Simultaneous Diagonalization Using Baseband Beamforming in mmWave MU-MIMO Systems

Abstract

We consider the problem of simultaneous diagonalization of Hermitian matrices for the desired and co-channel interference terms of millimeter-wave (mmWave) multi-user multiple-input multiple-output systems. The joint unitary eigenvectors and the corresponding eigenvalues are known to assist in the mathematical tractability of key performance metrics, such as outage probability, ergodic capacity, and spectral efficiency. We formulate the signal-to-interference-plus-noise ratio in a canonical quadratic form and subsume the digital baseband beamforming vectors in the weight matrices of channels at the transmitter side. Next, a real scalar objective function is defined, which quantifies the correlation loss due to joint-diagonalization. The objective function is then maximized using baseband beamforming under the hardware constraints of the mmWave system. Through simulations, the proposed beamforming algorithm is evaluated by employing several non-linear optimization sub-routines, and it is shown that the “active-set” approach results in improved summary statistics both for the correlation metric and for the time complexity. We also reflect on the effect of optimization on the channel scatterers in mmWave systems.

1. Introduction

Millimeter-wave (mmWave) multi-user multiple-input multiple-output (MU-MIMO) is considered a key technology for the next generation of communication networks [1,2,3,4,5]. The spectrum of mmWave includes the frequencies in the range 30–300 GHz, whereas the large bandwidth can address the data throughput congestion due to the massive number of devices connected to the network. Rain attenuation and oxygen absorption greatly, albeit non-linearly, influence the signal propagation in this spectrum [6,7]. Nevertheless, mmWave enabled devices can house large antenna arrays and hence provide high multiplexing gains [8]. In this context, a large antenna array at the base station (BS) also helps in catering to multiple mobile stations (MSs) by means of the analog, digital, or hybrid precoding schemes.

Analog beamforming is operated in the radio frequency (RF) domain by means of the RF analog phase shifter at each antenna element of the BS [9,10,11]. Analog beamforming is, however, inefficient in the management of co-channel interference. Digital beamforming is highly useful for the conventional low frequency MU-MIMO systems [12] and has tremendous potential to efficiently mitigate co-channel interference. However, digital beamforming fails to meet the hardware and computational complexity constraints of the mmWave systems. Hence, a hybrid beamforming approach emerged as an efficient solution for the mmWave MU-MIMO systems in which the precoding includes both the digital baseband and analog RF beamforming [13,14]. A landmark work on hybrid beamforming is [15], wherein a low complexity, albeit highly efficient, hybrid beamforming method with limited feedback is proposed for the downlink mmWave MU-MIMO system.

We noticed an inherent quadratic formulation in the expression of the signal-to-interference-plus-noise ratio (SINR) in [15], and formulated the problem in a canonical structure of indefinite quadratic forms [16,17,18]. While such forms enable the exact closed-form expression of outage probability, other metrics such as ergodic capacity and spectral efficiency are rather involved from the characterization point of view. Nevertheless, it was reported in [17] that by jointly diagonalizing the weight matrices of the desired and co-channel interference terms, the mathematical tractability of the later metrics can also be achieved. This paper focuses on the effectiveness of simultaneous diagonalization in mmWave settings. We will show that by joint-diagonalization of weight matrices, spectral efficiency is not degraded as compared with zero-forcing (ZF) beamforming by adopting the digital baseband beamforming, albeit now we have simple eigenvalue based equations with which it is easier to work. It is hoped that closed-form expressions of relevant performance metrics such as outage probability and the achievable rate can be achieved in future works based on the tools and methods discussed herein.

In the context of joint-diagonalization, the weight matrices of the desired and co-channel interference terms are Hermitian and have a unitary joint diagonal structure; hence, techniques such as [19,20] emerge as the potential solutions. These techniques achieve simultaneous diagonalization by utilizing the Jacobi-like algorithm with plane rotations, while exact diagonalization is achieved for the commutative matrices. Specifically, Reference [20] has been widely used in the new generation of communication systems, e.g., in [21,22], and hence, it is selected as the technique of joint-diagonalization in our work.

