I Introduction
Millimeterwave (mmWave) is a key enabling technology for 5G and beyond [1], and its applications to vehicular communications have attracted significant attention in recent years [2]. Multipleinput multipleoutput (MIMO) system with lensarray is a costefficient way to facilitate mmWave communications [3, 4, 5, 7, 8, 9, 6]. By exploiting the energyfocusing property of largeaperture lens and small number of radio frequency (RF) chains, system can be implemented by the simple antenna switching network (ASN) instead of the bulky phase shifter network (PSN). Main benefit of this approach is that the spatial multiplexing can be achieved by using lensarrays with reduced power consumption and hardware cost [3, 4, 5].
However, major challenge of this approach is that we need to estimate the highdimensional channels from a limited number of RF chains [3, 4, 5, 7, 8, 9, 6]. In [3], the lensbased approach has been proposed. In this scheme, MIMO channel is divided into the multiple singleinput singleoutput channels, assuming that angles of arrival (AoAs) and departure (AoDs) are separated sufficiently. On that basis, a channel estimation (CE) scheme for the linear lensarray has been proposed in [4]. In [5], the idea in [4] has been further extended to the problem to estimate the channels between base station (BS) using fulldimensional (FD) lensarray and multiple users with analog precoding. However, it has been pointed out in [6] that the residual interference from different paths still exists, resulting in the degradation of performance. To address the problem, a beam selection scheme for the singleantenna users has been proposed. In [7, 8], more sophisticated approaches to estimate the channels between FD lensarray and users with one or multiple singleantenna has been proposed. To support the multiantenna users with analog precoding, a CE scheme utilizing the image reconstruction technique has been proposed [9]. Drawback of the approaches in [4, 5] is that they require a complicated ASN, where one RF chain needs to activate all transmit antennas. Moreover, solutions in [7, 8, 9] require an extra complicated PSN (see Fig. 1(a)), causing insertion loss, power consumption, and also extra hardware cost.
In this paper, we propose a compressive sensing (CS)based CE technique for mmWave FDMIMO with lensarray. In this scheme, we use a lowcost and energysaving ASN where each RF chain is activating at most one antenna (see Fig. 1(b)). First, we propose a framework of pilot training taking into account the constraint imposed by ASN. Then, we design a redundant dictionary tailored for FD lensarray to combat the power leakage caused by continuous AoAs/AoDs. Moreover, to minimize the total mutual coherence of the measurement matrix [12, 14, 13, 15], we design the transmit/receive pilot signals in the baseband (BB) part. Simulations are conducted to demonstrate the effectiveness of the proposed scheme over the conventional approaches.
Our contributions are summarized as follows.

We propose a CSbased CE approach that takes into account the constraint imposed by ASN. This is in contrast to the existing CSbased solutions in [7, 8, 9] where a randomized PSN is employed to generate pilot signals, so that entries of the measurement matrix are independent identical distributed (i.i.d.) with good restricted isometry property (RIP), at the cost of complicated RF hardware.

To combat the power leakage in the angulardomain sparse CE for MIMO systems, we consider the unique antenna structure of lensarray and design a redundant dictionary tailored for FD lensarray to sparsify the channel and improve the sparse CE performance. To the best of our knowledge, this is the first trial to design a redundant dictionary for the FD lensarray.

