arXiv:1912.10671v2 [cs.IT] 1 Jun 2020 t onepr ceeblna eeaie M (BiGAMP), ill-conditioned. AMP are matrices generalized channel bilinear than when better scheme especially performs closed-form counterpart prop element estimation the passive A its channel that show each user. LIS results of Numerical based the derived. 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MIMO surface, intelligent —Large .I I. NTRODUCTION aa Mirza, Jawad IONetworks MIMO eirMme,IEEE Member, Senior mthat em uality, tiple- IMO LIS) ular, osed bles ber ive ill- on ng he he in d o s - t t o h aeatnastig si 1] h uhr n[12] in authors the [11], in as settings of antenna ph training (MSE) same the error the in For squared estimation phases discrete channel mean using the the channels user, estimated minimizes Fo and [11] channels. AP in the at process estimate each to antenna wit it and single equipped enable AP-user a is that LIS the chains the estimate RF that to receive assumed is [6] It in channels. employed LIS-user is estima- channel method MIMO tion conventiona based carried The (TDD) work systems. duplexing communication estimation time-division assisted channel LIS the in out on of amount discussion H enormous a [20]. [19], provide an [16], attracted [10]–[13], recently has attention estimati channels research channel based the LIS reason, this of to co assisted Due LIS system. the munication of performance the determines the terminals in thus, required significantly. LIS, that cost without hardware of the schemes half decreasing beamforming to decreases hybrid AP antennas traditional transmit the of at number ha minimum required LIS the [ of [17], [17], scenarios to use (MU) According the multiuser for scenarios, investigated single-user been and th also to at shift power addition signal received In phase the user. maximize joint at to out (AP) a point carried the access is [16], improve optimization vector to in beamforming transmit [14]–[16] where in efficiency, aided the reflect considered LIS spectral as system, of are known single-user systems some also a MIMO For design discuss beamforming. shift briefly passive phase or We shi on phase important. studies LIS existing also systems appropriate in an MIMO is role that, based design to important LIS addition an In of plays [10]–[13]. performance terminals the the determining informatio at state available channel com of (CSI) quality wireless The technology. in multiple-ou (MIMO) multiple-input attention employing systems research munication considerable attracted vehicl [9]. acces (NOMA) multiple aerial non-orthogonal unmanned and wirele [7], [8] as LIS communication (UAV) transfer such with power gain, and integrated performance signal information offered being the received are investigate Man enhance to technologies [6]. to security wireless improve used future and interference be terrestri suppress can power, in reflectors which reconfigurable for networks, way the en- paved metamaterial propagation have and RF the (MEMS) in systems where advancements However, micro-electromechanical scenarios changing. rapidly the is in vironment useful not were sdsusderir h ult fCIaalbea the at available CSI of quality the earlier, discussed As has technology LIS the features, attractive its to Due n ahirAli Bakhtiar and r,we ere, 18]. tput ase. the m- on ss al ft h y n e e s s s 1 r - l 2 employ a cost efficient method with no receive RF chains at the channels is carried out in the second stage. These estimated LIS, where concatenated user-LIS-AP channels are estimated channels are used to design the phase shifts for LIS elements. at the AP. Moreover, the idea of grouping LIS elements to In particular, we derive a closed form expression for phase reduce the complexity of the channel estimation process is shifts that maximizes the total channel gain at the user. We also introduced in [12]. assume that a programmable controller is fabricated with LIS Prior to [12], the estimation of concatenated user-LIS-AP which is controlled by the AP to adjust the phases of the channels was considered in [10], [13], where ON/OFF switch- LIS elements. The contributions of the paper are summarized ing of LIS elements was implemented. In [10], each passive el- below: ement controlled by the multi-antenna AP is powered ON one • We propose a two-stage channel estimation method for by one to capture the least square estimation of LIS channels. the LIS assisted MIMO communication system with The sparse nature of millimeter wave (mmWave) channels is multiple transmit and receive antennas at the AP and exploited in [16] to convert the