MmWave Channel Estimation via Atomic Norm Minimization for Multi-User Hybrid Precoding
Junquan Deng Olav Tirkkonen Christoph Studer
cs.IT math.IT

=1

  .pdf,.jpeg,.png

   .eps























./figs/


plain
theoremTheorem
corollaryCorollary[theorem]
lemma[theorem]Lemma
lem[]Lemma







#1#1_∞
#1#1_2
#1#1_1
#1#1_0
#1#1_F
#1#1_2,1
#1#1_2,∞
#1#1_
#1#1_Ω
#1#1_^*
#1#1_Ω^*
#1#1_,0
#1sign#1
#1conj#1
#1vec#1
#1Cov#1
#1|#1|
#1⟨#1⟩
→∞
→


subject to 

ℂ
𝕂
ℝ
𝕊
𝔽
𝕋
𝔼
ℙ
ℕ

ℤ

#1#1

𝓅
𝒜
𝒞
𝒟
ℰ
ℱ
ℋ
ℐ
𝒥
𝒦
ℒ
ℳ
𝒩
𝒫
𝒯
𝒬
ℛ
𝒮
𝒯
𝒰
𝒲
𝒳
𝒞𝒩

height6pt width 6pt depth 0pt

    a.e.
        
        
        
        
        
        
        
         #1
            #1
         #1
            #1
         #1(#1) #1[#1] #1{#1} #1diag#1 #1Tr#1 #1supp#1 #1rank#1 #1spark#1 s.t.  #1col#1(BP)(BP_ϵ)(QP_λ)(BP^o)(P_1^c)P_εP_RLS
        (SP_1)(SP_2)(P̅_̅1̅)
        
        
        
        
        
        
        
        
        
        
        
        MmWave Channel Estimation via Atomic Norm Minimization for Multi-User Hybrid Precoding
            Junquan Deng1, Olav Tirkkonen1 and Christoph Studer2
        1Department of Communications and Networking, Aalto University, Finland 
        2School of Electrical and Computer Engineering, Cornell University, NY, USA 
        
            December 30, 2023
        ===============================================================================================================================================================================================================
        
        
        To perform multi-user multiple-input and multiple-output transmission in millimeter-wave (mmWave) cellular systems, the high-dimensional channels need to be estimated for  designing the multi-user precoder. Conventional grid-based Compressed Sensing (CS) methods for mmWave channel estimation suffer from the basis mismatch problem, which prevents accurate channel reconstruction and degrades the precoding performance. This paper formulates mmWave channel estimation as an Atomic Norm Minimization (ANM) problem. In contrast to grid-based CS methods which use discrete dictionaries, ANM uses a continuous dictionary for representing the mmWave channel. We consider a continuous dictionary based on sub-sampling in the antenna domain via a small number of radio frequency chains. We show that mmWave channel estimation using ANM can be formulated as a Semidefinite Programming (SDP) problem, and the  channel can be accurately estimated via off-the-shelf SDP solvers in polynomial time. Simulation results indicate that ANM can achieve much better estimation accuracy compared to grid-based CS, and significantly improves the spectral efficiency provided by multi-user precoding.
        
        
        
        
        § INTRODUCTION
        
        In future millimeter-wave (mmWave) cellular networks, large antenna arrays are expected to be applied at the Base Station (BS) to serve multiple User Equipments (UE) in dense urban scenarios. To perform Multi-User Multiple-input and Multiple-Output (MU-MIMO) hybrid precoding <cit.>, accurate Channel State Information (CSI) is necessary. As mmWave channels typically compromise a few strong propagation paths, Compressed Sensing (CS) methods have been considered for mmWave channel estimation <cit.>. In order to apply CS to channel estimation, a discretization procedure is generally adopted to reduce the continuous angular or delay spaces to a finite set of grid points. Assuming that the Direction of Departure (DoD) and Direction of Arrival (DoA) lie on the grid, the virtual channel representation is sparse with few non-zero entries. The channel estimation problem then can be addressed with a specific measurement matrix and a recovery algorithm, e.g., the Orthogonal Matching Pursuit (OMP) <cit.>. However, as the actual signals are continuous and will not fall on the discrete points, basis mismatch will degrade recovery performance <cit.>. Although finer grids can reduce the reconstruction error, they require more computation resources.
        