In this paper, we develop a constrained optimization problem where a scalar correlation metric is set as an objective function and the optimization is performed using the digital baseband beamformer design. The main contributions are enumerated next:

• We formulate the SINR of the mmWave MU-MIMO system in a canonical quadratic form where the weight matrices of the channels are dependent on the digital baseband beamformers.
• We design an algorithm for the maximization of a proposed correlation metric, which nests the simultaneous joint-diagonalization approach in [20]. The quality-of-service (QoS) constraints are defined while the optimization variables are restricted to digital baseband beamformers only.
• A generic algorithm is developed by transforming a multi-objective problem to a single-objective problem using linear-scalarization. The algorithm encompasses several “of-the-shelf” optimization sub-routines, e.g., the interior-point method, sequential quadratic programming, and the active-set approach.

The proposed algorithm is evaluated and analyzed using a simulation environment, and the results indicate that simultaneous diagonalization of Hermitian matrices can be efficiently achieved without violating QoS constraints.

The organization of the rest of this paper is as follows. In Section 2, the system model is presented, and in Section 3, two approaches are outlined for the canonical quadratic formulation. In Section 4, we provide digital baseband beamformer design. We showcase in Section 5 the contrasting features of the algorithm sub-routines. In Section 6, we present a summary and conclusions.

Notations: Throughout this paper, vectors and matrices are indicated by lower-case and upper-case bold letters, respectively. I identifies the identity matrix. The notations $\left|a\right|$, ${∥\mathbf{a}∥}^2$, and $\left|\mathbf{A}\right|$ denote the absolute value of scalar a, the $L2$-norm of vector a, and the determinant of matrix A, respectively. The notations ${(·)}^T$ and ${(·)}^H$ represent transposition and conjugate transposition, respectively. Furthermore, ${(·)}^{\frac{1}{2}}$ denotes the matrix square root, and $\mathbb{E}(·)$ represents the expectation operator. Finally, the weighted norm ${\parallel \mathbf{a}\parallel }_{\mathbf{A}}^2$ is used to denote the quadratic form ${\mathbf{a}}^H\mathbf{A}\mathbf{a}$.

2. System Model

The system model considered in this work is based on [15] and shown in Figure 1 where a downlink mmWave MU-MIMO system with the BS having N antennas and $N_{RF}$ analog RF chains serves K users. The kth user, $k\in \{1,2,\cdots ,K\}$, has $M_k$ receive antenna elements, and it is linked to the kth stream, i.e., the number of streams $N_s=K$. Each stream has the same total average transmit power $P_t$. Furthermore, the following assumptions are considered: (i) the total number of users is bounded as $K\le N_{RF}$; (ii) the transmitted symbol $\mathbf{s}={[s_1,s_2,\cdots ,s_K]}^T$ follows $\mathbb{E}\left[\mathbf{s}{\mathbf{s}}^H\right]=\frac{1}{\eta }{\mathbf{I}}_K$, where $\eta =\frac{K}{P_t}$; and (iii) a geometric channel model of the kth user is considered, which incorporates the effect of minimum scattering, where each scatterer produces a single/dominant path and the power in other paths is negligible, i.e., ${\mathbf{H}}_k$ is dependent on L scatterers as outlined in the prior works [15,23,24].

