Two-Stage Channel Estimation Approach for Cell-Free IoT With Massive Random Access

As shown in Fig. 1, we consider a stochastic user-centric cell-free IoT network, which consists of randomly distributed APs and IoT devices. Each AP is equipped with $N$ antennas and connects to a central processing unit (CPU), via a back-haul network. We assume that the locations of APs follow a homogeneous Poisson point process (PPP) ${\Phi}_{A}$ with density $\lambda_{\rm A}$, while the locations of devices follow another independent homogeneous PPP ${\Phi}_{D}$ with higher density $\lambda_{\rm D}\rightarrow\infty$. In each time slot, each device independently transmit data to APs with probability $\epsilon$. Let $r_{m,k}$ denote the distance between the $m$-th AP and the $k$-th device. Let ${\boldsymbol{g}}_{m,k}\in\mathbb{C}^{N\times 1}$ denote the corresponding channel coefficients. Unlike a cellular network, each device is served by its adjacent APs within its coverage area. Here, the coverage area of each AP or each device is a circular area with sufficient large radius $R_{0}$. The $n$-th element of ${\boldsymbol{g}}_{m,k}$ is modeled as

where $h_{(m,n),k}\sim\mathcal{CN}(0,1)$ is the small-scale fading, and $\sqrt{\beta_{m,k}}$ denotes the path-loss which decreases monotonically with $r_{m,k}$. As shown in Fig. 1, the $k$-th device is served by a set of APs denoted by $\mathcal{A}_{k}$, which are located within its coverage area denoted by a red circle. Let $\mathcal{D}_{m}$ be the set of devices within the coverage area of the $m$-th AP denoted by the gray circle.

Let $\alpha_{k}$ be the activity indicator of the $k$-th device, i.e.,

Let ${\Pr}(\alpha_{k}=1)=\epsilon,\forall k$. We assume that the activity detection for the $k$-th device is conducted by APs located within the circular area of radius $R_{1}\leq R_{0}$ (the green circle in Fig. 1). We call $R_{1}$ the cooperative radius. We denote the cooperative APs by the set $\widetilde{\mathcal{A}}_{k}=\{m:r_{m,k}\leq R_{1}\}$.

In the channel estimation phase, each active device transmits the pilot sequence $\boldsymbol{\psi}_{k}\sim\mathcal{CN}\left(0,\frac{1}{\tau}{\boldsymbol{I}}_{\tau}\right)$ with a normalized power $\rho$, where $\tau$ is the length of each pilot sequence. For the $m$-th AP, we define ${\boldsymbol{\alpha}}_{m}={\rm diag}(\cdots,\alpha_{k},\cdots)$, ${\boldsymbol{\beta}}_{m}={\rm diag}(\cdots,\beta_{m,k},\cdots)$, and $\boldsymbol{\Psi}_{m}=\left[\cdots,\boldsymbol{\psi}_{k},\cdots\right]\sim\mathbb{C}^{\tau\times|\mathcal{D}_{m}|}$ where $k\in\mathcal{D}_{m}$, and $\boldsymbol{\Psi}_{m}$ and ${\boldsymbol{\beta}}_{m}$ are assumed to be available for the $m$-th AP. Let $E=\tau\rho$, the received signal at the $n$-th antenna of the $m$-th AP is given by

where ${\boldsymbol{w}}_{(m,n)}\sim\mathcal{CN}(0,\boldsymbol{I}_{\tau})$ is the normalized noise, and

denotes the channel coefficients vector between the $n$-th antenna of the $m$-th AP and the devices in $\mathcal{D}_{m}$. The channel coefficients between the $m$-th AP and the $k$-th device are given by

with $\boldsymbol{g}_{m,k}\sim\mathcal{CN}(0,\beta_{m,k}{\boldsymbol{I}}_{N}),~{}k\in\mathcal{D}_{m}$. We combine the channel coefficients between all the $N$ antennas of the $m$-th AP and all the devices in $\mathcal{D}_{m}$ into the matrix

According to (4), the received pilots at the $m$-th AP can be expressed as

where

and

with

According to (3), the distribution of ${\boldsymbol{x}}_{m,k}$ is

where $\delta_{\bf 0}$ denotes the point mass measure at zero as the $k$-th device is inactive, and $p_{{\boldsymbol{g}}_{m,k}}$ is the probability density function (PDF) of $\boldsymbol{g}_{m,k}$ defined in (5).