        In the literature, most of the multi-user hybrid precoding schemes assume perfect CSI at the transmitters (see e.g., <cit.>). However, grid-based CS for channel estimation will lead to the basis mismatch problem and hence channel estimation errors, which will degrade the performance of MU-MIMO precoding such as Zero-Forcing (ZF) <cit.>. Recently, a continuous basis pursuit technique with auxiliary interpolation points <cit.> was used for mmWave channel estimation in <cit.>, which shows that adding interpolation points to the original grids can improve channel estimation accuracy considerably.
        
        
        
        This paper considers a continuous dictionary for multi-user mmWave channel estimation in order to completely eliminate the basis mismatch error. We proposed a continuous dictionary based on antenna-domain sub-sampling via a much smaller number of Radio Frequency (RF) chains in the hybrid architecture. The sparse mmWave channel estimation is formulated as an Atomic Norm Minimization (ANM) problem <cit.>, which can be solved via Semidefinite Programming (SDP) in polynomial time. ANM has been considered for massive MIMO channel estimation  <cit.>, wherein a fully-digital architecture is assumed which has much higher complexity compared to the hybrid architectures. To show the advantage of ANM based on antenna-domain sub-sampling in the hybrid architecture, we investigate a low-complexity MU-MIMO hybrid precoding scheme using the estimated channel information. Finally, we evaluate the performance of ANM and the precoding scheme via simulations in realistic scenarios. Simulation results regarding the channel estimation error and user spectral efficiency are provided, which verify  the efficacy of our solutions.
        
        
        § SYSTEM MODEL
        
        
         §.§ Architectures of BS and UE
        
        We consider a mmWave cellular system where the BS has N antennas and Q RF chains to serve K = Q UEs in downlink (DL). In contrast to <cit.>, a BS architecture  equipped with both Phase Shifters (PS) and switches is proposed, as shown in Fig. <ref>. The BS antennas are grouped into Q sub-arrays. Antennas in each sub-array are associated with one particular RF chain. With the proposed architecture, the BS can adopt two working modes: 1) the PS mode with a partially-connected PS network (as A2 in <cit.>); 2) the SW mode with a partially-connected switch network (as A6 in <cit.>).
        When working in
        SW mode, the BS contains a fully-digital architecture with Q
        antennas. The
        
        switch network enables accessing
        the received signal from an individual antenna, which is useful for
        channel estimation, especially when the Signal-to-Noise
        Ratio (SNR) is high. The cost of adding dedicated switches to
        the phase-shifter network is moderate as the implementation complexity
        of switches is typically lower than that of phase
        shifters <cit.>.
        In contrast, the PS network is used to perform analog beamforming and combining for both beam training and data transmission.
        
        
        
        At the UE side, each UE is assumed to have a single RF chain with M antennas and M phase shifters. Such UEs are able to perform single-data-stream analog beamforming.
        This UE architecture has
        
        low hardware complexity and
        
        energy consumption, making it suitable for mobile UEs <cit.>.
        
        The BS has two kinds of RF precoder/combiner depending on the working modes. In SW mode, the sub-array RF precoder/combiner _q is an antenna selection vector with one non-zero element. In PS mode, _q is a sub-array steering vector. For UEs, the RF precoder/combiner w_k of UE k is a beam steering vector. We assume that BS and UE have Q_ps-bits quantized PSs <cit.>. The number of beams that can be steered by BS or UE depends on the PS resolution Q_ ps rather than the number of antennas, and is equal to 2^Q_ ps. The coefficient set available for each PS is denoted by ℙ = {ω^0,ω^1,…,ω^2^Q_ ps - 1} where ω=e^j2π/2^Q_ ps is the minimum angular domain separation.
        The quantized PSs can adjust the phases of the signals and have constant modulus. Specifically, the RF beamforming codewords are
        
            _q = 1_Ω_q⊙f_ω(c,N),   _k = f_ω(c,M),
        
        where Ω_q is the antenna index set for qth sub-array, 1_Ω_q is a binary valued vector with |Ω_q| ones indexed by Ω_q, f_ω(c, i) = [1,ω^c,ω^2c,…,ω^(i-1)c]^T with c ∈{0,1,…,2^Q_ ps-1}.
        