On the receiver side, the signal received at the kth user is given by:

$${\mathit{y}}_k={\mathbf{w}}_k^H{\mathbf{H}}_k{\mathbf{F}}_{RF}{\mathbf{f}}_k^{BB}{s}_k+\sum _{i=1,i\ne k}^K{\mathbf{w}}_k^H{\mathbf{H}}_k{\mathbf{F}}_{RF}{\mathbf{f}}_i^{BB}{s}_i+{\mathbf{w}}_k^H{z}_k.$$

(1)

Here, ${\mathbf{w}}_k\in {\mathbb{C}}^{N\times 1}$ is the analog RF combiner of the kth user, ${\mathbf{H}}_K$ is an $N\times M_k$ channel between the BS and kth user, ${\mathbf{F}}_{RF}$ is the analog RF precoder where ${\mathbf{F}}_{RF}=[{\mathbf{f}}_1^{RF},{\mathbf{f}}_2^{RF},\cdots ,{\mathbf{f}}_k^{RF}]$, ${\mathbf{f}}_k^{BB}$ is the digital baseband precoder, ${\mathbf{f}}_k^{BB}\in \{{\mathbf{f}}_1^{BB},{\mathbf{f}}_2^{BB},\cdots ,{\mathbf{f}}_K^{BB}\}$, and $z_k$ is white noise with zero mean and variance ${\sigma }_k^2$.

In (1), the first term identifies the desired signal, the second term the co-channel interference, and the third term the additive noise. Furthermore, the constraints on ${\mathbf{f}}_k^{RF}$ and ${\mathbf{f}}_k^{BB}$, $\forall k$ are similar to the ones identified in ([15], Equation (7)).

The instantaneous SINR for the kth user, denoted as ${\gamma }_k$, is formulated as:

$${\gamma }_k={\displaystyle \frac{|{\mathbf{w}}_k^H{\mathbf{H}}_k{\mathbf{F}}_{RF}{\mathbf{f}}_k^{BB}{|}^2}{{\sum }_{i=1,i\ne k}^K{\left|{\mathbf{w}}_k^H{\mathbf{H}}_k{\mathbf{F}}_{RF}{\mathbf{f}}_i^{BB}\right|}^2+\eta {\sigma }_k^2}}.$$

(2)

3. Problem Formulation

In this section, we outline an approach that can simplify the SINR expression in (2). The approach is based on the quadratic formulation, and it employs channel transformations to achieve an equivalent channel model. In this model, we absorb the analog RF precoder and combiner in the channel matrix, i.e., ${\mathbf{w}}_k^H{\mathbf{H}}_k{\mathbf{F}}_{RF}\equiv {\mathbf{h}}_k^H$. This transformation yields systematic simplification of an equivalent ${\gamma }_k$ as follows:

$$\begin{array}{ccc}\hfill {\gamma }_k& =& {\displaystyle \frac{|{\mathbf{h}}_k^H{\mathbf{f}}_k^{BB}{|}^2}{{\sum }_{i=1,i\ne k}^K{\left|{\mathbf{h}}_k^H{\mathbf{f}}_i^{BB}\right|}^2+\eta {\sigma }_k^2}}\hfill \\ & =& {\displaystyle \frac{\left({\mathbf{h}}_k^H{\mathbf{f}}_k^{BB}\right){\left({\mathbf{h}}_k^H{\mathbf{f}}_k^{BB}\right)}^H}{{\sum }_{i=1,i\ne k}^K\left({\mathbf{h}}_k^H{\mathbf{f}}_i^{BB}\right){\left({\mathbf{h}}_k^H{\mathbf{f}}_i^{BB}\right)}^H+\eta {\sigma }_k^2}}\hfill \\ & =& {\displaystyle \frac{{\mathbf{h}}_k^H{\mathbf{f}}_k^{BB}{\mathbf{f}}_k^{BB;H}{\mathbf{h}}_k}{{\sum }_{i=1,i\ne k}^K{\mathbf{h}}_k^H{\mathbf{f}}_i^{BB}{\mathbf{f}}_i^{BB;H}{\mathbf{h}}_k+\eta {\sigma }_k^2}}\hfill \\ & =& {\displaystyle \frac{{∥{\mathbf{h}}_k∥}_{{\mathbf{A}}_k}^2}{{∥{\mathbf{h}}_k∥}_{{\mathbf{B}}_k}^2+\eta {\sigma }_k^2}}.\hfill \end{array}$$

