METHOD OF JOINT USER ACTIVITY DETECTION AND CHANNEL INFORMATION ESTIMATION IN EXTRA-LARGE MIMO (XL-MIMO) SYSTEMS WITH NON-STATIONARITIES

Abstract

Joint user activity detection and channel information estimation in extra-large MIMO (XL-MIMO) systems with non-stationarities proceeds by: receiving signals and initialization of a channel estimation; performing soft interference cancel check, whereby, if the soft interference cancel check result is negative, the calculation of residual mean and variance is performed; whereby, if the soft interference cancel check result is positive, the extrinsic mean and variance is calculated directly; whereby, if the soft interference cancel check result is negative, the calculation of residual mean and variance is used, whereby the calculation of tentative estimation includes activity factors calculation; if reaching the maximum is not fulfilled, soft interference cancel check is performed; whereby, if reaching the maximum is fulfilled, the proceeding ends.

Claims

1. A method of joint user activity detection and channel information estimation in extra-large Multiple-Input Multiple-Output (XL-MIMO) systems with non-stationarities, comprising: receiving signals and initialization of a channel estimation; performing a soft interference cancel check, whereby, if the soft interference cancel check result is negative, the calculation of residual mean and variance is performed and whereby, if the soft interference cancel check result is positive, the extrinsic mean and variance is calculated directly; calculating the extrinsic mean and variance, whereby, if the soft interference cancel check result is negative, the calculation of residual mean and variance is used; calculating tentative estimates, whereby the calculation of tentative estimation includes activity factors calculation; determining whether a condition of reaching a maximum is fulfilled whereby, if the condition of reaching the maximum fulfilled check result is negative, soft interference cancel check is performed and whereby, if the condition of reaching the maximum fulfilled check result is positive, performance of the method ends.

2. The method of claim 1, wherein after ending a preamble and uplink data transmission via a grant-free random access is proceeded.

3. The method of claim 1, wherein a point process, which models randomly located clusters within a given area, defines an area where antenna arrays are embedded.

4. The method of claim 1 characterized by a modification to an iterative shrinkage-thresholding algorithm (ISTA) via boxing with range limiting and hard-thresholding is proceeded.

5. The method of claim 1, using a boxed-hard iterative shrinkage-thresholding algorithm (ISTA), a greedy selection of the positions of the antennas index and the symbol estimates, and their independent decoding of the corresponding antenna modulated and symbol modulated bits is determined.

6. The method of claim 1, wherein process working parallel to the greedy detections, to ensure valid estimates of the index vectors from the given finite set of index vectors are produced at the output of the method, and to apply interference cancellation with the confirmed values, while keeping track of which indices have been retrieved from the greedy selections, before every iteration a check is performed whether from the currently decoded indices, a final confirmation can be made, and, if the final confirmation cannot be made, remove the interference by the previous greedy selection and the next iteration is proceeded.

7. A receiver of a communication system having a processor, volatile and/or non-volatile memory, at least one interface adapted to receive a signal in a communication channel, wherein the non-volatile memory stores computer program instructions which, when executed by the microprocessor, configure the receiver to perform operations comprising: receiving signals and initialization of a channel estimation; performing a soft interference cancel check, whereby, if the soft interference cancel check result is negative, the calculation of residual mean and variance is performed and whereby, if the soft interference cancel check result is positive, the extrinsic mean and variance is calculated directly; calculating the extrinsic mean and variance, whereby, if the soft interference cancel check result is negative, the calculation of residual mean and variance is used; calculating tentative estimates, whereby the calculation of tentative estimation includes activity factors calculation; determining whether a condition of reaching a maximum is fulfilled whereby, if the condition of reaching the maximum fulfilled check result is negative, soft interference cancel check is performed and whereby, if the condition of reaching the maximum fulfilled check result is positive, performance of the operations ends.

8. (canceled)

9. (canceled)

10. The receiver of claim 7, wherein after ending a preamble and uplink data transmission via a grant-free random access is proceeded.

11. The receiver of claim 7, wherein a point process, which models randomly located clusters within a given area, defines an area where antenna arrays are embedded.

12. The receiver of claim 7 characterized by a modification to an iterative shrinkage-thresholding algorithm (ISTA) via boxing with range limiting and hard-thresholding is proceeded.

13. The receiver of claim 7, using a boxed-hard iterative shrinkage-thresholding algorithm (ISTA), a greedy selection of the positions of the antennas index and the symbol estimates, and their independent decoding of the corresponding antenna modulated and symbol modulated bits is determined.

14. The receiver of claim 7, wherein process working parallel to the greedy detections, to ensure valid estimates of the index vectors from the given finite set of index vectors are produced at the output of the method, and to apply interference cancellation with the confirmed values, while keeping track of which indices have been retrieved from the greedy selections, before every iteration a check is performed whether from the currently decoded indices, a final confirmation can be made, and, if the final confirmation cannot be made, remove the interference by the previous greedy selection and the next iteration is proceeded.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

[0038] For a fuller understanding of the nature of the present invention, reference should be had to the following detailed description taken in connection with the accompanying drawings. So that the manner in which the above-recited features of the present disclosure can be understood in detail, a more particular description, briefly summarized above, may be had by reference to aspects, some of which are illustrated in the drawings. It is to be noted, however, that the appended drawings illustrate only certain aspects of this disclosure and are therefore not to be considered limiting of its scope, for the description may admit to other equally effective aspects.

[0039] FIG. 1: Illustration of the uplink of a multiuser XL-MIMO system with spatial non-stationarity, whereby each user independently activates different subarrays of the XL-MIMO array depending propagation conditions.

[0040] FIG. 2: NMSE performance with respect to SNR with N=400 and M=200 for different pilot lengths.

[0041] FIG. 3: AER performance with respect to SNR with N=400 and M=200 for different pilot lengths.

[0042] FIG. 4: Resilience of the proposed algorithm against different sub-array activity indicators in perce.

[0043] FIG. 5: Convergence behavior of the proposed algorithm with respect to the number of algorithmic iterations.

[0044] FIG. 6: MCPP-based subarray activity.

[0045] FIG. 7: Uniformly random subarray activity.

[0046] FIG. 8: NMSE Performance with respect to SNR with N=400, M=200, and L=70 with MCPP for different .

[0047] FIG. 9: AER Performance with respect to SNR with N=400, M=200, and sL=70 with MCPP for different .

[0048] FIG. 10: NMSE performance of the proposed method for different cluster intensities. The actual array activity ratios for =3, 6 and 9 are annotated by allows.