        
        
         §.§ Channel Model
        
        MmWave channels are highly directional and contain a few dominant paths in the angular domain <cit.>. Due to the extremely short wave-length, diffraction effects in mmWave frequencies are small; only Line-of-Sight (LoS) and reflected paths are significant. As a consequence, the number of paths is smaller than the number of BS antennas. Suppressing the vertical directivity of the channel, for the kth UE, the DL channel is modeled as <cit.>,
        
            =∑_l=1^Lα_l𝐚_UE(θ_l)  𝐚_BS(ϕ_l)^ H∈ℂ^M× N,
        
        where L represents the number of paths, and α_l denotes the complex gain of the lth path. In addition, 𝐚_BS(ϕ_l) and 𝐚_UE(θ_l) represent the BS and UE array response vectors for the lth path, where ϕ_l is the DoD at BS, and θ_l is DoA at UE.
        Considering plane-wave model and Uniform Linear Array (ULA) for both BS and UE, the array steering vectors can be written as
        
            a_ BS(ϕ)   =[1, e^j 2π/λdsin(ϕ),…,e^j(N-1)2π/λdsin(ϕ) ]^ T,   
             a_ UE(θ)   =[1, e^j 2π/λdsin(θ),…,e^j(M-1)2π/λdsin(θ) ]^ T,
        
        where λ is carrier wavelength and the antenna spacing here is assumed to be d=λ/2. We assume that the system is working in the Time Division Duplex (TDD) mode and the DL and uplink (UL) channels are reciprocal.  Denote the BS baseband combiner or precoder as 𝐏_𝐁𝐁=[ p_1,…, p_K] ∈ℂ^ Q × K,
        the BS RF combiner or precoder
        as 𝐅_𝐑𝐅 = [_1,…, _Q ] ∈ℂ^ N × Q,
        and the RF combiner or precoder for all K UEs as
        W = [ w_1,…, w_K]∈ℂ^M × K.
        The DL and the UL received signals including interference and noise for UE k are
        
            z_ u   = w_k^ H( p_k x_k +∑_j≠ k p_j x_j)+ w_k^ Hn_ u,
            
            z_ b   = p^ H_k( w_k y_k+∑_j≠ k w_j y_j+ n_ b),
        
        where x_k and y_k are the DL and UL signals, satisfying
        𝔼(x_k^ x_k^*) = ρ_BS/K and 𝔼(y_k^ y_k^*) = ρ_UE/M with transmit powers as ρ_ BS  and ρ_ UE. In addition, n_ u∈ℂ^M × 1 is the 𝒞𝒩 ( 0,σ^2_ u) UE noise vector with noise power σ^2_ u per antenna, and n_ b∈ℂ^N × 1 is the 𝒞𝒩 ( 0,σ^2_ b) BS noise vector with noise power σ^2_ b per antenna. For DL transmission, F = 𝐅_𝐑𝐅𝐏_𝐁𝐁 satisfies power constraint tr( FF^ H ) ≤ 1. For UL transmission, w_k satisfies w_k_2^2≤ 1.
        
        
        
        § PROBLEM FORMULATION FOR CHANNEL ESTIMATION
        
        For channel training in UL, we assume that the kth UE transmits a pilot sequence s_k ∈ℂ^1 × T_ s using UE beam 𝐰_k. Assuming 𝐏_𝐁𝐁 = 𝐈_Q, the received training signal at BS is
        
            𝐘
            
                          =   (∑_k=1^K𝐡_k 𝐬_k + 𝐍_ b).
        
