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Hybrid Beamforming for Millimeter Wave Systems Using the MMSE Criterion

Hybrid analog and digital beamforming (HBF) has recently emerged as an attractive technique for millimeter-wave (mmWave) communication systems. It well balances the demand for sufficient beamforming gains to overcome the propagation loss and the desire to reduce the hardware cost and power consumption. In this paper, the mean square error (MSE) is chosen as the performance metric to characterize the transmission reliability. Using the minimum sum-MSE criterion, we investigate the HBF design for broadband mmWave transmissions. To overcome the difficulty of solving the multi-variable design problem, the alternating minimization method is adopted to optimize the hybrid transmit and receive beamformers alternatively. Specifically, a manifold optimization based HBF algorithm is firstly proposed, which directly handles the constant modulus constraint of the analog component. Its convergence is then proved. To reduce the computational complexity, we then propose a low-complexity general eigenvalue decomposition based HBF algorithm in the narrowband scenario and three algorithms via the eigenvalue decomposition and orthogonal matching pursuit methods in the broadband scenario. A particular innovation in our proposed alternating minimization algorithms is a carefully designed initialization method, which leads to faster convergence. Furthermore, we extend the sum-MSE based design to that with weighted sum-MSE, which is then connected to the spectral efficiency based design. Simulation results show that the proposed HBF algorithms achieve significant performance improvement over existing ones, and perform close to full-digital beamforming.


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I Introduction

Millimeter-wave (mmWave) communications is a key technology for 5G, which can address the bandwidth shortage problem in current mobile systems [1, 2, 3, 4, 5]. The large-scale antenna array is needed to compensate for the severe path loss and penetration loss at the mmWave wavelengths [6, 7]. However, the substantial increase in the number of antennas leads to non-trivial practical constraints. The traditional full-digital multiple-input and multiple-output (MIMO) beamforming which requires one dedicated radio frequency (RF) chain per antenna element is prohibitive in mmWave systems due to the unaffordable hardware cost and power consumption of a large number of antenna elements [8, 9]. By separating the whole beamformer into a low-dimensional baseband digital one and a high-dimensional analog one implemented with phase shifters, the hybrid analog and digital beamforming (HBF) architecture has been shown to dramatically reduce the number of RF chains while guaranteeing a sufficient beamforming gain [10, 9, 11, 12, 13, 15, 14].

I-a Related Works and Motivations

Compared with the traditional full-digital beamforming design, in HBF, besides the difficulty of the joint optimization over the four beamforming variables (the transmit and receive analog and digital beamformers), the constant modulus constraints of the analog beamformers due to the phase shifters make the problem highly non-convex and difficult to solve [9, 16, 19]. Most existing works overcome the difficulty by first decoupling the original problem into hybrid precoding and combining sub-problems and then focusing on the constant modulus constraint in solving the sub-problems. One effective and widely used approach is to regard the HBF design as a matrix factorization problem and to minimize the Euclidean distance between the hybrid beamformer with a full-digital beamformer [9, 17, 18]. To solve this matrix factorization problem, in [9], the authors exploited the spatial structure of the mmWave propagation channels and proposed spatially sparse precoding and combining algorithms via the orthogonal matching pursuit (OMP) method. In [18], a manifold optimization (MO) based HBF algorithm, as well as some low-complexity algorithms, was proposed. Besides the matrix factorization approach, another idea for HBF design is to tackle the original problem directly. In [19, 20], the closed-form solution of digital beamformers was first derived according to the original objective, followed by several iterative algorithms for the analog ones with the constant modulus constraint.

All the above works, as well as most of the other previous studies, design the HBF with the objective of maximizing the spectral efficiency. By recalling the joint precoding and combining designs in conventional full-digital MIMO systems, besides spectral efficiency, the mean square error (MSE) is another important metric [21, 22, 23, 24]. One direct motivation to consider MSE is that a practical system is normally constrained to some particular modulation and coding scheme instead of the Gaussian code [22], and thus MSE is a direct performance measure to characterize the transmission reliability. Furthermore, it has been shown that the variants of the MSE such as sum-MSE, minmax MSE, modified MSE, weighted MSE, etc., are related to other important performance measures (e.g., signal to interference plus noise ratio (SINR) and symbol error rate) [21, 22, 23, 24, 25]. For example, it has been shown in [21, 22] that the MSE is related to the SINR and SER (BER) metrics in the beamforming design for the full-digital MIMO systems with multiple data streams. Thus, it is of great interest to take MSE as an alternative optimization objective for HBF. Actually, even in some existing HBF designs with the spectral efficiency as the objective, the hybrid receive combining matrices were optimized by minimizing the MSE instead [9, 19, 20, 28]. Moreover, in [26, 30, 31], it was illustrated that precoding design based on the minimum MSE (MMSE) criterion can also achieve good performance in spectral efficiency.

