A Framework on Hybrid MIMO Transceiver Design based on Matrix-Monotonic Optimization

08/26/2018 ∙ by Chengwen Xing, et al. ∙ University of Victoria IEEE Georgia Institute of Technology 0

Hybrid transceiver can strike a balance between complexity and performance of multiple-input multiple-output (MIMO) systems. In this paper, we develop a unified framework on hybrid MIMO transceiver design using matrix-monotonic optimization. The proposed framework addresses general hybrid transceiver design, rather than just limiting to certain high frequency bands, such as millimeter wave (mmWave) or terahertz bands or relying on the sparsity of some specific wireless channels. In the proposed framework, analog and digital parts of a transceiver, either linear or nonlinear, are jointly optimized. Based on matrix-monotonic optimization, we demonstrate that the combination of the optimal analog precoders and processors are equivalent to eigenchannel selection for various optimal hybrid MIMO transceivers. From the optimal structure, several effective algorithms are derived to compute the analog transceivers under unit modulus constraints. Furthermore, in order to reduce computation complexity, a simple random algorithm is introduced for analog transceiver optimization. Once the analog part of a transceiver is determined, the closed-form digital part can be obtained. Numerical results verify the advantages of the proposed design.

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I Introductions

The great success of multiple-input multiple-output (MIMO) technology makes it widely accepted for current and future high data-rate communication systems[1]. Acting as a pillar to satisfy data hungry applications, a natural question is how to reduce the cost of MIMO technology, especially that of large scale antenna arrays. The traditional setting of one radio-frequency (RF) chain per antenna element is too expensive for large-scale MIMO systems, especially at high frequencies, such as millimeter wave bands or Terahertz bands. Hybrid analog-digital architecture is promising to alleviate the straits and strike a balance between the cost and the performance of practical MIMO systems.

A typical hybrid analog-digital MIMO transceiver consists of four components, i.e., digital precoder, analog precoder, analog processor, and digital processor [2]. In the early transceiver design, hybrid MIMO technology is often referred to antenna selection [3, 4] to reap spatial diversity. In these works, analog switches are used in the radio-frequency domain. Phase-shifter based soft antenna selection [5, 6, 7] has been proposed to improve performance for correlated MIMO channels. Nowadays, the phase-shifter based hybrid structure has been widely used.

For a phase shifter, only signal phase, instead of both magnitude and phase, can be adjusted. Thus, the optimization of a MIMO transceiver with phase shifters becomes complicated due to the constant-modulus constraints on analog precoder and analog processor. It has been shown in [8] that the performance of a full-digital system can be achieved when the number of shifters is doubled in a phase-shifter based hybrid structure. However, this can hardly be practical due to the requirement on a large number of phase shifters, especially in large-scale MIMO systems. As a matter of fact, the phase shifters in large-scale MIMO systems have been considered to be a burden sometimes. Thus, sub-connected hybrid structure has emerged as an alternative option [9, 10] and it has received much attention recently [9, 11, 12, 10, 13, 14].

Unit modulus and discrete phase make the optimization of analog transceivers nonconvex and thus difficult to address [3, 4]. There have been some works on hybrid transceiver optimization considering different design limitations and requirements. Their motivation is to exploit the underlying structures of the hybrid transceiver to achieve high performance but with low complexity.

Early hybrid transceiver design is based on approximating digital transceivers in terms of the norm difference between all-digital design and the hybrid counterpart. For the millimeter wave (mmWave) band channels, which are usually with sparsity, an orthogonal matching pursuit (OMP) algorithm has been used in signal recovery for the hybrid transceiver [15]. In order to overcome the non-convexity in hybrid transceiver optimization, some distinct characteristics of mmWave channels must be exploited [15]. This methodology is a compromise on the constant-modulus constraint, which has been validated in different environments, including multiuser and relay scenarios [16, 17]

. However, it has been found later on that the OMP algorithm cannot achieve the optimal solution sometimes. A singular-value-decomposition (SVD) based descent algorithm

[18] has been proposed, which is nearly optimal. An alternative fast constant-modulus algorithm [19] has also been developed to reduce the gap between the analog and digital precoders. The above methods are hard for complex scenarios due to high computation complexity [20, 21, 22]

. Therefore, based on the idea of unitary matrix rotation, several algorithms

[23, 24] have been proposed to improve the approximation performance while maintaining a relative low complexity at the same time.