(3)

Here, ${\mathbf{A}}_k$ and ${\mathbf{B}}_k$ are the weight matrices of the desired and co-channel interference terms defined as, respectively,

$$\begin{array}{ccc}\hfill {\mathbf{A}}_k& =& {\mathbf{f}}_k^{BB}{\mathbf{f}}_k^{BB;H},\hfill \end{array}$$

(4)

$$\begin{array}{ccc}\hfill \mathrm{and}{\mathbf{B}}_{\mathrm{k}}& =& \sum _{i=1,i\ne k}^K{\mathbf{f}}_i^{BB}{\mathbf{f}}_i^{BB;H}.\hfill \end{array}$$

(5)

Note that ${\mathbf{A}}_k$ and ${\mathbf{B}}_k$ are Hermitian matrices. The rank of ${\mathbf{A}}_k$ is unity, while the rank of ${\mathbf{B}}_k$ is dependent on the total summations, i.e., K−1.

Next, we utilize the SINR in (3) and select the cumulative density function (CDF) as a probability metric to formulate the problem. The CDF of the kth user, ${\mathit{F}}_k\left({\gamma }_{th}\right)$, is defined as the probability that the instantaneous SINR ${\gamma }_k$ falls below a predefined threshold value ${\gamma }_{th}$, i.e.,

$$\begin{array}{ccc}\hfill {\mathit{F}}_k\left({\gamma }_{th}\right)& =& \mathrm{Pr}({\gamma }_k<{\gamma }_{th})\hfill \\ & =& \mathrm{Pr}\left({\displaystyle \frac{{∥{\mathbf{h}}_k∥}_{\mathbf{A}}^2}{{∥{\mathbf{h}}_k∥}_{\mathbf{B}}^2+\eta {\sigma }_k^2}}<{\gamma }_{th}\right)\hfill \\ & =& \mathrm{Pr}\left(\eta {\sigma }_k^2{\gamma }_{th}+{\gamma }_{th}{∥{\mathbf{h}}_k∥}_{\mathbf{B}}^2-{∥{\mathbf{h}}_k∥}_{\mathbf{A}}^2>0\right)\hfill \\ & =& \mathrm{Pr}\left(\eta {\sigma }_k^2{\gamma }_{th}-{∥{\mathbf{h}}_k∥}_{\mathbf{P}}^2>0\right),\hfill \end{array}$$

(6)

where $\mathbf{P}=\mathbf{A}+\mathbf{C}$, and $\mathbf{C}=-\mathbf{B}{\gamma }_{th}$, and we dropped the subscript k from the matrices for notational brevity. Furthermore, we used the difference and scaling properties of weighted quadratic norms in the fourth equality in (6).

Performing eigenvalue decomposition on the weight matrix P yields $\mathbf{P}={\mathbf{U}}_{\mathbf{P}}{\Lambda }_{\mathbf{P}}{\mathbf{U}}_{\mathbf{P}}^H$, where ${\mathbf{U}}_{\mathbf{P}}$ is a unitary matrix containing eigenvectors, while ${\Lambda }_{\mathbf{P}}$ is a diagonal matrix furnished with the corresponding eigenvalues. Furthermore, since P is the sum of Hermitian matrices A and C, therefore, all the eigenvalues are real, albeit indefinite. While eigenvalue decomposition of the composite matrix P is used directly in some related work (see, e.g., [12,16]), an alternate approach, i.e., the joint-diagonalization of the sum of Hermitian matrices, emerged as a method to solve complicated expressions of outage probability and ergodic capacity in [17,18], respectively. For the joint-diagonalization approach, we are interested in effectively remodeling (6) to:

$$\begin{array}{ccc}\hfill {\mathit{F}}_k\left({\gamma }_{th}\right)& =& \mathrm{Pr}\left(\eta {\sigma }_k^2{\gamma }_{th}-{∥{\tilde{\mathbf{h}}}_k∥}_{{\Lambda }_{\mathbf{A}}+{\Lambda }_{\mathbf{C}}}^2>0\right),\hfill \end{array}$$