There have been some works on the HBF design using the MMSE criterion for mmWave systems. In [26], the authors focused on the hybrid MMSE precoding at the transmitter side and proposed an OMP-based algorithm. To improve the system performance, in our previous work [27], we tackled the MMSE precoding problem directly and proposed an algorithm based on the general eigen-decomposition (GEVD) method. In [17], the authors replaced the hybrid MMSE precoding problem by the one of factorizing the optimal full-digital MMSE precoder. In their later work [28], the hybrid MMSE combiner was further considered with a similar approach to that in [9, 16], aiming at minimizing the weighted approximation gap between the hybrid combiner and a full-digital combiner. However, all of these works considered the narrowband scenario and cannot be straightforwardly extended to the broadband scenario, which is more relevant for mmWave communication systems.

I-B Contributions and Paper Organization

In this paper, we investigate the joint transmit and receive HBF optimization for broadband point-to-point mmWave systems, aiming at minimizing the modified MSE [24]. Besides the aforementioned challenges in the joint optimization of the four beamforming variables and the constant modulus constraint on the analog beamformers, it is also worth noting that in the broadband scenario, yet another challenge is that the digital beamformers should be optimized for different subcarriers while the analog one is invariant for the whole frequency band. Aiming at these challenges in the MMSE based HBF design for broadband mmWave MIMO systems, the contributions in this paper can be summarized as follows.

  • Instead of factorizing the optimal full-digital beamformer in the indirect HBF design approach [9, 17, 18], we optimize the hybrid beamformers by directly targeting the MMSE objective for better performance. Different from the conventional MMSE based HBF designs [17, 26, 28] which only considered the narrowband scenario, we propose a general HBF design approach for both the narrowband and broadband mmWave MIMO systems. In particular, we decompose the original sum-MSE minimization problem into the transmit hybrid precoding and receive combining sub-problems, and show that the two sub-problems can be unified in almost the same formulation and solved through the same procedure. The alternating minimization method is adopted to solve the overall HBF problem, for which a novel initialization method is proposed to reduce the number of iterations. Furthermore, following the approach of extending the sum-MSE minimization problem to the weighted sum-MSE minimization (WMMSE) problem and connecting it to the spectral efficiency maximization problem in the narrowband scenario [28], we show that in the broadband scenario the proposed MMSE based HBF algorithms can be generalized to the ones for maximizing the spectral efficiency.

  • To deal with the constant modulus constraint in the analog beamforming optimization, we apply the manifold optimization (MO) method [18, 33]. In contrast to the application of the MO method in [18] for minimizing the Euclidean distance between the hybrid beamformer and the target full-digital beamformer, in this study, the MO method is applied to directly minimize the sum-MSE and the new contribution is to derive the more complicated Euclidean conjugate gradient of the sum-MSE with some skilled derivations so that the Riemannian gradient can be computed. This provides a direct approach with guaranteed convergence to solve the MMSE HBF problem instead of the indirect approach in [18].

  • To avoid the high complexity in the MO-HBF algorithm, we propose several low-complexity algorithms. In the narrowband scenario, we show that the analog beamforming matrix can be optimized column-by-column with the GEVD method. In the broadband scenario, we derive both upper and lower bounds of the original objective and then propose two eigen-decomposition (EVD) based HBF algorithms. Compared with the existing algorithms based on the OMP method [26, 28, 17], the proposed algorithms directly tackle the original sum-MSE objective without the restriction of the space of feasible solutions and thus result in better performance.

The rest of the paper is organized as follows. For the ease of presentation, we start with the narrowband scenario and introduce the system model along with the HBF problem formulation in Section II. In Section III, we present the basic idea and the optimization procedure, and propose the MO-HBF and GEVD-HBF algorithms. In Section IV, we extend the problem formulation and design procedure to the broadband scenario, and propose three HBF algorithms. In Section V, we extend the MMSE based HBF design to the WMMSE one for maximizing the spectral efficiency. We discuss the convergence property and analyze the computational complexity for all the proposed HBF algorithms in Section VI. We demonstrate various numerical results in Section VII. Finally, we conclude the paper in Section VIII.