On the other hand, some works for hybrid precoding design are based on codebooks, which relax the problem into a convex optimization problem [25]. However, the codebook-based algorithm suffers performance loss if channel state information (CSI) is inaccurate [26]. In order to reduce the complexity of codebook design and the impact of partial CSI, special structures of massive MIMO channels [27, 28], can be exploited. Recently, an angle-domain based method has been proposed from the viewpoint of array signal processing [29, 30], which provides a useful insight on hybrid analog and digital signal processing. Based on the concept of the angle-domain design, some mathematical approaches, such as matrix decomposition algorithm, have been developed [31, 32]. Energy efficient hybrid transceiver design for Rayleigh fading channels has been investigated in [33]. Hybrid transceiver optimization with partial CSI and with discrete phases has been discussed in [34] and [35], respectively.

Hybrid MIMO transceivers are not only limited to mmWave frequency bands or terahertz frequency bands but also potentially work in other frequency bands. The transceiver itself could either be linear or nonlinear. Moreover, the performance metrics for MIMO transceiver could be different, including capacity, mean-squared error (MSE), bit-error rate (BER), etc. A unified framework on hybrid MIMO transceiver optimization will be of great interest. In this paper, we will develop a unified framework for hybrid linear and nonlinear MIMO transceiver optimization. Our main contributions are summarized as follows.

  • Both linear and nonlinear transceivers with Tomlinson-Harashima precoding (THP) or deci-sion-feedback detection (DFD) are taken into account in the proposed framework for hybrid MIMO transceiver optimization.

  • Different from the existing works in which a single performance metric is considered for hybrid MIMO transceiver designs, more general performance metrics are considered.

  • Based on matrix-monotonic optimization framework, the optimal structures of both digital and analog transceivers with respect to different performance metrics have been analytically derived. From the optimal structures, the optimal analog precoder and processor correspond to selecting eigenchannels, which facilitates the analog transceiver design. Furthermore, several effective analog design algorithms have been proposed.

The rest of this paper is organized as follows. In Section II, a general hybrid system model and the MSE matrices corresponding to different transceivers are introduced. In Section III, a unified hybrid transceiver is discussed in detail and the related transceiver optimization is present. In Section IV, the optimal structure of digital transceivers is derived based on matrix-monotonic optimization. In Section V, basic properties of the optimal analog precoder and processor are investigated, based on which effective algorithms to compute the analog transceiver are proposed. Next, in Section VI, simulation results are provided to demonstrate the performance advantages of the proposed algorithms. Finally,

Fig. 1: General hybrid MIMO transceiver.

conclusions are drawn in Section VII.

Notations

: In this paper, scalars, vectors, and matrices are denoted by non-bold, bold lower-case, and bold upper-case letters, respectively. The notations

and denote the Hermitian and the trace of a complex matrix , respectively. Matrix is the Hermitian square root of a positive semi-definite matrix . The expression denotes a square diagonal matrix with the same diagonal elements as matrix . The th row and the th column of a matrix are denoted as and , respectively, and the element in the th row and the th column of a matrix is denoted as . In the following derivations, always denotes a diagonal matrix (square or rectangular diagonal matrix) with diagonal elements arranged in a nonincreasing order. Representation means that the matrix is positive semidefinite. The real and imaginary parts of a complex variable are represented by and , respectively, and statistical expectation is denoted by .

Ii General Structure of Hybrid MIMO Transceiver

In this section, we will first introduce the system model of MIMO hybrid transceiver designs. Then a general signal model is introduced, which includes nonlinear transceiver with THP or DFD and linear transceiver as its special cases. Based on the general signal model, the general linear minimum mean-squared error (LMMSE) processor and data estimation mean-squared error (MSE) matrix are derived, which are the basis for the subsequent hybrid MIMO transceiver design.

Ii-a System Model

As shown in Fig. 1, we consider a point-to-point hybrid MIMO system where the source and the destination are equipped with and antennas, respectively. Without loss of generality, it is assumed that both the source and the destination have RF chains. A transmit data vector is first processed by a unit with feedback operation and then goes through a digital precoder and an analog precoder . This is a more general model as it includes both linear precoder and nonlinear precoder as its special cases. For the nonlinear transceiver with THP at source, the feedback matrix is strictly lower triangular. The key idea behind THP is to exploit feedback operations to pre-eliminate mutual interference between different data streams. In order to control transmit signals in a predefined region, a modulo operation is introduced for the feedback operation [36]. Based on lattice theory, it can be proved that the modulo operation is equivalent to adding an auxiliary complex vector whose element is with integer imaginary and real parts [36, 37]. The vector makes sure in a predefined region [36, 37]. Based on this fact, the output vector of the feedback unit satisfies the following equation

(1)

that is

(2)

It is worth noting that can be perfectly removed by a modulo operation [36, 37] and thus recovering is equivalent to recovering . On the other hand, for linear precoder, there is no feedback operation, i.e., and [38]. Moreover, based on (2) we have .

