mmWave/THz Channel Estimation Using Frequency-Selective Atomic Norm Minimization
aa r X i v : . [ ee ss . SP ] J un mmWave/THz Channel Estimation UsingFrequency-Selective Atomic Norm Minimization Yicheng Xu, Hongyun Chu, Xiaodong Wang
Abstract —We propose a MIMO channel estimation methodfor millimeter-wave (mmWave) and terahertz (THz) systemsbased on frequency-selective atomic norm minimization (FS-ANM). For the strong line-of-sight property of the channelin such high-frequency bands, prior knowledge on the rangesof angles of departure/arrival (AoD/AoA) can be obtained asthe prior knowledge, which can be exploited by the proposedchannel estimator to improve the estimation accuracy. Simulationresults show that the proposed method can achieve considerableperformance gain when compared with the existing approacheswithout incorporating the the strong line-of-sight property.
Index Terms —mmWave/THz channel estimation, frequency-selective atomic norm, ultra-massive MIMO.
I. I
NTRODUCTION
The mmWave/THz communication has been deemed as apotential solution for future wireless communication systems[1], [2]. To compensate for the severe signal propagation lossat mmWave/THz band, the systems are expected to configurewith ultra-massive antenna arrays at transceivers to achievesufficient beamforming gains. For such MIMO systems, itis well known that the channel state information (CSI) isindispensable for reliable signal transmission and reception,and especially useful for designing efficient beamformersin mmWave/THz band [3]. However, channel estimation ischallenging in mmWave/THz systems where the number ofantennas is large and the received signal-to-noise ratio (SNR)is low. Most of the existing estimation methods are based onthe rich scattering assumption of the channel, which limitstheir applications due to high training overhead and computa-tional cost.The inherent sparsity [4] of the ultra-massive MIMO chan-nel has been used for reducing the training overhead and/orimproving the estimation accuracy [5]–[15], which are overallclassified into the closed-loop methods, e.g., [5], [6], andthe open-loop methods, e.g., [7]–[15], both using grid-basedcompressive sensing (CS). To be specific, the closed-loopschemes estimate the channels via the multistage beam searchoperation, and their performance is limited by the resolutionof the pre-determined codebook, i.e., the better performancewith the cost of larger storage, more complex parsing process
Yicheng Xu is with the National Mobile Communications Re-search Laboratory, Southeast University, Nanjing 210096, China (e-mail:[email protected]). (Corresponding author: Yicheng Xu.)
Hongyun Chu is with the School of Communications and InformationEngineering, Xi’an University of Posts and Telecommunications, Xi’an,710121, China (e-mail: hy [email protected]).Xiaodong Wang is with the Department of Electrical Engi-neering, Columbia University, New York 10027, USA (e-mail:[email protected]). and longer time delay; the open-loop techniques performchannel estimation without the feedback process by using thepilot-based multiple signal classification (MUSIC) methods.In particular, these on-grid methods suffer from the basismismatch problem. In view of the continuous valued anglesof channel, a gridless approach [16], which uses atomicnorm minimization (ANM) to manifest the signal sparsityin the continuous parameter domain, has been proposed forangular estimation [17], [18]. Under certain conditions, ANMbased denoising methods can achieve exact sparse signalsreconstruction, avoiding the effects of basis mismatch whichcan plague grid-based CS techniques.In mmWave/THz systems, the channels exhibit strong line-of-sight. Hence it is possible to obtain prior knowledge onthe ranges of AoD/AoAs which can push the solution to thereduced feasible regions. In this paper, we propose a channelestimator that can incorporate such prior knowledge, based onfrequency-selective atomic norm minimization.II. S
