Uplink Transmission Design for Crowded Correlated Cell-Free Massive MIMO-OFDM Systems

Nowadays, many works focus on the study of cell-free massive MIMO. For example, it was verified that cell-free massive MIMO can take advantage of the basic properties of co-located massive MIMO, including channel hardening and favorable propagation, when the number of antennas per AP was large and UEs were spatially well separated in independent and identically distributed (i.i.d.) fading channels [12]. In [5, 13], the uplink and downlink SE of cell-free massive MIMO were derived with i.i.d. small-scale fading channels. In this case, cell-free massive MIMO can significantly outperform the small-cell operation in terms of the 95%-likely per-user throughput [13], and can outperform the conventional single-cell massive MIMO in terms of the SE of cell-edge UEs [14, 4]. The uplink SE of different cell-free implementations are analyzed with spatially correlated fading channels [15]. Taking the fronthaul power consumption into account, the EE was optimized via power control in i.i.d. fading channels [16].

Some papers also study the pilot assignment in cell-free massive MIMO systems, but the vast majority of them consider i.i.d. fading channels and orthogonal pilot set. The most straightforward scheme is random pilot assignment (RPA), where each UE is randomly assigned a pilot from the orthogonal pilot set. It can lead to serious pilot contamination due to less consideration of the minimum distance among copilot UEs [17]. In [5], a greedy pilot assignment scheme was utilized, where pilots were randomly allocated to UEs and then iteratively updated to improve the lowest rate among UEs. A structured pilot assignment (SPA) scheme was proposed in [18] to maximize the minimum distance among copilot UEs. A tabu-based scheme was utilized to assign each UE a pilot from an orthogonal pilot set [19]. An efficient pilot assignment scheme based on graph coloring was proposed to mitigate the severe pilot contamination [20].

To mitigate severe pilot contamination, most of existed works consider how to assign a pilot for each UE from the orthogonal pilot set in i.i.d. fading channels. However, in crowded scenarios where UEs are active in an intermittent pattern, it is unnecessary and inefficient to allocate each UE with a pilot from the orthogonal pilot set. Besides, in realistic outdoor wireless propagation environments, the channel angle-delay spread is limited due to the finite number of scattering clusters, i.e., practical channels between antennas of an AP and a UE over all subcarriers are spatially and frequently correlated [11]. Hence, how to design uplink transmission scheme to identify active UEs, reduce channel estimation error, and efficiently transmit data in crowded cell-free massive MIMO-OFDM systems with spatially and frequently correlated channels becomes an important issue.

Motivated by the aforementioned discussion, we consider the uplink pilot and data transmission in crowded cell-free massive MIMO-OFDM systems with spatially and frequently correlated channels. The main contributions of this paper are as follows:

• •

  We establish a comprehensive channel model with spatial and frequency correlation between antennas of an AP and a UE over all subcarriers resulting from finite angle-delay spread. We formulate the relationship between the space-frequency domain channel covariance matrix and the angle-delay domain channel power spectrum for cell-free massive MIMO-OFDM systems.

• •

  We achieve the identification of active UEs based on non-orthogonal pilot sequences and non-orthogonal pilot phase shift hopping patterns.

• •

  We derive the expressions of mean square error (MSE) of channel estimation (MSE-CE) and obtain the condition of minimizing MSE-CE. Based on this condition, we propose the adjustable phase shift pilot (APSP) set allocation scheme. Simulation results indicate its performance gain over other schemes.

• •

  We devise a max-min power control algorithm to maximize the minimum SE lower bound of active UEs. We obtain the globally optimal solutions by iteratively solving linear programs.

The rest of paper is organized as follows. Section 2 describes the system model for cell-free massive MIMO-OFDM systems. Section 3 presents the channel estimation, the identification of active UEs, and the APSP set allocation scheme. The expressions of SE and a lower bound of SE are derived in Section 4. Here, the power control scheme is developed. Simulation results and discussions are given in Section 5. Section 6 concludes the work.