[0049] FIG. 11: NMSE performance of the proposed method for different cluster radius sizes. The actual array activity ratios for =3, 6 and 9 are annotated by allows.

[0050] FIG. 12: Describing the Flowchart of the inventive method.

[0051] FIG. 13: Illustrates the Grand Free Random Access.

DETAILED DESCRIPTION

[0052] Extra large MIMO (XL-MIMO) systems are subject to spatial non-stationarity which leads to a doubly-sparse and user-specific structure of received signals, such that the activity of each user at each sub-array can be characterized by a nested Bernoulli-Gaussian distribution. This application considers the joint activity and channel estimation (JACE) problem in XL-MIMO systems subject to spatial non-stationarity, offering two major embodiments solving this problem.

[0053] The first is a novel bilinear Bayesian inference method capable of jointly estimating sub-array activity patterns (a.k.a. spatial non-stationarity), user activity patterns, and associated channel coefficients, boosted by expectation maximization (EM)-based auto-parameterization.

[0054] The second embodiment is the introduction of realistic Poisson point process (PPP) and Matern-cluster point process (MCPP) stochastic-geometry (SG) models to simulate sub-array activity patterns, which enables the performance assessment of both the proposed and state-of-the-art (SotA) XL-MIMO JACE solutions under different conditions in a structured manner. The efficiency of the proposed bilinear JACE method is confirmed by numerical simulations, which shows that the proposed method not only significantly outperforms the SotA but also can reach the performance of a genie-aided (theoretical) scheme over wide signal-to-noise-ratio (SNR) ranges.

[0055] A Matrn cluster point process is a type of cluster point process, meaning that its randomly located points tend to form random clusters. Using techniques from spatial statistics, it is possible to make the definition of clustering more precise. This point process is an example of a family of cluster point processes known as Neyman-Scott point processes, which have been used in spatial statistics and telecommunications.

[0056] The Matrn cluster point process should not be confused with the Matrn hard-core point process, which is a completely different type of point process. Bertril Matrn proposed at least four types of point processes, and his name also refers to a specific type of covariance function used to define Gaussian processes.

[0057] Simulating a Matrn cluster point process requires first simulating a homogeneous Poisson point process with an intensity >0 on some simulation window, such as a rectangle, which is the simulation window I will use here. Then for each point of this underlying point process, simulate a Poisson number of points with mean >0 uniformly on a disk with a constant radius r>0. The underlying point process is sometimes called the parent (point) process, and its points are centres of the cluster disks.

[0058] The subsequent point process on all the disks is called daughter (point) process and it forms the clusters. It has been known about simulating the homogeneous Poisson point processes on a rectangle and a disk, so those posts are good starting points, and it won't not be focused too much on details for these steps.

[0059] The main challenge behind sampling this point process, is that it's possible for daughter points to appear in the simulation window that come from parents points outside the simulation window. In other words, parents points outside the simulation window contribute to points inside the window. To remove these edge effects, the point processes must be simulated on an extended version of the simulation window. Then only the daughter points within the simulation window are kept and the rest are removed. Consequently, the points are simulated on an extended window, but we only see the points inside the simulation window.

[0060] To create the extended simulation window, you can add a strip of width r all around the simulation window. The distance r is the maximum distance from the simulation window that a possibly contributing parent point (outside the simulation window) can exist, while still having daughter points inside the simulation window. This means it is impossible for a hypothetical parent point beyond this distance (outside the extended window) to generate a daughter point that can fall inside the simulation window.

[0061] Simulate the underlying or parent Poisson point process on the rectangle with NP points. Then for each point, simulate a Poisson number of points, where each disk now has Di number of points. Then the total number of points is simply N=D1+ . . . +DP=NPi=1 Di. The random variables P and Di are Poisson random variables with respective means A and , where A is the area of the rectangular simulation window. To simulate these random variables in MATLAB, the poissrnd function is used. To do this in R, use the standard function rpois. In Python, it can be used either functions scipy.stats.poisson or numpy.random.poisson from the SciPy or NumPy libraries.

[0062] The points of the parent point process are randomly positioned by using Cartesian coordinates. For a homogeneous Poisson point process, the x and y coordinates of each point are independent uniform points, which is also the case for the binomial point process, covered in a previous post. The points of all the daughter point process are randomly positioned by using polar coordinates. For a homogeneous Poisson point process, the and coordinates of each point are independent variables, respectively with uniform and triangle distributions, which was covered in a previous post. Then we convert coordinates back to Cartesian form, which is easily done in MATLAB with the pol2cart function. In programming languages without such a function: x= cos () and y= sin ().

[0063] In practice, all the daughter points are simulated in a disk with its centre at the origin. Then for each cluster disk, all the points have to be shifted to the origin is the center of the cluster, which completes the simulation step.

[0064] To use vectorization in the code, the coordinates of each cluster point are repeated by the number of daughters in the corresponding cluster by using the functions repelem in MATLAB, rep in R, and repeat in Python.

[0065] FIG. 1 shows an illustration of the uplink of a multiuser XL-MIMO system with spatial non-stationarity, whereby each user independently activates different subarrays of the XL-MIMO array depending propagation conditions.

Sytem Model Description

[0066] Consider an XL-MIMO system consisting of S sub-arrays, each equipped with N.sub.s antenna elements, such that the total number of antenna array elements is given by N=.sub.s=1.sup.SN.sub.s, and let GC.sup.NM be the effective channel matrix between the XL-MIMO array and M single-antenna users, which jointly depicts user activities, sub-array VRs, and the fading gains. Then, the corresponding system model as shown in FIG. 1 is given by

[00001]$\begin{array}{cc}Y=\mathrm{GX}+W,& \left(1\right)\end{array}$ [0067] where XC.sup.ML is a pilot matrix collecting the L signals transmitted by each user, while WC.sup.NL denotes zero-mean unit-variance independent and identically-distributed (iid) additive white Gaussian noise (AWGN) such that vec(W)(0, .sup.2I).

[0068] In equation (1), it is assumed that only a small fraction of the M users is active, while the rest remains silent during the time interval of L transmissions. Letting K be a random variable that denotes the number of active users at a given time interval, the average user-activity probability can be expressed as [K/M].