Then, the received signal at the destination is

(3)

where is an additive Gaussian noise vector with zero mean and covariance , is an channel matrix, and is a general feedback matrix at source, which is determined by the types of precoders. It is worth noting that corresponds to linear precoder without feedback operation. As shown in Fig. 1, after analog and digital processing at the destination, the recovered signal is given by

(4)

where is an analog processor, is a digital processor, and is a general feedback matrix at the destination. Note that since the analog precoder and analog processor are implemented through phase shifters, they are restricted to constant-modulus matrices with constant magnitude elements. For DFD at the receiver, the decision feedback matrix in (4) is a strictly lower-triangular matrix. For linear detection, the feedback matrix in (4

) is an all-zero matrix, i.e.,

. Based on (3) and (4), the recovered signal vector can be rewritten as

(5)

This is a general signal model and includes nonlinear hybrid transceivers with THP or DFD and linear hybrid transceiver as its special cases.

More specifically, for a linear hybrid transceiver, there is no feedback, either at the source or at the destination, i.e., . Therefore, the recovered signal in (5) becomes

(6)

For the nonlinear transceiver with THP at the source and linear decision at the destination, i.e., [36, 37], the detected signal vector in (5) becomes

(7)

For the nonlinear transceiver with DFD at the destination and a linear precoder at the source, i.e., , the detected signal vector in (5) becomes

(8)

Ii-B Unified MSE Matrix for Different Precoders and Processors

Based on the general signal model in (5), the general MSE matrix of the recovered signal at the destination equals

(9)

where the third equality is based on given in (2).

Based on lattice theory, each element of is identical and independent distributed, i.e., [37]. Thus, for notational simplicity, we can assume in the following derivations. Denote , then

(10)

It is obvious that is a strictly lower-triangular matrix based on the definitions of and , which implies that using nonlinear precoding at transmitter and nonlinear detection at the receiver at the same time is equivalent to just one of two. Therefore, nonlinear precoding at the transmitter and nonlinear detection at the receiver are equivalent and only one is enough.

Direct matrix derivation [38] yields that the optimal will be

(11)

That is, the general MSE matrix can be further simplified into

(12)

for any .

If in (11) and (II-B), the results are reduced to linear transceiver. Specifically, the corresponding digital LMMSE processor for linear transceiver is given as follows

(13)

and the MSE matrix for linear transceiver is

(14)

where

(15)

, which is signal-to-noise ratio for single antenna case.

For the nolinear transceivers, for THP or for DFD in (10)-(II-B). Based on (II-B) and (II-B), the general MSE matrix for nonlinear transceivers can also be written in the following unified formula

(16)

which turns into the MSE matrix in (II-B) when .

In the following, we will investigate unified hybrid MIMO transceiver optimization, which is applicable to various objective functions based on on the general MSE matrix (II-B).

Iii The Unified Hybrid MIMO Transceiver Optimization

Because of the multi-objective optimization nature for MIMO systems with multiple data streams, there are different kinds of objectives that reflect different design preferences [39]. All can be regarded as a matrix monotonic function of the data estimation MSE matrix in (II-B) [40]. A function is a matrix monotone increasing function if for [40]. To avoid case-by-case discussion, we will investigate in depth hybrid MIMO transceiver optimization with different performance metrics from a unified viewpoint, in this section.

Based on the MSE matrix in (II-B), the unified hybrid MIMO transceiver design can be formulated in the following form

(17)

where is a matrix monotone increasing function [40]. The sets and are the feasible analog precoder set and analog processor set satisfying constant-modulus constraint, and denotes the maximum transmit power at the source.

Iii-a Specific Objective Functions

There are many ways to choose the matrix monotone increasing function. In this subsection, we will investigate the properties of different objective functions in (II-B).