Notation: We adopt uppercase and lowercase boldface letters to denote matrices and column vectors, respectively. We use $\left[\mathbf{A}\right]_{m,n}$, $\left[\mathbf{A}\right]_{m,:}$, and $\left[\mathbf{A}\right]_{:,n}$ to denote the $\left(m,n\right)$-th element, the $m$-th row vector, and the $n$-th column vector of matrix $\mathbf{A}$, respectively. We adopt $\mathbf{I}_{N\times N}$ to denote the $N\times N$ dimensional identity matrix, and $\mathbf{I}_{N\times G}$ to denote the matrix comprising the first $G\left(\leq N\right)$ columns of $\mathbf{I}_{N\times N}$. We use $\textbf{1}_{N\times 1}$ to denote all-one vector, and 0 to denote all-zero vector or matrix. $\left\|\cdot\right\|$ denotes the Euclidean norm. The conjugate, transpose, and conjugate transpose are denoted by $\left(\cdot\right)^{*}$, $\left(\cdot\right)^{T}$, and $\left(\cdot\right)^{H}$, respectively. $\mathbb{E}\{\cdot\}$ denotes the expectation operator. The notation $\rm{vec}$, $\otimes$, and $\odot$ denote vectorization, Kronecker product, and Hadamard product, respectively. $\backslash$ denotes the set subtraction. $\left|\mathcal{A}\right|$ denotes the cardinal of set $\mathcal{A}$. Modulo-N is denoted by $\left\langle\cdot\right\rangle_{N}$. $\lfloor x\rfloor$ denotes the largest integer not greater than $x$. $z\sim\mathcal{CN}(0,\sigma^{2})$ denotes a circularly symmetric complex Gaussian random variable (RV) with mean $0$ and variance $\sigma^{2}$. Let $\delta(x)=\left\{\begin{array}[]{ll}1,&x=0\\ 0,&x\neq 0\end{array}\right.$. Let $\operatorname{diag}\left\{\mathbf{a}\right\}$ denote a diagonal matrix with vector $\mathbf{a}$ comprising its diagonal elements. $\operatorname{diag}\left\{\mathbf{A},\mathbf{B}\right\}$ denotes a block diagonal matrix with $\mathbf{A}$ and $\mathbf{B}$ in diagonal blocks.

In realistic outdoor wireless propagation environments, the channel angle spread is limited due to finite scattering clusters. Hence, we assume two ULAs in an AP separately receive waves from UEs with the mean angle of arrival (AoA) in a range of $180$ degrees, i.e., the waves from a UE are received by directional antennas of a ULA in each AP. Let $\mathbf{g}_{k,l,s}^{\beta}$ denote the uplink channel response vector between the l-th AP and the k-th UE over the s-th subcarrier. We employ $\theta$ to denote the incidence angle. $\mathbf{g}_{k,l,s}^{\beta}$ can be modeled as [21, 23, 24]

where $\beta_{k,l}$ denotes the large-scale fading between the l-th AP and the k-th UE, $\mathbf{g}_{k,l,s}$ denotes the small-scale fading, and $a_{k,l}^{\beta}(\theta,\tau)\triangleq\sqrt{\beta_{k,l}}a_{k,l}(\theta,\tau)$ represents the angle-delay domain channel gain function corresponding to the incidence angle $\theta$ and delay $\tau$. If AP antennas are spaced with half of wavelength, the steering vector $\mathbf{v}_{N}(\theta)=[1,e^{-\overline{\jmath}\pi\sin\theta},\cdots,e^{-\overline{\jmath}\pi(N-1)\sin\theta}]^{T}$. The space-frequency domain channel response matrix between UE k and all APs over $N_{c}$ subcarriers is denoted as

where ${\mathbf{G}_{k,l}^{\beta}}\in\mathbb{C}^{N\times N_{c}}$ and ${\mathbf{g}_{k,s}^{\beta}}\in\mathbb{C}^{M\times 1}$ are the space-frequency domain channel response matrix between the l-th AP and the k-th UE over $N_{c}$ subcarriers, and the vector between all APs and the k-th UE over the s-th subcarrier, respectively. It is assumed that $\operatorname{vec}\left\{\mathbf{G}_{k}^{\beta}\right\}\sim\mathcal{CN}\left(0,\mathbf{R}_{k}^{\beta}\right)$.