[0069] Furthermore, owing to non-stationarities observed in the XL-MIMO setting, the channel matrix G possesses block-sparsity that captures both user activity and the sub-arrays in their VRs (i.e. active sub-arrays), such that the m-th column of G, relative to the m-th user, can be modeled as

[00002]$\begin{array}{cc}{g}_m=a_m.Math.{\stackrel{}{g}}_m{p}_m,& \left(2\right)\end{array}$ [0070] where denotes the Hadamard (element-wise) product, a.sub.m{0,1} is the user activity indicator, {tilde over (g)}.sub.m is the channel response vector, and p.sub.m[p.sub.1m, . . . , p.sub.Sm] denotes a sub-array activity indicator defined by

[00003]$\begin{array}{cc}{p}_{\mathrm{sm}}\stackrel{}{=}\{\begin{array}{cc}{1}_{N_{s}1}& \mathrm{if}\mathrm{the}s-\mathrm{th}\mathrm{sub}-\mathrm{array}\mathrm{is}\mathrm{in}\mathrm{the}\mathrm{VR}\mathrm{of}\mathrm{the}m-\mathrm{th}\mathrm{user},\\ 0_{N_{s}1}& \mathrm{if}\mathrm{the}s-\mathrm{th}\mathrm{sub}-\mathrm{array}\mathrm{is}\mathrm{outside}\mathrm{the}\mathrm{VR}\mathrm{of}\mathrm{the}m-\mathrm{th}\mathrm{user}.\end{array}& \left(3\right)\end{array}$

[0071] Assuming that {tilde over (g)}.sub.m is Gaussian, as typically, the distribution of g.sub.m can be written as

[00004]$\begin{array}{cc}{g}_m~p_{g_m}\left(g_m\right)\stackrel{}{=}\left(1-\right)\left(g_m\right)+\stackrel{\mathrm{jointing}\mathrm{all}\mathrm{sub}-\mathrm{arrays}}{\stackrel{}{{.Math.}_{s=1}^S\underset{\mathrm{captures}\mathrm{fading}\mathrm{and}\mathrm{sub}-\mathrm{array}\mathrm{activity}}{\underset{}{f\left(g_{\left(s\right)m}.Math.{}_{\mathrm{sm}},0,{}_{\mathrm{sm}}\right)}},}}& \left(4\right)\end{array}$ [0072] where () denotes the Dirac delta function, .sub.sm is the covariance matrix of the m-th user's channel to the s-th sub-array, .sub.sm depicts the mean activity of the s-th sub-array, with respect to the m-th user and

[00005]$\begin{array}{cc}f\left(\right)|,,)\stackrel{}{=}\left(1-\right)\left(z\right)+.Math.\left(,\right).& \left(5\right)\end{array}$ [0073] where is an active probability, denotes a certain mean, and is a given covariance matrix.

Joint Estimation Strategy

[0074] In this section, we propose a novel bilinear message passing method for joint activity and channel estimation in XL-MIMO systems subjected to spatial non-stationarity. To this end, a decomposition of the system model given in equation (1) is carried out, followed by detailed derivations of message passing rules.

[0075] We remark that for the sake of generality, throughout the section it is assumed that the elements of the channel vectors {tilde over (g)}.sub.m are independently but not identically distributed, which is equivalent to saying that the covariance matrices .sub.sm are all diagonal, but have different norms. This is motivated by the fact that the VRs of each user at the XL-MIMO array in general result from the impinging of signals from different propagation paths, as illustrated in FIG. 1.

[0076] For the sake of future convenience, let us first reformulate equation (4) as

[00006]$\begin{array}{cc}Y=\mathrm{HAX}+W,& \left(6\right)\end{array}$ [0077] with GHA, where H is the block fading channel matrix and the MM diagonal matrix A, with diag(A)=[a.sub.1, a.sub.2, . . . , a.sub.M]{0,1}.sup.M, captures user activity indicators.

[0078] For notation simplicity, we hereafter introduce the quantity (s).sub.i=1.sup.sN.sub.i, with (0)0, to denote the cumulative collection of sub-array antenna indices at the s-th sub-array. In order to gain a closer insight into the effect of non-stationarity onto the column vectors of the channel matrix H, consider the anatomized m-th column of H, which is given by

[00007]$\begin{array}{cc}{h}_m=\stackrel{\underset{}{1\mathrm{st}\mathrm{sub}-\mathrm{array}}}{[p_{1m}{\stackrel{\_}{h}}_{\left(1\right)m}^{{}_T}},\stackrel{\underset{}{2\mathrm{nd}\mathrm{sub}-\mathrm{array}}}{p_{2m}{\stackrel{\_}{h}}_{\left(2\right)m}^{{}_T}},.Math.,{\left(\stackrel{\underset{}{S-\mathrm{th}\mathrm{sub}-\mathrm{array}}}{p_{\mathrm{Sm}}{\stackrel{\_}{h}}_{\left(S\right)m}^{{}_T}]}\right)}^T,& \left(7\right)\end{array}$ [0079] where p.sub.ms{0,1}, with s{1,2, . . . , S}, denotes the sub-array activity indicator, the vectors h.sub.(s)m(0, .sub.sm), and (s){(s1)+1, (s1)+2, . . . , (s)} is a set of antenna indices corresponding to the s-th sub-array.

[0080] More conveniently, the m-th column and s-th sub-array of the channel matrix can be modeled as the Bernoulli-Gaussian random variable, that is,

[00008]$\begin{array}{cc}{h}_{\left(s\right)m}{p}_{h_{\left(s\right)m}}\left(h_{\left(s\right)m}\right)\stackrel{}{=}\left(1-{}_{\left(s\right)m}\right)\left(h_{\left(s\right)m}\right)+{}_{\mathrm{sm}}.Math.\left(0,{}_{\mathrm{sm}}\right),& \left(8\right)\end{array}$ [0081] where .sub.sm denotes the mean of p.sub.ms.

[0082] From the above, one can readily notice that the problem of jointly estimating users and subarrays activity indicators, as well as channel coefficients, belongs to the class of bilinear inference problems. More precisely, given the received signal matrix Y and the predetermined reference signal matrix X, our goal is to jointly estimate a.sub.m, p.sub.ms and h.sub.(s)m for all m{1,2, . . . , M} and s{1,2, . . . , S}, which are linearly multiplied by one another. In the next subsections we proceed to derive message passing rules designed to tackle this challenging problem, proposing a new joint activity and channel estimation for XL-MIMO subject to non-stationarity.