Throughout this paper, bold-faced upper case letters, bold-faced lower case letters, and light-faced lower case letters are used to denote matrices, column vectors, and scalar quantities, respectively. The superscripts

, , and represent matrix (vector) transpose, complex conjugate, and complex conjugate transpose, respectively. denotes the Euclidean norm of a vector. , and denote the trace and the Frobenius norm of a matrix. denotes the conjugate gradient of a function. denotes the expectation operator. denotes the absolute value or the magnitude of a complex number. denotes the -th entry of a matrix .

Ii System Model and Problem Formulation

Ii-a System Model

For the ease of presentation, we first consider a point-to-point narrowband mmWave MIMO system with HBF as in Fig. 1, where data streams are sent and collected by transmit antennas and receive antennas, respectively. Both the transmitter and receiver are equipped with RF chains, where . The original symbol vector, denoted by with , is firstly precoded through an digital beamforming matrix , and then an analog beamforming matrix which is implemented in the analog circuitry using phase shifters. From the equivalent baseband representation point of view, the precoded signal vector at the transmit antenna array can be represented as . Without loss of generality, the normalized transmit power constraint is set to .

Similar to that in [9, 19], the mmWave propagation channel is characterized by a geometry-based channel model with clusters and rays within each cluster. Considering the mmWave system with a half-wave spaced uniform linear array (ULA) at both the transmitter and the receiver, the channel matrix can be represented as


where denotes the complex gain of the th ray in the th propagation cluster, and and denote the normalized responses of the transmit and receive antenna arrays to the th ray in the th cluster, respectively, where and denote the angles of arrival and departure.

With a similar HBF at the receiver, i.e., an analog combiner followed by an digital baseband combiner , we finally have the processed signal as


where denotes the additive noise vector at the

receive antennas satisfying the complex circularly symmetric Gaussian distribution with zero mean and covariance matrix

, i.e., . Similar to existing works on the HBF design (e.g. [10, 18, 19, 20]), in this paper, it is assumed that perfect channel state information (CSI) is available at both the transmitter and receiver and that there is perfect synchronization between them.

Fig. 1: Diagram of a point-to-point narrowband mmWave MIMO system with HBF.

Ii-B Problem Formulation

In this work, we take the modified MSE [24] as the performance measure and optimization objective for the joint transmit and receive HBF design, which is defined as


where is a scaling factor to be jointly optimized with the hybrid beamformers. By substituting (2) into (3) and after some mathematical manipulations, we have


where , are defined as the overall hybrid transmit and receive beamformers, respectively. Notice that since the analog beamformers are assumed to be implemented with phase shifters which only adjust the phases of the input signals, the elements of analog beamformers should satisfy the constant modulus constraint, namely for and , and for and . With the derived MSE expression in (4), the transmit power constraint and the constant modulus constraint of the phase shifters, the HBF optimization problem in the narrowband scenario can be formulated as


It is worth noting that there are mainly three reasons or advantages for introducing the scaling factor and taking the modified MSE as the objective function. First, as the joint transmit and receive HBF problem will be decoupled into the hybrid precoding and combining sub-problems, adjusting achieves a better performance for the precoding optimization by considering the noise effect (which is also referred to as the transmit Wiener filter) [24]. Second, is also helpful in dealing with the total transmit power constraint and thus simplifies the precoding optimization procedure [30, 31]. Finally, by introducing , the hybrid precoding and combining sub-problems can be unified and solved in the same way aiming at the same modified MSE objective. These advantages will be elaborated in more details in the following sections.

Iii Hybrid Beamforming Design Based on The MMSE Criterion

Since the HBF problem in (5) involves a joint optimization over five variables, along with non-convex constraints, it is unlikely to find the optimal solution. A sub-optimal but efficient way to overcome the difficulties is to separate the original problem into two sub-problems corresponding to the optimization for the hybrid transmit precoder and receive combiner, respectively, and solve each independently [9, 19, 20, 28]. Taking this approach, we propose several HBF algorithms in the following two subsections. Finally, we develop the whole alternating minimization algorithm for the HBF optimization based on the MMSE criterion.

