Channel Estimation for RIS Assisted Wireless Communications: Part II -- An Improved Solution Based on Double-Structured Sparsity
aa r X i v : . [ c s . I T ] J a n Channel Estimation for RIS Assisted WirelessCommunications: Part II - An Improved SolutionBased on Double-Structured Sparsity (Invited Paper)
Xiuhong Wei, Decai Shen, and Linglong Dai
Abstract —Reconfigurable intelligent surface (RIS) can manip-ulate the wireless communication environment by controlling thecoefficients of RIS elements. However, due to the large number ofpassive RIS elements without signal processing capability, chan-nel estimation in RIS assisted wireless communication systemrequires high pilot overhead. In the second part of this invitedpaper, we propose to exploit the double-structured sparsityof the angular cascaded channels among users to reduce thepilot overhead. Specifically, we first reveal the double-structuredsparsity, i.e., different angular cascaded channels for differentusers enjoy the completely common non-zero rows and thepartially common non-zero columns. By exploiting this double-structured sparsity, we further propose the double-structuredorthogonal matching pursuit (DS-OMP) algorithm, where thecompletely common non-zero rows and the partially commonnon-zero columns are jointly estimated for all users. Simulationresults show that the pilot overhead required by the proposedscheme is lower than existing schemes.
Index Terms —Reconfigurable intelligent surface (RIS), channelestimation, compressive sensing.
I. I
NTRODUCTION
In the first part of this two-part invited paper, we have in-troduced the fundamentals, solutions, and future opportunitiesof channel estimation in the reconfigurable intelligent surface(RIS) assisted wireless communication system. One of themost important challenges of channel estimation is that, thepilot overhead is high, since the RIS consists of a large numberof passive elements without signal processing capability [1],[2]. By exploiting the sparsity of the angular cascaded channel,i.e., the cascade of the channel from the user to the RISand the channel from the RIS to the base station (BS), thechannel estimation problem can be formulated as a sparsesignal recovery problem, which can be solved by compressivesensing (CS) algorithms with reduced pilot overhead [3], [4].However, the pilot overhead of most existing solutions is stillhigh.In the second part of this paper, in order to further reducethe pilot overhead, we propose a double-structured orthogonal
All authors are with the Beijing National Research Center forInformation Science and Technology (BNRist) as well as the Departmentof Electronic Engineering, Tsinghua University, Beijing 100084, China(e-mails: [email protected], [email protected],[email protected]).This work was supported in part by the National Key Research andDevelopment Program of China (Grant No. 2020YFB1807201) and in partby the National Natural Science Foundation of China (Grant No. 62031019). matching pursuit (DS-OMP) based cascaded channel estima-tion scheme by leveraging the double-structured sparsity ofthe angular cascaded channels . Specifically, we reveal thatthe angular cascaded channels associated with different usersenjoy the completely common non-zero rows and the par-tially common non-zero columns, which is called as “double-structured sparsity” in this paper. Then, by exploiting thisdouble-structured sparsity, we propose the DS-OMP algorithmbased on the classical OMP algorithm to realize channelestimation. In the proposed DS-OMP algorithm, the com-pletely common row support and the partially common columnsupport for different users are jointly estimated, and the user-specific column supports for different users are individuallyestimated. After detecting all supports mentioned above, theleast square (LS) algorithm can be utilized to obtain the esti-mated angular cascaded channels. Since the double-structuredsparsity is exploited, the proposed DS-OMP based channelestimation scheme is able to further reduce the pilot overhead.The rest of the paper is organized as follows. In Section II,we introduce the channel model and formulate the cascadedchannel estimation problem. In Section III, we first reveal thedouble-structured sparsity of the angular cascaded channels,and then propose the DS-OMP based cascaded channel estima-tion scheme. Simulation results and conclusions are providedin Section IV and Section V, respectively. Notation : Lower-case and upper-case boldface letters a and A denote a vector and a matrix, respectively; a T denotes theconjugate of vector a ; A T and A H denote the transpose andconjugate transpose of matrix A , respectively; k A k F denotesthe Frobenius norm of matrix A ; diag ( x ) denotes the diagonalmatrix with the vector x on its diagonal; a ⊗ b denotes theKronecker product of a and b . Finally, CN ( µ, σ ) denotesthe probability density function of the circularly symmetriccomplex Gaussian distribution with mean µ and variance σ .II. S YSTEM M ODEL
In this section, we will first introduce the cascaded channelin the RIS assisted communication system. Then, the cascadedchannel estimation problem will be formulated.