YSTEM M ODEL
We consider a downlink ultra-massive MIMO communica-tion system working at mmWave/THz band, where a basestation (BS) equipped with N t antennas transmits data to auser equipment (UE) equipped with N r antennas. The BS-UEchannel H can be expressed as [19] H = X Ll =1 α l a ( N r , φ l ) a H ( N t , θ l ) , (1)where α l ∼ CN (0 , ¯ P / √ ρ ) is the complex gain of the l thpath, l = 1 , ..., L , with ¯ P and ρ denoting the average powergain and the average path loss between the BS and the UErespectively. a ( N t , θ l ) and a ( N r , φ l ) denote the antenna arrayresponse vectors of the BS and the UE respectively. In thispaper, we consider the uniform linear arrays (ULA), wherearray response is in the form of a ( N, φ ) = [1 , e i πφ , ..., e i π ( N − φ ] T , (2)In (1), φ l = ( d/λ ) sin( ¯ φ l ) and θ l = ( d/λ ) sin(¯ θ l ) , with λ denoting the signal wavelength, d denoting the intervalbetween adjacent antenna elements, and ¯ φ l , ¯ θ l being the UE’sazimuth AoA and the BS’ azimuth AoD of the l th pathrespectively.In this paper, we assume that the ranges of the AoD/AoAsare known a priori, i.e., ∀ l , ¯ θ l ∈ Ω , ¯ φ l ∈ Ω with Ω , Ω ⊂ [0 , π ] . Thus we have θ l ∈ I and φ l ∈ I , with I , I ⊂ [ − d/λ, d/λ ] , l = 1 , ..., L . Without loss of generality, we set d/λ = 1 / in this paper. The channel H in (1) can be rewritten in the matrix formas H = A r ΛA Ht , θ l ∈ I , φ l ∈ I , (3)where Λ = diag( α , ..., α L ) , and the matrices A t = [ a ( N t , θ ) , ..., a ( N t , θ L )] and A r =[ a ( N r , φ ) , ..., a ( N r , φ L )] contain the array response ofthe BS and the UE respectively.To estimate the channel matrix, the transmitter sends S distinct beams during S successive time slots, i.e., in the s -thtime slot, the beamforming vector f s ∈ C N t is selected from acodebook [21]. Thus the received signal of the s -th time slotcan be expressed as y s = Hf s x s + n s , (4)where n s ∼ CN ( , σ n I N r ) is the additive white Gaussiannoise with I N r denoting the N r × N r identity matrix, and x s denotes the pilot symbol in the s -th time slot. After thereceiver collecting y s ∈ C N r for s = 1 , ..., S , the obtainedsignal matrix Y = [ y , ..., y S ] = HFX + N , (5)where F = [ f , ..., f S ] ∈ C N t × S consists of the beamformingvectors of the S time slots, X = diag( x , ..., x S ) ∈ C S × S ,and N = [ n , ..., n S ] ∈ C N r × S . Our aim is to estimate H from Y .III. C HANNEL ESTIMATION USING
FS-ANMIn this section, we first present the channel estimator for thecase that the UE has only one antenna, and then for the casethat the UE has multiple antennas.
A. Single Rx Antenna
When the UE has only one antenna, i.e., N t > , N r = 1 ,then y s is a scalar. We denote ˜ y = [ y , ..., y S ] H = X H F H ˜ h + ˜ n , (6)where ˜ h = H H | N r =1 = P Ll =1 α ∗ l a ( N t , θ l ) , ˜ n =[ n , ..., n S ] H .To solve the off-grid problem, we employ the FS atomicnorm to enforce the sparsity of ˜ h . First, we briefly introducethe concept of FS Vandermonde decomposition and FS atomicnorm [20].Define I = ( f L , f H ) ⊂ [ − , ] as a frequency interval, andtrigonometric polynomial β ( f ) = r z − + r + r − z, (7)where z = e i πf , r = e iπ ( f L + f H ) sgn( f H − f L ) , r = − π ( f H − f L ))sgn( f H − f L ) , r − = r ∗ . Then β ( f ) isalways positive for f ∈ I , and negative for f ∈ [ − , ] \ I .Given I ⊂ [ − , ] , a Toeplitz matrix T ∈ C N × N with r = rank( T ) ≤ N − admits a