Robust Linear Hybrid Beamforming Designs Relying on Imperfect CSI in mmWave MIMO IoT Networks

In a typical linear decentralized estimation setup, a large number of IoTNos are deployed in an area of interest which one wants to monitor. The IoTNos transmit their precoded observations over a coherent multiple access channel (MAC) to the FC for receiver combining (RC) and final estimation [16]. Coherent MAC requires that all the IoTNos in the network are synchronized and their individual transmitted signals arrive as a coherent sum at the FC. The FC determines the transmit precoders (TPCs) for all IoTNos in the IoTNe and subsequently, feeds back to each IoTNo its own TPC weights through a feedback link. The pioneering paper by Xiao et al. [17] proposed an optimal linear transceiver design for the efficient decentralized estimation of a vector parameter for a coherent MAC-based MIMO IoTNe. However, the analysis of [17] assumed a simplistic diagonal model for the MIMO channel between each IoTNo and the FC, which restricts the applicability of their results. By extending this framework, Behbahani et al. [18] proposed a ground-breaking iterative procedure for a joint transceiver design relying on mean square error (MSE) minimization for vector parameter estimation at the FC. Along similar lines, Liu et al. [19] developed decentralized and distributed schemes for the efficient estimation of a parameter vector. However, the algorithms developed in [18] and [19] are based on the iterative block coordinate descent (BCD) framework, which results in a high computational complexity, especially in a MIMO IoTNe associated with a large number of IoTNos. The authors of [20, 21, 22] proposed an optimal IoTNo collaboration strategy, wherein each IoTNo shares its observations with its neighbouring IoTNos and only a subset of the IoTNos transmit their observations to the FC for the final estimation of the unknown parameter, which leads to an improved estimation performance. However, the inter-IoTNo communication required for supporting this can potentially impose high overheads.

Another major drawback of all the above contributions is their reliance on perfect channel state information (CSI), which can be a significant stumbling block in the practical implementation of such systems. Given the stringent bandwidth constraints, it is also desirable that each node transmits very few pilot symbols to the FC, which often leads to a very coarse estimate of the channel state information (CSI). Therefore, this forms the motivation to develop robust hybrid TPC designs incorporating the uncertainty in the available estimate of the CSI. To address this limitation of the existing literature, Venkategowda et al. [23] developed robust TPC designs by considering both the so-called norm ball and the ellipsoidal CSI uncertainty models. By further extending this idea, Liu et al.[24] developed centralized and distributed robust transceiver designs by relying on MSE minimization subject to per IoTNo power constraints, and total power minimization under a specific MSE constraint. However, their work considered only the ellipsoidal CSI uncertainty model. Iterative schemes were also conceived for robust transceiver design for vector parameter estimation in [25] by considering the norm ball CSI uncertainty model. At this juncture, it is important to note that while the above treatises have comprehensively addressed the problem of TPC/ RC design for IoTNes, they cannot be directly extended to a 5G mmWave-based MIMO IoTNe relying on hybrid beamforming. Hence, the contributions specifically related to mmWave MIMO beamforming are reviewed in detail next.

