I Introduction
Millimeterwave (mmWave) communication has become a key technique for nextgeneration wireless communication systems owing to its substantial bandwidth[1, 2, 3], but unfortunately it has a high freespace path loss [4]. A promising technique for mitigating this problem is to involve massive multipleinputmultipleoutput (MIMO) techniques [5, 6, 7]. Fortunately the mmscale wavelengths allow for example 256 antenna elements to be packed in a relatively small physical size [4], which is capable of compensating for the high path loss with the aid of transmit precoder (TPC) [5]. Despite this potential, a range of practical challenges hamper the implementation of mmWave massive MIMO systems. Traditionally, MIMO systems tend to rely on the fully digital precoding, where each antenna is supported by a dedicated radio frequency (RF) chain. This leads to high hardware cost and high power consumption for mmWave massive MIMO systems relying on large antenna arrays[8].
Hence, hybrid TPC solutions have been proposed for circumventing this problem [9, 10, 11, 12, 13], where the TPC is decomposed into the digital precoding requiring a reduced number of RF chains and the large analog precoding, realized by analog phase shifters. Hybrid precoding combined with beam selection has been discussed in [14] and [15], when the phase shifters are used for analog precoding. A switching network is incorporated between the RF chains and phase shifters in [14] for reducing the hardware cost, while retaining good performance. The authors of [15] propose to add a fully/sub connected switching network between the phase shifters and the antennas for reducing the number of phase shifters without degrading the spectral efficiency. A particularly promising way of implementing the analog precoding is using the lens antenna array [16, 17, 18, 19, 20], which includes a lens and an antenna array located on the focal surface of the lens. Lenses focus the incident mmWave beams (signals) on different antennas. In this way, the traditional spatial channel is transformed into the socalled “beamspace channel.” Due to the limited scattering experienced at mmWave frequencies, the number of focusedenergy beams of the beamspace channel is small [21]. Thus the transmitter can select a subset of focusedenergy beams by switches for ensuring that the number of RF chains is reduced without significant performance loss[22, 23]. Then, the digital precoding is performed on the reduceddimensional beamspace channel, which is termed as “beamspace precoding” in this paper.
However, designing the TPC consisting of beam selection and beamspace precoding is not a trivial task for wideband mmWave MIMO systems. Due to the effect of beam squint in wideband systems [24], the focusedenergy beams of beamspace channels become frequencydependent. However, the traditional beam selection network is frequencyindependent [11], which will lead to considerable performance loss due to the power leakage of beamspace channel at certain frequencies. To overcome this problem, an intuitive technique is to select more beams to cover the entire channel bandwidth. However, this will unfortunately increase the number of RF chains, hence the hardware cost and power consumption as well. In conclusion, it is challenging to design the beam selection and beamspace precoding schemes for wideband systems to maintain satisfactory performance across the entire channel bandwidth without increasing the number of RF chains.
Ia Prior work
For the family of wideband mmWave MIMO systems relying on lens antenna arrays, the authors of [25, 26] adopted a singlecarrier transmission scheme relying on a path delaycompensation technique, where the frequencyselective multiuser MIMO channels are transformed to several lowdimensional parallel frequencyflat MIMO channels for different users. Then, a joint antenna selection and beamforming scheme was proposed for eliminating the interuser interference. The authors of [27] analyzed the effect of beam squint on the system’s performance loss and quantified the number of dominant beams required for maintaining a satisfactory performance. The results were obtained for singleinput multipleoutput systems communicating over lineofsight channels. Gao et al. [28]
proposed an adaptive beam selection network, which consists of a small number of 1bit phase shifters. This architecture is used adaptively as a random combiner during the channel estimation while during data transmission as a traditional beam selection network. However, the performance erosion caused by the effect of beam squint was still not alleviated.