unique FS Vandermondedecomposition as T = P rk =1 c k a ( N, f k ) a H ( N, f k ) with f k ∈I , if and only if ( T (cid:23) T β (cid:23) , (8) where T = Toep( t ) is generated by a complex sequence t = [ t − N +1 , t − N +2 , ..., t N − ] T , where Toep( · ) denotes theToeplitz matrix whose first column is the last N elements ofthe input vector, c k > , T β is a Toeplitz matrix defined as [ T β ] mn = P j = − r j t m − n − j , ≤ m, n ≤ N − .From (6), the class of signals is ˜ h = P Ll =1 α ∗ l a ( N t , θ l ) .Therefore the FS atom is of the form a ( N t , θ ) ∈ C N t × . TheFS atomic set is defined as A I = { a ( N t , θ ) | θ ∈ I} . The FSatomic norm is then k ˜ h k A I = inf a ( N t ,θ l ) ∈A I α l ∈ C ( L X l =1 | α l | : ˜ h = L X l =1 α ∗ l a ( N t , θ l ) ) . (9)Note that (9) is equivalent to the following semi-definiteprogram (SDP) [22] k ˜ h k A I = inf t ∈ C (2 Nt − × t ∈ R N t Tr(Toep( t )) + t s.t. (cid:20) Toep( t ) ˜ h ˜ h H t (cid:21) (cid:23) , T β (cid:23) , (10)where Tr( · ) denotes the trace, (cid:23) indicates a semidefinitematrix, t = P Ll =1 | α l | , and T β is defined in (8).According to (6), the 1D channel estimation can be formu-lated as the following optimization problem: ˆ h = arg min ˜ h ∈ C Nt × k ˜ y − X H F H ˜ h k + µ k ˜ h k A I , (11)where µ > is the weight factor. In practice, we set µ ≃ σ n p N t log( N t ) .The problem in (11) has n = O ( N t ) free variables and m = 2 linear matrix inequations (LMI), and the i -th LMI hassize of k i × k i with k i = O ( N t ) . It follows from [25] that aprimal-dual algorithm for (11) has a computational complexityon the order of m X i =1 k i ! n n + n m X i =1 k i + m X i =1 k i ! = O ( N . t ) . (12)By arguments similar to those above, the original atomic normmethod [23] in the absence of prior knowledge has the samecomputational complexity of O ( N . t ) . B. Multiple Rx Antennas
For the case of N t , N r > , Y in (5) is vectorized as ˜ y = vec( Y ) = ( X T F T ⊗ I )˜ h + ˜ n , (13)where I is the identity matrix of size N r , ˜ h = vec( H ) = P Ll =1 α l a ∗ ( N t , θ l ) ⊗ a ( N r , φ l ) and ˜ n = vec( N ) , with ⊗ beingthe Kronecker product.Before solving the problem, we first extend the FS atomicnorm in [20] to the 2D case as follows.For the 2D case, we define I = ( f L , f H ) ⊂ [ − , ] , I = ( f L , f H ) ⊂ [ − , ] and a 2-level Toeplitz matrix T ( V ) ∈ C N t N r × N t N r formed by the elements of V , where V is defined as V = [ v − N r +1 , v − N r +2 , ..., v N r − ] , with v j =[ v j ( − N t + 1) , v j ( − N t + 2) , ..., v j ( N t − T ∈ C (2 N t − × , j = − N r + 1 , − N r + 2 , ..., N r − . More specifically, T ( V ) is in the form of [ T ( V )] pq = Toep( v p − q ) , [Toep( v j )] mn = v j ( m − n ) , (14)where Toep( · ) denotes the Toeplitz matrix whose first columnis the last N t elements of the input vector, j = − N r +1 , − N r + 2 , ..., N r − , with ≤ p, q ≤ N r denoting the blockindices and ≤ m, n ≤ N t denoting the element indices.Similar to (7), we can write β ( f ) and β ( f ) accordingto I and I , whose parameters are denoted as { r ,j } j = − and { r ,j } j = − , respectively. Then the corresponding 2-levelToeplitz matrices, i.e., T β ∈ C ( N t − N r × ( N t − N r and T β ∈ C N t ( N r − × N t ( N r − , are given by [[ T β ] pq ] mn = X j = − r ,j v p − q ( m − n − j ) , (15)where ≤ p, q ≤ N r , ≤ m, n ≤ N t − , and [ T β ] pq = X j = − r ,j Toep( v p − q − j ) , (16)where ≤ p, q ≤ N r − .Define the 2D FS atom as b ( θ, φ ) = a ( N t , θ ) ⊗ a ( N r , φ ) ∈ C N t N r , and the set of 2D FS atoms as A I , I = { b ( θ, φ ) | θ ∈I , φ ∈ I } . Then the 2D FS atomic norm of any signal p with respect to A I , I is defined as k p k A I , I = inf α k ∈ C θ k ∈I ,φ k ∈I { K X k =1 | α k | : p = K X k =1 α k b ( θ k , φ k ) } . (17) Lemma 1