Iii-a Hybrid Transmit Design

This section focuses on the hybrid precoder design (including ) in (5) by fixing the receive combining matrices and . As shown in [24, 26, 31], the original precoder can be separated as , where is an unnormalized baseband precoder. With this separation, the precoder optimization problem can be formulated as


where denotes the equivalent channel of the concatenation of the air interface channel and the hybrid receive combiner. Our optimization approach is to first derive the optimal digital precoding matrix and the scaling factor by fixing , then derive the resulting objective as a function of , and finally optimize by further minimizing the objective with the constant modulus constraint. Due to the transmit power constraint, it can be proved by contradiction that the optimal solution must be achieved with the maximum total transmit power, i.e., the optimal is given by


Then according to the Karush-Kuhn-Tucker (KKT) conditions, the closed-form solution of the optimal is given by


where is defined for notational brevity. Substituting the optimal and into (6) and after some mathematical derivation, the resulting MSE is given by111Note that the above derivations benefit from the introduction of . To show this, it can be checked that if we remove from (6) (or just set in (6)), it is highly challenging to get a closed-form expression of via the KKT conditions and further get a closed-form expression of the MSE as a function of for the optimization of the analog precoder.


The optimizing problem in (6) is now reduced to the following one for the optimization of


Here we propose two algorithms for optimizing the analog precoding matrix with the constant modulus constraint, which are based on the MO and GEVD methods, respectively.

Iii-A1 Analog Precoder Design Based on the MO Method

To deal with the constant modulus constraint, the MO method [18][33] can be applied to obtain a local optimal . The basic idea is to define a Riemannian manifold for with the consideration of the constant modulus constraint, and iteratively update this optimization variable on the direction of the Riemannian gradient (i.e., a projection of the Euclidean conjugate gradient onto the tangent space of a point on the Riemannian manifold) in a similar way to that in the conventional Euclidean gradient descent algorithm (the details can be referred to [18]). However, the application of the MO method is not straightforward, and the most difficult part is the derivation of the conjugate gradient in the Euclidean space, in order to obtain the associated Riemannian gradient. It should be mentioned that for the scalar function associated with a complex-valued variable , the conjugate gradient [32] is defined as . By defining in (9) for notational brevity, we have the following lemma for the conjugate gradient.

Lemma 1.

The conjugate gradient of the function with respect to is given by


Proof: According to some basic differentiation rules for complex-value matrices [32], the differential of can be expressed as


where denotes the differential with respect to while taking as a constant matrix during the derivation of the conjugate gradient . The second equality in (12) holds due to the properties of and .

On the other hand, we can directly compute from (9). According to some differentiation rules for differentiating a matrix’s trace and inverse, we express as


It can be further derived that




By substituting (15) and (14) into (13) and using again , we have


By comparing (16) with (12), the proof is completed.

With the derived Euclidean conjugate gradient, the manifold optimization can be applied to solve the problem with the constant modulus constraints [33]. The overall MO-HBF algorithm is summarized in Algorithm 1, where the iteration index is denoted in the subscript of . In particular, the detailed operation in the 4th step is given as follows. First, project the Euclidean gradient onto the tangent space to obtain the Riemannian gradient. Second, search a point in the tangent space along the Riemannian gradient and use the Armijo-Goldstein condition to determine the step size. Finally, retract the searched point back to the manifold.

Input: , , Output: , ,

1:Initialize randomly and set ;
3:Compute according to (11);
4:Use the manifold optimization method to compute ;
6:Until a stopping condition is satisfied;
7:Compute and according to (7) and (8).
Algorithm 1 The MO-HBF Algorithm

Iii-A2 Analog Precoder Design Based on the GEVD Method

The above algorithm for optimizing the analog precoding matrix in (10) is essentially a gradient based algorithm, where the computational complexity is proportional to the number of iterations and is related to the form of the objective function and the stop condition. In this part, we propose a low-complexity algorithm based on GEVD. According to [19], for large-scale MIMO systems, it can be approximated that based on the fact that the optimized analog beamforming vectors for different streams are likely orthogonal to each other. With this approximation, (9) can be simplified as


With this simplified form, it can be shown that the analog precoding matrix can be optimized column-by-column. Specifically, define as the remaining sub-matrix of after removing the th column . Further define . Then, using the fact that for a full-rank matrix and a rank-one matrix , the MSE expression in (17) can be written as


where and are both Hermitian matrices. It is seen from (18) that the MSE expression is separated into two terms which are related to and , respectively. By fixing , becomes a function on in the second term in (18). As both and are Hermitian and is positive definite, according to [35], the optimal in the sense of maximizing the last term in (18) or minimizing the whole term in (18

) is the eigenvector associated with the maximum generalized eigenvalue between

and , which can be obtained via the GEVD operation. To further take the constant modulus constraint into account, a simple but effective way is to only extract the phase of each element in the generalized eigenvector. By applying the above GEVD and phase extraction operations for each column and repeating them for the whole matrix until the stop condition is satisfied, we finally get the optimized analog precoding matrix. The overall GEVD-HBF algorithm is summarized in Algorithm 2.