A. Cascaded Channel
We consider that the BS and the RIS respectively employthe M -antenna and the N -element uniform planer array (UPA) Simulation codes are provided to reproduce the results presented in thispaper: http://oa.ee.tsinghua.edu.cn/dailinglong/publications/publications.html. to simultaneously serve K single-antenna users. Let G of size M × N denote the channel from the RIS to the BS, and h r,k of size N × denote the channel from the k th user to the RIS ( k = 1 , , · · · , K ) . The widely used Saleh-Valenzuela channelmodel is adopted to represent G as [5] G = r M NL
G L G X l =1 α Gl b (cid:16) ϑ G r l , ψ G r l (cid:17) a (cid:16) ϑ G t l , ψ G t l (cid:17) T , (1)where L G represents the number of paths between the RIS andthe BS, α Gl , ϑ G r l ( ψ G r l ), and ϑ G t l ( ψ G t l ) represent the complexgain consisting of path loss, the azimuth (elevation) angle atthe BS, and the azimuth (elevation) angle at the RIS for the l th path. Similarly, the channel h r,k can be represented by h r,k = s NL r,k L r,k X l =1 α r,kl a (cid:16) ϑ r,kl , ψ r,kl (cid:17) , (2)where L r,k represents the number of paths between the k thuser and the RIS, α r,kl , ϑ r,kl ( ψ r,kl ) represent the complex gainconsisting of path loss, the azimuth (elevation) angle at theRIS for the l th path. b ( ϑ, ψ ) ∈ C M × and a ( ϑ, ψ ) ∈ C N × represent the normalized array steering vector associated tothe BS and the RIS, respectively. For a typical N × N ( N = N × N ) UPA, a ( ϑ, ψ ) can be represented by [5] a ( ϑ, ψ ) = 1 √ N h e − j πd sin( ϑ )cos( ψ ) n /λ i ⊗ h e − j πd sin( ψ ) n /λ i , (3)where n = [0 , , · · · , N − and n = [0 , , · · · , N − , λ is the carrier wavelength, and d is the antenna spacing usuallysatisfying d = λ/ .Further, we denote H k , G diag ( h r,k ) as the M × N cascaded channel for the k th user. Using the virtual angular-domain representation, H k ∈ C M × N can be decomposed as H k = U M ˜ H k U TN , (4)where ˜ H k denotes the M × N angular cascaded channel, U M and U N are respectively the M × M and N × N dictionaryunitary matrices at the BS and the RIS [5]. Since thereare limited scatters around the BS and the RIS, the angularcascaded channel ˜ H k has a few non-zero elements, whichexhibits the sparsity. B. Problem Formulation
In this paper, we assume that the direct channel betweenthe BS and the user is known for BS, which can be easilyestimated as these in conventional wireless communicationsystems [5]. Therefore, we only focus on the cascaded channelestimation problem.By adopting the widely used orthogonal pilot transmissionstrategy, all users transmit the known pilot symbols to theBS via the RIS over Q time slots for the uplink channelestimation. Specifically, in the q th ( q = 1 , , · · · , Q ) time slot,the effective received signal y k,q ∈ C M × at the BS for the k th user after removing the impact of the direct channel canbe represented as y k,q = G diag ( θ q ) h r,k s k,q + w k,q = G diag ( h r,k ) θ q s k,q + w k,q , (5) where s k,q is the pilot symbol sent by the k th user, θ q =[ θ q, , · · · , θ q,N ] T is the N × reflecting vector at the RISwith θ q,n representing the reflecting coefficient at the n thRIS element ( n = 1 , · · · , N ) in the q th time slot, w k,q ∼CN (cid:0) , σ I M (cid:1) is the M × received noise with σ repre-senting the noise power. According to the cascaded channel H k = G diag ( h r,k ) , we can rewrite (5) as y k,q = H k θ q s k,q + w k,q . (6)After Q time slots of pilot transmission, we can obtain the M × Q overall measurement matrix Y k = [ y k, , · · · , y k,Q ] by assuming s k,q = 1 as Y k = H k Θ + W k , (7)where Θ = [ θ , · · · , θ Q ] and W k = [ w k, , · · · , w k,Q ] . Bysubstituting (4) into (7), we can obtain Y k = U M ˜ H k U TN Θ + W k . (8)Let denote ˜ Y k = (cid:0) U HM Y k (cid:1) H as the Q × M effectivemeasurement matrix, and ˜ W k = (cid:0) U HM W k (cid:1) H as the Q × M effective noise matrix, (7) can be rewritten as a CS model: ˜ Y k = ˜ Θ ˜ H Hk + ˜ W k , (9)where ˜ Θ = (cid:0) U TN Θ (cid:1) H is the Q × N sensing matrix. Basedon (9), we can respectively estimate the angular cascadedchannel for each user k by conventional CS algorithms, suchas OMP algorithm. However, under the premise of ensuringthe estimation accuracy, the pilot overhead required by theconventional CS algorithms is still high.III. J OINT C HANNEL E STIMATION FOR
RIS A
SSISTED W IRELESS C OMMUNICATION S YSTEMS
In this section, we will first reveal the double-structuredsparsity of the angular cascaded channels. Then, by exploitingthis important channel characteristic, we will propose a DS-OMP based cascaded channel estimation scheme to reducethe pilot overhead. Finally, the computational complexity ofthe proposed scheme will be analyzed.
A. Double-Structured Sparsity of Angular Cascaded Channels
In order to further explore the sparsity of the angular cas-caded channel both in row and column, the angular cascadedchannel ˜ H k in (4) can be expressed as ˜ H k = s M NL G L r,k L G X l =1 L r,k X l =1 α Gl α r,kl ˜ b (cid:16) ϑ G r l , ψ G r l (cid:17) ˜ a T (cid:16) ϑ G t l + ϑ r,kl , ψ G t l + ψ r,kl (cid:17) , (10)where both ˜ b ( ϑ, ψ ) = U HM b ( ϑ, ψ ) and ˜ a ( ϑ, ψ ) = U HN a ( ϑ, ψ ) have only one non-zero element, which lie onthe position of array steering vector at the direction ( ϑ, ψ ) in U M and U N . Based on (10), we can find that each completereflecting path ( l , l ) can provide one non-zero element for ˜ H k , whose row index depends on (cid:16) ϑ G r l , ψ G r l (cid:17) and columnindex depends on (cid:16) ϑ G t l + ϑ r,kl , ψ G t l + ψ r,kl (cid:17) . Therefore, ˜ H k has L G non-zero rows, where each non-zero row has L r,k non-zero columns. The total number of non-zero elements is L G L r,k , which is usually much smaller than M N . BS RISuser 1 user 2 G ,1 r h ,2 r h H H Fig. 1. Double-structured sparsity of the angular cascaded channels.