In this work, we consider the space-frequency domain correlation between the antennas of an AP and a UE over $N_{c}$ subcarriers. We assume different UEs or APs are spatially separated by a few wavelengths, and the channel realizations between different UEs and an AP and the channel realizations between a UE and different APs are uncorrelated [22]. Besides, we assume the angle-delay domain channel gain function ${a_{k,l}(\theta,\tau)}$ is uncorrelated [12, 23], i.e.,

where $P^{A}_{k,l}(\theta)$ and $P^{D}_{k,l}(\tau)$ represent the channel power azimuth spectrum (PAS) and power delay spectrum (PDS), respectively. Let $\varsigma_{k,l}$, $\theta_{k,l}$, and $\zeta_{k,l}$ denote the angle spread, mean AoA, and delay spread between the k-th UE and the l-th AP, respectively. PAS and PDS are given by [25]

We assume the channel angle spreads between different UEs and APs are equal, i.e., $\varsigma_{k,l}=\varsigma$ for $k\in\mathcal{K}$ and $l\in\mathcal{L}$. Similarly, the delay spreads satisfy $\zeta_{k,l}=\zeta$ for $k\in\mathcal{K}$ and $l\in\mathcal{L}$.

Let $\mathbf{H}^{\beta}_{k,l}\triangleq\sqrt{\beta_{k,l}}\mathbf{H}_{k,l}\in\mathbb{C}^{N\times N_{\mathrm{cp}}}$ denote the angle-delay domain channel response matrix between the k-th UE and the l-th AP. Let $\mathbf{H}^{\beta}_{k}\triangleq\left[\left(\mathbf{H}^{\beta}_{k,0}\right)^{T}\left(\mathbf{H}^{\beta}_{k,1}\right)^{T}\cdots\left(\mathbf{H}^{\beta}_{k,L-1}\right)^{T}\right]^{T}$ and $\mathbf{H}_{k}\triangleq\left[\mathbf{H}_{k,0}^{T}\;\mathbf{H}_{k,1}^{T}\cdots\mathbf{H}_{k,L-1}^{T}\right]^{T}$ denote the angle-delay domain channel response matrices between UE k and all APs with and without large-scale fading, respectively. Different elements in $\mathbf{H}^{\beta}_{k}$ correspond to the channels for different incidence angles, delays or APs, which satisfy [12, 23]

where $\bm{\Upsilon}_{k}^{\beta}\!\triangleq\!\begin{bmatrix}\left(\bm{\Upsilon}^{\beta}_{k,0}\right)^{\!T}\!\!\!&\left(\bm{\Upsilon}^{\beta}_{k,1}\right)^{\!T}\!\!\!&\!\cdots\!\!\!&\left(\bm{\Upsilon}^{\beta}_{k,L-1}\right)^{\!T}\end{bmatrix}^{\!T}\!\in\!\mathbb{R}^{M\times N_{\mathrm{cp}}}$ is the angle-delay domain channel power spectrum between UE k and all APs. We define $\bm{\Upsilon}_{k}\!\triangleq\!\begin{bmatrix}\bm{\Upsilon}^{T}_{k,0}\!&\bm{\Upsilon}^{T}_{k,1}\!\!&\!\cdots\!\!&\!\bm{\Upsilon}^{T}_{k,L-1}\end{bmatrix}^{\!T}\!\in\!\mathbb{R}^{M\times N_{\mathrm{cp}}}$.

Wireless channels are sparse in many typical scenarios [26, 27]. In this work, we consider the angle-delay domain sparsity from a UE to all APs in a cell-free massive MIMO-OFDM system. Most elements in $\bm{\Upsilon}_{k}^{\beta}$ are approximately 0 due to finite angle-delay spread.

