# Channel Estimation for Switch-Based Millimeter-Wave Communications via Atomic Norm

Authors: Tao Ma; Xiangning Fan; Xiaohuan Wu
Journal: International Journal of Antennas and Propagation
 (2021)

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## Abstract

Channel estimation is a challenging issue in millimeter-wave massive multiple-input-multiple-output (MIMO) communication systems due to the large number of antennas in the transceiver. Existing methods are usually based on phase shifters which may not be a simple circuit at mmWave band. In this paper, we construct a switch-based architecture for analog processors from the coarray point of view and then propose an atomicℓ0-norm minimization problem. We then propose an efficient algorithm to solve this problem based on Wirtinger projection. Since the proposed method requires no angle discretization, it does not suffer from grid mismatch effect that greatly deteriorates the estimation performance of grid-based channel estimation methods. Compared to the atomic norm minimization (ANM) method, our method does not involve vectorization of the channel matrix and hence the dimensionality of the problem is much less than that of ANM. We show that our method is able to provide comparable estimation performance to ANM but with much less computational time. Extensive simulations are carried out to verify the effectiveness of our proposed method.


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## Body

## 1. Introduction

Millimeter-wave (mmWave) communications is a key technology for the 5th generation (5G) mobile communication system. Compared to the sub-6 GHz communication system, the main differentiating factor for mmWave communication systems is the tenfold increase in carrier frequency. This difference can provide multigigabit services which are able to meet future traffic demand [1]. However, the mmWave signals in the high-frequency band suffer from large pathloss; thus, the power of the received signal to be detected by the receiver can be negligible and reliable communication cannot be achieved. To solve this problem, high-resolution beamforming by using massive multiple-input-multiple-output (MIMO) is essential in combating the large pathloss for mmWave communication systems. By using massive MIMO, the transmitter can concentrate the transmitted power on a specific direction to highly improve the power of the received signal. On the other hand, the small wavelength of mmWave signal allows hundreds of antennas to be accommodated within a reasonable physical size, e.g., the 8×8 antenna array in a hand-hold unit, making high-resolution beamforming with large-scale antenna arrays possible.Nevertheless, high-resolution beamforming requires accurate full channel state information (CSI) which is difficult to obtain due to the large number of antennas. Conventional channel estimation methods suffer from high training overhead and complexity [2, 3]. To address this issue, a codebook-based beam searching strategy is proposed to find the correct pair between the beamformer and combiner [4, 5]. Although the hierarchical search can be incorporated to reduce complexity to some extent, the performance heavily depends on the predefined training beam codebook [6]. Another approach is to exploit the sparse nature of the mmWave channel. Due to large pathloss, there only exist a few ray components between the transmitter and receiver, i.e., the channel is sparse in space [7, 8]. With a high degree of freedom provided by a large number of antennas, we are able to find the angle of arrival (AoA) and angle of departure (AoD) as well as the complex gain of each ray [9, 10]. Then, the channel estimation problem can be formulated as an angle estimation one. Existing methods include compressive sensing- (CS-) based methods [11–13] and subspace-based methods [14–17]. CS-based methods formulate the channel estimation problem as a sparse signal recovery one and utilize the CS recovery methods such as orthogonal matching pursuit (OMP) [18] to retrieve the sparse signal where the indices of nonzero elements indicate the AoAs and AoDs. The subspace method in [14, 19] employs beamspace 2-D MUSIC to estimate the channel. However, for CS-based methods, formulating the sparse model requires discretizing the angle space into a set of predefined angle grids and then assuming that the AoAs and AoDs exactly lie on the grids [20, 21]. Since the angle space is continuous rather than discrete, this discretizing procedure will bring in nonnegligible bias between the true angle and the closest grid. We call this the grid mismatch effect. Thus, a dense grid set is appealing since its bias can be small. However, since the dimensionality of the sparse model is proportional to the size of the grid set, a dense grid set may suffer from high computational cost. Moreover, a dense grid may conflict the restricted isometry property (RIP), and thus, it is not easy to find the balance between accuracy and efficiency. For 2-D MUSIC, finding the angles also require discretizing, and thus, it may also encounter computational issues when the grid set is large.Recently, a gridless method which does not require angle discretization is proposed [22–26]. It employs the atomic norm minimization (ANM) concept into angle estimation and then reformulates the channel estimation problem as the semidefinite programming (SDP) which can be solved by CVX [27]. The ANM method does not suffer from the accuracy and efficiency issues caused by gridding. Theoretical analysis shows that the ANM method is an asymptotic maximum likelihood (ML) estimator [28] and its complexity is also immune to the size of the grid set. Although ANM shows excellent estimation performance in angle estimation [29] or channel estimation [24, 25], the main obstacle of the ANM estimator is the computational issue since solving SDP by CVX is time-consuming. Especially, in the full-dimensional MIMO case, ANM requires to solve an n2-dimensional SDP, where n denotes the number of antennas on the transmitter or receiver which may be large [24]. Thus, it is urgent to derive a new algorithm for ANM methods to reduce computational complexity.The ANM-based channel estimation method [24] only considers the phase shifters in hybrid architecture of the mmWave system. The hybrid architecture can achieve near-optimal performance compared to the fully digital transceivers [1]. However, the phase shifter-based network is not a simple circuit at the mmWave band [30]. Another type of architecture is to employ switch-based networks [31]. It is shown that the switch-based network is preferred in a range of operating conditions [30]. From the viewpoint of array structures, antenna selection is relative to sparse arrays. Different selection strategies result in different sparse array architectures. From the coarray perspective [32], the antenna selection strategy having the longest uniform coarray part enjoys the best estimation performance. Several recently proposed sparse arrays such as coprime array [32], nested array [33, 34], and fractal array [35] have good coarray property and can be used in competitive antenna selection strategies for mmWave channel estimation. The coprime array has been incorporated into mmWave channel estimation in [36], and the nested array has been used in channel estimation and tracking in [37]. However, these papers only consider a single user with one antenna rather than multiuser or multiantenna.In this paper, we consider the switch architecture for the analog processor in channel estimation of mmWave massive MIMO systems. We first exploit the antenna selection from the coarray point of view and then propose an atomicℓ0-norm minimization problem. Compared to ANM, our method has much less dimensionality of the problem and hence is much more efficient than ANM. We also propose an efficient algorithm to solve this problem based on Wirtinger projection. Our method requires no angle discretization and hence is immune to the grid mismatch effect. We also carry out simulations to show the superiority of our method.Notations:ℂ and ℤ denote the sets of complex numbers and integers, respectively. A∗, AT, and AH denote the conjugate, transpose, and conjugate transpose of matrix A, respectively. vecA denotes the vectorization operator that stacks matrix A column by column. A⊙B and A⊗B are the Khatri–Rao and Kronecker products of matrices A and B, respectively. tr• and rank• denote the trace and rank operators. IN denotes the identity matrix of size N×N. A1, A2, and AF denote the ℓ1-norm, ℓ2-norm, and Frobenius norm of A, respectively. A≥0 means that matrix A is positive semidefinite (PSD). For a vector x, diagx denotes a diagonal matrix with the diagonal elements being the elements of vector x in turn.The rest of this paper is organized as follows: Section2 introduces the coarray concept and the system model. Section 3 provides our proposed method. Extensive simulations are provided in Section 4, and Section 5 concludes the whole paper.