IB Contributions
In this paper, we propose a phase shifteraided selection network combined with an efficient TPC design for wideband mmWave MIMO systems relying on lens antenna arrays. The main contributions of this paper are summarized as follows:

We propose a phase shifteraided selection network for mitigating the effect of beam squint. The wideband beamspace MIMO channel is frequencydependent due to the beam squint, while the beam selection network is frequencyindependent, which will lead to the power leakage of beamspace channel at certain frequencies. In the proposed selection network, we capture most of the channel’s output energy over the entire bandwidth without increasing the number of RF chains, where each RF chain is designed for supporting multiple focusedenergy beams via a subarray connected phase shifter network. Upon relying on a carefully designed TPC composed of beamspace precoding and beam selection, the proposed architecture achieves a nearunimpaired sumrate despite relying on a reduced number of RF chains.

To design an efficient TPC maximizing the sumrate, we first decouple the design problems of beamspace precoding and beam selection by exploiting the fact that the number of candidate beam selection matrices is limited. For a given beam selection matrix, we propose a successive interference cancellation (SIC)based beamspace precoding scheme, which is capable of achieving the maximum mutual information (MI) of wideband mmWave MIMO channels. Specifically, the beamspace precoding is realized by intrinsically amalgamating our baseband precoding and the phase shifter network. Given the subarray connected structure of the phase shifter network, the optimization problem of MI maximization may be readily decomposed into several subproblems. Then, by appropriately adapting the classic concept of SIC signal detection [12], we propose a SICbased beamspace precoding design, where each subproblem is solved after removing the contributions of the previously solved subproblems.

Once the beamspace precoding method has been designed for a given beam selection matrix, the TPC design problem is reduced to the beam selection design problem, which can be solved by the optimal exhaustive search, but its complexity may still be excessive. To avoid the high complexity of exhaustive search, we develop an energymax beam selection method. Specifically, we first prove that the energymax beam selection design is capable of approaching the maximum MI, which means that the selection matrix should be designed to select the focusedenergy beams of the wideband beamspace channel. Then, we derive the number of beams required for striking an attractive hardware cost/power consumption vs sumrate tradeoff. Extensive simulations verify the superior performance of the proposed beamspace precoding and beam selection methods.
The rest of the paper is organized as follows. In Section II, we present both our system and channel models. In Section III, we first propose a phase shifteraided selection network for mmWave MIMO systems relying on lens antenna arrays. Then the SICbased beamspace precoding and energymax beam selection methods are developed. Our simulation results are provided in Section IV. Finally, our conclusions are drawn in Section V.
Notation
: Lowercase and uppercase boldface letters denote vectors and matrices, respectively.
, , , and denote the transpose, conjugate transpose, conjugate, and inverse of a matrix, respectively. denotes the determinant of a matrix. denotes rounding toward its nearest higher integer. is the trace of a matrix. is the Frobenius norm of a matrix. Finally,denotes the identity matrix of size
.Ii System and Channel Models
In this section, we describe our wideband mmWave MIMO system relying on a lens antenna array and wideband beamspace channel model. The effect of beam squint over different frequencies is also highlighted.
Iia System model
We commence by briefly introducing the wideband mmWave MIMO system model relying on a lens antenna array. We consider a MIMO aided orthogonalfrequencydivisionmultiplexing (OFDM) system using subcarriers. There are transmit antennas (TAs) and RF chains at the transmitter (). At the receiver side, there are receive antennas (RAs) and RF chains (). The transmitter conveys data streams at each subcarrier to the receiver, so that we have and .