Since the channel power lies in a finite number of delays due to limited scattering, channels are frequently correlated [23]. Considering most channel power is concentrated in a finite region of angles, channels between the antennas of an AP and a UE are spatially correlated [27]. Proposition 2.3 shows the relationship between the space-frequency domain channel covariance matrix and the angle-delay domain channel power spectrum in cell-free massive MIMO-OFDM systems. \proposition[] Define $\theta_{n}\!=\!\arcsin\left(\frac{2n}{N}-1\right)$ for $n=0,1,\cdots\!,N$. Let $\left[\mathbf{V}_{N\times N}\right]_{i,j}\triangleq\frac{1}{\sqrt{N}}\exp\left\{-\overline{\jmath}\pi i\sin\theta_{n}\right\}$, $\mathbf{V}_{M\times M}\triangleq\mathbf{I}_{L\times L}\otimes\mathbf{V}_{N\times N}$, and $\left[\mathbf{F}_{N_{c}\times N_{\mathrm{cp}}}\right]_{s,q}\triangleq\frac{1}{\sqrt{N_{c}}}\exp\left\{-\overline{\jmath}2\pi\frac{s}{N_{c}}q\right\}$. We have $\mathbf{V}_{N\times N}^{H}\mathbf{V}_{N\times N}\stackrel{{\scriptstyle N\rightarrow\infty}}{{=}}\mathbf{I}_{N\times N}$ and $\mathbf{V}_{M\times M}^{H}\mathbf{V}_{M\times M}\stackrel{{\scriptstyle N\rightarrow\infty}}{{=}}\mathbf{I}_{M\times M}$. The channel power spectrum between the k-th UE and the l-th AP is denoted as $\bm{\Upsilon}_{k,l}^{\beta}\in\mathbb{R}^{N\times N_{\mathrm{cp}}}$ satisfying

where $\tau_{q}=qT_{s}$ for $q=0,1,\cdots,N_{\mathrm{cp}}-1$. The relationship between the space-frequency domain channel covariance matrix and angle-delay domain channel power spectrum is given by

The space-frequency domain channel response matrix $\mathbf{G}_{k}^{\beta}$ can be decomposed as

We assume channels are piece-wise constant over a coherence interval. The time coherence interval is assumed to be equal to the length of $Z_{a}$ OFDM symbols. The frequency coherence interval is assumed to be equal to $\frac{1}{2\zeta}$. We assume $Z$ OFDM symbols are used for pilot transmission and the remaining $Z_{a}-Z$ OFDM symbols are for uplink data transmission.

During the training phase, all active UEs simultaneously transmit pilots to APs. The space-frequency domain signal received at the l-th AP is given by

where $\rho_{p}$ is the normalized signal-to-noise ratio (SNR) in the training phase, $\mathbf{W}_{l}$ is the additive white Gaussian noise (AWGN) matrix including i.i.d. $\mathcal{CN}(0,1)$ elements, $\mathbf{G}_{k^{\prime},l}^{\beta}=\mathbf{V}_{N\times N}\mathbf{H}_{k^{\prime},l}^{\beta}\mathbf{F}^{T}_{N_{c}\times N_{\mathrm{cp}}}$, and $\mathbf{\Phi}_{k^{\prime}}$ is the pilot signal given in (11). We assume large-scale fading coefficients are known wherever required.