        Here h_k = _k is the effective  channel for UE k. The user pilots are assumed to satisfy
        𝐬_k𝐬_k'^ H =  T_ sρ_ UEδ_k,k', with δ_i,j
        the Kronecker delta function. Correlating 𝐘 with  s_k, we get
        
            𝐲_k    = 𝐘𝐬^ H_k=  T_ sρ_ UE𝐡_k  + 𝐍_ b𝐬^ H_k.
        
        Assuming that the BS takes T snapshots of measurements with different combining matrices, stacking the W = T × Q measurement samples gives
        
            z =[ y_1,k; y_2,k;     ⋮; y_T,k;       ]=
            T_ sρ_ UE[ F_1, RF^ H; F_2, RF^ H;          ⋮; F_T, RF^ H;            ]_Φ𝐡_k  +
            [ F_1, RF^ H𝐍_1, b; F_2, RF^ H𝐍_2, b;                ⋮; F_T, RF^ H𝐍_T, b;                  ]𝐬^*_k
            _ n
            ,
        
        where Φ is the sensing matrix and n is the noise after combining.
        The objective of BS channel estimation is to estimate the effective channel. From (<ref>), we have
        
            𝐡_k=_k=∑_l=1^Lβ_l𝐚_BS(ϕ_l),
        
        where β_l = (α_l (𝐰_k)^ H𝐚_UE(θ_l))^*. The effective channel 𝐡_k contains fewer significant paths compared to the full MIMO channel H_k. To estimate 𝐡_k, one can estimate DoAs {ϕ_l} and the complex gains {β_l} for the paths and then reconstruct the channel. In grid-based CS methods, a discrete dictionary Ψ_ BS= [   a_ BS(ϕ_1),…, a_ BS(ϕ_G_ b) ] with G_ b bases is used to represent the channel h_k. We consider a grid in which
        {sin(ϕ_1),…,sin(ϕ_G_ b)} are evenly distributed in (-1,1] <cit.>. The channel vector is represented as
        
            𝐡_k = Ψ_ BS,
        
        where ∈ℂ^ G_ b× 1 is the sparse virtual channel in dictionary Ψ_ BS. Denoting  𝐀 =ΦΨ_ BS, to estimate the sparse virtual channel , one can formulate the following optimization problem:
        
            minimize‖‖_0
               s.t.       z_k - 𝐀_2^2 ≤η.
        
        where η is an optimization parameter and typically set as the noise power after combining. In contrast to <cit.>, we consider a switch-based method for constructing the sensing matrix, in which only Q individual antennas are sampled during one measurement  snapshot. We call this method Antenna Domain Sub-Sampling (ADSS). The ADSS  sensing matrix has the following structure:
        
            Φ
                          =T_ sρ_ UE  [ e_i_1,1, e_i_1,2, …,  e_i_1,Q, …,  e_i_T,Q]^ H,
        
        where the unit vector e_i_t,q has one entry equal to 1 at position i_t,q, and i_t,q is the antenna index for the qth RF chain in the tth measurement snapshot. Furthermore, if we define the antenna index set Ω = {i_1,1, i_1,2, …, i_1,Q, …, i_T,Q} for all sampled antennas, we have
        
            z = Φ h_k +  n = T_ sρ_ UE   ( h_k)_Ω +  n.
        
        A variety of algorithms have been proposed for obtaining an
        approximate solution to (<ref>) at polynomial complexity. OMP
        
        is a preferred method due to its simplicity and fast implementation <cit.>. To solve (<ref>) based on the measurement data z
        in (<ref>), we use OMP as in Algorithm <ref>. The
        computational complexity of OMP is proportional to the number G_
          b of grid points. Step 3 in Algorithm <ref> needs
        𝒪(WG_ b) computations and solving the Least
        Square (LS) problem in step 5 requires 𝒪(W
        |ℐ_t|^2 ).
        