We design the BB pilot signals for further improvement of CE performance. The stateoftheart pilot design in [7, 8, 9, 11] depends on the randomized PSN to design the measurement matrix according to the RIP. However, these solutions are no longer applicable for lensarray with simple ASN, and RIPbased pilot design is very difficult in practical scenario [12]. Hence, we design the pilot under a more tractable total mutual coherence minimization criterion [12, 14, 13, 15], whereby the closedform solution to optimize the BB pilot signals can be derived.
Notations
: Vectors and matrices are denoted by lower and uppercase boldface letters, respectively.
, , , and denote the conjugate, transpose, conjugate transpose and trace of a matrix, respectively. and are the sets of complex numbers and integers, respectively.denotes the complex Gaussian distribution.
and represent the th element of a vector and th row, th column element of a matrix, respectively. represents the identity matrix. and denote the flooring function and ceiling function, respectively. The “sinc” function is defined by . , , and represent the norm, Frobenius norm, and (block) diagonalization, respectively. is the Kronecker product and is the vectorization operation according to the columns of the matrix.Ii System Model
We consider a mmWave FDMIMO system with lensarrays at both the transmitter and the receiver. In this work, we employ a simple and practical ASN, where each RF chain can activate at most one antenna, as shown in Fig. 1(b). Compared to the fullconnected PSN (see Fig. 1(a)) [7, 8, 9], the proposed ASN is simple to implement and also powerefficient so that it is a more appealing model for MIMO communication under the limited RF power resource [16]. On the focal surface of electromagnetic (EM) lens, we use the antenna distribution for FD angular coverage proposed in [5]. In this model, the total number of transmit antennas is given by^{1}^{1}1Our distribution is slightly different from that in [5] to make be even without degrading the spatial resolution of lensarray for ease of analysis.
(1) 
where and are the normalized apertures of the transmit FD lens in the horizontal and vertical dimensions, respectively. Similarly, for the normalized apertures of receive FD lens, the number of receive antennas can also be calculated using (1). The numbers of RF chains at the transmitter and the receiver are denoted by and , respectively. Moreover, the associated channel matrix can be modeled as [9]
(2) 
where is the number of multipath components, is the complex gain corresponding to the th path, () and () are the vertical angle and horizontal angle of AoD (AoA) of the th path, respectively, is the normalization factor, and and are the steering vectors of lensarrays at the transmitter and receiver, respectively. The steering vectors for lensarrays are different from those for phasedarrays. Taking the transmitter for instance, the steering vector can be expressed as
(3) 
where (,) is the angular coordinate of the th antenna dependent on the location of the antenna on the focal surface [5] and is a normalization factor guaranteeing .
Iii Proposed Channel Estimation Technique
Iiia Proposed CS Approach Based on Pilot Training
Considering the hardware property of lensarray with ASN, we design a pilot training framework based on CS. Specifically, the transmit pilot signal in the th time block can be expressed as a product of the RF part and the BB part as
(4) 
We assume , without loss of generality, and . This implies that every successive transmit BB pilot signals will share the same transmit RF pilot signal . In our scheme, antennas are simultaneously activated as a transmit (Tx) group to form directional transmit beams, as illustrated in Fig. 2.
At the receiver, we assume that and all RF chains are used. Thus, each time block can be divided into equallength time slots (see Fig. 2). The received signal in the th time slot from the th time block can be expressed as
(5) 
where and are the RF and BB parts of the receive pilot signals, respectively, is the additive white Gaussian noise vector, and . Similar to the transmitter, antennas are activated as a receive (Rx) group and directional receive beams are formed. For successive time slots at the receiver, we obtain by collecting . That is,
(6) 
where , , , and .
Further, by collecting , we obtain an aggregate observation of the channel given by
(7) 
where , , , , and .
In a nutshell, in each Tx (Rx) group, () nonoverlapping transmit (receive) training beams will be simultaneously formed by activating all RF chains at the transmitter (receiver). With the pilot signals received by all Tx groups and all Rx groups, the FD angulardomain channels will be sounded. For lensarray with simple ASN, our proposed CSbased approach takes into account the constraint imposed by practical ASN, so that the RF training beams are directional, which is different from the omnidirectional RF training beams in [4, 5, 7, 8, 9]. Note also that in each Tx group, the number of BB pilot signals is smaller than that of RF beams as we set , i.e., , and the smaller indicates the smaller training overhead.
We note that a signal transmitted in mmWave band suffers from severely high path loss and blockage effect. Consequently, there exists only a few multipath components between the transmitter and the receiver in mmWave MIMO systems (i.e., or ). This property is often referred to as the angular sparsity [15, 1, 17]. Thanks to the angledependent energy focusing property of lensarray [3, 4, 5], along with the angular sparsity of mmWave channel, the channel matrix can be readily modeled as a sparse matrix, indicating that only a small number of channel elements have the dominated channel energy. By vectorizing , therefore, a problem to estimate from (7) can be formulated as the sparse signal recovery problem as
(8) 
where is the vectorized received signal, is the measurement matrix, and . Since is a sparse vector, we can basically use any sparse signal recovery algorithms such as orthogonal matching pursuit (OMP) to efficiently estimate . Since is in general significantly smaller than , pilot overhead of our approach is much lower than that required by conventional approaches such as least square (LS) estimation technique [4, 5]. Also note that the measurement matrix in (IIIA) is dedicated to the proposed framework of pilot training, which is essential to the design of BB pilot in Section IIIC.
IiiB Redundant Dictionary Design to Sparsify Channels
The accuracy of CSbased CE depends heavily on the sparsity of in (IIIA) [12]. However, the power leakage caused by the mismatch between continuous AoAs/AoDs and discrete dictionary with limited resolution may weaken the sparsity of [7]. To mitigate this behaviour, we design a redundant dictionary by quantizing both the virtually vertical and horizontal angles with a finer resolution, where the sets of the quantized virtual angles can be expressed as
(9) 
Here and are the sets of quantized vertical and horizontal angles, respectively, and . Under this setting, the channel matrix can be expressed as
(10) 
where
are the dedicated redundant dictionaries for lensarray, is the sparse channel approximation in the quantized virtual angular domain, is the quantization error matrix treated as a random noise. Note that the redundant dictionary design in (10) is tailored for FD lensarray according to (3), which is essentially different from the dictionary design for phased uniform linear array in [10].
By substituting (10) into (IIIA), we have
(11) 
where is the redundant dictionary matrix, is the enhanced sparse channel vector after the transformation by , and is the effective noise vector. Usually, the quantization error and the degradation of the sparsity of can be mitigated by increasing and . Thus, to obtain a better performance, we first use (11) to estimate and then get the estimate of using (10).