In cell-free massive MIMO-OFDM systems, the number of phase shift orthogonal pilots (PSOPs) is approximately $\lfloor ZN_{c}/N_{cp}\rfloor$. Their phase shift differences are no less than the maximum channel delay of UEs to promise orthogonality of different pilots [29]. However, pilot contamination can be serious since UEs are far more than PSOPs. Hence, APSP is adopted in this work [24]. The pilot signal for the k-th UE over $N_{c}$ subcarriers and $Z$ OFDM symbols is

where $\mathbf{U}$ is an arbitrary $Z\times Z$ dimensional unitary matrix, $\phi_{k}\in\mathbf{\Psi}=\left\{0,1,\cdots,N_{c}Z-1\right\}$, $\mathbf{D}_{i}=\operatorname{diag}\left\{\mathbf{f}_{N_{c},i}\right\}$, $\mathbf{f}_{N_{c},i}=\left[1\;\exp\left\{-\overline{\jmath}2\pi\frac{i}{N_{c}}\right\}\cdots\exp\left\{-\overline{\jmath}2\pi\frac{\left(N_{c}-1\right)i}{N_{c}}\right\}\right]^{T}$, and $\mathbf{\Phi}_{b}$ is a diagonal matrix satisfying $\mathbf{\Phi}_{b}\mathbf{\Phi}_{b}^{H}=\mathbf{I}_{N_{c}\times N_{c}}$, which is the basic pilot matrix shared by all UEs. Then we have, for $\forall k,k^{\prime}\in\mathcal{K}$,

where (a) follows from the properties of Kronecker product, i.e., $(\mathbf{A}\otimes\mathbf{B})(\mathbf{C}\otimes\mathbf{D})=(\mathbf{A}\mathbf{C})\otimes(\mathbf{BD})$ and $(\mathbf{A}\otimes\mathbf{B})^{H}=\mathbf{A}^{H}\otimes\mathbf{B}^{H}$. For the APSP, phase shifts are divided into $Z$ groups. The group index is $\left\langle\phi_{k}\right\rangle_{Z}$. There is no pilot interference if UEs use phase shifts in different groups.

Based on (10), (11), and (12), after decorrelation, we have

where ${\check{\mathbf{W}}_{k,l}}$ includes i.i.d. $\mathcal{CN}(0,1)$ entrys due to unitary transformation. Define the permutation matrix $\mathbf{\Gamma}^{i}_{N_{c}}=\left[\begin{array}[]{cc}{\mathbf{0}}&{\mathbf{I}_{\left(N_{c}-\left\langle i\right\rangle_{N_{c}}\right)\times\left(N_{c}-\left\langle i\right\rangle_{N_{c}}\right)}}\\ {\mathbf{I}_{\left\langle i\right\rangle_{N_{c}}\times\left\langle i\right\rangle_{N_{c}}}}&{\mathbf{0}}\end{array}\right]$. Then, $\mathbf{H}_{{k}^{\prime},l}^{\beta,\left\lfloor{\phi_{k^{\prime}}}/{Z}\right\rfloor-\left\lfloor{\phi_{k}}/{Z}\right\rfloor}$ can be expressed as

The power spectrum of (14) is given by

Define ${\mathbf{\Upsilon}}_{k^{\prime}}^{\beta,{\left\lfloor{\phi_{k^{\prime}}}/{Z}\right\rfloor-\left\lfloor{\phi_{k}}/{Z}\right\rfloor}}\triangleq\begin{bmatrix}\left({\mathbf{\Upsilon}}_{k^{\prime},0}^{\beta,{\left\lfloor{\phi_{k^{\prime}}}/{Z}\right\rfloor-\left\lfloor{\phi_{k}}/{Z}\right\rfloor}}\right)^{T}&\left({\mathbf{\Upsilon}}_{k^{\prime},1}^{\beta,{\left\lfloor{\phi_{k^{\prime}}}/{Z}\right\rfloor-\left\lfloor{\phi_{k}}/{Z}\right\rfloor}}\right)^{T}&\cdots&\left({\mathbf{\Upsilon}}_{k^{\prime},L-1}^{\beta,{\left\lfloor{\phi_{k^{\prime}}}/{Z}\right\rfloor-\left\lfloor{\phi_{k}}/{Z}\right\rfloor}}\right)^{T}\end{bmatrix}^{T}$.