Input: , , Output: , ,

1:Initialize randomly and set = 0;
2:for  do
3:     Compute , , defined in Section III-A2;
4:     Compute the maximum generalized eigenvector of and ;
5:     Set , i.e., extract the phase of each element of ;
6:end for
8:Compute and according to (7) and (8) .
Algorithm 2 The GEVD-HBF Algorithm

Iii-B Hybrid Receive Combiner Design

By fixing the updated precoding matrices and along with the scaling factor , the optimizing problem in (5) can be reduced to the following one for the hybrid receive combiner


where and . Similarly, by differentiating the objective function of (19) with respect to and setting the result to zero, we have the optimal as follows


Substituting (20) back into the problem in (19), we have


Comparing (20) with (8) and (21) with (10), it can be seen that they have almost the same form respectively. Thus, the MO-HBF and GEVD-HBF algorithms, which were introduced in Section III-A, can be directly applied to optimize the hybrid combiner.

Iii-C Alternating Minimization for Hybrid Beamforming

Iii-C1 Alternating Optimization

A joint hybrid precoding and combining design based on the MMSE criterion can be developed by iteratively and alternatively using the hybrid precoding design in Section III-A and the hybrid combining design in Section III-B. Specifically, during the th iteration, first for the optimization of the hybrid precoder, by updating the problem in (6) with the optimized combiners and in the th iteration, the hybrid precoding matrices , and the scaling factor are optimized via the MO-HBF or the GEVD-HBF algorithm. Similarly, with the new hybrid precoder, the hybrid combining optimization problem in (19) is then updated and solved via the same algorithm. This alternating optimization is repeated until a stop condition is satisfied.

Iii-C2 Stopping Condition

To distinguish the iteration of the alternating minimization between the transmitter and receiver beamforming optimization and the iteration in the optimization of the analog beamformer (i.e., Algorithms 1 and 2), we refer to the former as the outer iteration and the latter as the inner iteration. The stopping condition of these two iterations can be set as either the number of iterations exceeding a specified value , or the relative difference between the MSE values of two consecutive iterations becoming smaller than a specified value . For example, considering a typical system configuration in Section VII, according to the observation in simulations, good performance can be achieved when we set for both the inner and outer iterations for the alternating MO-HBF algorithm and set for the inner iteration and for the outer iteration for the alternating GEVD-HBF algorithm.

Iii-C3 Beamforming Initialization

It is worth noting that the number of iterations for the alternating optimization highly depends on the initialization of the beamformers. One simple idea is to randomly generate a hybrid combiner (precoder) for the optimization of the precoder (combiner) at the beginning of the alternating minimization. This method does not require extra information, but may need a lot of iterations to converge to a local optimal point with certain performance loss. As the concatenation of the hybrid beamformer will gradually approach the full-digital one during the iterations, we propose to take a full-digital beamformer as the initialization for better convergence. Specifically, the optimal full-digital precoder based on the MMSE criterion proposed in [21] can be used here. Assuming without loss of generality that the alternating optimization starts from the transmit beamforming problem in (6), we assume that there is a virtual full-digital combiner at the receiver in the initialization step. Namely, we initialize the concatenation of the hybrid combiner, , as the optimal full-digital combiner in [21] and substitute it into (6) for the precoder optimization in the first iteration. We refer to the proposed initialization method as the virtual full-digital beamformer (VFD) method. It is worth noting that as the VFD initialization method assumes a virtual full-digital beamformer at one side, which generally cannot be directly implemented using the HBF structure, at least one outer iteration is needed to obtain the hybrid beamformers for both sides. As the VFD initialization does not require the alternating optimization to obtain the full-digital beamformer, its additional complexity is much lower compared with that of the main HBF algorithms. Simulations in Section VII will show that with the VFD method, the convergence speed improves significantly with even some MSE performance improvement, in comparison with random initialization.