The transmitter is illustrated in Fig. 1, while the receiver relies on the inverse architecture. For the th subcarrier, the transmit data () is first precoded for reducing the interference between the different data streams. The baseband precoding matrix can adjust both the amplitude and phase of each data stream. Then the precoded data is transformed into the time domain using the
point inverse fast Fourier transform (IFFT). Then the cyclic prefix (CP) is added to each data block to eliminate the intersymbol interference (ISI). A selection network
selects TAs to be coupled to RF chains through mmWave switches^{1}^{1}1Note that the switches are passive devices, which will lead to inevitable insertion loss compared to active phase shifters. Fortunately, mmWave switches exhibit low insertion loss (1dB) and good isolation properties [29].. It is worth noting that the selection network is frequencyindependent. The transmitted discretetime complex baseband signal at the th subcarrier is given by [30](1) 
where is the transmit power. The transmit data has the normalized power of . The baseband precoding matrix satisfies the transmit power constraint of . The selection matrix has one and only one nonzero element “1” in each column, so that the RF signal generated by a single RF chain is transmitted by a selected antenna. The transmit signals are passed through a lens to form several focusedenergy beams. The lens transforms the spatial channel into the beamspace channel. We will discuss the lens and the beamspace channel later in the following Subsection IIB.
At the receiver, the incident beams (signals) are focused on a subset of the RAs by a lens. Those antennas associated with focusedenergy beams are selected by a selection network to be supported by
RF chains. After passing through the RF chains, the received baseband signals are transformed back to the frequency domain using the
point FFT. Then, the symbols at each subcarrier are combined by a baseband combining matrix . The constraints imposed on the selection matrix and combining matrix at the receiver are similar to those of the transmitter. Assuming that the frame synchronization/timing offset (TO) synchronization and carrier frequency offset (CFO) synchronization are perfect at the receiver, the received discretetime complex baseband signal at the th subcarrier is then given by(2) 
where is the beamspace channel at the th subcarrier generated by the lens, which will be discussed in the following Subsection IIB. Finally, is the additive white Gaussian noise (AWGN), where is the noise power.
IiB Beamspace channel model
To incorporate the multipath structure of wideband mmWave MIMO channels, we adopt a raybased channel model having clusters of scatterers[31, 32]. Each cluster has a limited angleofdeparture/arrival (AoD/AoA) spread and . The th cluster is assumed to be contributed by propagation subpaths. Each subpath has a time delay , a physical AoD , a physical AoA , and a complex path gain (). The spatial AoD and AoA at the th subcarrier are defined as and , where is the antenna spacing and is the wavelength of the th subcarrier. Under this model, the delay tap of the wideband mmWave MIMO channel can be expressed as [32, 11]
(3) 
where is a bandlimited pulseshaping filter evaluated at , with the system’s sampling period given by . Furthermore, and are the antenna array responses at the transmitter and receiver, which can be respectively presented as follows, where the uniform linear arrays (ULAs) are employed
(4)  
(5) 
Given the in (3), the spatial wideband mmWave MIMO channel at the th subcarrier is described as
(6)  
where is the complex gain at the th subcarrier. In contrast to narrowband systems, the spatial AoD/AoA (/) varies with the subcarrier index in wideband systems. Since we have , where is the speed of light and is the frequency of the th subcarrier, the spatial AoD can be rewritten as and the spatial AoA can be rewritten as . The value is given by with and being the central carrier frequency and the system’s bandwidth, respectively.
Due to the frequencydependent spatial AoD/AoA, the beamspace channel of wideband systems is different from that in narrowband systems. Specifically, by exploiting the fact that a lens computes a spatial Fourier transformation of incident signals, the spatial channel is transformed into its beamspace representation as [30, 28]
(7) 
where and are discrete Fourier transform (DFT) matrices, and for and for . By substituting (6) into (7), we have
(8) 
where the antenna array response of the beamspace channel at the transmitter is given by
(9)  
where . Similarly, the antenna array response of the beamspace channel at the receiver is given by
(10)  
Observe that has the following characteristics: when [33]. Thus and can be regarded as sparse vectors. As a result, the beamspace channel in (IIB) is a sparse matrix when the number of clusters is limited and the AoD/AoA spread of each cluster is small, which are commonly assumed for mmWave channels [4]. In other words, the number of focusedenergy beams in the beamspace channel is limited. Thus the selection network can select a subset of focusedenergy beams, so that the number of RF chains and effective MIMO dimension is reduced without any substantial performance loss. However, as we have mentioned above, the spatial AoD and spatial AoA are frequencydependent. Therefore, the beamspace channel is also subcarrierdependent, where both the values and positions of the nonzero elements of vary with the subcarrier index. This effect of the frequencydependent beamspace channel is termed as ‘beam squint’ [27, 28, 34]. This is an important feature of the wideband beamspace channel that differentiates it from the traditional narrowband beamspace channel. Fig. 2 shows the power distribution of a singlepath beamspace channel at different subcarriers with the physical AoD of . The central carrier frequency is 28 GHz and the system bandwidth is 4 GHz. We observe that the beamspace channel is frequencydependent.