[0083] Focusing on the received signal element y.sub.nl at the n-th row and l-th column of Y, the received signal after soft interference cancellation (Soft-IC) using tentative estimates can be written as

[00009]$\begin{array}{cc}=-\stackrel{\underset{}{\mathrm{Soft}-\mathrm{IC}}}{\underset{lm}{\overset{M}{.Math.}},}=& \left(9\right)\end{array}$$h_{\mathrm{nm}}{a}_m+\stackrel{\underset{}{\mathrm{Residual}\mathrm{interference}\mathrm{plus}\mathrm{noise}}}{\underset{lm}{\overset{M}{.Math.}}\left(h_{\mathrm{nl}}{a}_l-\widehat{a_{n,l}}\right)+},$ [0084] where the soft estimates and are generated in variable nodes at the previous iteration, while denotes the noise element at the n-th row and -th column of the AWGN matrix W.

[0085] Assuming that the residual interference plus noise component of equation (9) can be approximated as a complex Gaussian random variable in conformity to the central limit theorem, the conditional probability density function (PDF) of equation (9) for given h.sub.nm can be written as

[00010]$\begin{array}{cc}{|h_{\mathrm{nm}}}_{}\left(|h_{\mathrm{nm}}\right)\exp \left(-\frac{{.Math.{\tilde{y}}_{m,n}-h_{\mathrm{nm}}.Math.}^2}{}\right),& \left(10\right)\end{array}$ [0086] where the error variance is given by

[00011]$\begin{array}{cc}=\left[{.Math.\left(a_m-\right)h_{\mathrm{nm}}+\underset{im}{\overset{M}{.Math.}}\left(h_{\mathrm{ni}}{a}_i-\right)+.Math.}^2\right]={}_{\mathrm{nm}}{.Math..Math.}^2+\underset{\stackrel{}{}}{\underset{im}{\overset{M}{.Math.}}\left({.Math..Math.}^2+\left({.Math..Math.}^2+\right)\right){.Math..Math.}^2+{}^2,}& \left(11\right)\end{array}$ [0087] with .sub.nm denoting the variance of the n-th row and m-th column of H, and where we implicitly defined the residual error variance for future convenience.

[0088] Similarly, the conditional PDF of given a.sub.m can be approximated as

[00012]$\begin{array}{cc}{.Math.a_m}_{}\left(.Math.a_m\right)\exp \left(-\frac{{.Math.-.Math.}^2}{}\right),& \left(12\right)\end{array}$ [0089] with variance given by

[00013]$\begin{array}{cc}{=[|\left(h_{\mathrm{nm}}-\right)a_m{x}_m+\underset{im}{\overset{M}{.Math.}}\left(h_{ni}{a}_i-\right)+.Math.}^2]={.Math..Math.}^2+\underset{im}{\overset{M}{.Math.}}\left({.Math..Math.}^2+\left({.Math..Math.}^2+\right)\right){.Math..Math.}^2+{}^2,& \left(13\right)\end{array}$ [0090] where we utilized the fact that E[a.sup.2.sub.m]=E[a.sub.m]=, since a.sub.m{0, 1}.

Variable Node

[0091] Taking advantage of the Soft-IC mechanism and its resultant statistics shown above, the beliefs corresponding to the s-th sub-array combined over all available time resources except the -th time index, yields the PDF of the extrinsic belief given h.sub.(s)m, which is given by

[00014]$\begin{array}{cc}{p}_{,\left(s\right)m|h_{\left(s\right)m}}\left(|h_{\left(s\right)m}\right)=\underset{n=v\left(s-1\right)+1}{\overset{v\left(s\right)}{.Math.}}\underset{L}{\overset{}{.Math.}}{p}_{{\tilde{y}}_{m,\mathrm{ni}|h_{\mathrm{nm}}}}\left({\tilde{y}}_{m,\mathrm{ni}}|h_{\mathrm{nm}}\right)\underset{n=v\left(s-1\right)+1}{\overset{v\left(s\right)}{.Math.}}\exp \left(-\frac{{.Math.h_{\mathrm{nm}}-.Math.}^2}{}\right)\left(,\right)& \left(14\right)\end{array}$ [0092] with

[00015]$\begin{array}{cc}{,\left(s\right)m}_{}\stackrel{}{=}{\left[{,\left(v\left(s-1\right)+1\right)}_{},.Math.,{\left(s\right)m}_{}\right]}^T,& \left(15a\right)\end{array}$$\begin{array}{cc}\stackrel{}{=}\mathrm{diag}\left(\left[{\left(v\left(s-1\right)+1\right)m}_{},.Math.,\right]\right),& \left(15b\right)\end{array}$ [0093] where

[00016]$\begin{array}{cc}\stackrel{}{=}{\left(\underset{i}{\overset{L}{.Math.}}\frac{|{}_{\mathrm{ni},m}{x}_{\mathrm{mi}}{|}^2}{v_{m,\mathrm{ni}}^h}\right)}^{-1}& \left(16a\right)\end{array}$$\begin{array}{cc}\stackrel{}{=}\underset{i}{\overset{L}{.Math.}}\frac{{{\stackrel{}{y}}_{m,\mathrm{ni}}{}_{ni,m}^{*}{X}_{mi}^{*}}_{}}{v_{m,\mathrm{ni}}^h}& \left(16b\right)\end{array}$

[0094] In turn, the PDF of the extrinsic belief .sup.a.sub.nl,m given a.sub.m can be similarly obtained as

[00017]$\begin{array}{cc}{|a_m}_{}\left(|a_m\right)=\underset{j=n}{\overset{N}{.Math.}}\underset{L}{\overset{}{.Math.}}{p}_{{\tilde{y}}_{m,\mathrm{ji}|a_m}}\left({\tilde{y}}_{m,\mathrm{ji}}|a_{\mathrm{nm}}\right)\exp \left(-\frac{{.Math.a_m-.Math.}^2}{}\right),\left(,\right),& \left(17\right)\end{array}$ [0095] where

[00018]$\begin{array}{cc}\stackrel{}{=}{\left(\underset{jn}{\overset{N}{.Math.}}\underset{i}{\overset{L}{.Math.}}\frac{{.Math.{\hat{h}}_{i,\mathrm{jm}}{x}_{\mathrm{mi}}.Math.}^2}{v_{m,\mathrm{ji}}^a}\right)}^{-1}& \left(18a\right)\end{array}$$\begin{array}{cc}\stackrel{}{=}\underset{jn}{\overset{N}{.Math.}}\underset{L}{\overset{}{.Math.}}\frac{{\stackrel{}{y}}_{m,\mathrm{ji}}{\stackrel{}{h}}_{i,\mathrm{jm}}^{*}{x}_{mi}^{*}}{v_{m,\mathrm{ji}}^a}& \left(18b\right)\end{array}$