channel estimation problem user, respectively. In the first stage, the direct MIMO into a sparse signal recovery problem, which is then solved channel between the AP and user is estimated, whereas, by using compressed sensing algorithms such as orthogonal the LIS channels are estimated in the second stage. The matching pursuit (OMP) and expectation maximization gener- LIS channel matrices are considered to be ill-conditioned, alized approximate message passing (EMGAMP). According which is a realistic propagation scenario in near-field to [16], the training overhead with the EMGAMP algorithm is LOS communications. For the LIS channels, the channel lower than the classical OMP algorithm. To reduce the training estimation problem is transformed into the well known overhead, a three phase channel estimation process is proposed dictionary learning problem and the recently developed in [19] for MU scenarios, where the correlation among LIS BAdVAMP algorithm is used to learn the ill-conditioned reflected channels is exploited. channels. The aforementioned studies on the channel estimation prob- • The LIS channels estimated via the BAdVAMP algorithm lem either assume a single antenna at both AP and user or have inherent ambiguities out of which the permutation multiple antennas at the AP only. In this study, we investigate ambiguity has the most destructive effects. Therefore, to LIS channel estimation for the system where both AP and get rid of the permutation ambiguity from the recovered user are equipped with multiple antennas. Since, LIS will be channels, we propose an effective method that exploits densely deployed in indoor and outdoor locations, the near- the sparsity of the phase shift matrix. field communication channels are likely to experience line-of- • Based on the estimated channels, an optimization prob- sight (LOS) propagation conditions [21], [22]. This motivates lem is formulated to obtain the phase shift matrix for us to investigate the channel estimation of ill-conditioned1 the passive elements of LIS which maximizes the total LIS channels in MIMO systems. For this purpose, we em- channel gain at the user. Here, we assume a continuous ploy a bilinear adaptive vector approximate message-passing high-resolution phase shift design and derive a closed- (BAdVAMP) scheme developed in [23] which was originally form expression to compute the phase shifts of LIS designed for reconstructing ill-conditioned and mean shifted passive elements. matrices in dictionary learning problems. This recently devel- The rest of the paper is organized as follows. Section oped algorithm has not yet been applied to channel estimation II introduces the LIS assisted MIMO system model. Sec- problems in wireless communication systems, therefore, this tion III presents the proposed two-stage channel estimation is the first study to use the BAdVAMP algorithm to estimate scheme. The ambiguity elimination and the recovery of a LIS based ill-conditioned MIMO channels. Previously, for LIS unique solution from the estimated channels are discussed in assisted MIMO systems, the authors in [13] have proposed Section IV. In Section V, we derive the phase shift design for to estimate LIS channels using the approximate message- LIS elements. Numerical results are provided in Section VI. passing (AMP) based algorithm known as bilinear generalized Finally, Section VII concludes the paper. AMP (BiGAMP). However, the BiGAMP method is designed Notations: We use (·)H , (·)∗, (·)T , (·)−1 and (·)⊥ to denote only for reconstructing well-conditioned matrices, whereas, the conjugate transpose, the conjugate, the transpose, the the BAdVAMP algorithm has shown to be more robust in both inverse and the pseudoinverse operations, respectively. E[·] ill-conditioned and well-conditioned matrices [23]. Moreover, denotes expectation. The Hadamard product is represented by the BiGAMP method has higher computational complexity ⊙. For any given matrix A, the quantity ai,j denotes the entry and less accuracy as compared to the BAdVAMP method [23]. of the matrix A corresponding to the ith row and jth column. In this paper, the proposed channel estimation technique th Similarly, al represents the l column of the matrix A. is divided into two stages. In the first phase, the direct MIMO channel between user and AP is estimated using the II. SYSTEM MODEL conventional TDD based MIMO channel estimation scheme. Whereas, the BAdVAMP based channel estimation for LIS Consider a single-user MIMO communication system as shown in Fig. 1, where an AP equipped with M antennas 1In an ill-conditioned channel matrix, some of the singular values of the serves a user having N antennas. An LIS is installed in a sur- channel are negligible compared to the largest singular value of the channel. rounding area, which consists of L low-cost passive reflecting The term condition number is generally used to characterize the ill-conditioned channel matrix, which is the ratio of the largest singular value of that channel elements. The LIS is connected to the AP via a programmable matrix to the smallest singular value. controller, where the AP is capable of adjusting the phase of 3

