β(t) = α(t) + kπ/2,k= 0,1,2, ... The associated estimation of the rotation angle θ in the optimization of 2-dimensional contrasts may be easily expressed as a closed function of the following complexvalued linear combinations (centroids) of the statistics of the outputs [34] ξγ=E[r4(t)ej4βpq(t)](13) ξη=E2[r4(t)ej2βpq(t)](14) γ=κr−8(15) Based on the so called ‘weighted estimators’ (WE) or WAML [35], [36] a general 2-dimensional estimator yields ˆ θGW E(ωγ, ωξ) = 1 4∠(ωξωγξγ+ (1 −ωξ)ξη)(16) 0< ωξ<1, ωγ=±1, γ This estimates yields EML estimator in [34] ˆ θEML = ˆ θGW E(sign(γ),1) or the [37], MK [38], [30], SKSE and SKDE [39], ML [40] estimates ˆ θGW E(±1,1),ˆ θMaSSF OC = ˆ θGW E(γ, 1/2) in [39], or the ˆ θAML =ˆ θGW E(γ, 1/3) [35]. The discussion on the optimum wξ[36] is open as it is difficult 4 to compute the statistics of these estimates and analyze them for all possible p.d.f’s. In this paper, we propose to use θSICA =ˆ θGW E(γ, 3/7) (17) as it may be proved that the SICA approach is as robust as the φME 24 (y), i.e, it provides a solution to the ICA problem with the only condition of no more than one source with null fourth order statistics in the mixture. The minimization of φSICA(θ)is immediate as the solution is minus four times the phase of the resulting function, a sinusoid. This may be easily implemented by a look-up table so that we minimize the number of operations needed. C. n-dimensional case The 2-dimensional case may be easily extended to ndimensions by using the ’Jacobi Optimization’ [15]. We have here rewritten the algorithm using φSICA(θ). Such an algorithm can be summarized as follows. Algorithm 1: n-dimensional SICA using Jacobi Optimization. 1) Whitenning. Compute a whitening matrix Wand the output vector z=W x. Set c= 1 and y=z. 2) Sweep c. For all g=n(n−1)/2pairs, i.e., for 1≤p < q≤n, do (a) Compute the Givens angle θpq θSICA(y)in (17) with [zp, zq]T= [yp, yq]T. (b) if θpq > θmin, do rotate the pair (yp, yq)by θpq according to (12). 3) End? If the number of sweeps csatisfies c≥K= 1 + √nor no angle θpq has been updated, stop. Otherwise go to step 2 for another sweep with c=c+ 1. D. The Natural Gradient The steepest descent method updates Caccording to the direction of the gradient ∇Lof a loss function L(C). The natural [23] or relative gradient [5] proposes to use e ∇L(C) = e ∇E[l(C)] = ∇L(C)CTC. The stochastic version uses the intantaneous value e ∇l(C). The learning law yields C←− C−λ∇l(C)CTC(18) In the BSS problem it is usually assumed statistical independence at the outputs y(t). Maximum likelihood (ML) is an extended technique to derive a loss function for this criteria [23], [5]. It can be shown that by forcing the estimating function K(y) = ∇l(C)·CT=ϕ(y)yT−Ito cancel, we achieve independence at the output. The learning law yields C←− C−λK(y)C(19) where in the ML approach ϕi(yi) = −q0 i(yi)/qi(yi), being qi(·)the probability density function of source bi. However, as source distributions are supposed unknown, each author introduces his own activation function ϕi(yi). A family of them may be found, for sources with negative or positive kurtoses, in [41]. By normalizing (19) the learning law may be written as follows C←− C−λϕ(y)yT−I 1 + λ|ϕT(y)y|C(20) In [16] the authors proposed a NG based algorithm to separate signals in digital communication, the M-EASI (MedianEquivariant Adaptive Separation via Independence). This method assumes zero-mean, symmetric, ‘circularly distributed’ signals and introduces the sign function to reduce the bias introduced by the noise. It estimating function yields K(y) = ysgn(y)∗−I 1 + λsgn(y)∗y+1 α ϕsgn(y)∗−sgn(y)ϕ∗ 1 + λ|y∗ϕ|(21) where sgn(y) = sgn(<(y)) + jsgn(=(y)),y∈C. With this method we improve the stability of the algorithm [29], provide the method with phase recovering properties and make the method more robust against noise. IV. HYBRID HIGH ORDER STATISTICS MULTIUSER DETECTORS A. Natural Gradient Based HyHOS We could use the natural gradient to compute a detector Cby imposing some criteria such as the MSE. But we may directly exploit the MMSE estructure in 5 for the centralized (i.e., base station, all codes available) and non-centralized case (i.e., user equipment, only user code available). First, we propose the WWH, centralized detector by computing Wvin CWWH =H∗HW ∗ vWvH∗(22) by using (20) with y=v=WvH∗xand activation functions ϕ(v) = v. The learning law yields the decorrelating algorithm Wv←− Wv+λI−(Wvv)(Wvv)T 1 + λ|(Wvv)T(Wvv)|Wv(23) Now, we propose a non-centralized MUD, the HWW detector. The non-centralized MMSE-MUD was given in (5) as CHW W =H∗W∗ xWx(24) If we redefine y=Wxxand rewrite the natural gradient blind source separator in (18) as we did in (23), it follows that Wx←− Wx+λI−(Wxx)(Wxx)T 1 + λ|(Wxx)T(Wxx)|Wx(25) Following the same structure as before, we have centralized and non-centralized BSS based detector algorithms. In the case of the BSS centralized MUD, the BH detector, we can proceed as follows It is posible to substitute the matrix product H∗HW ∗ vWvin (22) by a separating matrix B. The new detector yields CBH =BH∗=QW vH∗ v(26) where Bis computed as a blind source separator, that is, a whitening-rotator. Notice that in the centralized case the dimensional reduction is carried out by the matched filter H∗, 5 thus J=Iin (6). Matrix in B(31) is computed by using the M-EASI algorithm B←− B−λKB(y)·B(27) where KB(y) = ysgn(y)∗−I 1 + λsgn(y)∗y+1 α ϕsgn(y)∗−sgn(y)ϕ∗ 1 + λ|y∗ϕ|, (28) y=Bv =BH∗x, and ϕ(y)may be chosen as described in [41]. Thus, the MMSE structure is further enhanced by the properties of the M-EASI algorithm. On the other hand, it is not possible to define an architecture for the non-centralized MUD as in (31). Suppose the candidate now to be the detector CHB =H∗B=H∗QW x(29) Here, matrix CB=Bis already a solution, as independence is imposed at the outputs. Thus CHB is not a detector as H6=I. Computing matrix CBis basically a blind separation problem and it is out of the scope of this paper. Notice that this detector is fully blind as it does not use the spreading codes [16], [7], [17], but rather constructs them. Now, we include a discussion on some theoretical aspects of the methods above. We focus on the near-far problem, noise, convergence and complexity. The main point in using the natural gradient in MUD is that it is equivariant [5], i.e., the convergence has a uniform performance. Let’s define D=CS and right multiply (19) by Hto obtain D←− D+λK(Db)D(30) Matrix S=AH is only present at the initial value D0= C0AH. Thus, the convergence does not depend on matrix AH. It can be concluded that the methods proposed in this paper are near-far resistance, as convergence is independent of the user’s amplitudes. Algorithms in BSS usually do not cope with the noisy case whenever the number of sources and mixtures are the same m=n. By using the M-EASI algorithm we combat the effect of noise in the separation process for digital communication. On the other hand if m>na signal subspace projection allows noise reduction. Previous BSS approaches to fully blind MUD’s use a SVD decomposition, a MPLL,[17] ... However, this involves a higher computational complexity. As the spreading codes are usually available (at least at the base station), it is straightforward to introduce them as a subspace algorithm at a null complexity cost [4]. Besides, the structure of the MMSE detector has been used in (22) and (24) to cope with noise. Another important characteristic of natural gradient BSS techniques is that of superefficiency [42]. In this sense, provided E[ϕ(y)] = 0 (an usual case), the covariance between two outputs decreases of the order of 1/t2in batch estimation and of the order of λ2in on-line learning. Futhermore, λ= 1/t gives, asymptotically, the best performance, which is the same as the optimal batch estimator. Thus, with λ= 1/t we achieve an on-line algorithm with batch features at every t. On the other hand, if adaptive features are needed, we may use λ < 1 to achieve an output covariance decreasing as λ2. On the question of complexity, the exact MMSE solution at every time teinvolves computing the eigenvalues of the autocorrelation matrix. Thus, the computational resources needed are significant. The methods proposed in this paper allow computing, as described in the last section devoted to superefficiency, the batch solution as an