[0096] Combining the PDF in equation (14) with the prior channel PDF in equation (8) yields the posterior distribution of the channel. Therefore, taking the expectation of h.sub.(s)m over the latter yields the corresponding soft estimate .sub.(s)m at the -th variable node, which is given by

[00019]$\begin{array}{cc}{\hat{h}}_{,\left(s\right)m}=& \left(19\right)\end{array}$${}_{h_{\left(s\right)m}}{h}_{\left(s\right)m}\frac{{,\left(s\right)m|h_{\left(s\right)m}}_{}\left({,\left(s\right)m}_{}|h_{\left(s\right)m}\right)p_{h_{\left(s\right)m}}\left(h_{\left(s\right)m}\right)}{{}_{h_{\left(s\right)m}^{}}{,\left(s\right)m|h_{\left(s\right)m}}_{}\left({,\left(s\right)m}_{}|h_{\left(s\right)m}^{}\right)p_{h_{\left(s\right)m}^{}}\left(h_{\left(s\right)m}^{}\right)},$ [0097] where the denominator in the integrand is introduced for normalization purposes.

[0098] The error covariance associated with .sub.(s)m is given by

[00020]$\begin{array}{cc}\stackrel{}{=}\mathrm{diag}\left({}_{h_{\left(s\right)m}}{h}_{\left(s\right)m}{h}_{\left(s\right)m}^H\frac{{\left(s\right)m|h_{\left(s\right)m}}_{}\left({,\left(s\right)m}_{}|h_{\left(s\right)m}\right)p_{h_{\left(s\right)m}}\left(h_{\left(s\right)m}\right)}{{}_{h_{\left(s\right)m}^{}}{p}_{,\left(s\right)m|h_{\left(s\right)m}^{}}\left({}_{,\left(s\right)m}|h_{\left(s\right)m}^{}\right)p_{h_{\left(s\right)m}^{}\left(h_{\left(s\right)m}^{}\right)}^{}}\right)-\mathrm{diag}\left({,\left(s\right)m}_{}{,\left(s\right)m}_{}\right),\mathrm{such}\mathrm{that}=\mathrm{diag}\left({}_{,(v\left(s-1\right)+m}^h,{}_{,\left(v\left(s-1\right)+2\right)m}^h,.Math.,{,v\left(s\right)m}_{}\right).& \left(20\right)\end{array}$

[0099] In turn, the soft replica of the user activity indicator can be similarly obtained as

[00021]$\begin{array}{cc}{\hat{a}}_{n,m}=\underset{\left\{0,1\right\}}{.Math.}.Math.\frac{p_{{}_{n,m}|{}_m}\left({}_{n,m}.Math.\right)p_{a_m}\left(\right)}{{.Math.}_{a^{}\left\{0,1\right\}}{p}_{{}_{n,m|{}_m}}\left({}_{n,m}|{}^{}\right)p_{a_m}\left({}^{}\right)},& \left(21\right)\end{array}$ [0100] with p.sub.a.sub.m() denoting the Bernoulli probability mass function (PMF) with intensity , which is accompanied by its mean square error (MSE) given by

[00022]$\begin{array}{cc}{}_{n,m}^a=\underset{\left\{0,1\right\}}{.Math.}{}^2.Math.\frac{p_{{}_{n,m}|{}_m}\left({}_{n,m}.Math.\right)p_{a_m}\left(\right)}{{.Math.}_{a^{}\left\{0,1\right\}}{p}_{{}_{n,m}|{}_m}\left({}_{n,m}|{}^{}\right)p_{a_m}\left({}^{}\right)}-{\hat{a}}_{n,m}^2.& \left(22\right)\end{array}$

[0101] In order to compute equations (19)-(22) above, it is essential to analyze the effective distributions .sub.|h.sub.(s)m(|h.sub.(s)m)p.sub.h.sub.(s)m(h.sub.(s)m) and (|)p.sub.a.sub.m() such that the resultant calculations become tractable. To that end, plugging equations (8) and (14) into the effective distribution of h.sub.(s)m yields

[00023]$\begin{array}{cc}{p}_{{}_{,\left(s\right)m}|h_{\left(s\right)m}}\left({}_{,\left(s\right)m}.Math.h_{\left(s\right)m}\right)p_{h_{\left(s\right)m}}\left(h_{\left(s\right)m}\right)=\left(\left(1-{}_{sm}\right)\left(h_{\left(s\right)m}\right)+{}_{sm}.Math.\left(0,{}_{\mathrm{sm}}\right)\right)\left({}_{,\left(s\right)m}^h,{}_{,\left(s\right)m}^h\right)=\left(1-{}_{sm}\right)\left(h_{\left(s\right)m}\right)\frac{\exp \left(-{}_{,\left(s\right)m}^h,{}_{,\left(s\right)m}^h{}_{,\left(s\right)m}^h\right)}{{}^{N_s.Math.{}_{,\left(s\right)m}^h.Math.}}.Math.+{}_{\mathrm{sm}}\frac{\exp \left(-{{}_{,\left(s\right)m}^h\left({}_{\mathrm{sm}}+{}_{,\left(s\right)m}^h\right)}^{-1}{}_{,\left(s\right)m}^h\right)}{{}^{N_s.Math.{}_{\mathrm{sm}}+{}_{,\left(s\right)m}^h.Math.}}\underset{\stackrel{}{=}f\left(h_{\left(s\right)m}.Math.{}_{,\left(s\right)m}^h,{}_{,\left(s\right)m}^h,{}_{\mathrm{sm}}\right)}{\underset{}{\left({{}_{\mathrm{sm}}\left({}_{\mathrm{sm}}+{}_{,\left(s\right)m}^h\right)}^{-1}{}_{,\left(s\right)m}^h,{{}_{\mathrm{sm}}\left({}_{\mathrm{sm}}+{}_{,\left(s\right)m}^h\right)}^{-1}{}_{,\left(s\right)m}^h\right)}}& \left(23\right)\end{array}$