Iii Proposed Beamspace Precoding and Beam Selection
In this section, we first propose a phase shifteraided selection network for coping with the effect of beam squint in wideband mmWave MIMO systems. Based on this architecture, a SICbased beamspace precoding technique is presented for a given beam selection design. Then, a lowcomplexity energymax beam selection method is proposed.
Iiia Phase shifteraided selection network
Note that the beamspace MIMO channel is sparse associated with frequencydependent nonzero elements (both the locations and values vary over frequencies). However, the selection network realized in the timedomain is frequencyindependent, which will lead to power leakage at certain frequencies. To address this problem, we propose a phase shifteraided selection network for mmWave MIMO systems relying on a lens antenna array, as shown in Fig. 3.
Fig. 3 illustrates the transmitter, while the receiver obeys the inverse architecture. The main difference between the proposed phase shifteraided selection network and the traditional selection network seen in Fig. 1 is that RF chains are connected to switches through a subarray connected phase shifter network. Each switch is associated with two phase shifters and each RF chain is associated with switches. The twophaseshifter structure is adopted for facilitating the amplitude variations of beamspace precoding [35], which will be explained in detail later. Within this architecture, the beamspace precoding is realized by the baseband precoding and the subarray connected phase shifter network. Through careful design of beam selection and beamspace precoding, the channel’s output energy across the entire bandwidth can be captured without increasing the number of RF chains. The design of beam selection and beamspace precoding methods will be discussed in the following Subsection IIIB and Subsection IIIC.
The model of the proposed phase shifteraided mmWave MIMO system can be formulated as a modification of (2). Specifically, the received discretetime complex baseband signal at the th subcarrier is given by
(11) 
where is the precoding matrix realized by phase shifters, while is the number of beams selected at the transmitter. Similarly, is the combining matrix realized by phase shifters at the receiver, while is the number of beams selected at the receiver. Note that and are realized relying on the subarray connected phase shifters, i.e., each RF chain is associated with phase shifters as shown in Fig. 3. Thus, and are blockdiagonal matrices. Still referring to (IIIA), and are selection matrices at the transmitter and the receiver, respectively. The baseband precoding matrix and the phase shifteraided precoding matrix satisfy the transmit power constraint of . The baseband combining matrix and the phase shifteraided combining matrix satisfy the similar power constraint of . In the following subsections, we will focus our attention on the TPC design, which is composed of beamspace precoding and beam selection.
IiiB SICbased beamspace precoding
To obtain an efficient TPC maximizing the sumrate, we first decouple the design problems of beam selection and beamspace precoding, based on the fact that the selection matrix is taken from a limited number of candidates. We consider the throughput optimization problem that captures the fact that the beam selection matrices and are taken from the sets of candidates and ,
(12) 
The throughput of the wideband channels is given by [36]
(13) 
where . The reduceddimensional beamspace channel after beam selection is defined as
(14) 
Since the selection matrices are taken from a limited number of candidates, the throughput optimization problem in (12) can be equivalently expressed in the following form
(15) 
The outer maximization is over the legitimate selection matrix candidates ( and ), while the inner maximization is obtained by finding the beamspace precoding (, ) and beamspace combining (, ) methods, given the selection matrices and . In this way, we decouple the design problems of beam selection and beamspace precoding/combining.