Proposed Scheme with OMP  DCbased Support Detection Scheme [7]  


Correlation  
Project subspace 

Update residual  
Stop criteria  N/A  

IiiC Pilot Signals Design Based on CS Theory
To obtain a measurement matrix with a good RIP, many CSbased CE techniques employ the pilot signals randomized by the phaseshifters in the RF part [7, 9, 8]. To do so, an extra PSN, as shown in Fig. 1(a), is required. Nevertheless, designing pilot signals for lensarray with simple ASN to achieve the i.i.d. entries of measurement matrix with good RIP is not possible due to the hardware constraint resulted from ASN. This motivates us to design the pilot signals constructing the measurement matrix in (11), based on the more tractable total mutual coherence minimization criterion [12, 14, 13, 15] for reliable sparse CE. Since we adopt the simple ASN, the RF parts of the pilot signal can be expressed as
(12) 
where is the identity matrix after the random permutation among its columns, and the elements “1” and “0” denote switching on and off, respectively. Note that we randomly permute the columns of and to improve the CE performance.
Moreover, given and , we minimize the total mutual coherence [12, 13, 14, 15] of the matrix by formulating the design problem of and as
(13) 
where is the total mutual coherence of , and we assume that the transmit power is normalized and for ease of analysis, respectively. According to [10], satisfies the following inequality
(14) 
which sheds light on how we decouple the problem (13) to avoid the difficulty in joint optimization. To be specific, we minimize and over and , respectively. Taking as an example, we have
(15) 
where (a) is based on and the normalized norm assumption for each column of , (b) follows from and ^{2}^{2}2This approximation is empirically reasonable for the dictionary matrix and we can use the metric with to justify it. In our simulations, we calculate that the value of will be smaller than , which is sufficiently small to ensure the validity of this approximation., and (c) is due to . One can readily observe that columns from , should be mutually orthogonal to minimize (15). Therefore, the solution minimizing (15) can be expressed as
(16) 
where are the matrices satisfying . In our simulations, we use the specific solution , where is the
discrete Fourier transformation (DFT) matrix and the notation
denotes the submatrix of constructed from the first columns of . Similarly, the obtained solution for is given by(17) 
where can be arbitrary unitary matrices. In our simulations, we set as the DFT matrix.
IiiD Computational Complexity Analysis
In this subsection, we focus on the computational complexity of the proposed scheme. Since the CE problem has been formulated as the sparse signal recovery problem in Section IIIB, various offtheshelf CS algorithms can be used to estimate the channels for the proposed scheme. Clearly, the computational complexity of the proposed scheme heavily depends on the adopted sparse signal recovery algorithms. In this paper, we employ the wellknown OMP algorithm [10, 12] to solve in (11). The computational complexity of our scheme is summarized in the left column of Table I. In Table I, the notation stands for “on order of ”, and the index of iteration in OMP algorithm is denoted by . In essence, the OMP algorithm consists of four major steps: correlation, project subspace, residual update, and stop criteria identification, and the computational complexity of each step is summarized in the table. Taking a typical system configuration with , and as an example, we can observe that the most significant computational complexity burden comes from the step of correlation due to the large and . However, the computational complexity of other steps, especially for the step of project subspace requiring matrix inversion operation, is irrelevant to and , which indicates that the complexity of the proposed scheme is acceptable even though large and are choosen for better resolution of the redundant dictionary.
To compare the proposed scheme with existing techniques, we consider the stateoftheart dual crossing (DC)based support detection algorithm [7]. The DCbased support detection algorithm is an extension of OMP algorithm, and its computational complexity is provided in the right column of Table I. In [7], is the geometric size of transmit lensarray, the reciever has one antenna with , and is a predefined parameter. Note that can be very large compared with the number of multipath (e.g., was set to when in [7]). Therefore, the DCbased support detection algorithm suffers from high computational complexity due to highdimensional matrix inversion operation in project subspace step. By contrast, although the computational complexity of the proposed scheme increases with and , it can provide a more robust performance and a tradeoff between the performance and the complexity, which will be detailed in the next section.
Iv Simulation Results