Channels are estimated in a decentralized fashion in cell-free massive MIMO-OFDM systems, i.e., each AP autonomously estimates channels without interchanging information [5]. For an AP, the channel estimation is performed in an element-wise manner in the angle-delay domain. The minimum mean-square error (MMSE) estimate of $\left[{{\mathbf{H}}_{k,l}}\right]_{i,j}$ is

Denote by ${{\widetilde{\mathbf{H}}}_{k,l}}={{{\mathbf{H}}}_{k,l}}-{{\widehat{\mathbf{H}}}_{k,l}}$ the channel estimation error. Based on the property of MMSE estimation, ${{\widetilde{\mathbf{H}}}_{k,l}}$ is independent of ${{\widehat{\mathbf{H}}}_{k,l}}$, and the MSE-CE is shown as

Define $\bm{\Xi}_{k}^{\beta}\triangleq\begin{bmatrix}\beta_{k,0}\bm{\Xi}_{k,0}^{T}&\beta_{k,1}\bm{\Xi}_{k,1}^{T}&\cdots&\beta_{k,L-1}\bm{\Xi}_{k,L-1}^{T}\end{bmatrix}^{T}$.

Pilot contamination is unavoidable in crowded scenarios. In this case, transmitting UEs cannot be identified in a slot, and we have to resort to the information among some slots. In this work, the uplink transmission phase is divided into frames consisting of $R$ slots, as shown in Fig. 2. Each slot comprises $Z_{a}$ OFDM symbols. The phase shifts of pilot signals among $R$ slots, i.e., the hopping patterns, form the identification basis of active UEs. When PSOPs are utilized, the number of UEs that can be identified is $\lfloor ZN_{c}/N_{cp}\rfloor^{R}$. It can be further increased if APSPs are adopted due to channel sparsity. Hence, we assume each UE has a unique and non-orthogonal pseudo-random hopping pattern, which are known by APs and CPU in advance.

Assuming the activation probability of UEs is $p_{a}$, the probability of having $K_{a}$ active UEs among K UEs is given by

When a UE is active, it selects the pilot phase shift based on its hopping pattern. Next, it formulates corresponding pilot over $N_{c}$ subcarriers based on (11) and transmits this pilot during the pilot transmission phase. Data codewords are transmitted afterwards. At the receiver, APs identify hopping patterns by running a correlation decoder across $R$ slots to detect transmitting UEs. Normalized Maximum Ratio Combining (MRC) is applied to the data at APs and the outputs are merged based on the identified hopping patterns. Since APs can know which UEs are active and what they transmit after $R$ slots, this scheme is suitable for delay-tolerant mMTC.

In cell-free massive MIMO-OFDM systems, for a UE, the APs far from it cannot contribute significantly to the channel estimation performance and spatial diversity gains due to heavy path loss. Besides, the transfer of data between APs and CPU causes much fronthaul energy consumption. Hence, in order to improve channel estimation performance and save fronthaul energy consumption, each UE should be served by a group of APs but not all APs within the network [30]. Hence, before allocating APSP set for each UE, AP selection is performed at CPU [16]. The set of APs selected to serve the k-th UE is denoted as $\mathcal{B}_{k}\subseteq\mathcal{L}$ satisfying $\sum_{b=0}^{\left|\mathcal{B}_{k}\right|-1}\overline{\beta}_{k,b}\geq\sum_{l=0}^{L-1}\beta_{k,l}\lambda$, where $\left\{\overline{\beta}_{k,0},\overline{\beta}_{k,1},\cdots,\overline{\beta}_{k,L-1}\right\}$ includes large-scale fading coefficients between UE k and APs in descending order, and $\lambda$ is assumed to be equal for all UEs. The APs belonging to $\mathcal{B}_{k}$ need to estimate channels and decode signals of UE $k$. The set of UEs served by AP l is denoted as $\mathcal{D}_{l}$.