passive elements in the LIS terminal. The downlink channels Downlink Reconfigurable Intelligent Surface Uplink from the AP to the LIS and from the LIS to the user are L elements T CL×M T CN×L denoted by H ∈ and G ∈ , respectively. Programmable In this paper, we consider a TDD transmission mode and Controller assume perfect channel reciprocity between the uplink and downlink. Therefore, the uplink channels from the user to the M antennas HT LIS and from the LIS to the AP are given by G ∈ CL×N and H ∈ CM×L, respectively. The direct downlink and H GT uplink channels between the AP and the user are denoted by G T CM×N CN×M T Z ∈ and Z ∈ , respectively. Z N antennas We adopt a narrowband geometric channel model based on extended Saleh-Valenzuela representation to characterize the Z uplink and downlink channels. Note that the geometric channel Access Point User model for LIS assisted MIMO/MISO systems has been widely Fig. 1: An illustration of an LIS-enabled MIMO system. used in previous studies [13], [24], [25]. We can express the direct uplink channel between the user and AP as [26]

L NM Z element, respectively, then we can write the phase shift vector r r H t t jφ1 jφ2 jφL T Z = αlar (ϕl, ϑl) at ϕl, ϑl , (1) for any given time as s = [s1e ,s2e ,...,sLe ] . Note LZ r l=1 that the switched off LIS elements may still produce a residual X
where LZ denotes the number of propagation paths and αl signal, degrading the overall SNR of the signal. However, it is is the complex gain of the lth path for the channel matrix assumed that the ON/OFF operation at the LIS unit is perfect, r r t t and therefore, we ignore the imperfections at the LIS unit. Z. ar (ϕl, ϑl) and at (ϕl, ϑl) represent the normalized receive and transmit array response vectors with azimuth (elevation) For the given T samples or channel uses, the total number r r t t of training vectors (channel uses) sent by the user in the angles of arrival and departure ϕl(ϑl) and ϕl(ϑl), respectively. In the case of a uniform planar array (UPA) in the yz-plane first and second stage is denoted by Td and Tr, respectively. with W and H elements on the y and z axes respectively, an Therefore, during the uplink channel estimation process, the N = WH array response vector is received signal at the AP for the first and second stage are

1 ya [t]= Zxa [t]+ na [t] , t =1,...,Td (3) a (ϕ, ϑ)= 1,...,ejkd(m sin (ϕ)sin(ϑ)+n cos (ϑ)),..., N and r T ejkd((W−1) sin (ϕ)sin(ϑ)+(H−1) cos (ϑ)) , (2) yb [t]= H (s [t] ⊙ Gxb [t]) + Zxb [t]+ nb [t] , (4) where d denotes the inter-element spacing. Denoting λ as the CN×1 CN×1 wavelength, we have k =2π/λ. The complex gain of the lth respectively, where, xa[t] ∈ and xb[t] ∈ are path is assumed to be distributed according to independent the uplink training vectors transmitted by the user in the first 2 and second channel estimation stages, respectively. na[t] and and identically distributed (i.i.d.) CN (0, σαl ). The value of 2 E Z 2 nb[t] are the complex additive white Gaussian noise (AWGN) σαl is set in order to achieve k kF = NM. The same narrowband geometric channel model is assumed for the vectors at the AP having zero mean. The covariance matrix of H G   the noise vectors at the first and second stages are given by uplink channels and . For the channel model in (1), when 2 2 σ IM and σ IM , respectively. We can write the received the number of propagation paths are small (which is the case na nb in high frequency communication links), then the resulting observation at the AP for the first stage as channel is low-rank and also ill-conditioned [27]. In addition Ya = ZXa + Na, (5) to this, inadequate antenna spacing at the terminals and small CM×Td link distance are also key factors that makes the channel where, Ya ∈ , Xa = [xa[1], xa[2],..., xa[Td]] ∈ CN×Td CM×Td ill-conditioned [25], [28]. We assume that all the channels and Na ∈ . Similarly, for the second stage follow a flat quasi-static block fading model2. We denote the of the channel estimation, where the LIS unit is switched ON, condition