Hybrid TPC/ RC designs have been proposed in[12, 13, 14, 15, 26, 27] for cellular mmWave MIMO systems. The authors of [28] developed a novel signal processing strategy for the reconfigurable intelligent surface (RIS)-aided MIMO uplink for energy and spectral efficiency maximization. An interesting analysis has been carried out therein to study the trade off between the energy and spectral efficiency. Furthermore, Huang et al. [29] have proposed an interesting deep reinforcement learning-based hybrid beamforming design for an RIS-aided Terahertz system. Recently, the authors of [30] have proposed both the centralized (C)- and the asynchronous distributed (AD)-alternating direction method of multipliers (ADMM)-based schemes for linear minimum MSE (LMMSE) hybrid transceivers employed in coherent MAC-based IoTNe. Furthermore, Liu et al. [7] proposed a sophisticated technique for hybrid transceiver designs relying on perfect CSI in the context of an orthogonal MAC-based IoTNe. Briefly, hybrid transceiver designs were developed relying on the ADMM and steepest descent (SD) techniques. Since, the designs proposed in [30] and [7] are iterative, they potentially lead to a high complexity. Additionally, it must also be borne in mind that the orthogonal MAC employed in [7] for IoTNo communication has a low bandwidth efficiency in comparison to the coherent MAC [30, 17], especially in modern IoTNes having a large number of nodes. Similar to sub-6 GHz systems, the availability of perfect CSI is also a serious limiting factor in the design of mmWave MIMO systems, which has only been addressed by a few studies [31, 32, 33]. Luo et al. [31] conceived a hybrid transceiver for a scenario having a single source and multiple relays. The proposed designs target the maximization of the average signal-to-noise (SNR) power ratio at the destination using realistic imperfect CSI of the source-relay and relay-destination links. The cutting-edge paper of Jiang and Jafarkhani [32] designed a low complexity robust hybrid transceiver for a single-relay cooperative communication system, leveraging the principle of mutual information maximization. As a further advance, Zhao et al. [33] proposed a scheme for robust joint hybrid transceiver design in a full-duplex (FD) multi-cell mmWave network to maximize the sum-rate. However, the majority of these solutions cannot be incorporated in a IoTNe framework, since they do not address the associated stringent bandwidth and power limitations. Furthermore, the authors of [31, 32, 33] consider only the stochastic CSI uncertainty model, which can lead to a poor worst-case performance. To the best of our knowledge, none of the existing treatises derived the hybrid beamforming techniques for vector parameter estimation in a coherent MAC- based mmWave MIMO IoTNe, accounting also for the imperfect nature of the CSI in a realistic implementation. Hence we are inspired to fill this knowledge gap in the existing literature. The main contributions of this study are itemized next and they are also boldly contrasted to the literature in Table-I.

• •

  Non-iterative hybrid linear beamforming techniques are proposed for minimizing the MSE of vector parameter estimation at the FC in a mmWave MIMO IoTNe, initially assuming perfect CSI availability. Sophisticated optimization paradigms are employed for incorporating both total and per-IoTNo power constraints into our TPC design.

• •

  A two-step procedure is proposed for overcoming the non-convexity associated with the MSE optimization problem, arising due to the constant gain constraint for the elements of the RF TPC and RC. The optimality of this procedure is rigorously justified via a mathematical proof of the proposition. Closed-form expressions are derived for the fully digital minimum MSE (MMSE) TPCs, employing a suitable setting for the RF combiner.

• •

  Subsequently, a low-complexity simultaneous orthogonal matching pursuit (SOMP)-based technique is proposed for designing the RF and baseband components of the fully-digital MMSE precoder.

• •

  In contrast to the existing works [7], [30], we consider the realistic scenario of practical CSI uncertainty, and subsequently develop robust linear hybrid TPCs by modelling the CSI uncertainty using both the stochastic and the norm ball CSI uncertainty models for the above system. This reduces the pilot overhead and allows the IoTNe to work well even when the channel estimate is not very accurate.

• •

  The centralized MMSE bound is derived, which acts as a benchmark for characterizing the estimation performance of the proposed designs.

• •

  Finally, our simulation results illustrate the performance of the various analytical solutions derived in this paper.

The remainder of the paper is organized as follows. Section-II presents the mmWave MIMO IoTNe system model and formulates the MSE minimization problem for parameter estimation. Section-III describes the detailed procedure of obtaining the fully digital MMSE TPCs and the algorithm of determining the RF/ baseband TPC weights. Section-IV describes a pair of novel robust hybrid pre-processing designs accounting for the CSI uncertainty using both the stochastic and the norm ball CSI uncertainty models. Our simulation results are provided in Section-V, while our conclusions are given in Section-VI. The proofs of the various propositions are provided in the Appendices.