(7)

where ${\tilde{\mathbf{h}}}_k={\widehat{\mathbf{U}}}_{\mathbf{P}}^H{\mathbf{h}}_k$, and we need to compute matrix $\widehat{\mathbf{P}}$, such that, $\widehat{\mathbf{P}}={\widehat{\mathbf{U}}}_{\mathbf{P}}\left[{\Lambda }_{\mathbf{A}}+{\Lambda }_{\mathbf{C}}\right]{\widehat{\mathbf{U}}}_{\mathbf{P}}^H$, where ${\widehat{\mathbf{U}}}_{\mathbf{P}}$ is the joint unitary matrix containing eigenvectors, while ${\Lambda }_{\mathbf{A}}$ and ${\Lambda }_{\mathbf{C}}$ are diagonal matrices having the eigenvalues of matrix A and C, respectively.

Since P is dependent on the digital baseband precoder, we will find the appropriate $\widehat{\mathbf{P}}$ by optimizing ${\mathbf{f}}_k^{BB},\forall k$.

4. Baseband Beamformer Design

This section proposes a solution of finding eigenvectors that can jointly diagonalize weight matrices A and C by optimizing ${\mathbf{f}}_k^{BB},\forall k$. The baseband beamformer can also be incorporated into minimizing the overall outage probability in (7); however, this is not the focus of this paper. Since we consider MU-MIMO system, wherein each user has distinct weight matrices, hence the problem is a multi-objective optimization problem. Furthermore, the following considerations are taken into account in the digital baseband beamformers:

• defining the objective function such that $\widehat{\mathbf{P}}$ and $\mathbf{P}$ have a high correlation metric,
• constraining the norms of the digital baseband beamformer for each user,
• maintaining the QoS in terms of sum spectral efficiency, and
• formulating a generic problem on which several “off-the-shelf” non-linear optimization techniques can be employed.

Next, we elaborate each of the above points. Firstly, the weight matrix $\widehat{\mathbf{P}}$ can be achieved by jointly diagonalizing matrices A and C by using the landmark work in [20]. The joint-diagonalization approach obtains the orthonormal change of basis making these weight matrices as diagonal as possible. Furthermore, $\mathbf{P}=\widehat{\mathbf{P}}$ if A and C are commuting matrices. However, this is not possible for the problem at hand. Secondly, constraining the norm of ${\mathbf{f}}_k^{BB},\forall k$ limits the overall transmit power. Thirdly, we transform the multi-objective problem to a single-objective problem by means of linear-scalarization. This entails that the solution of the transformed objective function lies on the Pareto front. Lastly, optimization routines such as the “interior-point method” (IPM), “sequential quadratic programming” (SQP), and the “active-set method” (ASM) can be utilized by nesting the joint-diagonalization algorithm [20] in the proposed algorithm. Using these methods, good local solutions can be obtained on the Pareto front from multiple starting points. While the proposed search method can be initialized using random initial beamvectors, we start the search by initializing the digital baseband beamformer using zero-forcing vectors.

Now, we define a single-objective constrained optimization problem as follows:

$$\begin{array}{cccc}\hfill & \underset{[{\mathbf{f}}_1^{BB},{\mathbf{f}}_2^{BB},\cdots ,{\mathbf{f}}_K^{BB}]}{\mathrm{maximize}}\hfill & \hfill & {\rho }^{\left(i\right)}({\mathbf{f}}_1^{BB},{\mathbf{f}}_2^{BB},\cdots ,{\mathbf{f}}_K^{BB}),\hfill \\ \hfill & \mathrm{subject}\mathrm{to}\hfill & \hfill & {\left\{{∥{\mathbf{f}}_k^{BB}∥}^2\right\}}^{\left(i\right)}\le {\left\{{∥{\mathbf{f}}_k^{BB}∥}^2\right\}}^{(i-1)},\forall k\hfill \end{array}$$