However, unfortunately the joint optimization of the throughput over the TPC and receiver combiner is generally intractable [9, 32]. Hence, we follow the common assumption that the receiver combiner is optimal and focus our attention on the TPC design. The design ideas of TPC given in this paper can be directly used for the receiver combiner design [10]. Without considering the combining, the throughput tends towards the MI, and we design both as well as to maximize the MI , which can be presented as [9, 10]
(16)  
The overall problem of MI maximization is modeled as the following outerinner problem form based on (15)
(17) 
This subsection focuses on the inner problem formulated in (17). Our goal is to obtain the beamspace precoding scheme including the phase shifteraided precoding matrix and the baseband precoding matrix , that maximizes for a given and . This problem is solved in two steps. The first step is to design the baseband precoding matrix for a fixed . The optimal baseband precoding matrix is given as [36]. Here gathers the right singular vectors corresponding to the
largest singular values of
, andis a diagonal matrix of the power allocated to data streams according to the water filling solution. In the high signaltonoise ratio (SNR) regime, we have
[36]. Note that is a blockdiagonal matrix due to the subarray connected structure of the proposed phase shifteraided selection network. Therefore, by normalizing the precoding vector of each subarray of phase shifters^{2}^{2}2As we will discuss later in this subsection, the precoding vector of each subarray of phase shifters, i.e., the nonzero part of the column vectors of is a singular vector., we have , where is a normalization coefficient ensuring the transmit power constraint of . Substituting into this constraint, we have . This property of ensures that the optimal baseband precoding matrix for typically satisfies in the highSNR regime^{3}^{3}3For wideband mmWave MIMO systems with beam squint, the beams selected for a given subcarrier may have zerochannel power values. However, the baseband precoding is performed based on the effective channel , which is of column full rank, since each RF chain at the transmitter is associated with multiple beams. Thus, for wideband systems with beam squint, we still have in the highSNR regime..By substituting into (16), the problem of MI maximization only depends on . Thus, the second step of solving the maximization problem of (16) is to design for maximizing the MI expressed as
(18)  
Employing Jensen’s inequality, (18) can be upperbounded as [33]
(19)  
Upon defining , the desired phase shifteraided precoding matrix maximizing the upper bound of in (19) is given by
(20) 
Once has been obtained, the baseband precoding matrix is correspondingly given by . Next, we present a SICbased algorithm to solve (20).
Since is a Hermitian positive definite matrix, it can be decomposed as . Then, the optimization target of (20) can be presented as
(21)  
By expressing , where is the th column vector of and is the submatrix of by removing , (21) is rewritten as [12]
(22) 
where . Note that has the same form as the optimization target in (21). Thus, by defining , where is the th column vector of and is the submatrix of by removing (), the optimization target in (21) can be further decomposed as
(23)  
where and . Thus, the optimization problem of (20) can be decomposed into subproblems, where the th subproblem is
(24) 
where .
Inspired by the SIC concept [12], we propose to successively solve the above subproblems. The proposed SICbased beamspace precoding scheme is summarized in Algorithm 1 and explained as follows. The algorithm starts by solving the first subproblem (24). The first column of is obtained as . Then, it is used to update the matrices and , where . The second column of is obtained by solving the second subproblem. We then repeat this procedure until the desired precoding matrix is obtained. Then, the baseband precoding matrix is designed as .
Finally, we discuss how to solve (24) for obtaining . Defining a vector , which gathers the nonzero elements of , (24) can be rewritten as
(25) 
where is the submatrix composed of the elements of corresponding to the nonzero elements of . The solution of the subproblem (25) is given by the first right singular vector corresponding to the largest singular value of [12]. Note the phase shifteraided precoding matrix is realized by a subarray connected phase shifter network and each element of is realized by a pair of phase shifters, as shown in Fig. 3. Given that the amplitude of each element of is not larger than 1, since , we can assume that each element of is expressed as (). Our target is to design two phase shifters with the phases of and to satisfy , which is equivalently expressed as
(26) 
Solving (26), we have
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