More importantly, we can find that different sparse channels { ˜ H k } Kk =1 exhibit the double-structured sparsity, as shown inFig. 1. Firstly, since different users communicate with the BSvia the common RIS, the channel G from the RIS to theBS is common for all users. From (10), we can also findthat (cid:26) (cid:16) ϑ G r l , ψ G r l (cid:17) (cid:27) L G l =1 is independent of the user index k . Therefore, the non-zero elements of { ˜ H k } Kk =1 lie on thecompletely common L G rows. Secondly, since different userswill share part of the scatters between the RIS and users, { h r,k } Kk =1 may enjoy partially common paths with the sameangles at the RIS. Let L c ( L c ≤ L r,k , ∀ k ) denote the numberof common paths for { h r,k } Kk =1 , then we can find that for ∀ l ,there always exists (cid:26) (cid:16) ϑ G t l − ϑ r,kl , ψ G t l − ψ r,kl (cid:17) (cid:27) L c l =1 sharedby { ˜ H k } Kk =1 . That is to say, for each common non-zero rows l ( l = 1 , , · · · , L G ), { ˜ H k } Kk =1 enjoy L c common non-zero columns. This double-structured sparsity of the angularcascaded channels can be summarized as follows from theperspective of row and column, respectively. • Row-structured sparsity: Let Ω kr denote the row set ofnon-zero elements for ˜ H k , then we have Ω r = Ω r = · · · = Ω Kr = Ω r , (11)where Ω r represents the completely common row supportfor { ˜ H k } Kk =1 . • Partially column-structured sparsity: Let Ω l,kc denote thecolumn set of non-zero elements for the l th non-zerorow of ˜ H k , then we have Ω l , c ∩ Ω l , c ∩· · ·∩ Ω l ,Kc = Ω l , Com c , l = 1 , , · · · , L G , (12)where Ω l, Com c represents the partially common columnsupport for the l th non-zero row of { ˜ H k } Kk =1 .Based on the above double-structured sparsity, the cascadedchannels for different users can be jointly estimated to improvethe channel estimation accuracy. B. Proposed DS-OMP Based Cascaded Channel Estimation
In this subsection, we propose the DS-OMP based cas-caded channel estimation scheme by integrating the double-structured sparsity into the classical OMP algorithm. Thespecific algorithm can be summarized in
Algorithm 1 , whichincludes three key stages to detect supports of angular cas-caded channels.
Algorithm 1:
DS-OMP based cascaded channel esti-mation
Input : ˜ Y k : ∀ k , ˜ Θ , L G , L r,k : ∀ k , L c . Initialization : ˆ˜ H k = M × N , ∀ k .1. Stage 1:
Return estimated completely common rowsupport ˆΩ r by Algorithm 2 .2.
Stage 2:
Return estimated partially common columnsupports { ˆΩ l , Com c } L G l =1 based on ˆΩ r by Algorithm 3 .3.
Stage 3:
Return estimated column supports {{ ˆΩ l ,kc } L G l =1 } Kk =1 based on ˆΩ r and { ˆΩ l , Com c } L G l =1 by Algorithm 4 .4. for l = 1 , , · · · , L G do for k = 1 , , · · · , K do ˆ˜ H Hk ( ˆΩ l ,kc , ˆΩ r ( l )) = ˜ Θ † (: , ˆΩ l ,kc ) ˜ Y k (: , ˆΩ r ( l )) end for end for ˆ H k = U HM ˆ˜ H k U N , ∀ k Output : Estimated cascaded channel matrices ˆ H k , ∀ k .The main procedure of Algorithm 1 can be explainedas follows. Firstly, the completely common row support Ω r is jointly estimated thanks to the row-structured sparsity inStep 1, where Ω r consists of L G row indexes associatedwith L G non-zero rows. Secondly, for the l th non-zero row,the partially common column support Ω l , Com c can be furtherjointly estimated thanks to the partially column-structuredsparsity in Step 2. Thirdly, the user-specific column supportsfor each user k can be individually estimated in Step 3.After detecting supports of all sparse matrices, we adoptthe LS algorithm to obtain corresponding estimated matrices { ˆ˜ H k } Kk =1 in Steps 4-8. It should be noted that the sparse signalin (9) is ˜ H Hk , thus the sparse matrix estimated by the LSalgorithm in Step 6 is ˆ˜ H Hk . Finally, we can obtain the estimatedcascaded channels { ˆ H k } Kk =1 by transforming angular channelsinto spatial channels in Step 9.In the following part, we will introduce how to estimatethe completely common row support, the partially commoncolumn supports, and the individual column supports forthe first three stages in detail.