(8)

where ${\rho }^{\left(i\right)}({\mathbf{f}}_1^{BB},{\mathbf{f}}_2^{BB},\cdots ,{\mathbf{f}}_K^{BB})$ is a linearly-scalarized objective function at iteration index (i), and it is defined as:

$${\rho }^{\left(i\right)}=\frac{1}{K}\sum _k^{}\sum _p^{}\sum _q^{}{\left[Cov.(\mathbf{P},\widehat{\mathbf{P}})\right]}_{k,p,q}.$$

(9)

Here, we dropped the argument of the objective function for the brevity of the notations, while $Cov.(\mathbf{P},\widehat{\mathbf{P}})$ is a tensor of dimension $K\times P\times Q$. The first summation transforms the multi-objective optimization problem to a single-objective one, whereas the second and third summations are used to achieve a real scalar value from the elements of covariance matrix of a given user. Herein, $k\in \{1,2,\cdots ,K\}$, $p\in \{1,2,\cdots ,P\}$, $q\in \{1,2,\cdots ,Q\}$, and we restricted the range of ${\rho }^{(.)}$ to $-1\le \rho \le 1$ by normalization with $1/K$.

Since we are concerned only with the digital baseband beamformer design, therefore, other constraints pertinent to analog RF precoders and combiners are not violated. In Algorithm 1, the pseudocode based on the maximization problem given in (8) is presented.

Algorithm 1 Baseband beamformer design

1: Input: Optimization sub-routine: either IPM, SQP, or ASM. 2:            Termination conditions: “max-iteration” and “max-function evaluations”. 3:            ${ϵ}_a$: precision level of the optimization sub-routine. 4:            ${ϵ}_{JD}$: precision level of the joint-diagonalization algorithm. 5: Output: ${\left\{{\mathbf{f}}_k^{BB}\right\}}^{Opt.},\forall k$: optimized digital baseband beamformers. 6: Initialize the time index (i) and the baseband beamvectors ${\mathbf{f}}_k^{BB},\forall k$ using the zero-forcing approach. 7: Initialize the termination condition, i.e., condition = true. 8: repeat 9:       Compute $\mathbf{P}=\mathbf{A}+\mathbf{C}$ and $\widehat{\mathbf{P}}={\widehat{\mathbf{U}}}_{\mathbf{P}}\left[{\Lambda }_{\mathbf{A}}+{\Lambda }_{\mathbf{C}}\right]{\widehat{\mathbf{U}}}_{\mathbf{P}}^H$ using the joint-diagonalization algorithm. 10:     Compute ${\rho }^{\left(i\right)}$ using (8). 11:     $i=i+1$. 12:     Compute ${\rho }^{\left(i\right)}$ from the updated ${\left\{{\mathbf{f}}_k^{BB}\right\}}^{\left(i\right)},\forall k$ by using Steps 9 and 10. 13:     if ${\rho }^{\left(i\right)}-{\rho }^{(i-1)}\ge {ϵ}_a$ then 14:          Update digital baseband beamvectors ${\left\{{\mathbf{f}}_k^{BB}\right\}}^{Opt.}={\left\{{\mathbf{f}}_k^{BB}\right\}}^{\left(i\right)},\forall k$. 15:          Condition = false. 16:     else 17:          Condition = true. 18:     end if 19: until Condition = true

5. Results and Discussion

In this section, the performance of joint-diagonalization by means of the proposed digital baseband beamformer design is provided using computer simulations by using an Intel(R) Core(TM) i7-6500U CPU @2.50 GHz processor with 8 GB RAMs, and the significant results are described and discussed. Since the present work is concerned with the digital beamformers initialized using the zero-forcing method, the rest of the system parameters are the same as in [15]. Specifically, we used $8\times 8$ and $4\times 4$ uniform planar arrays at the BS and MSs, respectively. Single path channels are considered wherein the angles of arrival and departure (AoAs and AoDs) for azimuth and elevation are uniformly distributed in $[0,2\pi ]$ and $[-\frac{\pi }{2},\frac{\pi }{2}]$, respectively.

