Smart Hybrid Beamforming and Pilot Assignment for 6G Cell-Free Massive MIMO

A portion of the total number of resource units, the latter denoted by $\tau_{c}$, is used for channel estimation. During $\tau(\leq\tau_{c})$ channel uses, UE $k$ is assigned a pilot $\boldsymbol{\phi}_{k}\in\mathbb{C}^{\tau\times 1}$ with $||\boldsymbol{\phi}_{k}||^{2}=\tau$ and the pilot matrix is denoted by $\boldsymbol{\Phi}=(\boldsymbol{\phi}_{1},\dots,\boldsymbol{\phi}_{K})\in\mathbb{C}^{\tau\times K}$. Upon pilot transmission at a certain power $p^{(t)}$, the observations at the $m$th AP are

with $\boldsymbol{Z}_{m}\sim\mathcal{N}_{\mathbb{C}}(\boldsymbol{0},\sigma^{2}\boldsymbol{I}_{N})$ for $\sigma^{2}$ being the noise power. Standard MMSE estimation leads to the next estimates [11]

with

for $\boldsymbol{R}_{m}^{(g)}=\mathrm{diag}\{\boldsymbol{R}_{m,k}^{(g)}\mspace{4.0mu}\mathrm{for}\mspace{4.0mu}k=1,\dots,K\}$. It can be verified that $\boldsymbol{g}_{m,k}=\boldsymbol{\hat{g}}_{m,k}+\boldsymbol{\tilde{g}}_{m,k}$ with $\boldsymbol{\tilde{g}}_{m,k}$ denoting the error, uncorrelated with the estimate. More concretely, $\boldsymbol{\hat{g}}_{m,k}\sim\mathcal{N}_{\mathbb{C}}(\boldsymbol{0},\boldsymbol{\Gamma}_{m,k}^{(g)})$ with $\boldsymbol{\Gamma}_{m,k}^{(g)}$ defined by

and the channel error following $\boldsymbol{\tilde{g}}_{m,k}\sim\mathcal{N}_{\mathbb{C}}(\boldsymbol{0},\boldsymbol{C}_{m,k}^{(g)})$ with $\boldsymbol{C}_{m,k}^{(g)}=\boldsymbol{R}_{m,k}^{(g)}-\boldsymbol{\Gamma}_{m,k}^{(g)}$.

Although CF networks allow users to establish connectivity to multiple APs, scalability must be taken into account. Therefore only a subset of APs jointly serve a particular user. Hence, we define by $\mathcal{F}_{k}$ the subset of APs involved in the decoding of the $k$th UE and by $\mathcal{U}_{m}$ the subset of UEs treated as signal by AP $m$. Thus, the binary matrix $\boldsymbol{M}=(\boldsymbol{m}_{1},\dots,\boldsymbol{m}_{K})\in\mathbb{Z}_{2}^{M\times K}$ whose entries are

accounts for scalability. Provided that each AP observes an $L$-dimensional signal after the hybrid beamforming stage, the expanded version of $\boldsymbol{M}$ is $\boldsymbol{{M}}^{(s)}=\boldsymbol{M}\otimes\boldsymbol{1}_{L}$ with $\boldsymbol{1}_{L}$ an $L$-dimensional vector of ones. The complementary matrix $\boldsymbol{{M}}^{(i)}=\boldsymbol{1}-\boldsymbol{{M}}^{(s)}$ accounts for the disregarded UEs per AP.

After data transmission, the signal collected by the $M$ APs is $\boldsymbol{y}=(\boldsymbol{y}_{1},\dots,\boldsymbol{y}_{M})^{\text{T}}\in\mathbb{C}^{ML\times 1}$ with $\boldsymbol{y}_{m}\in\mathbb{C}^{L\times 1}$

with $\circ$ denoting the Hadamard product, $\boldsymbol{G}\in\mathbb{C}^{ML\times K}$ being the effective channel matrix whose entries are $(\boldsymbol{G})_{m,k}=\boldsymbol{g}_{m,k}\in\mathbb{C}^{L\times 1}$. Vector $\boldsymbol{x}=(\sqrt{p_{1}}s_{1},\dots,\sqrt{p_{K}}s_{K})^{\rm T}$ for given UE transmit powers and symbols, denoted by $p_{k}$ and $s_{k}$, respectively. Finally, $\boldsymbol{W}=\textrm{diag}\{\boldsymbol{W}_{m}\mspace{4.0mu}\textrm{for}\mspace{4.0mu}m=1,\dots,M\}$ and $\boldsymbol{n}=(\boldsymbol{n}_{1},\dots,\boldsymbol{n}_{M})^{\mathrm{T}}$ where $\boldsymbol{n}_{m}\sim\mathcal{N}_{\mathbb{C}}(\boldsymbol{0},\sigma^{2}\boldsymbol{I}_{N})$.

In the downlink, the APs jointly precode the users data. More particularly, the precoder intended for UE $k$ is denoted by $\boldsymbol{v}_{k}\in\mathbb{C}^{ML\times 1}$ and after data transmission, the signal collected at UE $k$ is

where $n_{k}\sim\mathcal{N}_{\mathbb{C}}({0},\sigma^{2})$.