on-line algorithm at a low computational burden. B. SICA Based HyHOS In the section above, it is proposed to substitute the matrix product H∗HW ∗ vWvin (5) by a matrix Bthat makes the outputs yias statistically independent as possible. The new detector yields CBH =BH∗=QW vH∗ v(31) where Bis a whitening–rotator computed by using the SICA algorithm [19]. Notice that in the centralized case the dimensional reduction is carried out by the matched filter H∗, thus J=Iin (6). In [19] authors prove that under whiteness constrains some fourth-order contrasts may be approximated by a sinusoid. Thus, the minimization of the contrast reduces to computing its phase. The starting point is the Minimum Entropy (ME) ICA contrast given by Comon in [15] so that we use the ’Jacobi optimization’ to cope with higher dimensions. This method, called SICA (Sinusoidal ICA,) has a good performance along with a low computational cost, outperforming the ME by Comon and the JADE methods [43]. Applying SICA to our problem, assuming inputs vihave been decorrelated, this algorithm computes y=Bv minimizing the following function, denoted as contrast, φME [y]≈φ4[y] = 1 48 X ijkl6=iiii (Cumy ijkl)2(32) where Cumy ijkl is the fourth order cumulant of outputs yi, yj, ykand yl. Thus, this ICA algorithm may be seen as a fourth order decorrelation, and we go further than the second order based MMSE to find a new solution to the problem. The algorithm SICA is a block method that computes matrix Bfrom a whole set of observations, similar to the MMSE method. As the spreading codes are usually available (at least at the base station), it is straightforward to introduce them as a subspace algorithm at a null complexity cost [3]. Besides, the structure of the MMSE detector has been used to cope with noise. We will compare the performance of this algorithm to that of the M-EASI algorithm, an adaptive method to compute matrix Bthat try to minimize a similar fourth order based contrast by using the natural gradient [16]. V. EXPERIMENTAL RESULTS The test environment used here is a synchronous CDMA system in which the users here are spread using Gold sequences with spreading factor 31. We will present here two 6 main results: first, the typical evaluation of the performance in a digital communications system, i.e., Bit Error Rates from 10−3to 10−6and the equivalent SNR ≥8dBs, a Low Noise Environment (LNE); second, an evaluation in a High Noise Environment (HNE) where we test the noise resistance capabilities of the HyHOS algorithm with Bit Error Rates as low as 2×10−1≤BER <5×10−1and equivalent Signal to Noise Ratio of −15 ≤SNR ≤0dBs. As the BSS problem is posed in the absence of noise, one of the main issues is to test the performance of the resulting algorithms in noisy environments. The comparison of the various algorithms presented here must be done with equal convergence speed (i.e. learning speed) on one hand, and equal final variance on the other, to faithfully characterize their performance. The methods described in [44], [45] and [46] are to be used to analyze the final variance and the convergence speed of each algorithm. Those methods are useful and straightforward in the case of linear algorithms such as the MMSE but, in the case of the nonlinear functionals of the HyHOS algorithms, this is a much difficult task and, as such, merits a separate analysis which is currently under way. In the present work, and as a means to compare the algorithms on an equal footing, the training length for each of the detectors is 2000 samples, the convergence speed (i.e. step-size) and the final variance are fixed throghout the experiments. The number of Monte Carlo simulations averaged is 100 in the HNE and 500 in the LNE, which amounts to a confidence margin of at least 95% for any BER estimation presented in this work. A. Low Noise Environment In Low Noise Environment we present, for clarity, the convergence performance of the HyHOS I and the MMSE algorithms both with 15 and 20 users. We see a clear difference in the convergence behaviour of the HyHOS I and the MMSE algorithm. That of the HyHOS II algorithms closely resembles that of the HyHOS I. The average decrease