Iv Hybrid BeamForming Design for Broadband MmWave Systems

Fig. 2: Diagram of a broadband mmWave MIMO system with HBF.

Due to the large available bandwidth of mmWave systems, frequency selective fading will be encountered. Therefore, in this section we generalize the previous hybrid beamformer design to broadband mmWave systems. In particular, we point out that similar to the narrowband scenario, the optimization of precoder and combiner in the broadband scenario can also be unified and solved through the same procedure. Thus, we focus on the hybrid transmit precoder design in the broadband scenario and propose three algorithms.

Iv-a System Model in the Broadband Scenario

To overcome the channel frequency selectivity, we assume that the orthogonal frequency division multiplexing (OFDM) technology is applied so that the channel fading on each subcarrier can be regarded as being flat. To facilitate the following system design, the broadband mmWave MIMO channel model with half-wavelength spaced ULAs at both the transmitter and the receiver in [20] is adopted here, where the channel matrix at the th subcarrier, for with being the total number of subcarriers, is given by


where the other parameters are defined in the same way as that in (1). It is worth noting that although the geometry-based spatial channel model is applied in simulations, all proposed HBF algorithms are compatible for other general models.

As shown in Fig. 2, the processed signal vector at the th subcarrier after the hybrid receive combining can be expressed as


where and denote the transmitted symbol vector and the additive noise vector at the th subcarrier, respectively, and denote the digital precoder and combiner at the th subcarrier, respectively, and and denote the analog precoder and combiner, respectively. It is worth noting that in the broadband scenario, the digital beamformers and must be optimized for different subcarriers while the analog precoder or combiner is invariant for all subcarriers due to the post-IFFT or pre-FFT processing. Similar to [18, 20], we assume equal power allocation among subcarriers at the transmitter, namely . To deal with the new difficulty in the HBF design for broadband mmWave MIMO systems, we take the sum-MSE of all the subcarriers and all the streams as the objective function. That is, define , where denotes the modified MSE on the th subcarrier and is given by


where , , and is a scaling factor for the th subcarrier as similar to that in the narrowband scenario. Then, the optimization problem in the broadband scenario can be formulated as


By comparing the problem in (25) with that in (5) for the narrowband scenario, it can be found that they have almost the same form except that the digital beamformers need to be optimized for different subcarriers in (25). Thus, the alternating minimization principle is also applicable here. In particular, it can be shown that in the broadband scenario the two sub-problems associated with (25) for the optimization of precoding and combining can also be solved through the same procedure. Therefore, in the following we focus on the hybrid transmit precoder design.

Iv-B Broadband Hybrid Transmitter Design

Analogous to that in Section III-A, the original precoder is also separated as , where is an unnormalized precoder for the th subcarrier. It can also be proved by contradiction that the optimal is given by . Then, by fixing the hybrid receive combiner and based on the KKT conditions, the optimal can be derived as a function of . That is,


where and . Now the original problem in (25) is reduced to the one for as follows




Iv-B1 Analog Precoding Based on the MO Method

Although the objective function for analog precoding optimization in the broadband scenario is more complicated than that in the narrowband scenario, the MO method can still be applied here. Using some differentiation rules for complex-valued matrices, the conjugate gradient of the function with respect to can be expressed as


where can be derived as follows according to Lemma 1


with defined for notational brevity. Thus, the Riemannian gradient can be computed by projecting the above Euclidean gradient onto the tangent space of the Riemannian manifold [18]. According to the property of the gradient descent method, with a proper selection of the step size, is guaranteed to converge to a feasible local optimal solution via MO.

Iv-B2 Analog Precoding Based on the EVD Method

Note that since the objective function in the broadband scenario is the sum-MSE of all the subcarriers, the variant channel matrices and digital beamformers at different subcarriers prevent us from rewriting the original problem in an GEVD-available formulation as that in Section III-A2. Nevertheless, the approximation of can still be utilized here for developing other low-complexity algorithms. The basic idea is to ignore the constant modulus constraint in (27) temporarily and add a new constraint of . Then, we have the following new problem


It turns out that the above optimization problem is still difficult to solve. Therefore, we devote to derive its lower bound first with the help of the following lemma.

Lemma 2.