        The joint design of the sensing matrix Φ and dictionary
        matrix Ψ plays a key role in achieving accurate channel
        estimation. To reconstruct ,  should have low
        mutual coherence μ(), which
        
        is the largest normalized inner product for two different columns of
        . Specifically, when the path directions lie on the defined grid, it can be shown that OMP can recover  in the
        noiseless case if <cit.>
            μ() = i≠ jmax |a_i^ H a_j|/a_i_ 2·a_j_ 2 < 1/2L-1,
        
        where a_i and a_j are two different columns of
        . The lower bound of μ() is given by the Welch bound <cit.>, which
        decrease as the number of measurement W increases.
        
        
        
        § ATOMIC NORM MINIMIZATION FOR EQUIVALENT CHANNEL ESTIMATION
        
        When using the dictionary Ψ_ BS for channel estimation
        via OMP, the estimated path angles are considered to be on a discrete
        grid {ϕ_1,…,ϕ_G_ b}, which introduces the basis
        mismatch problem <cit.>. To avoid this, we consider
        
        a continuous dictionary for estimating the effective channel.
        
        
        To estimate the effective channel based on UL measurements, the
        following constrained optimization problem can be formulated:
        
        hminimize h   z - Φh_2^2≤η.
          
        Here (h) denotes a sparse
        metric to be minimized for a channel vector h. The noise is
        assumed to be bounded by n^2_2 ≤η, and h can
        be treated as the signal of interest which needs to be estimated based
        on the observed data z in the context of line spectral
        estimation <cit.>.
        
        Problem (<ref>) differs from (<ref>) as it is
        formulated without a discrete dictionary. Instead
        of (<ref>), one can solve a regularized optimization
        problem as hminimize  ξh + 1/2z - Φh^2,
           where ξ > 0 is a regularization
        parameter related to η. There are different choices for (
        h) <cit.>. Here, we consider the atomic norm ( h)
        =h proposed in <cit.>. The atomic norm
        has been  used in grid-less compressive sensing for a range of
        applications <cit.>,
        including DoA estimation, line spectral estimation and massive MIMO
        channel estimation. The continuous set of atoms used to represent
        h is defined as = a_ BSϕα:  ϕ∈(-π/2,π/2],α∈,
          α=1. The atomic norm is the gauge
          function <cit.> of the convex hull
          conv. Formally, the atomic norm can be written as
          h   =inf  g>0: h∈ g ·conv
           =inf{∑_i b_i: h = b_i∑_i a_i, b_i>0, a_i ∈}. 
        
        The atomic norm is a continuous counterpart of the ℓ_1-norm used in on-grid CS methods.
        
        It can be computed efficiently via semidefinite programming <cit.>. Specifically, h_ defined in (<ref>) is the optimal value of the following matrix trace minimization problem:
        
        u,tminimize  1/2(t+u_1)   [ 𝒯(u)    h; h^ H    t ]≽0, 
        where 𝒯(u) is a Hermitian Toeplitz matrix with the first row as u = [u_1,…,u_N]^ T.
        
        
        
        
        In the noisy case, using the atomic norm, we can rewrite the optimization problem (<ref>) as
        
            D ≽0minimize    ξ/2(t+u_1) +1/2z-Φh^2     
                D = [ 𝒯(u)    h; h^ H    t ].
        
        The above problem is a SDP which can be solved by off-the-shelf convex optimization tools in polynomial time <cit.>. The computational complexity is 𝒪((N+1)^6) in each iteration using the interior point method.
        
        To guarantee that h can be exactly recovered via ANM, the measurement matrix Φ and the number of atoms in h (i.e., the number of paths L in the user channel) should satisfy a condition which is similar to the mutual coherence condition in on-grid CS. To define this condition for ANM, a concept called spark for the continuous set of atoms is introduced. Let us define the transformed continuous dictionary based on the measurement matrix Φ as
        
            _Φ = {Φa_ BS(ϕ): ϕ∈(-π/2,π/2] }.
        
        Similar to the definition of matrix spark, spark(_Φ) is defined as the smallest number of atoms which are linearly dependent in _Φ. From <cit.>, the problem (<ref>) with η = 0 in the noiseless setting has a unique solution if the number of paths satisfies
        
            L < spark(_Φ)/2.
        