For the ease of understanding, some notations of this paper are summarized in Table I. To facilitate analysis, we have the following assumptions made throughout this paper:

• •

  We consider the cell-free IoT with massive connectivity, i.e., $\lambda_{D}\rightarrow\infty$, which implies the number of active devices within the coverage of $m$-th AP $|\Omega_{m}|/|\mathcal{D}_{m}|\rightarrow\epsilon$,  $\forall m$.

• •

  Similar to [10], we consider the asymptotic regime with $T,\tau,|\mathcal{D}_{m}|\rightarrow\infty$, while keeping the total transmit energy of pilot $E=\tau\rho$, and the ratios $\tau/T$ and $\tau/|\mathcal{D}_{m}|$ fixed.

For the $m$-th AP, the path-loss coefficients $\beta_{m,k},k\in\mathcal{D}_{m}$ follow independent identical distribution for different geometry realizations, i.e., $\beta_{m,k}\sim p_{\beta},\forall k\in\mathcal{D}_{m}$. For a snap shot, each geometry realization of $\{\beta_{m,k}:k\in\mathcal{D}_{m}\}$ is equivalent to $|\mathcal{D}_{m}|$ multiplied by the realization of any $\beta_{m,k}$ with $k\in\mathcal{D}_{m}$. Thus, the PDF of each geometry realization of $\{\beta_{m,k}:k\in\mathcal{D}_{m}\}$ converges to $p_{\beta}$ as $\lambda_{D}\rightarrow\infty$.

For the distributed and user-centric architecture, it is nontrivial to detect the user activity and estimate the CSI for cell-free IoT networks due to the following reasons: (1) The activity $\alpha_{k}$ is jointly determined by its adjacent APs, which means the traditional centralized detection techniques [10] for cellular networks are unavailable. (2) Considering the overhead over back-haul, the distributed APs cannot fully cooperate to detect user activities. (3) Compared with [15], the performance analysis for LMMSE channel estimation is more challenging due to the possible errors of activity detection.

In Section III, we will propose a two-stage channel estimation approach based on the AMP algorithm and the LMMSE method.

• •

  Stage I: According to the signals $\boldsymbol{Y}_{m}$ defined in (6), each AP first estimates $\boldsymbol{X}_{m}$ using the AMP algorithm. Then, the activity $\widehat{\alpha}_{k}$ of the $k$-th device is jointly determined by fusing the estimates $\{\widehat{\boldsymbol{x}}_{m,k}:m\in\widetilde{\mathcal{A}}_{k}\}$ from the adjacent APs $\widetilde{\mathcal{A}}_{k}$.

• •

  Stage II: The channel vectors $\{\widehat{\boldsymbol{g}}_{m,k}:k\in\mathcal{D}_{m}\}$ are re-estimated by the $m$-th AP using the low-complexity LMMSE method based on $\widehat{\boldsymbol{\alpha}}_{m}={\rm diag}(\cdots,\alpha_{k},\cdots),k\in\mathcal{D}_{m}$ and $\boldsymbol{Y}_{m}$.

In Section IV, the activity detection error probability and the mean-squared error of channel estimate are derived in closed-form using the asymptotic analysis. In addition, we investigate how the number of antennas and the density of APs affect the system performance.

In Section V, we define the coverage probability as a metric to evaluate the performance for the entire stochastic cell-free IoT network. Then, we compare our user-centric based channel estimation approach with the collocated massive MIMO and small-cell in terms of coverage probability .

First, the $m$-th AP independently estimates $\boldsymbol{X}_{m}$ based on $\boldsymbol{Y}_{m}$ using the vector AMP algorithm. Similarly to [10], the vector AMP algorithm is updated as follows with initialization $\boldsymbol{X}_{m,{(0)}}=\boldsymbol{0}$ and $\boldsymbol{R}_{m,{(0)}}=\boldsymbol{Y}_{m}$,

where $t=0,1,\cdots$ is the index of iteration, and

$\boldsymbol{X}_{m,{(t)}}=[\cdots,\boldsymbol{x}_{{m,k},{(t)}},\cdots]^{T},~{}k\in\mathcal{D}_{m}$ is the estimate of $\boldsymbol{X}_{m}$ at the $t$-th iteration, and $\boldsymbol{R}_{m,{(t)}}=[\cdots,\boldsymbol{r}_{{m,k},{(t)}},\cdots]^{T},~{}k\in\mathcal{D}_{m}$ represents the corresponding residual. $\eta(\cdot)$ and $\eta^{\prime}(\cdot)$ represent an appropriately designed denoiser and its first-order derivative, respectively. Using the state evolution [10], $\widehat{\boldsymbol{x}}_{{m,k},{(t)}}$ can be modeled as signal $\boldsymbol{x}_{m,k}$ plus independent Gaussian noise $\boldsymbol{v}_{m,k}\sim\mathcal{CN}(\textbf{0},{\boldsymbol{I}}_{N})$, i.e.,