Given I , I ⊂ [ − , ] , a 2-level Toeplitz matrix T ( V ) ∈ C N t N r × N t N r with r = rank( T ( V )) < min( N t , N r ) admits a unique FS Vandermonde decomposition as T ( V ) = P rk =1 c k b ( θ k , φ k ) b H ( θ k , φ k ) with θ k ∈ I and φ k ∈ I , ifand only if T ( V ) (cid:23) T β (cid:23) T β (cid:23) , (18) where T ( V ) , T β and T β are defined in (14) , (15) and (16) respectively. The proof of lemma 1 can be found in the appendix.For the 2D FS atomic norm, we have the correspondingSDP formulation as follow:
Lemma 2
It holds that k p k A I , I = min V ,t N t N r Tr( T ( V )) + 12 t,s.t. (cid:20) T ( V ) pp H t (cid:21) (cid:23) , T β (cid:23) , T β (cid:23) . (19)The proof of lemma 2 can be found in the appendix.From (13), the 2D FS atomic norm for ˜ h is then k ˜ h k A I , I = inf α l ∈ C θ l ∈I φ l ∈I { L X l =1 | α l | : ˜ h = L X l =1 α l a ∗ ( N t , θ l ) ⊗ a ( N r , φ l ) } . (20) According to lemma 2 , the corresponding SDP formulationof k ˜ h k A I , I is given by k ˜ h k A I , I = inf V ∈ C (2 Nt − × (2 Nr − t ∈ R N t N r Tr( T ( V )) + t s.t. (cid:20) T ( V ) ˜ h ˜ h H t (cid:21) (cid:23) , T β (cid:23) , T β (cid:23) , (21)where the T β and T β are defined by (15) and (16), respec-tively.Based on (13), the 2D channel estimation can be formulatedas the following optimization problem: ˆ h = arg min ˜ h ∈ C NtNr × k ˜ y − ( X T F T ⊗ I )˜ h k + µ k ˜ h k A I , I , (22)where µ > is the weight factor. In practice, we set µ ≃ σ n p N t N r log( N t N r ) .Finally the estimated channel matrix is given by ˆ H =vec − (ˆ h ) , with ˆ h being the solution to (22).The problem in (22) has n = O ( N t N r ) free variables and m = 3 LMIs, and the i -th LMI has size of k i × k i with k i = O ( N t N r ) . It follows from (12) that a primal-dual algorithm for(22) has a computational complexity of O ( N . t N . r ) , whichequals to that of the original 2D atomic norm method in [23].IV. SIMULATION RESULTS -10 -8 -6 -4 -2 0 2 4 6 8 10 SNR (dB) -25-20-15-10-505 N M S E o f c h a nn e l e s t i m a t i o n ( d B ) FS ANM limited in 180/360FS ANM limited in 120/360FS ANM limited in 60/360FS ANM limited in 30/360original ANMOMP G=0.5NtOMP G=0.75NtOMP G=1Nt (a) NMSE of 128 × -10 -8 -6 -4 -2 0 2 4 6 8 10 SNR (dB) -20-15-10-50510 N M S E o f c h a nn e l e s t i m a t i o n ( d B ) original 2D ANMOMP 2D G=0.5NtOMP 2D G=0.75NtOMP 2D G=1NtFS 2D ANM limited in 180/360FS 2D ANM limited in 120/360FS 2D ANM limited in 60/360FS 2D ANM limited in 30/360 (b) NMSE of 16 × × × ◦ , ◦ , ◦ and ◦ ) and different grids ( . N t , . N t and N t ) respectively. In this section, we use simulation to illustrate the perfor-mance of the proposed algorithm. The pilot matrix is set asan identity matrix of size S . The wavelength of the signalis set as λ = 1 × − m, i.e., the system is working on . . Note that the scattering at THz frequencies inducesmore than 20 dB attenuation, which means that almost only theLoS component can be used for reliable high-rate transmissionin THz communications. It is also worth pointing out thatthe performance loss induced by the consideration that onlyLoS component exists is negligible, since the number ofNLoS components is quite limited and the power of NLoScomponents is much weaker (more than 20 dB) than that ofLoS component in THz communications [24]. Therefore, thenumber of paths is set as L = 2 , with { α l } Ll =1 generatedby CN (0 , and CN (0 , . respectively. The interval of theULA is set as d = λ/ . For the 1D case, we set N t = 128 , N r = 1 and S = 50 . { φ l } Ll =1 is generated according to thedistribution U ( − , ) , with min i = j | φ i − φ j | > /N t . Forthe 2D case, we set N t = 16 , N r = 8 and S = 16 . Thefrequencies { θ l } Ll =1 and { φ l } Ll =1 are generated by U ( − , ) ,with min i = j ( | φ i − φ j | , | θ i − θ j | ) > /N t N r . The fre-quency constraints are set as Ω ∈ π ◦ { ◦ , ◦ , ◦ , ◦ } .The corresponding I = [min sin(˜ θ ) , max sin(˜ θ )] , I =[min sin( ˜ φ ) , max sin( ˜ φ )] , where ˜ θ ∈ [¯ θ − Ω2 , ¯ θ + Ω2 ] , ˜ φ ∈ [ ¯ φ − Ω2 , ¯ φ + Ω2 ] .For the first experiment, as shown in Fig. 1(a), to comparewith the FS-ANM algorithm, we consider two existing meth-ods, i.e., the OMP method and the original ANM algorithm in[23]. The result indicates that the proposed FS-ANM alwaysoutperforms the mentioned two existing method with a gap ofabout . As an off-grid algorithm, the original ANM showsa higher performance than that of the on-grid OMP method,which is limited by the density of grid setting. However, theoriginal ANM performs still worse than the proposed FS-ANM. For the second experiment, as shown in Fig. 1(b), wealso consider the OMP method and the original ANM in [23]as the reference. It can be seen that the precision of FS-ANMis generally higher than the original ANM, with the gap ofabout and at SNR = − and SNR = 10dB respectively. On the other hand, prior knowledge is moreinformed, i.e., the frequency range becomes smaller, the FS-ANM algorithm achieves higher accuracy.V. CONCLUSIONSIn this paper, we have proposed a new off-grid MIMOchannel estimation method for mmWave/THz systems that canexploit the prior knowledge on the ranges of AoD/AoA, basedon frequency-selective atomic norm minimization. Simulationresults indicates that the proposed algorithm significantlyoutperforms the existing on-grid/off-grid channel estimators.A
PPENDIX
Proof of Lemma 1.
We first show the “if” part. It followsfrom T ( V ) (cid:23) that T ( V ) admits a unique Vandermondedecomposition [20]. Therefore, it suffices to show θ k ∈ I and φ k ∈ I under the condition T β (cid:23) and T β (cid:23) .According to (14), the element of T ( V ) is given by [[ T ( V )] pq ] mn = v p − q ( m − n )= X rk =1 c k e i π ( m − n ) θ k e i π ( p − q ) φ k . (23) Then we have [[ T β ] pq ] mn = X j = − r ,j v p − q ( m − n − j )= X j = − r ,j r X k =1 c k e i π ( m − n − j ) θ k e i π ( p − q ) φ k = r X k =1 c k e i π ( m − n ) θ k e i π ( p − q ) φ k X j = − r ,j e − i πjθ k | {z } β ( θ k ) = r X k =1 c k β ( θ k ) e i π ( m − n ) θ k e i π ( p − q ) φ k , (24)and hence T β = X rk =1 c k β ( θ k ) b ( θ k , φ k ) b ( θ k , φ k ) H = B diag( c β ( θ ) , ..., c r β ( θ r )) B H , (25)where b ( θ k , φ k ) = a ( N t − , θ k ) ⊗ a ( N r , φ k ) , B =[ b ( θ , φ ) , ..., b ( θ r , φ r )] .According to (16) and (14), we have [[ T β ] pq ] mn = X j = − r ,j v p − q − j ( m − n ) . (26)Similarly, we can get T β = B diag( c β ( φ ) , ..., c r β ( φ r )) B H , (27)where B = [ b ( θ , φ ) , ..., b ( θ r , φ r )] , with b ( θ k , φ k ) = a ( N t , θ k ) ⊗ a ( N r − , φ k ) .Since r ≤ min( N t ( N r − , ( N t − N r ) , B and B havefull column ranks. Using (18), (25) and (27), we have ( diag( c β ( θ ) , ..., c r β ( θ r )) = B † T β B † H (cid:23) , diag( c β ( φ ) , ..., c r β ( φ r )) = B † T β B † H (cid:23) , (28)where † denotes the matrix pseudo-inverse operator. Thus ∀ k , c k β ( θ k ) ≥ , c k β ( φ k ) ≥ . Since c k > , we have β ( θ k ) ≥ and β ( φ k ) ≥ . By the property of β ( f ) , we finally have θ k ∈ I , φ k ∈ I , k = 1 , ..., r .The “only if” part can be shown by similar arguments.Given T ( V ) = P rk =1 c k b ( θ k , φ k ) b H ( θ k , φ k ) , it is evidentthat T ( V ) (cid:23) . Then on the basis of (25), (27) and theproperty of β ( f ) , we have T β (cid:23) and T β (cid:23) . Proof of Lemma 2.