        Note that the spark in (<ref>) satisfies spark() ≥ 1 + (μ())^-1, so the  condition above  is equivalent to (<ref>) in the case of discrete atoms. Thus the spark(_Φ) should be as large as possible so that ANM can recover a channel with as many paths as possible.
        For a specific BS array, one may notice that spark(_Φ) depends only on Φ. For ADSS via switches, spark(_Φ) depends on the number of sampled antennas |Ω| and the sampled antenna indexes. It was shown in <cit.> that 2 ≤spark(_Φ) ≤ |Ω|+1, and a random Ω or a Ω with |Ω| consecutive integers can achieve a sufficiently large spark.
        
        
        
        § MULTI-USER DOWNLINK PRECODING/COMBINING  BASED ON THE ESTIMATED CHANNELS
        
        In this section, we consider the designs of downlink BS precoder and UE combiners. To design the BS precoder {,} and UE combiners W optimally, full CSI ℋ = { H_1,  H_2,…, H_K} needs to be known. However, acquiring ℋ would require each UE to send huge amount of pilots which consumes lots of resources. A sub-optimal method, which works well when user channels are sparse, is that each UE first finds its best beam which aligns with the most significant path, then UEs transmit their pilots to BS for multi-user channel estimation as discussed in sections <ref> and <ref>. As a result, the BS only needs to estimate the user effective channel h_k which has much fewer coefficients and significant paths to be estimated. The BS will then design the its MU-MIMO precoder based on those estimated effective channels. For UEs, the beam codebook is
        
            𝒰={√(1/M)f_ω(c,M): c ∈{0,…,2^Q_ ps-1}}.
        
        For BS, the DL training beam codebook used by the qth sub-array is
        
            ℱ_q ={1_Ω_q⊙f_ω(c,N): c ∈{0,…,2^Q_ ps-1}}.
        
        Sub-array q will use one beam from ℱ_q for UE beam training.
        
        The objective of the UE beam training is to find a UE beam which maximizes the norm of the effective channel
        
            _k∈𝒰maximize 𝐰_k^ H^2_ 2,
        
        The problem in (<ref>) is equivalent to finding the best steering vector _k∈𝒰 that aligns with the paths that have large path gains. In principle, to find the best UE codeword, BS needs to transmit DL training beams which cover all the paths. However, this would increase the training overhead.
        We propose using all sub-arrays to transmit Q DL pilots with Q sub-array beams simultaneously which are evenly distributed in [-π/2,π/2 ). Fig. <ref> (a) shows the designed DL beams used for the BS sub-arrays and Fig. <ref> (b) shows the UE beam codebook we used.
        
        The MU-MIMO precoding scheme based on the UE beam training and the effective channel estimation is summarised and shown in Algorithm <ref>. Noticing that DoAs and DoDs of the propagation paths change much slower than the channel coefficients <cit.>, the best UE beam remains the same during a period which is much longer than channel coherence time, and the UE beam tracking can be performed less frequently.
        
        
        
        § NUMERICAL RESULTS
        
        In this section, the performance of the proposed channel estimation method and the precoding scheme are evaluated via numerical simulations. We assume M=8 for UEs, and N=64, Q=8 for BS. The BS antennas are grouped into Q sub-arrays with equal size. We consider an outdoor small cell where BS is lower than buildings, and signals will be reflected or blocked by the walls. A ray-tracing channel model<cit.> is considered to generate the channels. The reflection coefficients are computed based on the Fresnel equation and up to 3rd order reflections are taken into account. The relative permittivities of building walls are uniformly distributed between 3 and 7. The BS is at the center while UEs are dropped outdoor with distances to BS smaller than 150 meters, as shown in Fig. <ref>. Blocked UEs which cannot find a LOS or a reflected path are not considered in the simulation.
        The path coefficient in (<ref>) is
        
            α_l = e^jψ√(G_0 x^-2 g_1(θ_l) g_2(ϕ_l)∏_i=0^R |r_i|^2),
        
        where G_0 = 10^-6.14 is the omnidirectional path gain <cit.> at reference distance  one meter, x is the path propagation distance in meter, ψ∼ U(0,2π) is a random phase, g_1(θ_l) and g_2(ϕ_l) are UE and BS antenna element patterns, respectively, R is the reflection order and r_i is the ith reflection coefficient. For a LOS path, we have R=0, r_0 = 1. More details of simulation parameters are given in Tabe <ref>.
        