Notation: Small and capital boldface letters $\mathbf{d}$ and $\mathbf{D}$ denote vectors and matrices, respectively. The transpose and Hermitian transpose operations are denoted by $(.)^{T}$ and $(.)^{H}$, respectively. Furthermore, $\left[\mathbf{D}\right]_{i,j}$ denotes the $(i,j)$th element of a matrix $\mathbf{D}$. The operator $\mathbb{E}\{.\}$ denotes the statistical expectation operator, while $\mathrm{diag}(\mathbf{D}_{1},\mathbf{D}_{2},\ldots,\mathbf{D}_{L})$ denotes a block diagonal matrix constituted by the matrices $\mathbf{D}_{1}$, $\mathbf{D}_{2}$ upto $\mathbf{D}_{L}$ on its principal diagonal. The $l_{2}$ and Frobenius norms of vectors and matrices, respectively, are represented by $||.||$ and $||.||_{F}$, respectively, whereas $l_{0}$ norm is represented by $||.||_{0}$. The symbol $\otimes$ denotes the matrix Kronecker product and $\lambda_{\text{max}}(\mathbf{D})$ represents the maximum eigenvalue of the matrix $\mathbf{D}$. The operator $\mathrm{vec}(\mathbf{D})$ stacks the columns of a matrix $\mathbf{D}$ into a vector, while the operator $\mathrm{vec}^{-1}_{m}(\mathbf{d})$ converts a vector $\mathbf{d}$ into a matrix of $m$ rows and the appropriate number of columns. $\mathbf{X}=\mathbf{D}\left(:,\mathcal{K}\right)$ represents a sub-matrix $\mathbf{X}$ consisting of those columns of the matrix $\mathbf{D}$ whose information is provided by the set $\mathcal{K}$.

The average transmit power of the $m$th IoTNo can be expressed as

Therefore, the expression for the total average transmit power of the IoTNe is given by

Exploiting now the identity in (12), the average transmit power expression of the $m$th IoTNo in (III-A) can be recast as

where we have $\mathbf{\Gamma}_{m}=\left[\left(\mathbf{A}_{m}\mathbf{R}_{\theta}\mathbf{A}^{H}_{m}+\mathbf{R}_{m}\right)^{T}\otimes\mathbf{I}_{N_{T}}\right]\in\mathbb{C}^{qN_{T}\times qN_{T}}$. Hence, the compact expression for the total power of the IoTNe can be formulated as

where we define the matrix $\mathbf{\Gamma}=\mathrm{diag}\left(\mathbf{\Gamma}_{1},\mathbf{\Gamma}_{2},\ldots,\mathbf{\Gamma}_{M}\right)\in\mathbb{C}^{MqN_{T}\times MqN_{T}}$. Therefore, the optimization problem that minimizes the resultant MSE at the FC constrained by the zero-forcing and total network power constraints in (16) and (22), respectively, is given by

where the quantity $\rho_{T}$ denotes the total IoTNe transmit power level, which can be utilized for transmitting observations to the FC. Once again, upon applying the KKT framework[39], the optimal TPC vector $\mathbf{f}^{\mathrm{opt}}$ is obtained as

where $\lambda^{\mathrm{opt}}$ denotes the optimal dual variable corresponding to the inequality constraint. One can follow the procedure given after Eq. (18) to determine the optimal TPC matrix for every IoTNo in the network from the optimal TPC vector $\mathbf{f}^{{\mathrm{opt}}}$. The MSE of parameter estimation in (13) can also be minimized by taking each sensor’s transmit power level into account. The corresponding optimization paradigm can be formulated as

where the quantity $\rho_{m}$ denotes the transmit power constraint of the $m$th IoTNo. Upon exploiting the convex nature of the above optimization problem, standard convex optimization solvers, such as CVX [40], can be employed for finding the solution. The corresponding optimal TPC matrix $\mathbf{F}_{m}^{\mathrm{opt}}$ can be obtained, once again, using a procedure similar to the one described after Eq.(18). The next subsection describes the procedure of decomposing each TPC matrix $\mathbf{F}_{m}$ into the corresponding RF and baseband TPC matrices $\mathbf{F}_{\text{RF},m}$ and $\mathbf{F}_{\text{BB},m}$, respectively, for each IoTNo $m$ in the IoTNe.