Figure 2 shows the convergence of $\rho$ in (8) for the IPM, SQP, and ASM methods versus the iteration count for the number of users K set to four and eight. It is observed that as the number of users increase, the correlation metric finding the similarity of $\widehat{\mathbf{P}}$ and $\mathbf{P}$ decreases. Furthermore, Algorithm 1 based on the SQP and ASM methods has faster convergence and higher stability, and both are almost superimposed. Conversely, the IPM based approach has similar convergence at the 10th iteration, but it is not monotonic in nature; hence, it may yield undesired results.

Figure 2 is based on a single channel realization only. To have greater insight, we increase the channel realizations to 100, set $K=4$, and visualize the summary statistics of correlation metric $\rho$ and computational time $\tau$ with box plots in Figure 3. The median is indicated by the central mark on each box, and the 25th and 75th percentiles are indicated by the bottom and top edges of the box. The whiskers account for the coverage of approximately twice the standard deviation, and the “+” symbol identifies the outliers. Figure 3a visualizes the statistics of IPM, SQP, and ASM in terms of $\rho$. It is shown that Algorithm 1 with ASM has the best statistics followed by SQP and IPM. Next, in Figure 3b, we show the normalized time complexity $0\le \tau \le 1$ of Algorithm 1 using the three methods under similar computing power. Again, the performance of the ASM based approach is a clear winner among the competing approaches. The reason for this is because the ASM based approach is effective for nonsmooth constraints, and the convergence is faster because of the larger steps. The aforementioned methods/algorithms are generally classified into medium scale or large scale. For the former, full matrix structures are formulated using the involved linear algebra and hence need higher computational power and more memory elements. For the latter, sparse matrix structures are formulated using linear algebra, and hence need less computational power and fewer memory elements. In the existing algorithms, the “interior-point method” is a large-scale method, whereas “SQP” and the “active-set method” are medium-scale methods. Note that direct search methods such as “Nelder–Mead simplex” or “Fibonacci search” are not useful in the current problem because of the non-linear constraints in (8), whereas evolutionary multi-objective methods such as the “genetic algorithm with goal attainment” are computationally more demanding and hence not considered.

Next, we stick with ASM algorithm based on the aforementioned merits and test its utility in terms of spectral efficiency in bps/Hz versus the SNR in dB. In Figure 4, we set $K=4$ and analyze the performance by varying the scatterers L. The results are bench-marked with respect to baseband zero-forcing (BB ZF) precoding in [15]. For L set to one and four, it is observed that the ASM algorithm produces similar result as the BB ZF precoder, while at $L=8$, a slight gain is observed across the SNR range for the ASM algorithm. In Figure 5, we set $L=1$ and perform the experiment by varying the total number of users K. Again, the proposed algorithm improves the spectral efficiency for the considered cases and across the SNR range. Note that the objective function defined in (8) is specific to the maximization of the similarity metric $\rho$, which the proposed algorithm has done efficiently. Moreover, the joint-diagonalization is achieved without violating the QoS aspect outlined as the third point of the design consideration in Section 4.

6. Conclusions

We propose a joint-diagonalization algorithm for the sum of Hermitian matrices in mmWave systems by means of digital baseband beamformers. The algorithm finds solutions in the neighborhood of standard eigenvalue decomposition techniques if the total number of users is small. The similarity metric reduces as the total users increase in the system. For all cases, it is shown that the QoS either remains intact or shows slight improvement. The work opens the door for researchers interested in the characterization of the closed-form expression of mmWave systems. Digital baseband beamforming intended for the maximization of spectral efficiency constrained with enforcing weight matrices to be as diagonal as possible can be explored in the future.