number of the channel matrix Z by κ(Z). Moreover, we can write the received observation at the AP as it is assumed that the channel G is highly correlated due to a Yb = H (S ⊙ GXb)+ ZXb + Nb, (6) nearly perfect LOS propagation [13], [30], [31], resulting in a M×Tr low-rank structure of G. where, Yb ∈ C , Xb = N×Tr In this study, we consider a constant amplitude and continu- [xb[Td + 1], xb[Td + 2],..., xb[Td + Tr]] ∈ C , L×Tr ous phase shift design for reflection coefficients of LIS [5]. Let S = [s[Td + 1], s[Td + 2],..., s[Td + Tr]] ∈ C and CM×Tr si ∈ (0, 1) and φi ∈ (0, 2π] denote the amplitude reflection Nb ∈ . After the channel estimation process, the coefficient (ON/OFF state) and phase shift value at the ith downlink received signal at the user is given by T T T 2There are various path loss models developed for LIS-assisted wireless r = G ΦH + Z Wu + n, (7) communications [29]. However, the path loss model does not directly influ- jφ1 jφ2 jφL ences the performance of the proposed channel estimation scheme. Therefore, where, Φ = diag s1e ,s2e ,...,s
Le is the phase for simplicity, we do not consider effects of the path loss. shift matrix. The precoding matrix at the AP is given by
4

CM×Ns 2 W ∈ , such that kWkF = Ns, where the total Note that in the RMMSE channel estimation scheme a prior number of data streams is denoted by Ns. The data vector knowledge of the channel parameters are not needed, however, CNs×1 2 is u ∈ and n ∼ CN (0, σnIN ) is an AWGN vector for only the receiver noise power is required. the user. Denoting Gˇ T , Hˇ T and ZˇT as the estimates of GT , HT and ZT , respectively, the transmit precoding matrix, W, is B. BAdVAMP based channel estimation equal to the right singular vectors of the composite estimated channel matrix Gˇ T ΦHˇ T + ZˇT (obtained via a singular After the estimation of the direct MIMO channel between value decomposition), corresponding to the Ns largest singular the AP and user, the second stage of the channel estimation
values. The user estimates the downlink channel during the process begins, where the user sends Tr training vectors to downlink training phase and uses the left singular vectors the AP. In this stage, the LIS terminal is switched ON and the of the estimated composite channel corresponding to the N received signal at the AP is given by (4). The AP removes s ˇ largest singular values, and uses them as a receive combining the direct channel part from (6) by using Z and Xb, such that CN×Ns 2 the resulting observation for all Tr training vectors at the AP matrix, V ∈ , such that kVkF = Ns. This is known as eigenmode transmission as it maximizes the signal-to-noise becomes ratio (SNR) and achieves full diversity order [32]. The received Y = H (S ⊙ GXb)+ N, (12) signal at the user can be written as where N is the noise matrix which is given by N = H T T T H ˇ ˜r = V G ΦH + Z Wu + V n. (8) Nb + ZXb − ZXb. Therefore, the entries of N follow the 2 2 2 2 CN (0, σnb + σe ) distribution, where σnb and σe denote the The achievable rate of the system is
given by noise and error variances, respectively. The phase shift design C = log det I + ρ WH ∆H VVH ∆W , (9) at the LIS unit also plays an important role during the channel 2 estimation process. For the second channel estimation stage, T T T ∆ 2

we assume that ON/OFF state values of LIS passive where ∆ = G ΦH + Z and SNR= ρ =1/σn. Estimated {si} channels are used to design the phase shift matrix, Φ. elements are independent and for the given training duration t, s[t] is a K-sparse vector, such that total K non-zero value are III. PROPOSED TWO-STAGE CHANNEL ESTIMATION in the vector s[t]. Note that the sparse nature of the matrix S in (12) is necessary for channel estimation via the BAdVAMP In this section, we present our proposed two stage channel algorithm. It is also assumed that X is a full rank matrix and estimation method to estimate the three channels: Z, H and b X 2 . G. In the first stage, the direct uplink MIMO channel from k b)kF = pu the user to the AP, given by Z, is estimated using the con- We propose to use the BAdVAMP algorithm to estimate ventional training-based MIMO channel estimation technique channels H and G from (12). The BAdVAMP algorithm is [33]. Whereas, in the second stage, the LIS channels H and known to be a computationally efficient method for estimating G are estimated at the AP using the BAdVAMP scheme the matrices H and D from the noisy signal Y = HD + N [23]. Here, we only provide the details of the uplink channel where, D = S ⊙ GXb. This is also known as the dictionary estimation at the AP via uplink training