The optimal TPC vector in (18) can be rewritten as,

where $\tilde{\mathbf{b}}=\mathbf{Z}^{H}\left(\mathbf{Z}\mathbf{\Psi}^{-1}\mathbf{Z}^{H}\right)^{-1}\mathbf{b}\in\mathbb{C}^{MqN_{T}\times 1}$. Furthermore, the MMSE digital TPC vector $\mathbf{f}^{\mathrm{opt}}$ can be rearranged as

It can be readily deduced from the above equation that the MMSE digital TPC vector corresponding to the $m$th IoTNo can be derived as $\mathbf{f}^{\mathrm{opt}}_{m}=\textrm{vec}\left(\mathbf{F}_{m}^{\mathrm{opt}}\right)=\mathbf{{\Psi}}_{m}^{-1}\tilde{\mathbf{b}}_{m}$. The quantity $\tilde{\mathbf{b}}_{m}$ can be obtained by extracting the $[(m-1)qN_{T}]$th element to the $[mqN_{T}]$th elements of the column vector $\tilde{\mathbf{b}}$. Furthermore, the optimal TPC matrix for the $m$th IoTNo can be determined as $\mathbf{F}_{m}^{\mathrm{opt}}=\mathrm{vec}^{-1}_{q}\left(\mathbf{f}_{m}\right)$. Subsequently, the problem of decomposing $\mathbf{F}_{m}^{\mathrm{opt}}$ into its baseband and RF components $\mathbf{F}_{\text{BB},m}$ and $\mathbf{F}_{\text{RF},m}$, respectively, can be formulated as the following best linear approximation problem:

Note that the constant magnitude constraint for the elements of the RF TPC $\mathbf{F}_{\text{RF},m}$ makes the above problem non-convex and intractable. One can overcome this problem by exploiting the following interesting relationship between the MMSE TPC matrix $\mathbf{F}_{m}^{\mathrm{opt}}$ and the corresponding transmit array response matrix $\mathbf{A}_{s,m}$.

Thus, one can choose the columns of $\mathbf{F}_{\text{RF},m}$ as the columns of the transmit array response matrix $\mathbf{A}_{s,m}$. The resultant optimization problem can be formulated as

where the constraint $||\mathrm{diag}(\widetilde{\mathbf{F}}_{\text{BB},m}\widetilde{\mathbf{F}}_{\text{BB},m}^{H})||_{0}=N^{m}_{\text{RF}}$ results from the fact that only $N^{m}_{\text{RF}}$ out of $N_{T}$ rows must be non-zero, since there are only $N^{m}_{\text{RF}}$ RF chains at the $m$th IoTNo in the hybrid mmWave MIMO IoTNe. Thus, the matrix $\widetilde{\mathbf{F}}_{\text{BB},m}\in\mathbb{C}^{N_{cl}\times q}$ is block sparse in nature. The sparse signal recovery TPC design problem in (30) can be readily solved by employing the popular SOMP algorithm. The algorithm terminates after a finite number of iterations, i.e. it requires $N^{m}_{\text{RF}}$ iterations to choose the $N^{m}_{\text{RF}}$ dominant transmit array response vectors $\mathbf{a}_{\text{s}}(\theta_{k,m})$ from the matrix $\mathbf{A}_{s,m}$. A brief summary of the algorithm follows.