[0102] Thus, the corresponding normalization factor can be written as

[00024]$\begin{array}{cc}{}_{h_{\left(s\right)m}}{p}_{{}_{,\left(s\right)m}|h_{\left(s\right)m}}\left({}_{,\left(s\right)m}.Math.h_{\left(s\right)m}\right)p_{h_{\left(s\right)m}}\left(h_{\left(s\right)m}\right)={}_{h_{\left(s\right)m}}\left(1-{}_{sm}\right)\left(h_{\left(s\right)m}\right)\frac{\exp \left(-{}_{,\left(s\right)m}^{h}H{}_{,\left(s\right)m}^h{}_{,\left(s\right)m}^h\right)}{{}^{N_s.Math.{}_{,\left(s\right)m}^h.Math.}}+{}_{\mathrm{sm}}\frac{\exp \left(-{{}_{,\left(s\right)m}^h\left({}_{\mathrm{sm}}+{}_{,\left(s\right)m}^h\right)}^{-1}{}_{,\left(s\right)m}^h\right)}{{}^{N_s\left|{}_{\mathrm{sm}}+{}_{,\left(s\right)m}^h\right|}}& \left(24\right)\end{array}$$\stackrel{=1}{\stackrel{}{{}_{h_{\left(s\right)m}}f\left(h_{\left(s\right)m}.Math.{}_{,\left(s\right)m}^h,{}_{,\left(s\right)m}^h,{}_{\mathrm{sm}}\right)}}={}_{\mathrm{sm}}\frac{\exp \left(-{}_{,\left(s\right)m}^h{H\left({}_{\mathrm{sm}}+{}_{,\left(s\right)m}^h\right)}^{-1}{}_{,\left(s\right)m}^h\right)}{{}^{N_s}.Math.{}_{\mathrm{sm}}+{}_{,\left(s\right)m}^h.Math.}{}_{,\mathrm{ms}},$ [0103] where the activity detection factor is given by

[00025]$\begin{array}{cc}{}_{,\mathrm{ms}}\stackrel{}{=}1+\frac{1-{}_{sm}}{{}_{sm}}\exp \left(-{}_{,\left(s\right)m}^{h^H}\left({}_{,\left(s\right)m}^{h^{-1}}-{\left({}_{\mathrm{sm}}+{}_{,\left(s\right)m}^h\right)}^{-1}\right){}_{,\left(s\right)m}^h+\log \left(.Math.{}_{,\left(s\right)m}^h{}_{\mathrm{sm}}+I_{N_s}.Math.\right)\right).& \left(25\right)\end{array}$

[0104] Taking advantage of equations (23) and (24), the soft replica of h.sub.(s)m at the -th node can be re-written as

[00026]$\begin{array}{cc}{\stackrel{\_}{h}}_{,\left(s\right)m}=\frac{{{}_{\mathrm{sm}}\left({}_{\mathrm{sm}}+{}_{,\left(s\right)m}^h\right)}^{-1}}{{}_{,\mathrm{ms}}}{}_{,\left(s\right)m}^h,& \left(26\right)\end{array}$ [0105] while its MSE diag (, . . . , ) can be expressed as

[00027]$\begin{array}{cc}{\stackrel{\_}{}}_{,\left(s\right)m}^h=\left({}_{,\mathrm{ms}}-1\right)\mathrm{diag}\left({\hat{h}}_{,\left(s\right)m}{\hat{h}}_{,\left(s\right)m}^H\right)+\frac{{}_{,\left(s\right)m}^h{{}_{\mathrm{sm}}\left({}_{\mathrm{sm}}+{}_{,\left(s\right)m}^h\right)}^{-1}}{{}_{,\mathrm{ms}}}.& \left(27\right)\end{array}$

[0106] In turn, the effective distribution (|)p.sub.a.sub.m() can be simplified to

[00028]$\begin{array}{cc}{p}_{{}_{n,m}|{}_m}\left({}_{n,m}.Math.\right)p_{a_m}\left(\right)=\exp \left(-\frac{{.Math.-{}_{n,m}^a.Math.}^2}{{}_{n,m}^a}\right)\stackrel{\mathrm{Bemoulli}\mathrm{PDF}\mathrm{with}\mathrm{intensity}}{\stackrel{}{\left(+\left(1-\right)\left(1-\right)\right),}}& \left(28\right)\end{array}$ [0107] where {0,1} and the associated normalization factor can then be written as

[00029]$\begin{array}{cc}\underset{\left\{0,1\right\}}{.Math.}{p}_{{}_{n,m}|{}_m}\left({}_{n,m}.Math.\right)p_{a_m}\left(\right)=\exp \left(-\frac{{.Math.1-{}_{n,m}^a.Math.}^2}{{}_{n,m}^a}\right)+\left(1-\right)\exp \left(-\frac{{.Math.{}_{n,m}^a.Math.}^2}{{}_{n,m}^a}\right)& \left(29\right)\end{array}$

[0108] Thus, the soft replica of .sub.m can be obtained as

[00030]$\begin{array}{cc}{\hat{a}}_{n,m}=.Math..Math.\frac{p_{{}_{n,m}|{}_m}\left({}_{n,m}|\right)p_{a_m}\left(\right)}{{.Math.}_{{}^{}\left\{0,1\right\}}{p}_{{}_{n,m}|{}_m}\left({}_{n,m}|{}^{}\right)p_{a_m}\left({}^{}\right)}=\frac{\exp \left(-\frac{{.Math.1-{}_{n,m}^a.Math.}^2}{{}_{n,m}^a}\right)}{\exp \left(-\frac{{.Math.1-{}_{n,m}^a.Math.}^2}{{}_{n,m}^a}\right)+\left(1-\right)\exp \left(-\frac{{.Math.{}_{n,m}^a.Math.}^2}{{}_{n,m}^a}\right)},=\stackrel{\mathrm{Weighted}\mathrm{Sigmoid}\mathrm{Function}}{\stackrel{}{\frac{1}{1+\frac{1-}{}\exp \left(-\left(\frac{{.Math.{}_{n,m}^a.Math.}^2}{{}_{n,m}^a}-\frac{{.Math.1-{}_{n,m}^a.Math.}^2}{{}_{n,m}^a}\right)\right)}}}& \left(30\right)\end{array}$ [0109] and its error variance can be written as

[00031]$\begin{array}{cc}{}_{n,m}^a=\underset{\left\{0,1\right\}}{.Math.}{}^2.Math.\frac{p_{{}_{n,m}|{}_m}\left({}_{n,m}.Math.\right)p_{a_m}\left(\right)}{{.Math.}_{{}^{}\left\{0,1\right\}}{p}_{{}_{n,m}|{}_m}\left({}_{n,m}.Math.{}^{}\right)p_{a_m}\left({}^{}\right)}-{\hat{a}}_{n,m}=\frac{\frac{1-}{}\exp \left(-\left(\frac{{.Math.{}_{n,m}^a.Math.}^2}{{}_{n,m}^a}-\frac{{.Math.1-{}_{n,m}^a.Math.}^2}{{}_{n,m}^a}\right)\right)}{{\left(1+\frac{1-}{}\exp \left(-\left(\frac{{.Math.{}_{n,m}^a.Math.}^2}{{}_{n,m}^a}-\frac{{.Math.1-{}_{n,m}^a.Math.}^2}{{}_{n,m}^a}\right)\right)\right)}^2},& \left(31\right)\end{array}$ [0110] from which it is readily found that 0.