where $\boldsymbol{v}_{m,k}$ is independent of $\boldsymbol{x}_{m,k}$, and the state $\boldsymbol{\Theta}_{m,{(t)}}$ can be predicted according to the state evolution. Starting from the initial state

the state evolution of the AMP algorithm at each iteration is updated as [10], i.e.,

where $\boldsymbol{X}_{m,k}$, $\boldsymbol{V}_{m,k}$ and $\widehat{\boldsymbol{X}}_{{m,k},{(t)}}=\boldsymbol{X}_{m,k}+\boldsymbol{\Theta}_{m,{(t)}}^{\frac{1}{2}}\boldsymbol{V}_{m,k}$ are used to capture the distribution of $\boldsymbol{x}_{m,k}$, $\boldsymbol{v}_{m,k}$, and $\widehat{\boldsymbol{x}}_{{m,k},{(t)}}$, respectively. The expectation is taken over all of the devices in $\mathcal{D}_{m}$.

For a given $\widehat{\boldsymbol{x}}_{{m,k},{(t)}}$, the MMSE denoiser $\eta(\cdot)$ is designed the same as in [10], given by

where

Since ${\boldsymbol{\Theta}}_{m,(t)}$ is a weighted identity matrix in each iteration, by substituting (15) into (14), the MMSE denoiser can be simplified as

where $\phi_{{m,k},{(t)}}$ in $\xi_{m,k}^{(t)}$ is defined in (14) with

After the $t$-th iteration, the $m$-th AP obtains the output $\{\widehat{\boldsymbol{x}}_{{m,k},{(t)}},k\in\mathcal{D}_{m}\}$. According to the model (11), the entities of $\widehat{\boldsymbol{x}}_{{m,k},{(t)}}$ are i.i.d. zero-mean complex Gaussian variables with variances $\beta_{m,k}+\theta_{m,{(t)}}$ and $\theta_{m,{(t)}}$, and the corresponding PDFs $f_{1}(\widehat{\boldsymbol{x}}_{{m,k},{(t)}})$ and $f_{0}(\widehat{\boldsymbol{x}}_{{m,k},{(t)}})$, for $\alpha_{k}=1$ and $\alpha_{k}=0$, respectively. Similar to the cellular massive MIMO [10], each AP can separately solve the following hypothesis testing problem

whose optimal decision rule is the log-likelihood ratio given by

which can be further simplified as

To reap the benefits from multiple APs and the user-centric architecture, the distributed APs, which are denoted by $\widetilde{\mathcal{A}}_{k}$ within a circular area with radius $R_{1}\leq R_{0}$, jointly detect the activity ${\widehat{\alpha}}_{k}$. Each AP $m\in\mathcal{A}_{k}$ sends $\mu_{m,k,{(t)}}$ to the CPU, which determines ${\widehat{\alpha}}_{k}$ according to the following weighted rule,

where

with the fusing weights ${\boldsymbol{\zeta}}_{k}=(\cdots,\zeta_{m,k},\cdots),m\in\widetilde{\mathcal{A}}_{k}$ satisfying $\sum\nolimits_{m\in\widetilde{\mathcal{A}}_{k}}\zeta_{m,k}=1$.

Then, the CPU sends the estimated activity ${\widehat{\alpha}}_{k}$ to the APs $m\in\mathcal{A}_{k}$ within the coverage area of the $k$-th device for the following channel re-estimation. It is noted that the backhaul overhead can be neglected, because each AP only needs to deliver a scalar $\mu_{m,k,{(t)}}$ to the CPU.

The LMMSE channel estimation in cell-free systems have been already considered in [14, 15]. After obtaining $\widehat{\boldsymbol{\alpha}}_{m}$, we re-estimate the channel $\widehat{\boldsymbol{g}}_{(m,n)}$ using the LMMSE method to further improve the accuracy of channel estimation. From (4), we have