The sporadic and independent activation of UEs and the construction of pseudo-random hopping patterns can be modeled as the process that each active UE randomly selects a pilot phase shift from its allocated set in each slot [11]. We obtain the probability of having $K_{a}$ active UEs in (18), but it is still uncertain which $K_{a}$ UEs are active and which phase shifts they select. To quantify these uncertainties, we employ ${{\mathcal{U}}_{{{\mathcal{K}}_{a}}}}=\left\{\mathcal{U}_{{{\mathcal{K}}_{a}}}^{0},\mathcal{U}_{{{\mathcal{K}}_{a}}}^{1},\cdots\!,\mathcal{U}_{{{\mathcal{K}}_{a}}}^{{{N}_{{{\mathcal{K}}_{a}}}}-1}\right\}$ to denote all possible sets of $K_{a}$ active UEs. The j-th element in ${{\mathcal{U}}^{i}_{{{\mathcal{K}}_{a}}}}$ is denoted as ${{\mathcal{K}}^{i,j}_{a}}$. In a slot, all possible choices of phase shifts for UEs in ${{\mathcal{U}}^{i}_{{{\mathcal{K}}_{a}}}}$ are denoted as ${{\mathcal{P}}_{i}}=\left\{\mathcal{P}_{i}^{0},\mathcal{P}_{i}^{1},\cdots,\mathcal{P}_{i}^{{{N}_{\phi}-1}}\right\}$. Assuming the $p$-th choice of shifts is considered, the phase shift of UE ${{\mathcal{K}}^{i,j}_{a}}$ is denoted as $\mathcal{P}_{i,j}^{p}$. We assume $\nu_{k,l}$ is equal to $1$ when the l-th AP is selected to serve the $k$-th UE and is equal to $0$ otherwise. Let $\bm{\Upsilon}_{k}^{\beta,0}\triangleq\begin{bmatrix}\nu_{k,0}\beta_{k,0}\bm{\Upsilon}^{T}_{k,0}&\nu_{k,1}\beta_{k,1}\bm{\Upsilon}^{T}_{k,1}&\cdots&\nu_{k,L-1}\beta_{k,L-1}\bm{\Upsilon}^{T}_{k,L-1}\end{bmatrix}^{T}$ and $\bm{\Upsilon}_{k}^{0}\triangleq\begin{bmatrix}\nu_{k,0}\bm{\Upsilon}^{T}_{k,0}&\nu_{k,1}\bm{\Upsilon}^{T}_{k,1}&\cdots&\nu_{k,L-1}\bm{\Upsilon}^{T}_{k,L-1}\end{bmatrix}^{T}$. Based on the channel estimation analysis in Section 3.1, we obtain the MSE-CE of UE ${{\mathcal{K}}^{i,j}_{a}}$ averaged over its serving APs and $N_{c}$ subcarriers

We average (19) over all possible sets of active UEs, all UEs in the set, and all types of phase shift selection. We obtain

Then we calculate the expected value of (20) accounting for the number of active UEs and obtain

The channel estimation performance suffers from pilot interference. In Proposition 3.3, we show that the effect of pilot interference can be eliminated by proper APSP set allocation for UEs. Phase shifts in the APSP set of each UE are used to form the unique pilot phase shift hopping pattern for it. In correlated channels, many elements in $\mathbf{\Upsilon}_{k}^{\beta}$ are approximately $0$, which is the physical basis of the APSP set allocation scheme. \proposition[] The minimum value of ${{\mathbb{E}}_{\mathcal{U},{{\mathcal{K}}_{{a}}},\mathcal{P}}}(\varepsilon^{0})$ and $\overline{\varepsilon}^{0}$ are given by

and the minimum is achieved under the condition that for $\forall k,k^{\prime}\in\mathcal{K}$ and $k\neq k^{\prime}$,

In the uplink data transmission phase, $K_{a}$ active UEs simultaneously transmit data to APs. The data of UE k over the s-th subcarrier in an OFDM symbol is denoted by $x_{k,s}\sim\mathcal{CN}(0,1)$, which is weighted by a power control coefficient $\sqrt{\eta_{k}}$ satisfying $0\leq\eta_{k}\leq 1$. The received signal at AP l over the s-th subcarrier is given by