## 2. Preliminary and System Model

### 2.1. Coarray Concept

In array signal processing, the aperture of an array is an important factor for angle estimation. A larger array aperture can bring in high estimation accuracy and super resolution. But, increasing the inter-element spacing is not a positive way to extend the aperture because a uniform linear array (ULA) with interelement spacing being greater than half-wavelength suffers from angle ambiguity. In this case, pseudo AoAs or AoDs will prevent us from correctly identifying the true positions. Fortunately, we can exploit the coarray concept to solve this problem. It is shown that we can construct a sparse linear array (SLA) with much larger aperture if its coarray has a long uniform part without holes [32]. Denote Ω=Ω1,Ω2,…,ΩM as the antenna indices where Ω1<Ω2<⋯<ΩM and each element is a positive integer. Then, the coarray is defined as(1)D=m1−m2+1:m1,m2∈Ω,m1≥m2.For instance, denote the arrayΩ=1,2,5,7; then, its coarray D=1,2,3,4,5,6,7 which can be regarded as a 7-element ULA and has a large aperture without angle ambiguity. Thus, some special SLAs such as coprime array and nested array having a larger aperture can provide super resolution and satisfying performance.


### 2.2. System Model

Consider the mmWave massive MIMO communication system with a single user shown in Figure1, where the transmitter is equipped with Mt RF chains and Nt>Mt antennas and the receiver is equipped with Mr RF chains and Nr>Mr antennas. The interelement spacing of each array is set to half-wavelength to avoid angle ambiguity. The analog architectures of both the transmitter and receiver are implemented by using switches. In particular, each switch is connected to a specific RF chain and can build connection between the RF chain and any antenna. The antennas selected by the switches are activated to transmit data, while other antennas remain idle. The selection strategy can be specified by a 0,1 matrix denoted by F∈ℂNt×Mt for the transmitter and W∈ℂNr×Mr for the receiver. Let ΩF=Ω1,Ω2,…,ΩMt and ΦW=Φ1,Φ2,…,ΦMr denote the element indices of the transmit antennas and receiver antennas, respectively, where Ω1<Ω2<⋯<ΩMt≤Nt, Φ1<Φ2<⋯<ΦMr≤Nr, and each element is a positive integer. In the following, we provide a simple example to better demonstrate the selection matrix. For simplicity, we only consider the transmitter. Let Nt=7 and Mt=4; if we select the antenna indexed by ΩF=1,2,5,7, then the antenna array is an SLA and the selection matrix F is as follows:Figure 1
Block diagram of the mmWave massive MIMO communication system with an analog processor implemented using switches.(2)F=1000010000000000001000000001.From equation (2), it can be seen that the m-th column of matrix F contains all zeros but a single one at the Ωm-th position. Note that if we let ΩF=1,2,3,4, the transmitted antenna array is a short ULA having a shorter aperture than the SLA case. The discrete-time transmitted signal is therefore given by(3)x=Fs,where s∈ℂMt denotes the signal after the digital processor. For simplicity, we consider a narrowband block-fading propagation channel which gives that(4)ys=WHHFs+WHn,where ys∈ℂMr denotes the received signal at the RF chains, H∈ℂNr×Nt denotes the channel matrix, and n∈ℂNr is the additive Gaussian noise with zero mean. During Mt successive time slots, the received signal at the receiver can be given as(5)Y=WHHFS+WHN,where S is the transmitted signal and N is the noise matrix. For the training phase, we assume S=IMt, and therefore,(6)Y=WHHF+WHN.Our goal is to estimateH given Y.