where ${\boldsymbol{U}}_{m}={\rm diag}\left(\cdots,{\alpha}_{k}\beta_{m,k},\cdots\right),k\in\mathcal{D}_{m}$. Since only $\widehat{\boldsymbol{\alpha}}_{m}={\rm diag}\left(\cdots,\widehat{\alpha}_{k}\cdots\right),k\in\mathcal{D}_{m}$ and $\{\beta_{m,k},k\in\mathcal{D}_{m}\}$ are available at the $m$-th AP in (25) and (26), we use $\widehat{\boldsymbol{U}}_{m}={\rm diag}\left(\cdots,\widehat{\alpha}_{k}\beta_{m,k},\cdots\right),k\in\mathcal{D}_{m}$ instead of ${\boldsymbol{U}}_{m}$ for the LMMSE channel estimation. Thus, the LMMSE estimate of $\boldsymbol{g}_{(m,n)}$ is

where $\widehat{\boldsymbol{Z}}_{m}=E\boldsymbol{\Psi}_{m}\widehat{\boldsymbol{U}}_{m}\boldsymbol{\Psi}_{m}^{H}+\boldsymbol{I}$. Since ${\boldsymbol{U}}_{m}$ and $\widehat{\boldsymbol{U}}_{m}$ are diagonal matrices, ${\boldsymbol{Z}}_{m}$ and $\widehat{\boldsymbol{Z}}_{m}$ can be rewritten as

and

where $\widehat{\Omega}_{m}=\{i:\widehat{\alpha}_{i}=1,i\in\mathcal{D}_{m}\}$. We can easily observe that

where $\widehat{\Omega}_{m}^{\rm M}=\{i:\alpha_{i}=1,\widehat{\alpha}_{i}=0,i\in\mathcal{D}_{m}\}$ is the set of miss detected devices in $\mathcal{D}_{m}$, and $\widehat{\Omega}_{m}^{\rm F}=\{i:\alpha_{i}=0,\widehat{\alpha}_{i}=1,i\in\mathcal{D}_{m}\}$ is the set of false detected devices in $\mathcal{D}_{m}$.

Therefore, we have

where

Substituting (25) and (30) into (III-C), we can obtain

Then, the mean square of the channel estimate for the $k$-th device can be expressed as

and the corresponding mean-squared channel estimation error is

Remark 1: Noted that the activity $\alpha_{k}$ is jointly detected by the adjacent APs, while the CSI is estimated at each AP using the LMMSE method. Since the CSI of different antennas is independent, the antennas’ cooperation gains no benefits for the channel estimation when the active devices are known. It can be seen from [19] that, the optimal LMMSE channel estimation is performed at each antenna separately. In brief, the joint activity detection can improve the accuracy of activity detection and further improve the performance of channel estimation.

From (11), it follows that

where

According to (24a), $\mu_{k,(t)}$ is a sum of independent Gamma random variables with different scales, which implies the PDF of $\mu_{k,(t)}$ is too complex to analyze. To facilitate analysis, we approximate the distribution of $\mu_{k,(t)}$ by a Gamma distribution according to the well-known Welch-Satterthwaite approximation [20, 21], i.e., $\mu_{k,(t)}\sim{\rm Gamma}(s,\omega)$, with

where

According to the cumulative distribution function (CDF) of the Gamma distribution, the probability of missed detection ${\mathcal{P}}^{M}_{k}({\boldsymbol{\zeta}}_{k})$ can be approximated as

Similarly, the probability of false detection ${\mathcal{P}}^{F}_{k}({\boldsymbol{\zeta}}_{k})$ can be approximated as

According to (34), (37), and (37), the activity detection error probability ${\mathcal{P}}_{k}({\boldsymbol{\zeta}}_{k})$ can be approximated as

Simulation results will show that the error caused by the Welch-Satterthwaite approximation is negligible.

Since $\widetilde{{\mathcal{P}}}_{k}({\boldsymbol{\zeta}}_{k})$ is an accurate approximation of ${\mathcal{P}}_{k}({\boldsymbol{\zeta}}_{k})$, the optimal fusing weights for the $k$-th device can be expressed as

and it can be exhaustively searched through minimizing $\widetilde{\mathcal{P}}_{k}({\boldsymbol{\zeta}}_{k})$ which depends on the distribution of $\{\beta_{m,k},m\in\widetilde{\mathcal{A}}_{k}\}$.

Remark 2: The small cell system can be treated as a special case of the cell-free IoT, in which the activity of the $k$-th device $\alpha_{k}$ is only determined by the nearest AP with index $m^{o}$, i.e., $\beta_{m^{o},k}=\max_{m\in\mathcal{A}_{k}}\beta_{m,k}$. That is, only $\zeta^{\prime}_{m^{o},k}=1$ in the fusing weight is non-zero for the small cell system, i.e.,

Intuitively, the number of antennas $N$ of each AP and the density of APs $\lambda_{A}$ play important rules on the accuracy of activity detection. To show this, we first introduce the following Lemma.