Let F ⋆ be the optimal objective valueof (19). We need to show that k p k A I , I = F ⋆ .To begin with, we first show that F ⋆ ≤ k p k A I , I . Let p = P k c k b ( θ k , φ k ) ψ k be an 2D FS Vandermonde decompositionof p on I and I , with | ψ k | = 1 . Then let V conformto T ( V ) = P k c k b ( θ k , φ k ) b H ( θ k , φ k ) and t = P k c k . By lemma 1 , we have T β (cid:23) and T β (cid:23) . Furthermore, itholds that (cid:20) T ( V ) pp H t (cid:21) = X k c k (cid:20) b ( θ k , φ k )¯ ψ k (cid:21) (cid:20) b ( θ k , φ k )¯ ψ k (cid:21) H (cid:23) . (29)Thus the constructed t and V are a feasible solution to theproblem (19), with the objective value calculated as N t N r Tr( T ( V )) + 12 t = X k c k . (30) Therefore, it holds that F ⋆ ≤ P k c k . Since the inequalityholds for any FS atomic decomposition of p on I and I ,we have that F ⋆ ≤ k p k A I , I based on the definition of k p k A I , I .Next we will show that F ⋆ ≥ k p k A I , I . We supposethat ( t ⋆ , V ⋆ ) is the optimal solution to (19). By the fact that T ( V ⋆ ) (cid:23) , T ⋆β (cid:23) and T ⋆β (cid:23) , according to lemma 1 , T ( V ⋆ ) has an FS Vandermonde decomposition on I and I given by T ( V ⋆ ) = X r ⋆ k =1 c ⋆k b ( θ ⋆k , φ ⋆k ) b H ( θ ⋆k , φ ⋆k ) . (31)Since (cid:20) T ( V ⋆ ) pp H t ⋆ (cid:21) (cid:23) , p lies in the range space of T ( V ⋆ ) and thus has an FS atomic decomposition given by p = X r ⋆ k =1 c ⋆k b ( θ ⋆k , φ ⋆k ) ψ ⋆k , | ψ ⋆k | = 1 , θ ⋆k ∈ I , φ ⋆k ∈ I , (32)which achieves the FS atomic norm. Furthermore, it holds that t ⋆ ≥ p H [ T ( V ⋆ )] † p = X r ⋆ k =1 c ⋆k , N t N r Tr( T ( V ⋆ )) = X r ⋆ k =1 c ⋆k . (33)Thus we have F ⋆ = 12 N t N r Tr( T ( V ⋆ ))+ t ⋆ ≥ X k c ⋆k ≥ k p k A I , I . (34)Since that F ⋆ ≤ k p k A I , I and F ⋆ ≥ k p k A I , I have bothbeen shown, we conclude that F ⋆ = k p k A I , I . Thus lemma2 is proved. R EFERENCES[1] P. Wang, Y. Li, L. Song, B. Vucetic, “Multi-gigabit millimeter wavewireless communications for 5G: From fixed access to cellular net-works”,
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