### 2.3. Channel Model

In the mmWave massive MIMO system, the number of rays between the transmitter and receiver is limited. The channelH can be expressed as(7)H=∑k=1KαkarθkatHϕk,where αk, θk, and ϕk denote the complex gain, AoA, and AoD of the k-th ray, respectively, and atϕk and arθk denote the steering vectors of the transmitter and receiver arrays with respect to the k-th ray, respectively, and can be given as(8)atϕk=1,ej⋅1⋅2πd/λsinϕk,…,ej⋅Nt−1⋅2πd/λsinϕkT,arθk=1,ej⋅1⋅2πd/λsinθk,…,ej⋅Nr−1⋅2πd/λsinθkT,where λ and d denote the wavelength and the spacing between adjacent antennas, respectively. Equation (7) can be compactly rewritten as(9)H=ArΣAtH,where At=atϕ1,atϕ2,…,atϕK, Ar=arθ1,arθ2,…,arθK, and Σ=diagα1,…,αK. Substituting equation (9) into model (6), we have(10)Y=WHArΣAtHF+WHN.VectorizingY results in,(11)y=vecY=FT⊗WHh+vecWHN=FT⊗WHAt∗⊙Arz+vecWHN,where h=vecH and z=α1,…,αKT.


## 2.1. Coarray Concept

In array signal processing, the aperture of an array is an important factor for angle estimation. A larger array aperture can bring in high estimation accuracy and super resolution. But, increasing the inter-element spacing is not a positive way to extend the aperture because a uniform linear array (ULA) with interelement spacing being greater than half-wavelength suffers from angle ambiguity. In this case, pseudo AoAs or AoDs will prevent us from correctly identifying the true positions. Fortunately, we can exploit the coarray concept to solve this problem. It is shown that we can construct a sparse linear array (SLA) with much larger aperture if its coarray has a long uniform part without holes [32]. Denote Ω=Ω1,Ω2,…,ΩM as the antenna indices where Ω1<Ω2<⋯<ΩM and each element is a positive integer. Then, the coarray is defined as(1)D=m1−m2+1:m1,m2∈Ω,m1≥m2.For instance, denote the arrayΩ=1,2,5,7; then, its coarray D=1,2,3,4,5,6,7 which can be regarded as a 7-element ULA and has a large aperture without angle ambiguity. Thus, some special SLAs such as coprime array and nested array having a larger aperture can provide super resolution and satisfying performance.


## 2.2. System Model

Consider the mmWave massive MIMO communication system with a single user shown in Figure1, where the transmitter is equipped with Mt RF chains and Nt>Mt antennas and the receiver is equipped with Mr RF chains and Nr>Mr antennas. The interelement spacing of each array is set to half-wavelength to avoid angle ambiguity. The analog architectures of both the transmitter and receiver are implemented by using switches. In particular, each switch is connected to a specific RF chain and can build connection between the RF chain and any antenna. The antennas selected by the switches are activated to transmit data, while other antennas remain idle. The selection strategy can be specified by a 0,1 matrix denoted by F∈ℂNt×Mt for the transmitter and W∈ℂNr×Mr for the receiver. Let ΩF=Ω1,Ω2,…,ΩMt and ΦW=Φ1,Φ2,…,ΦMr denote the element indices of the transmit antennas and receiver antennas, respectively, where Ω1<Ω2<⋯<ΩMt≤Nt, Φ1<Φ2<⋯<ΦMr≤Nr, and each element is a positive integer. In the following, we provide a simple example to better demonstrate the selection matrix. For simplicity, we only consider the transmitter. Let Nt=7 and Mt=4; if we select the antenna indexed by ΩF=1,2,5,7, then the antenna array is an SLA and the selection matrix F is as follows:Figure 1
Block diagram of the mmWave massive MIMO communication system with an analog processor implemented using switches.(2)F=1000010000000000001000000001.From equation (2), it can be seen that the m-th column of matrix F contains all zeros but a single one at the Ωm-th position. Note that if we let ΩF=1,2,3,4, the transmitted antenna array is a short ULA having a shorter aperture than the SLA case. The discrete-time transmitted signal is therefore given by(3)x=Fs,where s∈ℂMt denotes the signal after the digital processor. For simplicity, we consider a narrowband block-fading propagation channel which gives that(4)ys=WHHFs+WHn,where ys∈ℂMr denotes the received signal at the RF chains, H∈ℂNr×Nt denotes the channel matrix, and n∈ℂNr is the additive Gaussian noise with zero mean. During Mt successive time slots, the received signal at the receiver can be given as(5)Y=WHHFS+WHN,where S is the transmitted signal and N is the noise matrix. For the training phase, we assume S=IMt, and therefore,(6)Y=WHHF+WHN.Our goal is to estimateH given Y.