Provided that for UE $k$ only $|\mathcal{F}_{k}|$ APs are relevant, taking the rows of $\boldsymbol{y}$ associated to $\mathcal{F}_{k}$ produces the following reduced signal model

where matrices in (11) are the reduced version of the original matrices which contain the rows related to $\mathcal{F}_{k}$ and all columns. Moreover, $\boldsymbol{z}_{k}\sim\mathcal{N}_{\mathbb{C}}(\boldsymbol{0},\boldsymbol{\Sigma}_{k})$ with $\boldsymbol{\Sigma}_{k}$ being a block diagonal matrix $\boldsymbol{\Sigma}_{k}=\mathrm{diag}\{\boldsymbol{\Sigma}_{k,m}\in\mathbb{C}^{L\times L}\mspace{4.0mu}\mathrm{for}\mspace{4.0mu}m\in\mathcal{F}_{k}\}$ where the diagonal terms are

In the uplink, the combiner maximizing the SINR is the MMSE, achieving a maximum value of

where $\boldsymbol{\hat{g}}_{k}$ and $\boldsymbol{\hat{g}}_{i}$ are the $k$th and $i$th columns of $\boldsymbol{\hat{G}}_{k}$, respectively, and a similar definition applies to $\boldsymbol{m}_{k,i}^{(s)}$. As a consequence, after accounting for the pilot overhead $\frac{\tau}{\tau_{c}}$, the ergodic spectral efficiency that the $k$th UE can achieve is

Various precoding strategies can be used to encode the users data. However, RZF provides an outstanding performance as studied in the literature. More particularly, the subset RZF precoding, denoted by $\boldsymbol{V}=(\boldsymbol{v}_{1},\dots,\boldsymbol{v}_{K})$, follows

with $\rho$ being the regularitzation parameter and $\boldsymbol{\Lambda}=\mathrm{diag}(\lambda_{1},\dots,\lambda_{K})$. Different formulations can be used for $\lambda_{k}$, such as to ensure (i) $\mathbb{E}\{||\boldsymbol{W}\boldsymbol{v}_{k}||^{2}\}\leq 1$ or (ii) $||\boldsymbol{W}\boldsymbol{v}_{k}||^{2}\leq 1$. In our case, since perfect CSI is not available, we use the former formulation. Once User $k$ receives $y_{k}$, as defined in Eq. (10), the following spectral efficiency can be achieved:

with

The design of $\boldsymbol{W}=\mathrm{diag}\{\boldsymbol{W}_{m}\mspace{4.0mu}\mathrm{for}\mspace{4.0mu}m=1,\dots,M\}$ is challenging given the complexity of the SINR. Therefore, directly solving (32) poses a major challenge. However, under perfect CSI, some algebraic properties on $\boldsymbol{W}_{m}$ can be extracted and therefore used for its design. Concretely, we first disregard the unit-modulus constraint and after SVD decomposition $\boldsymbol{W}$ factorizes as $\boldsymbol{W}=\boldsymbol{U}\boldsymbol{Q}$ with semi-unitary $\boldsymbol{U}$, i.e. $\boldsymbol{U}^{*}\boldsymbol{U}=\boldsymbol{I}_{ML}$.

According to [13], $\sum\limits_{i\neq k}^{K}|\boldsymbol{g}_{k}^{*}\boldsymbol{v}_{i}|^{2}p_{i}+\sigma^{2}\approx\sum\limits_{i\neq k}^{K}|\boldsymbol{g}_{i}^{*}\boldsymbol{v}_{k}|^{2}p_{i}+\sigma^{2}$. Under the condition that the previous approximation is tight, the following proposition, which is similar to the result obtained in [9] for another metric, can be obtained.

In order to full-fill both propositions, for UL and DL, $\boldsymbol{Q}$ can be set to $\boldsymbol{Q}=\boldsymbol{I}_{ML}$ and therefore $\boldsymbol{W}=\boldsymbol{U}$ meaning that the analog matrix should have orthogonal columns. The idea behind having orthogonal columns is that interference is reduced. To the best of our knowledge, there are two ways of smartly creating $\boldsymbol{W}$ explained in [8] and [9], respectively. While the latter is based on perfectly known channels, the former fails to capture the complete spectrum of the channel covariance matrices. In this work, we propose a method that takes into account all possible eigenvectors/eigenvalues of all $\boldsymbol{R}_{m,k}$ with the aim of maximizing the minimum average UE power signal, which is shown to maximize the minimum SINR in our simulations. More particularly, $\boldsymbol{R}_{m,k}=\boldsymbol{V}_{m,k}\boldsymbol{\Lambda}_{m,k}\boldsymbol{V}_{m,k}^{*}$ with $\boldsymbol{V}_{m,k}$ having orthonormal column vectors and $\boldsymbol{\Lambda}_{m,k}=\mathrm{diag}(\lambda_{m,k}^{(1)},\dots,\lambda_{m,k}^{(N)})$ containing the $N$ eigenvalues of $\boldsymbol{R}_{m,k}$. Note that the average signal power for UE $k$ is given by $\sum\limits_{m\in\mathcal{F}_{k}}\mathrm{tr}(\boldsymbol{W}_{m}^{*}\boldsymbol{R}_{m,k}\boldsymbol{W}_{m})$. Therefore, such a expression is maximized whenever the columns of $\boldsymbol{W}_{m}$ match the eigenvectors of $\boldsymbol{R}_{m,k}$. However, not all UEs and their respective eigenmodes can be captured by $\boldsymbol{W}_{m}$. A selection of $L$ out of $NK$ should be made. As a consequence, we define the UE average signal power as