In the simulation, for the channel model, we set the carrier frequency to GHz [11], , and the AoAs/AoDs , , ,
follow a uniform distribution
. First, we compare the performance of the proposed scheme and the DCbased support detection scheme [7]. For fairness, we consider a downlink system with a singleantenna user, i.e. , consistent with those in [7]. We set with according to (1), which is equivalent to an uniform rectangular array in [7], and . Note that the DCbased support detection scheme requires the bulky fullconnected PSN, so that it will cause a prohibitively large hardware cost.In Fig. 3(a), we plot the normalized mean square error (NMSE) performance of the DCbased support detection scheme and the proposed scheme with different
as a function of signaltonoise ratio (SNR), where
is considered. It can been seen that even with a simple ASN, the proposed scheme outperforms the DCbased support detection scheme when the appropriate parameters are chosen. We also observe that the performance of the proposed scheme improves with , which trades off the NMSE performance with the computational complexity. Moreover, when is larger than , the performance improvement is minor, but at the cost of prohibitive computational complexity. For this reason, we set in our experiments.We further investigate the robustness of different schemes as a function of the number of multipath in Fig. 3(b). Note that when increases, the power leakage becomes severe and the structured sparsity patterns of different paths leveraged in the DCbased support detection scheme are destroyed. As a result, we observe from Fig. 3(b) that the performance of the DCbased support detection scheme is degraded when increases. However, by leveraging the enhanced sparsity benefited from the designed redundant dictionary, the proposed scheme can effectively estimate the channels with more multipath components.
Another drawback of the DCbased support detection scheme in [7] is that it only considers the systems with singleantenna users, while the proposed scheme can be directly applied to the system with lensarray at both the transmitter and the receiver. We consider such a more general scenario and investigate the performance of the proposed scheme. In simulations, we consider with according to (1), (i.e., ) and .
For comparison, we also investigate the following schemes: 1) the proposed scheme with random BB pilot signals [11]; 2) the proposed scheme without using redundant dictionary, and 3) the conventional welldetermined LS estimator based on the beam training, i.e., using LS to estimate from (IIIA) when , and .
In Fig. 4(a), we plot the NMSE performance of different schemes as a function of SNR. We observe that the NMSE of the proposed scheme improves with . We also observe that the gain of the proposed BB pilot signal design and the redundant dictionary design is considerable within a wide range of SNR. For example, when and SNR dB, the proposed scheme has more than dB gain over the welldetermined LS scheme with only half the pilot resources. Note that for beam training based LS estimator, in (IIIA
) becomes a unitary matrix, so the NMSE performance of LS estimator achieves the minimum of CramérRao lower bound of linear estimators. However, by leveraging the sparsity of mmWave channels, the proposed CSbased CE scheme outperforms the LS estimator even with much smaller number of pilots, especially at low SNR. Considering that the SNR is usually low in most mmWave systems at the CE stage, the proposed scheme is effective in estimating the practical channels for mmWave FDMIMO with lensarray.
We further compare the biterrorrate (BER) performance of the proposed CE scheme and welldetermined LS approach. Based on the estimated channel, we apply an energybased antenna selection scheme in [4] for the data transmission. We consider the 64QAM modulation, SVD precoding, and turbo channel coding with 1/3 code rate. From Fig. 4(b) we see that when compared to the welldetermined LS scheme, the proposed scheme achieves improved BER performance with a reduced pilot overhead. When BER is , for example, the proposed scheme with achieves about dB gain over the LS scheme, while the pilot overhead is reduced by .
V Conclusions
In this paper, we proposed a CSbased CE scheme for mmWave FDMIMO with lensarray, which sheds light on the application of CS techniques to lensarray using simple and practical ASN. Specifically, we first proposed a framework of pilot training based on CS under the constraint imposed by ASN. Then, we designed the dedicated redundant dictionary tailored for FD lensarray. We also designed the transmit/receive pilot signals for improved CE performance. In particular, the BB pilot signals are designed to minimize the total mutual coherence of the measurement matrix. From the simulation results, we demonstrated the effectiveness of the proposed CE technique.
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