        
        
         §.§ Effective User Channel Estimation via OMP and ANM
        
        
        
        
        
        
        
        
        
        
        All UEs transmit orthogonal pilots to BS using their best beams, and BS receives them via ADSS. In these simulations, T=2 measurement snapshots are used, so we have T× Q = 16 antennas sampled pseudo-randomly via the switch network.
        The ANM problem (<ref>) is solved via the SDPT3 <cit.> solver.
        The OMP algorithm applys a non-redundant dictionary with G_b=N
        and redundant dictionaries with G_b= 2N,  4N.
        We estimate channel estimation performance, in terms
        of the Normalized Mean Square Error (NMSE) 𝔼{ h_k
          -ĥ_k^2_ 2/ h_k^2_ 2}.
        In total 8000 UEs are simulated. Results are presented as a function of
        SNR = ρ_ UE^2_ F/(MNσ^2_ b).
        As illustrated in Fig. <ref>, increasing G_ b
        directly results in better estimation accuracy for OMP. The ANM method outperforms
        OMP since it avoids basis mismatch by using the continuous dictionary.
        As G_ b increases, the gap between OMP and ANM decreases at the
        cost of increasing computational complexity.
        When a non-redundant dictionary with G_b = N is used for
        OMP, OMP suffers severely from basis mismatch;
        
        an error floor develops in the hign SNR regime.
        In comparison, ANM achieves much better estimation accuracy,
         especially in the high SNR regime, effectively removing the error floor.
        Inaccurate channel estimation degrades the interference mitigation performance in MU-MIMO precoding techniques. As a result, ANM would be a strong candidate for the hybrid precoding solution discussed in Section <ref>. We also found that in the simulated scenario, the user effective channel h_k is sparse in angular domain. In most cases, there are one to three significant paths even when the original channel H_k has much more paths. This ensures that the condition (<ref>) is satisfied and ANM can recover the channel with high probability. In the case that the condition (<ref>) is not satisfied, ANM aims to recover those strongest paths and moderate channel estimation accuracy is attainable. Compared to OMP, solving the ANM problem using SDPT3 entails higher complexity. In practice, an efficient solver based on the Alternating Direction Method of Multipliers (ADMM) <cit.> technique could be adopted to solve the ANM problem.
        
        
         §.§ Spectral Efficiency Performance of Multi-User Precoding
        
        To evaluate the multi-user precoding performance, 8000 UEs are randomly divided into 1000 groups and 8 UEs in each group are served simultaneously.
        The Cumulative Distribution Function (CDF) for user spectral efficiency is collected, where SE is estimated as log_2(1+γ)
        from the Shannon formula with γ the signal-to-interference-plus-noise ratio. As shown in Fig. <ref>, hybrid precoding with ANM-based channel estimation can provide much better performance compared to an OMP-based scheme since it provides accurate channel estimates for the effective user channel. The baseband ZF precoding is sensitive to channel estimation error and the error will damage the user orthogonality provided by ZF. On average, OMP with G_b = N, 2N and 4N loses 63%, 18% and 11% of the spectral efficiency with perfect CSI.  By comparison, ANM loses only 3.4%, which verify the efficacy of ANM.
        
        
        
        § CONCLUSION
        
        This paper has proposed a low-complexity hybrid architecture in which an inexpensive switch network is added to the subarrays to facilitate channel estimation. We have formulated the mmWave channel estimation as an ANM problem which can be solved via SDP with polynomial complexity. Compared to the grid-based CS methods, the proposed ANM approach uses a continuous dictionary with sub-sampling in antenna domain via the switch network. Simulation results show that ANM can achieve much better channel estimation accuracy compared to grid-based CS methods, and can help MU-MIMO hybrid precoding to achieve significantly better user spectral efficiency performance.
        IEEEtran