where $\alpha_{m,k}^{(n)}$ is a binary optimization variable scheduling the eigenvectors to the columns of $\boldsymbol{W}_{m}$. Therefore, the following optimization problem with respect to $\alpha_{m,k}^{(n)}$ can be formulated:

The reverse-delete algorithm is capable of efficiently solving (34) without the need of an exhaustive search. The surviving $\alpha_{m,k}^{(n)}$ determine which eigenvectors of which users will compose the columns of $\boldsymbol{W}_{m}$. However, note that $\boldsymbol{{W}}_{m}$ for $m=1,\dots,M$ does not necessarily have orthogonal columns given that, most likely, eigenvectors from multiple users will be used to construct the analog matrices. As a consequence, neither Prop. 1 nor Prop. 2 are satisfied. Thus, the final unconstrained analog beamformers are obtained by $\boldsymbol{W}_{m}^{(p)}=\mathcal{P}(\boldsymbol{{W}}_{m})$ where $\mathcal{P}(\boldsymbol{A}_{m})$ is the projection of matrix $\boldsymbol{A}_{m}$ into an orthonormal basis.

Still, $\boldsymbol{W}_{m}^{(p)}$ is not only composed by phase shifters, i.e. the entries are not roots of unity. Therefore, for given $\boldsymbol{W}_{m}^{(p)}$, we aim at solving the following optimization problem:

Although the optimal solution is obtained by taking the phase of the eigenvectors in $\boldsymbol{W}_{m}^{(p)}$, the orthogonality between columns achieved by $\mathcal{P}(\cdot)$ would be broken. Therefore, we modify our receiver. We add an orthogonality compensation matrix into our digital processing [9]. More concretely, for a constrained analog beamformer $\boldsymbol{\hat{W}}_{m}$, its SVD results in $\boldsymbol{\hat{W}}_{m}=\boldsymbol{\hat{U}}_{m}\boldsymbol{\hat{D}}_{m}\boldsymbol{\hat{V}}_{m}^{*}$. The orthogonality compensation matrix, denoted by $\boldsymbol{F}_{m}$, is defined as

Therefore, adding such a compensation matrix allows us to improve the design of the analog matrix exploiting the following proposition.

Using the previous proposition, the initial unconstrained beamformer $\boldsymbol{W}_{m}^{(p)}$ can be replaced by $\boldsymbol{W}_{m}^{(p)}\boldsymbol{A}_{m}$ without a performance degradation as long as $\boldsymbol{A}_{m}$ is nonsingular. As a consequence, we can formulate the following optimization problem:

Thanks to the degrees of freedom added by $\boldsymbol{A}_{m}$, the constrained analog beamformer $\boldsymbol{\hat{W}}_{m}$, can be made closer to the unconstrained one $\boldsymbol{W}_{m}^{(p)}$. By alternating minimization, we split the previous problem into two sub-problems: (i) find the optimal $\boldsymbol{A}_{m}$ for fixed $\boldsymbol{\hat{W}}_{m}$ and (ii) find the optimal $\boldsymbol{\hat{W}}_{m}$ for fixed $\boldsymbol{A}_{m}$. The solution to the previous subproblems is

An iterative process based on the block coordinate descend method follows until convergence is reached [14]. Therefore, a constrained analog matrix will be obtained and thus from Eq. (36) we can create $\boldsymbol{F}_{m}$ that goes into the baseband (or digital) part. As a consequence, the equivalent channel between AP $m$ and UE $k$ has an extra component:

The optimal solution to (32) with respect to $\boldsymbol{\Phi}$ requires an exhaustive search over the set of possible pilot sequences. However, based on the correlation between effective channels: $\Delta_{k,i}=\text{tr}(\boldsymbol{\Gamma}_{k}^{(g)}\boldsymbol{\Gamma}_{i}^{(g)})$ for $k\neq i$, an initial pilot assignment can be made, denoted by $\boldsymbol{\Phi}^{(0)}$. Particularly, a set of users is assigned the same pilot if their normalized cross-correlation, i.e. $\frac{\Delta_{k,i}}{\text{tr}(\boldsymbol{\Gamma}_{k}^{(g)})\text{tr}(\boldsymbol{\Gamma}_{i}^{(g)})}$, is minimized. Afterwards, the greedy algorithm proposed in Alg. 1 combined with the asymptotic approximations can be used to iteratively update the UE pilot assignment in a max-min SINR sense. Additionally, by construction, Alg. 1 converges provided that the cost function (i) is non-decreasing and (ii) is upper bounded.