## 2.3. Channel Model

In the mmWave massive MIMO system, the number of rays between the transmitter and receiver is limited. The channelH can be expressed as(7)H=∑k=1KαkarθkatHϕk,where αk, θk, and ϕk denote the complex gain, AoA, and AoD of the k-th ray, respectively, and atϕk and arθk denote the steering vectors of the transmitter and receiver arrays with respect to the k-th ray, respectively, and can be given as(8)atϕk=1,ej⋅1⋅2πd/λsinϕk,…,ej⋅Nt−1⋅2πd/λsinϕkT,arθk=1,ej⋅1⋅2πd/λsinθk,…,ej⋅Nr−1⋅2πd/λsinθkT,where λ and d denote the wavelength and the spacing between adjacent antennas, respectively. Equation (7) can be compactly rewritten as(9)H=ArΣAtH,where At=atϕ1,atϕ2,…,atϕK, Ar=arθ1,arθ2,…,arθK, and Σ=diagα1,…,αK. Substituting equation (9) into model (6), we have(10)Y=WHArΣAtHF+WHN.VectorizingY results in,(11)y=vecY=FT⊗WHh+vecWHN=FT⊗WHAt∗⊙Arz+vecWHN,where h=vecH and z=α1,…,αKT.


## 3. The Proposed Channel Estimation Method

### 3.1. The Proposed Method

Different from the ANM-based method in [23], we directly operate on model (10) rather than its vectorized version. First, we set up the following atom set:(12)A˜=arθatHϕ:θ,ϕ∈−90∘,90∘,based on which we can formulate the atomic ℓ0-norm of the channel matrix H as(13)HA˜,0=infK:H=∑k=1KαkA˜,A˜∈A˜,αk∈ℂ.We then propose the following optimization problem:(14)minHHA˜,0s.t.Y−WHHFF≤β.However, the above problem is a semi-infinite programming (SIP) which cannot be efficiently solved in polynomial time. To solve this problem, we have the following theorem.Theorem 1.
Assume thatK<minNt,Nr, then HA˜,0 equals the optimal value of the following rank minimization problem:(15)minu,v,WrankW,s.t.W=TvHHHTu≥0,where Tu and Tv are Toeplitz matrices.Proof.
First, for arbitrary decomposition ofH as H=∑k=1KαkarθkatHϕk, we can construct matrices Tu=∑k=1Kαk2arθkarHθk and Tv=∑k=1KatϕkatHϕk and then we have(16)W˜=TvHHHTu=∑k=1KatϕkatHϕkαkHatϕkarHθkαkarθkatHϕkαk2arθkarHθk=∑k=1KatϕkαkarθkatϕkαkarθkH≥0.
It follows that the optimal solution of (15) K°≤rankW˜=K=HA˜,0. On the other hand, if we find the optimal solution of (15) as H∘,u∘,v∘, then we have rankW∘=K∘≤K. It follows from the Vandermonde decomposition [38] that Tu∘=∑k=1K∘αk2arθk∘arHθk∘ and Tv∘=∑k=1K∘atϕk∘atHϕk∘. Then, since H∘ lies in the column space of Tu∘ and row space of Tv∘, there exists α˜∘ such that H∘=∑k=1K∘αk˜∘arθk∘atHϕk∘. Hence, HA˜,0≤K∘. Therefore, it can be concluded that HA˜,0=K∘.According to Theorem1, model (14) can be rewritten as(17)minH,u,v,WrankW,s.t.W=TvHHHTu≥0,Y−WHHFF≤β.It can be seen that, compared to the ANM model [23], the proposed model in (17) has much smaller problem dimensionality.However, directly solving this model is difficult due to the nonconvex rank operator. One possible approach is to relax the rank operator to the trace operator. The relaxed problem is convex and can be solved by CVX. Nevertheless, CVX is also an inefficient solver. In the following, we propose an efficient method to solve (17) based on Wirtinger projection.First, we define two matrices sets as(18)ℳ=M:rankM≤K,N=N:N=VHHHU≥0,Y−WHHFF≤β,where U and V have Toeplitz structure. Solving model (17) is equivalent to finding a matrix that both in ℳ and N or solving the following problem:(19)minM∈ℳ,N∈NM−NF2.One effective way is to alternatively updateM and N until convergence. Based on the Wirtinger strategy [39], we formulate the following update rule to solve problem (19):(20)Mt+1=PℳMt−δ1Mt−Nt,Nt+1=PNNt−δ2Nt−Mt+1,where the superscript t denotes the t-th iteration, Pℳ• and PN• denote the projection procedures onto matrices sets ℳ and N, respectively, and δ1 and δ2 are two user-defined parameters. The next target is to find the two projection operators Pℳ• and PN•.ProjectionPℳX is to find the best rank-K approximation that can be formulated as follows. We first apply singular value decomposition onto X as X=UXΣXVXH; then, projection PℳX is(21)MℳX=UKΣKVKH,where UK and VK are the first K columns of UX and VX, respectively, and ΣK is the corresponding singular value matrix.DenotingX˜=V˜amp;H˜HH˜amp;U˜, projection MNX˜ contains four sequential subprojections: PN1⟶PN2⟶PN3⟶PN4 where PN1 and PN2 are to project U˜ and V˜ onto the set of Toeplitz matrix, respectively, PN3 is to project H˜ onto the ball defined by H:Y−WHHFF≤β, and PN4 is to project X˜ onto a positive semidefinite matrix set. In particular, MN1U is defined as(22)PN1U=Tu,where the i-th element of u is ui=1/Nr−i+1∑iUm,n where i=n−m+1, n≥m, and m,n=1,…,Nr. Similarly, PN2V is(23)PN2V=Tv,where the i-th element of v is vi=1/Nt−i+1∑iVm,n where i=n−m+1, n≥m, and m,n=1,…,Nt. Projection PN3 is to project WHH˜F onto a ball with center Y and radius β. To realize PN3, we first define(24)H˜proj=WY+WHH˜F−YWHH˜F−YFβFH.Then,MN3 can be given as(25)PN3m,n=H˜projm,n,ifm∈ΦW,n∈ΩF,H˜m,n,otherwise.ForPN4, we first apply eigen-decomposition to X˜ and have X˜=UX˜diagσX˜UX˜H. Then, projection PN4 can be given as(26)PN4X˜=UX˜+diagσX˜+UX˜+H,where σX˜+ denotes the positive part of σX˜ and UX˜+ denotes the corresponding eigen-vectors.The proposed method converges whenNt+1−NtF/NtF≤ε, where ε is a predefined threshold.