1) Stage 1: Estimating thecompletely common row support.
Thanks to the row-structuredsparsity of the angular cascaded channels, we can jointlyestimate the completely common row support Ω r for { ˜ H k } Kk =1 by Algorithm 2 .From the virtual angular-domain channel representation (4),we can find that non-zero rows of { ˜ H k } Kk =1 are correspondingto columns with high power in the received pilots { ˜ Y k } Kk =1 .Since { ˜ H k } Kk =1 have the completely common non-zero rows, { ˜ Y k } Kk =1 can be jointly utilized to estimate the completely Algorithm 2:
Joint completely common row supportestimation
Input : ˜ Y k : ∀ k , L G . Initialization : g = M × .1. for k = 1 , , · · · , K do g ( m ) = g ( m ) + k ˜ Y k (: , m ) k F , ∀ m = 1 , , · · · , M end for ˆΩ r = Γ T ( g , L G ) Output : Estimated completely common row support ˆΩ r .common row support Ω r , which can resist the effect of noise.Specifically, we denote g of size M × to save the sum powerof columns of { ˜ Y k } Kk =1 , as in Step 2 of Algorithm 2 . Finally, L G indexes of elements with the largest amplitudes in g areselected as the estimated completely common row support ˆΩ r in Step 4, where T ( x , L ) denotes a prune operator on x thatsets all but L elements with the largest amplitudes to zero,and Γ( x ) denotes the support of x , i.e., Γ( x ) = { i, x ( i ) = 0 } .After obtaining L G non-zero rows by Algorithm 2 , wefocus on estimating the column support Ω l ,kc for each non-zero row l and each user k by the following Stage 2 and3.
2) Stage 2: Estimating the partially common column sup-ports.
Thanks to the partially column-structured sparsity of theangular cascaded channels, we can jointly estimate the par-tially common column supports { Ω l , Com c } Ll =1 for { ˜ H k } Kk =1 by Algorithm 3 . Algorithm 3:
Joint partially common column supportsestimation
Input : ˜ Y k : ∀ k , L G , ˜ Θ , L r,k : ∀ k , L c , ˆΩ r . Initialization : ˆΩ l ,kc = ∅ , ∀ l , k , c l = N × , ∀ l .1. for l = 1 , , · · · , L G do for k = 1 , , · · · , K do ˜ y k = ˜ Y k (: , ˆΩ r ( l )) , ˜ r k = ˜ y k for l = 1 , , · · · , L r,k do n ∗ = argmax n =1 , , ··· ,N k ˜ Θ H (: , n )˜ r k k F ˆΩ l ,kc = ˆΩ l ,kc S n ∗ ˆ˜ h k = N × ˆ˜ h k ( ˆΩ l ,kc ) = ˜ Θ † (: , ˆΩ l ,kc )˜ y k ,9. ˜ r k = ˜ y k − ˜ Θ ˆ˜ h k c l ( n ∗ ) = c l ( n ∗ ) + 1 end for end for ˆΩ l , Com c = Γ T ( c l ,P c ) end forOutput : Estimated completely common row support { ˆΩ l , Com c } L G l =1 .For the l th non-zero row, we only need to utilize theeffective measurement vector ˜ y k = ˜ Y k (: , ˆΩ r ( l )) to estimatethe partially common column support Ω l , Com c . The basic ideais that, we firstly estimate the column support Ω l ,kc with L r,k indexes for each user k , then we select L c indexes associatedwith the largest number of times from all { Ω l ,kc } Kk =1 as theestimated partially common column support ˆΩ l , Com c .In order to estimate the column supports for each user k ,the correlation between the sensing matrix ˜ Θ and the residualvector ˜ r k needs to be calculated. As shown in Step 5 of Algorithm 3 , the most correlative column index in ˜ Θ with ˜ r k is regarded as the newly found column support index n ∗ . Based on the updated column support ˆΩ l ,kc in Step 6,the estimated sparse vector ˆ˜ h k is obtained by using the LSalgorithm in Step 8. Then, the residual vector ˜ r k is updatedby removing the effect of non-zero elements that have beenestimated in Step 9. Particularly, the N × vector c l is usedto count the number of times for selected column indexes inStep 10. Finally, the L c indexes of elements with the largestvalue in c l are selected as the estimated partially commoncolumn support ˆΩ l , Com c in Step 13.
