Variational Bayes Inference for Data Detection in Cell-Free Massive MIMO

At this level, the APs do not process their received signals. Instead, the received signals are forwarded to the CPU for fully centralized processing, including the data detection task. The signals forwarded from the $L$ APs can be stacked into

where $\mathbf{y}=[\mathbf{y}_{1}^{T},\ldots,\mathbf{y}_{L}^{T}]^{T}$, $\mathbf{H}=[\mathbf{H}_{1}^{T},\ldots,\mathbf{H}_{L}^{T}]^{T}$, and $\mathbf{n}=[\mathbf{n}_{1}^{T},\ldots,\mathbf{n}_{L}^{T}]^{T}$. The processing for cell-free massive MIMO in this level is similar to the processing at a conventional co-located MIMO receiver. The CPU detects $\mathbf{x}=[x_{1},\ldots,x_{K}]^{T}$ using the received signal vector $\mathbf{y}$ and the channel matrix $\mathbf{H}$. Among the linear detectors, the LMMSE detector maximizes the signal-to-interference-and-noise ratio (SINR) and also achieves the best detection performance [4]. With the full knowledge of $\mathbf{H}$, the LMMSE estimate $\hat{\mathbf{x}}$ is formed as

which is then element-wise projected onto $\mathcal{S}$. We note that the LMMSE filter in the presented form requires the inverse of a $K\times K$-dimensional matrix.

At this level, each AP pre-processes its received signal by computing a local estimate of $\mathbf{x}$ that are forwarded to the CPU for final decoding [4]. Assuming full knowledge of channel matrix $\mathbf{H}_{\ell}$ at the $\ell$-th AP, the local LMMSE estimate $\check{\mathbf{x}}_{\ell}=[\check{x}_{i\ell},\ldots,\check{x}_{K\ell}]^{T}$ of $\mathbf{x}$ can be found as

We note that the LMMSE filter in this presented form requires the inverse of a $N\times N$-dimensional matrix. The CPU then can linearly combine the local estimates $\{\check{x}_{i\ell}\,:\,\ell=1,\ldots,L\}$ to obtain the estimate

which is eventually used to decode $x_{i}$. Here, the weighting coefficient vector $\mathbf{a}_{i}=[a_{i1},\ldots,a_{iL}]^{T}$ relies only on channel statistics and can be optimized by the CPU. This combining method is also known as the large-scale fading decoding (LSFB) strategy in the context of cellular massive MIMO. We note that no instantaneous CSI of any channel is required at the CPU.

At this level, the CPU forms an estimate of $x_{i}$ by simply taking the average of the local estimates [4]. This yield an estimate $\hat{x}_{i}$ as

We note that no statistical parameters of CSI are needed at the CPU at this level of centralized signal processing.

At this level, each user signal is decoded by only one AP that gives the highest spectral efficiency to the user, i.e., the highest SINR [4]. LMMSE filtering can be applied to obtain the local estimate of the user signal. Since only one estimate per user is forwarded to the CPU, no centralizing decoding is required.

We first present the background on VB for approximate inference that will be exploited for solving the data detection in cell-free systems. VB inference is a powerful framework from machine learning that approximates intractable posterior distributions of latent variables with a known family of simpler distributions through optimization. The goal of VB inference is to find an approximation for a computationally intractable posterior distribution $p(\mathbf{x}|\mathbf{y})$ given a probabilistic model that specifies the joint distribution $p(\mathbf{x},\mathbf{y})$, where $\mathbf{y}$ represents the set of all observed variables and $\mathbf{x}$ is a set of $m$ latent variables and parameters. The VB inference method aims at finding a density function $q(\mathbf{x})$ with its own setting of variational parameters within a family $\mathcal{Q}$ of density functions that makes $q(\mathbf{x})$ close to the posterior distribution of interest $p(\mathbf{x}|\mathbf{y})$. VB inference amounts to solving the following optimization problem:

where $\mathrm{KL}\big{(}q(\mathbf{x})\|p(\mathbf{x}|\mathbf{y})$ is the Kullback-Leibler (KL) divergence from $q(\mathbf{x})$ to $p(\mathbf{x}|\mathbf{y})$. Minimizing the KL divergence is equivalent to maximizing the evidence lower bound ($\mathrm{ELBO}$) [7], which is defined as

The maximum of $\mathrm{ELBO}(q)$ occurs when $q(\mathbf{x})=p(\mathbf{x}|\mathbf{y})$. Since working with the true posterior distribution is often intractable, it is more convenient to consider a restricted family of distributions $q(\mathbf{x})$. Among VB inference methods, the mean-field approximation enables efficient optimization of the variational distribution over a partition of the latent variables, while keeping the variational distributions over other partitions fixed [7]. The mean-field variational family is constructed such that

where the latent variables are mutually independent and each is governed by a distinct factor in the variational density. Among all mean-field distributions $q(\mathbf{x})$, the general expression for the optimal solution of the variational density $q_{i}(x_{i})$ that maximizes the ELBO can be obtained as [7]

where $\langle\cdot\rangle$ denotes the expectation with respect to all latent variables except $x_{i}$ using the currently fixed variational density $q_{-i}(\mathbf{x}_{-i})=\prod_{j\neq i}q_{j}(x_{j})$. By iterating the update of $q_{i}(x_{i})$ sequentially over all $j$, the $\mathrm{ELBO}(q)$ objective function can be monotonically improved. This is the basis behind the coordinate ascent variational inference algorithm, which guarantees convergence to at least a local optimum of $\mathrm{ELBO}(q)$ [7, 8]. To this send, we examine how the mean-field VB framework can be exploited for data detection at different levels of cooperation in a cell-free system.