The algorithm begins with step (5) that evaluates the projection of each column of $\mathbf{A}_{s,m}$ on every column of the residual matrix $\mathbf{F}_{\text{res}}$. Step (6) selects the specific column of $\mathbf{A}_{s,m}$ that is maximally correlated with the columns of $\mathbf{F}_{\text{res}}$. In step (7), the selected dominant vector is appended to the RF TPC matrix $\mathbf{F}_{\text{RF,m}}$. Upon employing this updated $\mathbf{F}_{\text{RF,m}}$, the classic least squares procedure is exploited for calculating the updated baseband TPC $\mathbf{F}_{\text{BB,m}}$ in step (8). The residual matrix $\mathbf{F}_{\text{res}}$ is updated in step (9). The above steps are repeated $N^{m}_{\text{RF}}$ times, followed by step (11) that scales the baseband TPC $\mathbf{F}_{\text{BB},m}$ to meet the transmit power constraint, to obtain the hybrid TPCs $\mathbf{F}_{\text{RF},m}$ and $\mathbf{F}_{\text{BB},m}$, corresponding to IoTNo $m$. Finally, the block diagonal $MN_{T}\times MN_{\text{RF}}$ RF TPC $\mathbf{F}_{\text{RF}}$ and the $MN_{\text{RF}}\times Mq$ baseband TPC $\mathbf{F}_{\text{BB}}$ are obtained using steps (13) and (14), respectively.

The computational complexities of the various steps of Algorithm 1 are shown in Table II. The complexity of Algorithm 1 given the fully-digital transmit precoder matrix $\mathbf{F}_{m}$ is of the order of $\mathcal{O}\left(N_{\text{RF}}^{m}N_{T}N_{cl}q+(N_{\text{RF}}^{m})^{2}N_{T}^{2}\right)$. As shown in Table III, the SOMP algorithm has a low complexity, i.e., $\mathcal{O}\bigg{(}\left(MqN_{T}\right)^{3}+M(N_{\text{RF}}^{m})^{2}N_{T}^{2}\bigg{)}$ in contrast to the manifold optimization [14] or ADMM-based [30] iterative design frameworks, which have a complexity order of $\mathcal{O}\left(\left(MqN_{T}\right)^{3}+MN_{\text{RF}}^{m}N_{T}^{3}q^{2}N_{\text{inn}}N_{\text{out}}\right)$ and $\mathcal{O}\left(M(N_{\text{RF}}^{m})^{2}q^{4}N_{\text{I}}+M(N_{\text{RF}}^{m})^{3}N_{T}^{3}N_{\text{I}}\right)$, respectively.
TABLE II: Computational Complexity of Algorithm 1 Expression Computational complexity $\boldsymbol{\Phi}$ $\mathcal{O}\left(N_{\text{RF}}^{m}N_{cl}N_{T}q\right)$ $\boldsymbol{\Phi}\boldsymbol{\Phi}^{H}$ $\mathcal{O}\left(N_{\text{RF}}^{m}N_{cl}^{2}q\right)$ $\mathbf{F}_{\text{RF},m}$ $\mathcal{O}\left(N_{\text{RF}}^{m}N_{cl}N_{T}q+N_{\text{RF}}^{m}N_{cl}^{2}q\right)$ $\mathbf{F}_{\text{BB},m}$ $\mathcal{O}\left(\left(N_{\text{RF}}^{m}\right)^{2}N_{T}^{2}+\left(N_{\text{RF}}^{m}\right)^{2}N_{T}q\right)$ $\mathbf{F}_{\text{res}}$ $\mathcal{O}\left(N_{\text{RF}}^{m}N_{T}q\right)$ TABLE III: Computational Complexity Comparison Algorithm Computational complexity Proposed $\mathcal{O}\bigg{(}\left(MqN_{T}\right)^{3}+M(N_{\text{RF}}^{m})^{2}N_{T}^{2}\bigg{)}$ MO-AltMin [15] $\mathcal{O}\left(\left(MqN_{T}\right)^{3}+MN_{\text{RF}}^{m}N_{T}^{3}q^{2}N_{\text{inn}}N_{\text{out}}\right)$ ADMM [30] $\mathcal{O}\left(M(N_{\text{RF}}^{m})^{2}q^{4}N_{\text{I}}+M(N_{\text{RF}}^{m})^{3}N_{T}^{3}N_{\text{I}}\right)$