Activity Detection Policy

[0111] Finally, in this subsection the last step is discussed, corresponding to the refinement of the user activity estimates according to a pre-defined user activity policy. Different approaches to perform this step can be considered. Here, the new log-likelihood ratio (LLR)-based approach is discussed taking both H{circumflex over ()} and A{circumflex over ()} into consideration.

[0112] To that end, first recognize that the user activity pattern captured by the binary quantities on the diagonal elements of A{circumflex over ()} can equivalently be expressed as column-sparsity of H{circumflex over ()}, which implies that an activity detection policy must jointly consider H{circumflex over ()} and A{circumflex over ()} for AUD. Denoting the estimated eftive channel matrix G{circumflex over ()}H{circumflex over ()}A{circumflex over ()}, the element-wise LLR can be written as

[00032]$\begin{array}{cc}{}_{\mathrm{nm}}=\log \frac{\left(0,{}_{\mathrm{nm}}+{}_{\mathrm{nm}}^h.Math.{\hat{g}}_{\mathrm{nm}}\right)}{\left(0,{}_{\mathrm{nm}}^h.Math.{\hat{g}}_{\mathrm{nm}}\right)}.& \left(39\right)\end{array}$

[0113] From the above, the user activity can be detected by combining the LLRs .sub.nm for all the receive antenna dimensions N. However, such a detection policy ignores the presence of the block-wise sparsity due to spatial non-stationarity, leading to detection performance degradation. To address this issue, the following sub-array activity aware AUD policy is considered.

[00033]$\begin{array}{cc}{\hat{a}}_m=\{\begin{array}{cc}1& \mathrm{if}\underset{s}{\max }\left({.Math.}_{n\left(s\right)}{}_{\mathrm{nm}}\right)0,\\ 0& \mathrm{otherwise}.\end{array}& \left(40\right)\end{array}$

Method 1 Bilinear Message Passing for JACE in XL-MIMO with Non-Stationarity

[0114] Inputs: Y and X, and initializers and {circumflex over ()}.sup.h, [0115] Outputs: and [0116] Internal Parameters: .sub.sm, {circumflex over ()}.sub.sm, .sup.2, , and t.sub.max