3) Stage 3: Estimating the individual column supports.
Based on the estimated completely common row support ˆΩ r and the estimated partially common column supports { ˆΩ l , Com c } Ll =1 , the column support Ω l ,kc for each non-zerorow l and each user k can be estimated by Algorithm 4 . Algorithm 4:
Individual column supports estimation
Input : ˜ Y k : ∀ k , ˜ Θ , L G L r,k : ∀ k , L c , ˆΩ r , { ˆΩ l , Com c } Ll =1 . Initialization : ˆΩ l ,kc = ˆΩ l , Com c , ∀ l , k .1. for l = 1 , , · · · , L G do for k = 1 , , · · · , K do ˜ y k = ˜ Y k (: , ˆΩ r ( l )) ˆ˜ h k = N × ˆ˜ h k ( ˆΩ l ,kc ) = ˜ Θ † (: , ˆΩ l , Com c )˜ y k r k = y k − ˜ Θ ˆ h k ,7. for l = 1 , , · · · , L r,k − L c do n ∗ = argmax n =1 , , ··· ,N k ˜ Θ H (: , n )˜ r k k F ˆΩ l ,kc = ˆΩ l ,kc S n ∗ ˆ˜ h k = N × ˆ˜ h k ( ˆΩ l ,kc ) = ˜ Θ † (: , ˆΩ l ,kc )˜ y k ˜ r k = ˜ y k − ˜ Θ ˆ˜ h k end for end for end forOutput : Estimated the individual column supports {{ ˆΩ l ,kc } L G l =1 } Kk =1 .For the l th non-zero row, we have estimated L c columnsupport indexes by Algorithm 3 . Thus, there are L r,k − L c user-specific column support indexes to be estimated for eachuser k . The column support ˆΩ l ,kc is initialized as ˆΩ l , Com c .Based on ˆΩ l , Com c , the estimated sparse vector ˆ˜ h k and residualvector ˜ r k are initialized in Step 5 and Step 6. Then, the columnsupport ˆΩ l ,kc for ∀ l and ∀ k can be estimated in Steps 7-13by following the same idea of Algorithm 3 .Through the above three stages, the supports of all angularcascaded channels are estimated by exploiting the double- structured sparsity. It should be pointed out that, if thereare no common scatters between the RIS and users, thedouble-structured sparse channel will be simplified as the row-structured sparse channel. In this case, the cascaded channelestimation can also be solved by the proposed DS-OMPalgorithm, where Stage 2 will be removed.