One group of matrix monotone increasing functions can be expressed as

(18)

where is a vector consisting of the diagonal elements of the matrix and is a function of a vector satisfying one the following four properties discussed in Appendix A:

  1. Multiplicatively Schur-convex

  2. Multiplicatively Schur-concave

  3. Additively Schur-convex

  4. Additively Schur-concave.

Many widely used metrics can be regarded as a special case of this group of functions [39, 37, 36].

Conclusion 1: For linear transceiver, the feedback matrix in (III) is an all-zero matrix, i.e., . For nonlinear transceiver, from Appendix B the optimal feedback matrix for is

(19)

where is a lower triangular matrix of the following Cholesky decomposition

(20)

It has been proved in [40] and [38] that for nonlinear transceiver design each data stream will have the same performance if in (III-A) is multiplicatively Schur-convex. On the other hand, if in (III-A) is multiplicatively Schur-concave, for nonlinear transceiver design the objective function includes geometrically weighted signal-to-noise-plus-interference-ratio (SINR) maximization as its special case.

If in (III-A) is additively Schur-convex, the objective function includes the the maximum MSE minimization and the minimum BER with the same constellation on each data stream as special cases. If in (III-A) is additively Schur-concave, the objective function includes weighted MSE minimization as its special case. Additive Schur functions are usually used for linear transceivers ( in (III-A)) since closed-form solutions can be obtained in this case.

Besides the above group of matrix monotone increasing functions, we can choose one to reflect capacity and MSE for linear transceivers. Capacity is one of the most popular performance metrics in MIMO transceiver optimization. It can be expressed as the form of MSE matrix considering the well-known relationship between the MSE matrix and capacity [40], i.e., . Then, the objective can be given as

(21)

MSE is another widely used performance metric that demonstrates how accurately a signal can be recovered. The corresponding weighted MSE minimization objective is

(22)

where is a general, not necessarily diagonal, weight matrix, even if it is often diagonal in many applications.

Iii-B Hybrid MIMO Transceiver Optimization

Denote

and

(23)

where is a unitary matrix to be determined by digital transceiver optimization in the next section. Then (15) can be rewritten as

(24)

where

(25)

The optimal is usually a function of , for all objective functions as demonstrated by (19) for in (III-A). From (24), we can conclude that the optimal is a function of . Therefore, using (II-B) and (24), the objective function of (III) can be expressed in terms of as

(26)

After introducing and a new auxiliary matrix , the objective function is transferred into rather than . Note that this new function notation, , is defined only for notational simplicity and it explicitly expresses the objective as a function of matrix variables and . Therefore, the optimization problem in (III) is further rewritten into the following one

(27)

We will discuss in detail how to solve the optimization problem (III-B) with respect to , ,, and subsequently. In (III-B), has been formulated as a function of , ,, and . When , ,, and are calculated, the optimal can be directly derived based on (19).

Iv Digital Transceiver Optimization

In the following, we focus on the digital transceiver optimization for the optimization problem (III-B). More specifically, we first derive the optimal unitary matrix and then find the optimal .

Iv-a Optimal

At the beginning of this section, two fundamental definitions are given based on the following eigenvalue decomposition (EVD) and SVD

(28)

where and denote a diagonal matrix with the diagonal elements in nondecreasing order.

Denote as the unitary matrix that makes the lower triangular matrix in (20) has the same diagonal elements. It has been shown in [39, 40, 38] that the optimal for the first group of matrix-monotonic functions can be expressed as

(29)

The above results are obtained by directly manipulating with the objective function in (III-B), and thus the optimal varies with the matrix-monotone increasing function in (III).

For the capacity maximization in (21), the objective function of (III-B) can be written as

(30)

Since the function in (30) is independent of as long as it is a unitary matrix, the optimal , namely , can be any unitary matrix with proper dimension.

For the weighted MSE minimization given by (22), the objective function of (III-B) can be rewritten as

(31)

Based on the EVD and SVD defined in (IV-A) and the matrix inequality in Appendix C, the optimal is

(32)

We have to stress that it is still hard to find the closed-form expression for the optimal for an arbitrary function . However, most of the meaningful and popular metric functions have been shown included in one of the above function families, and are with the closed-form expression for optimal .

Iv-B Optimal

After substituting the optimal into the objective function of (III-B), the objective function becomes a function of the eigenvalues of , i.e.,

(33)

where and is the th largest eigenvalue of . It is worth highlighting that for and based on (29) and (32) we can directly have (33). For the optimal can be an arbitrary unitary matrix, minimizing mathematically equals to minimizing for any . In other words, (33) always holds for these kinds of functions discussed above.