### 3.2. The Case ofNt>ΩMt

It should be noted that, given the number of RF chainsMt at the transmitter (we take the transmitter as an example.), the array aperture is limited by its maximum tag ΩMt which may not equal Nt. If Nt>ΩMt, finding the accurate channel matrix H from model (17) is difficult since there may exist too many variables to be determined in (17). Thus, instead of directly obtaining H, we alternatively first retrieve the angle information based on which we can find the channel gain. And then, the channel matrix H can be well estimated. Specifically, we first let Ω1=1 and replace F∈ℤNt×Mt by F¯∈ℤΩMt×Mt. Similarly, we obtain W¯∈ℤΦMr×Mr. Then, received signal model (10) can be rewritten as(27)Y=W¯HA¯rΣA¯tHF¯+W¯HN¯,where A¯t=a¯tϕ1,a¯tϕ2,…,a¯tϕK and A¯r=a¯rθ1,a¯rθ2,…,a¯rθK with(28)a¯tϕk=1,ej⋅1⋅2πdλsinϕk,…,ej⋅ΩMt−1⋅2πdλsinϕkT,a¯rθk=1,ej⋅1⋅2πdλsinθk,…,ej⋅ΦMr−1⋅2πdλsinθkT.It is easy to see thata¯tϕk and a¯rθk are subvectors of atϕk and arθk, respectively. Based on truncated model (27), we propose the following truncated problem:(29)minH¯,u¯,v¯,W¯rankW¯,s.t.W¯=Tv¯H¯HH¯Tu¯≥0,Y−W¯HH¯F¯F≤β.The optimal solutionH¯∘∈ℂΦMr×ΩMt is only a submatrix of the full channel matrix H. To estimate H, we should first find the AoAs and AoDs from (29). From the proof of Theorem 1, we can see that Tu¯ and Tv¯ contain the AoAs and AoDs information, respectively, and the two Toeplitz matrices can be regarded as the noiseless covariance matrices of a ULA. Thus, the traditional ESPRIT method can be applied to find the angle estimates. Alternatively, we can also find the angle estimates from the Vandermonde decomposition theorem [40]. The channel matrix can then be constructed according to (7) after finding the channel gain by the LS method.