At this level, the signals forwarded from the APs can be stacked into a single large-scale MIMO system as being shown in (3). In a recent work [9], we developed several VB-based methods for MIMO data detection. Among them, the LMMSE-VB algorithm showed superior performance in MIMO systems with non-i.i.d. channels. Certainly, the algorithm can be adopted for data detection in cell-free systems with fully centralized processing. In the following, we present key operations in the algorithm. For details of the algorithm, we refer the readers to [9].

The LMMSE-VB algorithm floats the background noise covariance matrix as an unknown random variable, instead of treating the noise’s variance $N_{0}$ as known. The postulated noise covariance matrix $\mathbf{C}^{\mathrm{post}}$ is estimated by the algorithm itself. For ease of computation, we use $\mathbf{W}=(\mathbf{C}^{\mathrm{post}})^{-1}$ to denote the precision matrix and assume a conjugate prior complex Wishart distribution $\mathcal{CW}(\mathbf{W}_{0},n)$ for $\mathbf{W}$, where $\mathbf{W}_{0}\succeq\mathbf{0}$ is the scale matrix and $n\geq NL$ indicates the degrees of freedom. The PDF of $\mathbf{W}\sim\mathcal{CW}(\mathbf{W}_{0},n)$ satisfies

The joint distribution $p(\mathbf{y},\mathbf{x},\mathbf{W};\mathbf{H})$ can be factored as

where $p(\mathbf{y}|\mathbf{x},\mathbf{W};\mathbf{H})=\mathcal{CN}(\mathbf{y};\mathbf{Hx},\mathbf{W}^{-1})$. Given the observation $\mathbf{y}$, we aim at obtaining the mean-field variational distribution $q(\mathbf{x},\mathbf{W})$ such that

The optimization of $q(\mathbf{x},\mathbf{W})$ is executed by iteratively updating $\{x_{i}\}$ and $\mathbf{W}$ as follows.

a) Updating $x_{i}$. The variational distribution $q_{i}(x_{i})$ is obtained by expanding the conditional in (13) and taking the expectation with respect to all latent variables except $x_{i}$ using the variational distribution $\prod_{j\neq i}^{K}q_{j}(x_{j})q(\mathbf{W})$:

where $z_{i}$ is a linear estimate of $x_{i}$ that is defined as

It is observed in (15) that $\mathcal{CN}\big{(}z_{i};x_{i},\hat{\sigma}_{i}^{2}\big{)}$ with $\hat{\sigma}_{i}^{2}=1/\big{(}\mathbf{h}_{i}^{H}\langle\mathbf{W}\rangle\mathbf{h}_{i}\big{)}$ can be interpreted as the likelihood function $p\big{(}z_{i}|x_{i};\hat{\sigma}_{i}^{2}\big{)}$. In other words, the mean-field VB approximation decouples the linear MIMO system into $K$ parallel AWGN channels $z_{i}=x_{i}+\mathcal{CN}\big{(}0,\hat{\sigma}_{i}^{2}\big{)}$.

The variational distribution $q_{i}(x_{i})$ is realized by normalizing $p(x_{i})\,\mathcal{CN}\big{(}z_{i};x_{i},\hat{\sigma}_{i}^{2}\big{)}$. The variational mean $\langle x_{i}\rangle=\mathbb{E}[x_{i}|z_{i}]$ and variance $\sigma_{x_{i}}^{2}$ are then computed accordingly.

b) Updating $\mathbf{W}$. The variational distribution $q(\mathbf{W})$ is obtained by taking the expectation of the conditional in (13) with respect to $q(\mathbf{x})$:

The variational distribution $q(\mathbf{W})$ is also complex Wishart with $n+1$ degrees of freedom [9]. The variational mean $\langle\mathbf{W}\rangle$ can be computed accordingly. In [9], we also proposed to use the estimator

where $\boldsymbol{\Sigma}_{\mathbf{x}}=\mathrm{diag}(\sigma_{x_{1}}^{2},\ldots,\sigma_{x_{K}}^{2})$.

By iteratively optimizing $\big{\{}q_{i}(x_{i})\big{\}}$ and $q(\mathbf{W})$ via the updates of $\{\langle x_{i}\rangle\}$ and $\langle\mathbf{W}\rangle$, we obtain the CAVI algorithm for estimating $\mathbf{x}$ and the precision matrix $\mathbf{W}$. We refer to this scheme as the LMMSE-VB algorithm since $z_{i}$ resembles an LMMSE estimate of $x_{i}$ due to the cancellation of the inter-user interference and the whitening with the postulated noise covariance matrix $\mathbf{C}^{\mathrm{post}}$.

At this level, our proposed VB-based method involves two operations: 1) Executing the LMMSE-VB algorithm independently at each AP to compute local estimates of $\mathbf{x}_{t}$ and 2) Aggregating the local estimates at the CPU for joint nonlinear decoding of $\mathbf{x}_{t}$. However, we make a minor modification to the LMMSE-VB algorithm which allow it to operate over the whole block of $T$ time slots.

At this level, only local estimates are fed back to the CPU. The LMMSE-VB mentioned in Level 3 signal processing can be used to generate the coarse local estimates. However, the local nonlinear MMSE estimates $\check{x}_{i\ell,t}$ is sent, instead of the LMMSE estimate $z_{i\ell,t}$ and the variance $\check{\sigma}_{i\ell}^{2}$. We note that $\check{x}_{i\ell,t}$ can be computed using $z_{i\ell,t}$ and $\check{\sigma}_{i\ell}^{2}$, but not the reverse.

A simple estimate of $x_{i,t}$ can be obtained by simply taking the average of all the estimates $\check{x}_{i\ell,t}$ as

The final detected symbol of $x_{i,t}$ is the constellation point that is closest to $\hat{x}_{i,t}$.