Note that the definition in (33) follows from the facts that the unitary matrix in has been removed by the optimal and only its eigenvalues remain to be optimized. Therefore, the unified hybrid MIMO transceiver optimization in (III-B) is simplified to

(34)

By applying the obtained results of and the fact that is a matrix-monotone increasing function, it can be concluded from the discussion in [39, 38] that is a vector-decreasing function for . Moreover, substituting the optimal into the objective function of (III-B), for and we have

(35)
(36)

respectively, which implies that is also vector-decreasing. In a nutshell, based on we can conclude that in (IV-B) is a vector-decreasing function. Thus, from (IV-B), the optimization becomes maximizing the eigenvalues of . Each eigenvalue of corresponds to SNR of an eigenchannel.

In problem (IV-B), the variables are still matrix variables. To simplify the optimization, we will first derive the diagonalizable structure of the optimal matrix variables. Based on the derived optimal structure, the dimensionality of the optimization problem are reduced significantly.

In order to derive the optimal structure and to avoid tedious case-by-case discussion, we consider a multi-objective optimization problem in the following. Its Pareto optimal solution set contains all the optimal solutions of different types of transceiver optimizations. In particular, as discussed in [40], the optimal solution of problem (IV-B) with a specific objective function, i.e., , , or , must be in the Pareto optimal solution set of the following vector optimization (multi-objective) problem

(37)

Equivalently, the vector optimization problem in (IV-B) can be rewritten as the following matrix-monotonic optimization problem

(38)

It is worth noting that optimization (IV-B) aims at maximizing a positive semi-definite matrix. Generally speaking, maximizing a positive semi-definite matrix includes two tasks, i.e., maximizing its eigenvalues and choosing a proper EVD unitary matrix. Note that in (IV-B) there is no need to optimize the EVD unitary matrix, because the constraints can remain satisfied if only EVD unitary matrix changes. Using the definitions in (23) and given analog precoder and analog processor , problem (IV-B) is a standard matrix-monotonic optimization with respect to . It follows

(39)

Based on the matrix-monotonic optimization theory developed in [40], the optimal solution of (IV-B) satisfies the following diagonalizable structure.

Conclusion 2: Defining the following SVD,

(40)

with the diagonal elements of in decreasing order, the optimal satisfies

(41)

where is a diagonal matrix determined by the specific objective functions, e.g., sum MSE, capacity maximization, etc., as discussed in the previous section. The unitary matrix can be an arbitrary unitary matrix.

Thus far by using Conclusion 2, the optimal can be obtained by conducting basic manipulations as in [40] on optimizing given a specific objective function. As a result, the remaining key task is to optimize the analog precoder and processor, which is the focus of the following section.

V Analog Transceiver Optimization

Based on the optimal solution of digital precoder given in the previous section, we optimize the analog precoder and processor under constant-modulus constraints. In the following, the optimal structure of the analog transceiver is first derived. Different from existing works, we show that the analog precoder and processor design can be decoupled by using the optimal transceiver structure. This optimal structure greatly simplifies the involved analog transceiver design.

For the analog transceiver optimization in (IV-B) and using (23), we have the following matrix-monotonic optimization problem

(42)

Denote the SVDs

(43)
(44)

In Appendix D, we prove the following conclusion on the optimal structure of and .

Conclusion 3: Let the SVD of be

(45)

The singular values in do not affect the objective function in (42), and the unitary matrix for the optimal satisfies

(46)

On the other hand, denote the SVD of as

(47)

The singular values in do not affect the objective in (42), and the unitary matrix for the optimal satisfies

(48)

Based on the optimal structure given in Conclusion 3, in the following two kinds of algorithms are proposed to compute the analog precoder and processor. The first one is based on phase projection, which provides better performance while the second one based on a heuristic random selection, is with low complexity.

V-a Phase Projection Based Algorithm

Analog Precoder Design

From Conclusion 3, the optimal analog precoder should select the first -best eigenchannels. It is challenging to directly optmize based on (46) because of the SVD of a constant-modulus matrix. Alternatively, we resort to finding a matrix in the constant-modulus space with the minimum distance to the space spanned by . Then, the corresponding optimization problem of analog precoder design can be formulated as