## 3.1. The Proposed Method

Different from the ANM-based method in [23], we directly operate on model (10) rather than its vectorized version. First, we set up the following atom set:(12)A˜=arθatHϕ:θ,ϕ∈−90∘,90∘,based on which we can formulate the atomic ℓ0-norm of the channel matrix H as(13)HA˜,0=infK:H=∑k=1KαkA˜,A˜∈A˜,αk∈ℂ.We then propose the following optimization problem:(14)minHHA˜,0s.t.Y−WHHFF≤β.However, the above problem is a semi-infinite programming (SIP) which cannot be efficiently solved in polynomial time. To solve this problem, we have the following theorem.Theorem 1.
Assume thatK<minNt,Nr, then HA˜,0 equals the optimal value of the following rank minimization problem:(15)minu,v,WrankW,s.t.W=TvHHHTu≥0,where Tu and Tv are Toeplitz matrices.Proof.
First, for arbitrary decomposition ofH as H=∑k=1KαkarθkatHϕk, we can construct matrices Tu=∑k=1Kαk2arθkarHθk and Tv=∑k=1KatϕkatHϕk and then we have(16)W˜=TvHHHTu=∑k=1KatϕkatHϕkαkHatϕkarHθkαkarθkatHϕkαk2arθkarHθk=∑k=1KatϕkαkarθkatϕkαkarθkH≥0.
It follows that the optimal solution of (15) K°≤rankW˜=K=HA˜,0. On the other hand, if we find the optimal solution of (15) as H∘,u∘,v∘, then we have rankW∘=K∘≤K. It follows from the Vandermonde decomposition [38] that Tu∘=∑k=1K∘αk2arθk∘arHθk∘ and Tv∘=∑k=1K∘atϕk∘atHϕk∘. Then, since H∘ lies in the column space of Tu∘ and row space of Tv∘, there exists α˜∘ such that H∘=∑k=1K∘αk˜∘arθk∘atHϕk∘. Hence, HA˜,0≤K∘. Therefore, it can be concluded that HA˜,0=K∘.According to Theorem1, model (14) can be rewritten as(17)minH,u,v,WrankW,s.t.W=TvHHHTu≥0,Y−WHHFF≤β.It can be seen that, compared to the ANM model [23], the proposed model in (17) has much smaller problem dimensionality.However, directly solving this model is difficult due to the nonconvex rank operator. One possible approach is to relax the rank operator to the trace operator. The relaxed problem is convex and can be solved by CVX. Nevertheless, CVX is also an inefficient solver. In the following, we propose an efficient method to solve (17) based on Wirtinger projection.First, we define two matrices sets as(18)ℳ=M:rankM≤K,N=N:N=VHHHU≥0,Y−WHHFF≤β,where U and V have Toeplitz structure. Solving model (17) is equivalent to finding a matrix that both in ℳ and N or solving the following problem:(19)minM∈ℳ,N∈NM−NF2.One effective way is to alternatively updateM and N until convergence. Based on the Wirtinger strategy [39], we formulate the following update rule to solve problem (19):(20)Mt+1=PℳMt−δ1Mt−Nt,Nt+1=PNNt−δ2Nt−Mt+1,where the superscript t denotes the t-th iteration, Pℳ• and PN• denote the projection procedures onto matrices sets ℳ and N, respectively, and δ1 and δ2 are two user-defined parameters. The next target is to find the two projection operators Pℳ• and PN•.ProjectionPℳX is to find the best rank-K approximation that can be formulated as follows. We first apply singular value decomposition onto X as X=UXΣXVXH; then, projection PℳX is(21)MℳX=UKΣKVKH,where UK and VK are the first K columns of UX and VX, respectively, and ΣK is the corresponding singular value matrix.DenotingX˜=V˜amp;H˜HH˜amp;U˜, projection MNX˜ contains four sequential subprojections: PN1⟶PN2⟶PN3⟶PN4 where PN1 and PN2 are to project U˜ and V˜ onto the set of Toeplitz matrix, respectively, PN3 is to project H˜ onto the ball defined by H:Y−WHHFF≤β, and PN4 is to project X˜ onto a positive semidefinite matrix set. In particular, MN1U is defined as(22)PN1U=Tu,where the i-th element of u is ui=1/Nr−i+1∑iUm,n where i=n−m+1, n≥m, and m,n=1,…,Nr. Similarly, PN2V is(23)PN2V=Tv,where the i-th element of v is vi=1/Nt−i+1∑iVm,n where i=n−m+1, n≥m, and m,n=1,…,Nt. Projection PN3 is to project WHH˜F onto a ball with center Y and radius β. To realize PN3, we first define(24)H˜proj=WY+WHH˜F−YWHH˜F−YFβFH.Then,MN3 can be given as(25)PN3m,n=H˜projm,n,ifm∈ΦW,n∈ΩF,H˜m,n,otherwise.ForPN4, we first apply eigen-decomposition to X˜ and have X˜=UX˜diagσX˜UX˜H. Then, projection PN4 can be given as(26)PN4X˜=UX˜+diagσX˜+UX˜+H,where σX˜+ denotes the positive part of σX˜ and UX˜+ denotes the corresponding eigen-vectors.The proposed method converges whenNt+1−NtF/NtF≤ε, where ε is a predefined threshold.


## 3.2. The Case ofNt>ΩMt

It should be noted that, given the number of RF chainsMt at the transmitter (we take the transmitter as an example.), the array aperture is limited by its maximum tag ΩMt which may not equal Nt. If Nt>ΩMt, finding the accurate channel matrix H from model (17) is difficult since there may exist too many variables to be determined in (17). Thus, instead of directly obtaining H, we alternatively first retrieve the angle information based on which we can find the channel gain. And then, the channel matrix H can be well estimated. Specifically, we first let Ω1=1 and replace F∈ℤNt×Mt by F¯∈ℤΩMt×Mt. Similarly, we obtain W¯∈ℤΦMr×Mr. Then, received signal model (10) can be rewritten as(27)Y=W¯HA¯rΣA¯tHF¯+W¯HN¯,where A¯t=a¯tϕ1,a¯tϕ2,…,a¯tϕK and A¯r=a¯rθ1,a¯rθ2,…,a¯rθK with(28)a¯tϕk=1,ej⋅1⋅2πdλsinϕk,…,ej⋅ΩMt−1⋅2πdλsinϕkT,a¯rθk=1,ej⋅1⋅2πdλsinθk,…,ej⋅ΦMr−1⋅2πdλsinθkT.It is easy to see thata¯tϕk and a¯rθk are subvectors of atϕk and arθk, respectively. Based on truncated model (27), we propose the following truncated problem:(29)minH¯,u¯,v¯,W¯rankW¯,s.t.W¯=Tv¯H¯HH¯Tu¯≥0,Y−W¯HH¯F¯F≤β.The optimal solutionH¯∘∈ℂΦMr×ΩMt is only a submatrix of the full channel matrix H. To estimate H, we should first find the AoAs and AoDs from (29). From the proof of Theorem 1, we can see that Tu¯ and Tv¯ contain the AoAs and AoDs information, respectively, and the two Toeplitz matrices can be regarded as the noiseless covariance matrices of a ULA. Thus, the traditional ESPRIT method can be applied to find the angle estimates. Alternatively, we can also find the angle estimates from the Vandermonde decomposition theorem [40]. The channel matrix can then be constructed according to (7) after finding the channel gain by the LS method.