in SINR of the HyHOS I algorithm whit an increase of 5users is half of the decrease suffered by the MMSE. In Figure 1 we can see that 2000 samples is more than enough training length for the HyHOS I algortihm whilst for the MMSE, convergence cannot be achieved in a highly occupied channel. In the evaluation of the Bit Error Rate (see Figure 2), this speed of convergence of the different algorithms is reflected in the performace of both algorithms, HyHOS I and II and that of the MMSE. In this Low Noise Environment (LNE), the sought features in multiuser detector is that of a controlled degradation of Bit Error Rate with an increase in the number of users. Both the SICA based HyHOS algorithm (HyHOS I) and the Natural Gradient based HyHOS (HyHOS II) show a performance close to that of the MMSE (if not better in the HyHOS I case) with a low occupation of the channel (5users) but the difference in behaviour arises in a highly occupied channel (25 users) , where the BER of the MMSE degrades from 10−6at approximately 14 dB of SNR to 10−5, while both HyHOS algorithms keep the BER to approximately 10−6in the same conditions. 0 500 1000 1500 2000 Samples 8 9 10 11 12 13 14 15 SINR (dB) HyHOS (15 users) HyHOS (20 users) MMSE (15 users) MMSE (20 users) Fig. 1. Signal to Interpference plus Noise Rate, with SNR of the User Of Interest (UOI) = 15 dB, for the HyHOS detector and the MMSE detector, both with 15 and 20 users. The powers of the interfering users is distributed homogeneously between 0and 30 dB above that of the UOI. 8 10 12 14 SNR (dB) 1×10-6 1×10-4 1×10-2 BER HyHOS I (5 users) HyHOS I (25 users) HyHOS II (5 users) MMSE (5 Users) MMSE (25 users) Fig. 2. Bit Error Rate in Low Noise Environment (LNE), i.e. SNR of the UOI ≥8dB, for the HyHOS detector and the MMSE detector, both with 5and 25 users. The powers of the interfering users is distributed homogeneously between 0and 30 dB above that of the UOI. B. High Noise Environment In the case of High Noise Environment, or HNE, the convergence speed (see Figure 3) of both HyHOS algorithms is completely similar but for higher variance in the HyHOS I algorithm. Both in a low or high occupation channel, the convergence is fairly similar. The MMSE shows a lack of convergence in a high-noise, low-interference channel, quit expected for the MMSE because the interference noise structure is completely masked by gaussian noise, avoiding this algorithm to “lock” on to the signal of the User of Interest. Quite the contrary is the case of the HyHOS algorithms. Their intrinsic nonlinear properties show here the limitations of linear functionals in linear detectors. The Bit Error Rate of the HyHOS algorithms show the noise resistance proper to the MMSE without showing its demeanors. The convergence to a Matched Filter solution in the asymptotic limit of σ→ ∞ is shown experimentally in Figure 4. 7 0 500 1000 1500 2000 Samples -26 -24 -22 -20 -18 -16 -14 SINR (dB) HyHOS I (5 users) HyHOS I (25 users) HyHOS II (5 users) HyHOS II (25 users) MMSE (5 users) MMSE (25 users) Matched Filter Fig. 3. Signal to Interpference plus Noise Rate, with SNR of the UOI =−15 dB, for the HyHOS detector and the MMSE detector, both with 5and 25 users. The powers of the interfering users is distributed homogeneously between 0 and 30 dB above that of the UOI. -15 -10 -5 0 SNR (dB) 2.0×10 -1 2.5×10 -1 3.1×10 -1 4.0×10 -1 5.0×10 -1 BER HyHOS I (5 users) HyHOS I (25 users) HyHOS II (5 users) HyHOS II (25 users) MMSE (5 users) MMSE (25 users) Matched Filter Fig. 4. Bit Error Rate in High Noise Environment (HNE), i.e. SNR of the UOI ≤0dB, for the HyHOS detector and the MMSE detector, both with 5 and 25 users. The powers of the interfering users is distributed homogeneously between 0and 30 dB above that of the UOI. VI. CONCLUSION In this paper the authors proposed both ICA and NG as valid techniques to exploit the matrix estructure of the parameters involved in the MUD problem, an application of the more general narrowband m-sensor linear-array case. Here, we propose a novel Hybrid MUD by introducing an ICA matrix into the structure of the blind centralized MMSE MUD. 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