[00034]$\mathrm{For}\mathrm{all}\left(n,m,\right)$$1:{\hat{a}}_{n,m}\left(1\right)0,{}_{n,m}^a\left(1\right)0$$2:{\hat{h}}_{,\mathrm{nm}}\left(1\right){\left[\stackrel{}{H}\right]}_{\mathrm{nm}},{}_{,\mathrm{nm}}^h\left(1\right){\left[{\stackrel{}{}}^h\right]}_{\mathrm{nm}}$$\mathrm{for}t=1,.Math.,t_{\max }$$\mathrm{For}\mathrm{all}\left(n,m,\right)$$\begin{array}{cc}3:{\stackrel{}{y}}_{m,n}{y}_n-{.Math.}_{im}^M{\hat{h}}_{,\mathrm{ni}}{\hat{a}}_{n,i}{x}_i& \mathrm{Eq}.\left(9\right)\end{array}$$\begin{array}{cc}4:v_{m,n}^y{.Math.}_{im}^M\left({.Math.{\stackrel{}{h}}_{,\mathrm{ni}}.Math.}^2{}_{n,i}^a+\left({.Math.{\hat{a}}_{n,i}.Math.}^2+{}_{n,i}^a\right){}_{,\mathrm{ni}}^h\right){.Math.x_i.Math.}^2+{}^2& \mathrm{Eq}.\left(11\right)\end{array}$$5:v_{m,n}^h{}_{n,m}^a{}_{\mathrm{nm}}{.Math.x_m.Math.}^2+v_{m,n}^y$$\begin{array}{cc}6:v_{m,n}^a{}_{,\mathrm{nm}}^h{.Math.x_m.Math.}^2+v_{m,n}^y& \mathrm{Eq}.\left(13\right)\end{array}$$\begin{array}{cc}7:{}_{,\mathrm{nm}}^h{\left({.Math.}_i^L\frac{|{\hat{a}}_{\mathrm{ni},m}{x}_{\mathrm{mi}}{|}^2}{v_{m,\mathrm{ni}}^h}\right)}^{-1}\mathrm{and}{}_{,\mathrm{nm}}^h{}_{,\mathrm{nm}}^h{.Math.}_i^L\frac{{\tilde{y}}_{m,\mathrm{ni}}{\stackrel{}{a}}_{\mathrm{ni},m}^{*}{x}_{\mathrm{mi}}^{*}}{v_{m,\mathrm{ni}}^h}& \mathrm{Eq}.\left(16\right)\end{array}$$\begin{array}{cc}8:{}_{n,m}^a{\left({.Math.}_{jn}^N{.Math.}_i^L\frac{{.Math.{\hat{h}}_{i,\mathrm{jm}}{x}_{mi}.Math.}^2}{v_{m,\mathrm{ji}}^a}\right)}^{-1}\mathrm{and}{}_{n,m}^a{}_{n,m}^a{.Math.}_{jn}^N{.Math.}_i^L\frac{{\tilde{y}}_{m,\mathrm{ji}}{h}_{i,\mathrm{jm}}^{*}{x}_{\mathrm{mi}}^{*}}{v_{m,\mathrm{ji}}^a}& \mathrm{Eq}.\left(18\right)\end{array}$$\begin{array}{cc}9:{\hat{a}}_{n,m}\left(1+\frac{1-}{}\exp \left(-\left(\frac{{.Math.{}_{n,m}^a.Math.}^2}{{}_{n,m}^a}-\frac{{.Math.1-{}_{n,m}^a.Math.}^2}{{}_{n,m}^a}\right)\right)\right)& \mathrm{Eq}.\left(21\right)\end{array}$$\begin{array}{cc}10:{}_{n,m}^a\frac{1-}{}\exp \left(-\left(\frac{{.Math.{}_{n,m}^a.Math.}^2}{{}_{n,m}^a}-\frac{{.Math.1-{}_{n,m}^a.Math.}^2}{{}_{n,m}^a}\right)\right){\left(1+\frac{1-}{}\exp \left(-\left(\frac{{.Math.{}_{n,m}^a.Math.}^2}{{}_{n,m}^a}-\frac{{.Math.1-{}_{n,m}^a.Math.}^2}{{}_{n,m}^a}\right)\right)\right)}^{-2}& \mathrm{Eq}.\left(22\right)\end{array}$$\mathrm{For}\mathrm{all}\left(m,s,\right)$$\begin{array}{cc}11:{}_{,\left(s\right)m}^h{\left[{}_{,\left(v\left(s-1\right)+1\right)m}^h,.Math.,{}_{,v\left(s\right)m}^h\right]}^T& \mathrm{Eq}.\left(15\right)\end{array}$$12:{}_{,\left(s\right)m}^h\mathrm{diag}\left[{}_{,\left(v\left(s-1\right)+1\right)m}^h,.Math.,{}_{,v\left(s\right)m}^h\right])$$\begin{array}{cc}13:{}_{,\mathrm{ms}}1+\frac{\left(1-{\stackrel{}{}}_{\mathrm{sm}}\right)}{{\stackrel{}{}}_{\mathrm{sm}}}\frac{.Math.{}_{\mathrm{sm}}+{}^h,\left(s\right)m.Math.}{.Math.{}^h,\left(s\right)m.Math.}\exp \left(-{}_{,\left(s\right)m}^h\left({}_{,\left(s\right)m}^{h-1}-{\left({}_{\mathrm{sm}}+{}_{,\left(s\right)m}^h\right)}^{-1}\right){}_{,\left(s\right)m}^h\right)& \mathrm{Eq}.\left(25\right)\end{array}$$\begin{array}{cc}14:{\stackrel{}{}}_{\mathrm{ms}}=\stackrel{}{}\frac{1}{\mathrm{MSL}}{.Math.}_{=1}^L{.Math.}_{m=1}^M{.Math.}_{s=1}^S\frac{1}{{}_{,\mathrm{ms}}}& \mathrm{Eq}.\left(32\right)\end{array}$$\begin{array}{cc}15:{\stackrel{}{h}}_{,\left(s\right)m}\frac{{{}_{\mathrm{sm}}\left({}_{\mathrm{sm}}+{}_{,\left(s\right)m}^h\right)}^{-1}}{{}_{,\mathrm{ms}}}{}_{,\left(s\right)m}^h& \mathrm{Eq}.\left(26\right)\end{array}$$\begin{array}{cc}16:{\stackrel{}{}}_{,\left(s\right)m}^h\left({}_{,\mathrm{ms}}-1\right)\mathrm{diag}\left({\stackrel{}{h}}_{,\left(s\right)m}{\stackrel{}{h}}_{,\left(s\right)m}^H\right)+\frac{{}_{,\left(s\right)m}^h{{}_{\mathrm{sm}}\left({}_{\mathrm{sm}}+{}_{,\left(s\right)m}^h\right)}^{-1}}{{}_{,\mathrm{ms}}}& \mathrm{Eq}.\left(27\right)\end{array}$$17:\mathrm{Damping}\left[30\right]:{\stackrel{}{h}}_{,\left(s\right)m}{\stackrel{}{h}}_{,\left(s\right)m}+\left(1-\right){\stackrel{}{h}}_{,\left(s\right)m}$$18:{\stackrel{}{}}_{,\left(s\right)m}^h{\stackrel{}{}}_{,\left(s\right)m}^h+\left(1-\right){\stackrel{}{}}_{,\left(s\right)m}^h$$\mathrm{end}\mathrm{for}$$19:\mathrm{For}\mathrm{all}\left(n,m,\right),\mathrm{update}{v}_{m,n}^h\mathrm{and}{v}_{m,n}^a\mathrm{by}\mathrm{repeating}\mathrm{lines}3\mathrm{to}6$$\begin{array}{cc}20:\mathrm{For}\mathrm{all}\left(n,m\right),{}_{\mathrm{nm}}^h{\left({.Math.}_{i=1}^L\frac{{.Math.{\stackrel{}{a}}_{\mathrm{ni},m}{x}_{mi}.Math.}^2}{v_{m,\mathrm{ni}}^h}\right)}^{-1}\mathrm{and}{}_{\mathrm{nm}}^h{}_{\mathrm{nm}}^h{.Math.}_{i=1}^L\frac{{\tilde{y}}_{m,\mathrm{ni}}{\stackrel{}{a}}_{\mathrm{ni},m}^{*}{x}_{\mathrm{mi}}^{*}}{v_{m,\mathrm{ni}}^h}& \mathrm{Eq}.\left(16\right)\end{array}$$\begin{array}{cc}21:\mathrm{For}\mathrm{all}\left(m,s\right),{\stackrel{}{h}}_{\left(s\right)m}\frac{{{}_{\mathrm{sm}}\left({}_{\mathrm{sm}}+{}_{\left(s\right)m}^h\right)}^{-1}}{{}_{\mathrm{ms}}}{}_{\left(s\right)m}^h.Math.\stackrel{}{H}& \mathrm{Eq}.\left(26\right)\end{array}$$\mathrm{For}\mathrm{all}m$$22:\mathrm{Refresh}{}_m^a{\left({.Math.}_{j=1}^N{.Math.}_{i=1}^L\frac{{.Math.{\hat{h}}_{i,\mathrm{jm}}{x}_{\mathrm{mi}}.Math.}^2}{v_{m,\mathrm{ji}}^a}\right)}^{-1}\mathrm{and}{}_m^a{}_m^a{.Math.}_{j=1}^N{.Math.}_{i=1}^L\frac{{\tilde{y}}_{m,\mathrm{ji}}{\stackrel{}{h}}_{i,\mathrm{jm}}^{*}{x}_{\mathrm{mi}}^{*}}{v_{m,\mathrm{ji}}^a}$$23:\mathrm{refresh}{\hat{a}}_m{\left(1+\frac{1-}{}\exp \left(-\left(\frac{{.Math.{}_m^a.Math.}^2}{{}_m^a}-\frac{{.Math.1-{}_m^a.Math.}^2}{{}_m^a}\right)\right)\right)}^{-1}$$\begin{array}{cc}24:\mathrm{make}\mathrm{hard}\mathrm{decision}\mathrm{on}{\hat{a}}_m.Math.\hat{A}.& \mathrm{Eq}.\left(40\right)\end{array}$