## 4. Simulation Results

In this section, we evaluate the channel estimation performance of our proposed method on the switch-based mmWave massive MIMO system. We also consider other methods including MUSIC [41], OMP [18], L1 minimization [42], ANM [22], and decoupled ANM (DANM) [43] for comparison. It should be pointed out that DANM is applied with the alternating direction method of multipliers (ADMM) [44] which is a fast solver. We assume that Nt=16, Nr=32, Mt=6, and Mr=10 and the antennas are placed with half-wavelength spacing. The channel estimation performance is evaluated by(30)NMSE=EH−H∘F2HF2,based on 400 independent trials. The number of paths is set to K=3. The AoAs and AoDs are randomly generated from −90∘,90∘ for each path. The SNR is defined as SNR=Pt/σ2, where Pt and σ2=1 denote the transmitted power and noise power, respectively. For our method, we assume δ1=δ2=0.8 and β=MtMr+2MtMrσ to the upper bound of noise energy.


### 4.1. Convergence Performance

We first evaluate the convergence of our method. DenoteΔUt+1=Ht+1−HtF2/HtF2 as the variation of the estimated channel matrices between the t+1-th and t-th iterations. The threshold ε is set to 10−3. We consider different SNR scenarios and show the relationship between ΔU and the iteration in Figure 2, from which we can observe that for different SNRs, ΔU decreases rapidly and our method converges after about 300 iterations.Figure 2
Convergence of the proposed method.


### 4.2. Different Antenna Selection Strategies

In this section, we compare the performance of different antenna selection strategies. We select two representative SLA structures, nested and coprime arrays, and random selection strategy is also taken into consideration. In particular, we setΩFnested=1,2,3,4,8,12 and ΦWnested=1,2,3,4,5,10,15,20,25,30 for the nested structure and ΩFnested=1,3,4,5,7,10 and ΦWnested=1,4,6,7,10,11,13,16,21,26 for the coprime structure. For random selection strategy, we always first select the first antenna and then randomly select Mt−1 antennas from 2,…,16 for the transmitter and Mr−1 antennas from 2,…,32 for the receiver. The NMSEs of our proposed method based on these three strategies are shown in Figure 3 with the SNR varying from −10 dB to 8 dB. It can be seen that the nested array enjoys the best estimation performance. The coprime array shows inferior performance compared to nested array since it has shorter successive uniform part in its coarray. The random strategy shows the worst performance.Figure 3
NMSE comparison of three different antenna selection strategies.


### 4.3. Performance Comparison

Next, we evaluate the channel estimation performance of our proposed method with other representative methods. For MUSIC, OMP, andL1 which require discretizing the angle space, we consider two grid resolutions 2∘ and 3∘. The NMSEs of these methods are shown in Figure 4(a). We can see that ANM enjoys the best estimation performance in the compared SNR region. The proposed method is superior to other methods except ANM in most cases. For the three grid-based methods with grid resolution 3∘, they show unsatisfying accuracy in the low SNR region while suffer from grid effect when SNR becomes large. Thus, we can see that when SNR is larger than 4 dB, the gaps between these methods and our method become large. For grid resolution 2∘, these methods show better performance than for the case of grid resolution 3∘. This is because reducing the grid interval can relieve the grid mismatch effect, yielding higher estimation accuracy. We also show the running time of these methods in Figure 4(b). Since ANM requires to solve a large-dimensional SDP, it has the largest running time as compared to other methods. The L1 method requires solving a BPDN problem by CVX, and hence, it also suffers from high computational complexity, with respect to the two grid resolutions. DANM, MUSIC, and OMP are much faster than the ANM and L1 method, especially in the case of grid resolution 2∘. Our method shows comparable computational efficiency to OMP and MUSIC with grid resolution 3∘ but has better estimation performance as shown in Figure 4(a).Figure 4
Channel estimation performance comparison with respect to the SNR. Two grid resolutions2∘ and 3∘ are considered. (a) NMSE comparison. (b) CPU time comparison.
(a)(b)We also evaluate the spectral efficiency of these methods and show the simulation results in Figure5 with different grid resolutions ranging from 0.5∘ to 8∘. The spectral efficiency with perfect CSI is also considered as the upper bound. From Figure 5(a), it can be seen that since ANM, DANM, and our method are immune to the angle discretization, they are not affected by the grid resolution and can coincide with the perfect CSI. For OMP, MUSIC, and L1 methods, their performance deteriorates as the grid becomes sparser. Although the spectral efficiency of the L1 method and OMP can approach our method, DANM and ANM in dense grid cases, from Figure 5(b), it can be seen that their computational times increase exponentially as the grid resolution decreases and are several times slower than our proposed method if the grid resolution is less than 1∘.Figure 5
Spectral efficiency comparison with respect to the grid resolution. SNR = 5 dB. (a) Spectral efficiency comparison. (b) CPU time comparison.
(a)(b)