A lower bound of the objective in (31) is given by


Proof: For notational brevity, we define and have


where denotes the th eigenvalue of and is positive because is positive definite. The inequalities (a) and (b) in (33) both come from the Jensen’s inequality, with equality of (a) satisfied if and equality of (b) satisfied if , respectively. Substituting the definition of , the proof is completed.

Then, instead of the objective function in problem (31), we devote to minimize its lower-bound, which is equivalent to maximizing . After omitting the constant terms, the optimization problem can be rewritten as


It can be proved that the optimal is times the isometric matrix containing the eigenvectors associated with the largest eigenvalues of [34], which can be obtained through EVD. To further make the constant modulus constraint satisfied, similar to that in Section III-A, we just extract the phase of each element of the optimal . The algorithm is referred to as the EVD-LB-HBF algorithm, where LB denotes the abbreviation for lower bound. In the following, we propose a better algorithm, where instead of minimizing a lower bound of (31) in EVD-LB-HBF an upper bound is derived for minimization.

Lemma 3.

For an positive definite and Hermitian matrix and an arbitrary

para-unitary matrix

, i.e., , define the eigenvalues of and in descending order as and , respectively. Then we have .

Proof: According to Courant-Fisher min-max theorem [36],

where is a non-zero vector, denotes a -dimension subspace of and

is a new subspace after a linear transform of

to . Similarly, as can be proved to be the th largest eigenvalue of , we have


Then we have . Since is positive definite, by Jensen’s inequality, holds for any non-zero vector . Thus, the proof is completed.

Then, denoting as , and using Lemma 3, the objective in (31) can be further upper bounded as


where (a) follows from the relationship between a matrix’s trace and eigenvalues and (b) follows from Lemma 3. Using the matrix inversion lemma [36], we have , where . The optimization problem in (31) can be converted to the one minimizing its upper bound in (36), or equivalently


Similar to that for (34), the solution can be obtained through the EVD and phase extraction operations. To distinguish, we refer to this algorithm as the EVD-UB-HBF algorithm, where UB denotes the abbreviation for upper bound.

Iv-B3 Analog Precoding Based on the OMP Method

By combining the OMP-MMSE-HBF algorithm for narrowband multiuser mmWave MIMO systems [26] and the OMP-based HBF algorithm aiming at maximizing the spectral efficiency for broadband mmWave MIMO systems [29], we come up with the low-complexity OMP-MMSE precoding algorithm for broadband mmWave systems. Specifically, by restricting the search range of within a set of basis vectors , the hybrid beamforming problem can be rewritten as


where denotes the zero norm of a matrix. and is an matrix having non-zero rows which constitute as defined in [9, 26, 29]. With the readily derived closed-form solution of the digital precoder in (26), the algorithm developed based on the OMP method can be applied to choose the columns of that are most strongly correlated with the residual error to form the analog precoder.

V Extend to Spectral Efficiency Based on the WMMSE Criterion

Spectral efficiency is another important performance metric for the HBF design [9, 18, 20]. Based on the full-digital beamforming design approach in [25], the authors in [28] investigated the HBF design with the WMMSE criterion and connected it to the one for sum-rate maximization. However, their HBF algorithm is based on the OMP method, which has a limited feasible set for the analog beamformers, and is only for the narrowband scenario. In this section, following the design approach in [25, 28], we first show that in the narrowband scenario our proposed HBF algorithms in Section III can be extended to the ones for achieving better spectral efficiency than the OMP based algorithm. We also extend the sum-MSE minimization problem to the WMMSE problem and connect it to the spectral efficiency maximization problem in the broadband scenario. It is shown that the proposed broadband HBF algorithms in Section IV can be generalized to the ones for maximizing the spectral efficiency. Simulation results in Section VII will show that the WMMSE HBF algorithms proposed in this section provide better or comparable spectral efficiency than the conventional ones [18, 19, 20, 28].

First start from the narrowband scenario. Assuming that the transmitted symbols follow a Gaussian distribution, the achievable spectral efficiency is then given by , where and denote the hybrid precoder and combiner, respectively. Inspired by [25, 28], a suboptimal but efficient HBF design for maximizing the spectral efficiency can be connected to the following WMMSE problem


where is an weighting matrix to be optimized, and denotes the modified MSE matrix. According to [25, 28], a three-step procedure is applied to solve (39). In the first step, is optimized by fixing and in (39). That is,


where . By comparing (40) with (19), it can be shown that the optimal has exactly the same form as that in (20). After substituting the optimal