The centralized MMSE bound can be derived as described next in order to benchmark the performance of the proposed hybrid beamforming designs. This is based on the principle that the best MSE performance is obtained, when the observations taken by every IoTNo in the IoTNe are available directly at the FC without any degradation. For such a hypothetical system, the MMSE estimator at the FC results in the best MSE performance of parameter estimation. The observations x for the centralized scenario can be expressed as

where the observation vector $\mathbf{x}\in\mathbb{C}^{Mq\times 1}$ and noise vector $\mathbf{v}\in\mathbb{C}^{Mq\times 1}$ are defined as

Note that the noise vector $\mathbf{v}$ is distributed as $\mathbf{v}\sim\mathcal{CN}\left(\mathbf{0},\mathbf{R}_{v}\right)$, where the covariance matrix $\mathbf{R}_{v}\in\mathbb{C}^{Mq\times Mq}$ and the overall observation matrix $\mathbf{A}\in\mathbb{C}^{Mq\times p}$ are defined as

The MSE of this centralized MMSE estimate is given as [41]

which is the centralized MMSE bound for the system under consideration. The next section develops robust hybrid beamformer design techniques in the presence of imperfect CSI.

Employing the stochastic/ Gaussian CSI uncertainty model[32, 42, 43], the channel matrix $\mathbf{H}_{m}$ can be modelled as

where the matrix $\widehat{\mathbf{H}}_{m}\in\mathbb{C}^{N_{R}\times N_{T}}$ denotes the available channel estimate, whereas $\Delta{\mathbf{H}}_{m}\in\mathbb{C}^{N_{R}\times N_{T}}$ represents the estimation error. The matrices $\mathbf{R}_{\text{FC}}\in\mathbb{C}^{N_{R}\times N_{R}}$ and $\mathbf{R}_{s,m}\in\mathbb{C}^{N_{T}\times N_{T}}$, represent the receive and transmit correlation matrices corresponding to the FC and the $m$th sensor, repectively[42, 43]. These can be modelled as $\mathbf{R}_{\text{FC}}(i,j)=\sigma_{H}^{2}\alpha^{|i-j|}$ and $\mathbf{R}_{s,m}(i,j)=\beta^{|i-j|}_{m}$, where $\alpha$ and $\beta_{m}$ represent the receive and transmit correlation, and $\sigma_{H}^{2}$ denotes the uncertainty variance of the CSI. Each element of the matrix $\mathbf{S}_{m}\in\mathbb{C}^{N_{R}\times N_{T}}$ is assumed to be independent and identically (i.i.d.) distributed with mean $0$ and variance one. Substituting $\mathbf{H}_{m}$ according to the channel model in (37), the equivalent received vector $\widetilde{\mathbf{y}}$ obtained after RF combining at the FC can be rewritten as

The estimation constraint under imperfect CSI availability can be reformulated as

Applying the $\mathrm{vec}$ operation at both sides and using the identity $\mathrm{vec}\left(\mathbf{N}\mathbf{P}\mathbf{Q}\right)=\left(\mathbf{Q}^{T}\otimes\mathbf{N}\right)\mathrm{vec}(\mathbf{P})$, the above constraint can be rewritten as

where we have $\mathbf{W}_{m}=\left[\mathbf{A}_{m}^{T}\otimes\mathbf{W}_{\text{RF}}^{H}\widehat{\mathbf{H}}_{m}\right]\in\mathbb{C}^{p^{2}\times qN_{T}}$ and $\mathbf{W}=[\mathbf{W}_{1},\mathbf{W}_{2},\ldots,\mathbf{W}_{M}]\in\mathbb{C}^{p^{2}\times MqN_{T}}$. The resultant MSE of parameter estimation is given by

The following lemma can now be used to derive the average-MSE defined as $\overline{\mathrm{MSE}}=\mathbb{E}\left[\mathrm{MSE}\right]$:

Employing the above lemma, one can simplify the MSE expression in (IV-A) to determine the average MSE as

By exploiting the property in (12), the above expression can be further simplified to:

The above expression can be recast as