## 4.1. Convergence Performance

We first evaluate the convergence of our method. DenoteΔUt+1=Ht+1−HtF2/HtF2 as the variation of the estimated channel matrices between the t+1-th and t-th iterations. The threshold ε is set to 10−3. We consider different SNR scenarios and show the relationship between ΔU and the iteration in Figure 2, from which we can observe that for different SNRs, ΔU decreases rapidly and our method converges after about 300 iterations.Figure 2
Convergence of the proposed method.


## 4.2. Different Antenna Selection Strategies

In this section, we compare the performance of different antenna selection strategies. We select two representative SLA structures, nested and coprime arrays, and random selection strategy is also taken into consideration. In particular, we setΩFnested=1,2,3,4,8,12 and ΦWnested=1,2,3,4,5,10,15,20,25,30 for the nested structure and ΩFnested=1,3,4,5,7,10 and ΦWnested=1,4,6,7,10,11,13,16,21,26 for the coprime structure. For random selection strategy, we always first select the first antenna and then randomly select Mt−1 antennas from 2,…,16 for the transmitter and Mr−1 antennas from 2,…,32 for the receiver. The NMSEs of our proposed method based on these three strategies are shown in Figure 3 with the SNR varying from −10 dB to 8 dB. It can be seen that the nested array enjoys the best estimation performance. The coprime array shows inferior performance compared to nested array since it has shorter successive uniform part in its coarray. The random strategy shows the worst performance.Figure 3
NMSE comparison of three different antenna selection strategies.


## 4.3. Performance Comparison

Next, we evaluate the channel estimation performance of our proposed method with other representative methods. For MUSIC, OMP, andL1 which require discretizing the angle space, we consider two grid resolutions 2∘ and 3∘. The NMSEs of these methods are shown in Figure 4(a). We can see that ANM enjoys the best estimation performance in the compared SNR region. The proposed method is superior to other methods except ANM in most cases. For the three grid-based methods with grid resolution 3∘, they show unsatisfying accuracy in the low SNR region while suffer from grid effect when SNR becomes large. Thus, we can see that when SNR is larger than 4 dB, the gaps between these methods and our method become large. For grid resolution 2∘, these methods show better performance than for the case of grid resolution 3∘. This is because reducing the grid interval can relieve the grid mismatch effect, yielding higher estimation accuracy. We also show the running time of these methods in Figure 4(b). Since ANM requires to solve a large-dimensional SDP, it has the largest running time as compared to other methods. The L1 method requires solving a BPDN problem by CVX, and hence, it also suffers from high computational complexity, with respect to the two grid resolutions. DANM, MUSIC, and OMP are much faster than the ANM and L1 method, especially in the case of grid resolution 2∘. Our method shows comparable computational efficiency to OMP and MUSIC with grid resolution 3∘ but has better estimation performance as shown in Figure 4(a).Figure 4
Channel estimation performance comparison with respect to the SNR. Two grid resolutions2∘ and 3∘ are considered. (a) NMSE comparison. (b) CPU time comparison.
(a)(b)We also evaluate the spectral efficiency of these methods and show the simulation results in Figure5 with different grid resolutions ranging from 0.5∘ to 8∘. The spectral efficiency with perfect CSI is also considered as the upper bound. From Figure 5(a), it can be seen that since ANM, DANM, and our method are immune to the angle discretization, they are not affected by the grid resolution and can coincide with the perfect CSI. For OMP, MUSIC, and L1 methods, their performance deteriorates as the grid becomes sparser. Although the spectral efficiency of the L1 method and OMP can approach our method, DANM and ANM in dense grid cases, from Figure 5(b), it can be seen that their computational times increase exponentially as the grid resolution decreases and are several times slower than our proposed method if the grid resolution is less than 1∘.Figure 5
Spectral efficiency comparison with respect to the grid resolution. SNR = 5 dB. (a) Spectral efficiency comparison. (b) CPU time comparison.
(a)(b)


## 5. Conclusion

In this paper, we proposed an atomicℓ0-norm-based channel estimation method for switch-based mmWave massive MIMO communication systems. The proposed method exploits the coarray property of sparse arrays to select antennas and then formulates an atomic ℓ0-norm minimization problem which is efficiently solved based on Wirtinger projection. The proposed method is shown to have higher computational efficiency than ANM at comparable estimation performance. Compared to grid-based methods such as OMP, MUSIC, and L1, our method does not require angle discretization and hence is immune to the grid mismatch effect, leading to higher estimation accuracy.



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*Source: 7356963-2021-09-13.xml*