C. Computational Complexity Analysis
In this subsection, the computational complexity of the pro-posed DS-OMP algorithm is analyzed in terms of three stagesof detecting supports. In Stage 1, the computational complex-ity mainly comes from Step 2 in
Algorithm 2 , which calcu-lates the power of M columns of ˜ Y k of size Q × M for k =1 , , · · · , K . The corresponding computational complexity is O ( KM Q ) . In Stage 2, for each non-zero row l and each user k in Algorithm 3 , the computational complexity O ( N QL r,k ) is the same as that of OMP algorithm [6]. Considering L G K iterations, the overall computational complexity of Algorithm3 is O ( L G KN QL r,k ) . Similarly, the overall computationalcomplexity of Algorithm 4 is O ( L G KN Q ( L r,k − L c ) ) .Therefore, the overall computational complexity of proposedDS-OMP algorithm is O ( KM Q ) + O ( L G KN QL r,k ) .IV. S IMULATION R ESULTS
In our simulation, we consider that the number of BSantennas, RIS elements and users are respectively M = 64 ( M = 8 , M = 8 ), N = 256 ( N = 16 , N = 16 ), and K = 16 . The number of paths between the RIS and the BSis L G = 5 , and the number of paths from the k th user to theRIS is set as L r,k = 8 for ∀ k . All spatial angles are assumedto be on the quantized grids. Each element of RIS reflectingmatrix Θ is selected from {− √ N , + √ N } by consideringdiscrete phase shifts of the RIS [7]. | α Gl | = 10 − d − . BR , where d BR denotes the distance between the BS and RIS and isassumed to be d BR = 10 m . | α r,kl | = 10 − d − . RU , where d RU denotes the distance between the RIS and user and isassumed to be d RU = 100 m for ∀ k [7]. The SNR is definedas E {|| ˜ Θ ˜ H Hk || F / || ˜ W k || F } in (14) and is set as dB.We compare the proposed DS-OMP based scheme withthe conventional CS based scheme [3] and the row-structuredsparsity based scheme [4]. In the conventional CS basedscheme, the OMP algorithm is used to estimate the sparsecascaded channel ˜ H k for ∀ k . In the row-structured sparsitybased scheme, the common row support Ω r with L G indexesare firstly estimated, and then for each user k and each non-zero row l , column supports are respectively estimated byfollowing the idea of the classical OMP algorithm. In addition,we consider the oracle LS scheme as our benchmark, wherethe supports of all sparse channels are assumed to be perfectlyknown.Fig. 2 shows the normalized mean square error (NMSE)performance comparison against the pilot overhead, i.e., thenumber of time slots Q for pilot transmission. As shownin Fig. 2, in order to achieve the same estimation accuracy,the pilot overhead required by the proposed DS-OMP basedscheme is lower than the other two existing schemes [3], [4].However, when there is no common path between the RIS
32 48 64 80 96 112 128
The pilot overhead Q for the cascaded channel estimation -25-20-15-10-5 N M SE ( d B ) Conventional CS based scheme [3]Row-structured sparsity based scheme [4]Proposed DS-OMP based scheme (Lc=0)Proposed DS-OMP based scheme (Lc=4)Proposed DS-OMP based scheme (Lc=6)Proposed DS-OMP based scheme (Lc=8)Oracle LS based scheme
Fig. 2. NMSE performance comparison against the pilot overhead Q . and all users, i.e., L c = 0 , the double-structured sparsitywill be simplified as the row-structured sparsity [4]. Thus theNMSE performance of the proposed DS-OMP based and therow-structured sparsity based scheme is the same. With theincreased number of common paths L c between the RIS andusers, the NMSE performance of the proposed scheme canbe improved to approach the benchmark of perfect channelsupports. V. C ONCLUSIONS
In this paper, we developed a low-overhead cascaded chan-nel estimation scheme in RIS assisted wireless communicationsystems. Specifically, we first analyzed the double-structuredsparsity of the angular cascaded channels among users. Basedon this double-structured sparsity, we then proposed a DS-OMP algorithm to reduce the pilot overhead. Simulationresults show that the pilot overhead required by the pro-posed DS-OMP algorithm is lower compared with existingalgorithms. For the future work, we will apply the double-structured sparsity to the super-resolution channel estimationproblem by considering the channel angles are continuous inpractice. R
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