vectors from the user. learning problem in the domain of compressive sensing [23], A similar procedure can be adapted at the user to estimate the [34]. The BAdVAMP algorithm [23] was shown to be superior downlink channels HT , GT and ZT . However, we skip the to its contenders both in complexity and performance, espe- details of the downlink channel estimation process. cially when the matrices H and D are ill-conditioned. We aim to estimate the matrix D ∈ CL×Tr , from the noisy received matrix Y ∈ CM×Tr in (12). As we are A. Conventional TDD based MIMO channel estimation dealing with the scenario where L>M and D is a sparse In the first stage of the channel estimation procedure at the matrix, our channel estimation problem becomes equivalent to AP, the LIS terminal is powered off, i.e., {si} = 0, ∀i, and the estimation of sparse vectors from underdetermined linear the AP estimates the direct uplink channel Z. In this stage, Td measurements. In addition, as the matrix H is also not known, training vectors are transmitted by the user, such that Td ≥ the problem becomes a bilinear recovery problem. We can N. We adopt relaxed MMSE (RMMSE) channel estimation express the channel matrix H as an unstructured matrix [23] at the AP, presented in [33]. Using the RMMSE criterion, the M L estimated direct channel can be expressed as T H(ψH )= ψH,m,lemel , (13) −1 Zˇ Y XH X 2 I XH (10) m=1 l=1 = a a a + σna Td a , X X where, e denotes the mth basis vector and ψ ∈ CM×L. where the training matrix satisfies kX
k2 = p . Similar m H a F u The vector e will have a value one at the th location and to [33], we rely on discrete Fourier transform (DFT) matrix m m zero elsewhere. We assume that the density of D, given by design for Xa, given by pD(.; θd), is parameterized by the vector θd. More specifically, th 1 1 ... 1 the n value of the vector θd represents the mean and variance − j2π j2π(Td 1) th T T values of the n column of D. In order to estimate H and D, p 1 e d ... e d X u   (11) the BAdVAMP aims to learn the parameters Θ , {ψ , θ } a = . .. . H d TdN 1 . . . from Y. In this study, it is assumed that the statistical r  − −   j2π(N−1) j2π(N 1)(Td 1)   e Td e Td  information θd is known at the AP and we focus on recovering 1 ...    5

ψH and D from the noisy measurement matrix Y in (12), Algorithm 1 Bilinear Adaptive VAMP (BAdVAMP) which can be rewritten as 1: Initialization: 0 0 0 ∀ : r1,t,γ1,t, θd, ψH Y = H(ψH )D + N. (14) 2: for i =0,...,Tmax do 1) Decoupling the Estimation Problem: To compute the 3: for τ1 =0,...,τ1,max do i i i parameters in Θ that maximize the received signal distribution, 4: ∀t : d1,t ← g1(r1,t,γ1,t, θd) i ′ i i i the BAdVAMP algorithm uses maximum likelihood (ML) 5: ∀t : 1/η1,t ← hg1(r1,t,γ1,t, θd)i/γ1,t i 1 i i 2 i estimation given by 6: ∀t : 1/γ1,t ← N kd1,t − r1,tk +1/η1,t 7: end for Θˆ ML = arg max pY(Y; Θ). (15) i i i Θ 8: ∀t : γ2,t = η1,t − γ1,t i i i i i i Based on these estimated parameters, the MMSE estimate of 9: ∀t : r2,t = (η1,td1,t − γ1,tr1,t)/γ2,t D can be expressed as [23] 10: for τ2 =0,...,τ2,max do i i i T i −1 11: ∀t : Ct ← (γ2,tIN + H(ψH ) H(ψH )) ˆ E ˆ i i i i i T DMMSE = D|Y; ΘML . (16) 12: ∀t : d2,t ← Cl(γ2,tr2,t + H(ψH ) yt) i i h i 13: ∀t : 1/η2,t ← tr{Ct}/N The BAdVAMP algorithm approximates the quantities in (15) i Tr i ˆ ˆ 14: C ← t=1 Ct and (16), i.e., ΘML and DMMSE, respectively. To find the T T 15: H(ψi ) ← YDi (Ci + Di Di )−1 expected value in (16), the mean of the posterior density HP 2 2 2 ˆ 16: end for pD|Y(D|Y; ΘML) is required. Using Bayes’ rule we can write t+1 t 17: H(ψH )= H(ψH ) the posterior density as i+1 i i 18: ∀t : γ1,t = (η2,t − γ2,t) i i p (Y|D; Θˆ ML)pD(D; Θˆ ML) +1 i i i i +1 ˆ Y|D 19: ∀t : r1,t = (η2,td2,t − γ2,tr2,t)/γ1,t pD|Y(D|Y; ΘML)= (17) pY(Y; Θˆ ML) 20: end for ˆ where pY|D(Y|D; ΘML) is the likelihood function of (16), pD(D; Θˆ ML) denotes the prior density and the likelihood Θ is the EM quantity. These quantities are computed and function of (15) is represented by pY(Y; Θˆ ML). Following the are updated iteratively by using Algorithm 1. In Algorithm decoupling of the posterior density (17) across the columns of 1, g1(r1,t,γ1,t; θd) represents a denoising function given by ˆ Tr D, as in [23], we can write pD(D; ΘML)= t=1 pd(dt; θd) Tr dtpd(dt; θd)ξ(dt)ddt and p (Y|D; Θˆ ) = p (y |d ; Θˆ ). Now, we g (r ,t,γ ,t; θd) , , (20) Y|D ML t=1 y|d t t QML 1 1 1 p (d ; θ )ξ(d )dd can express (17) as R d t d t t Q ξ d d r I Tr where ( t) ∼ CN ( t;R1,t, /γ1,t) and t represents the column of the matrix D. In Algorithm 1, the pD Y(D|Y; Θˆ ML)∝ pd(dt; Θˆ ML)py d(yt|dt; Θˆ ML) (18) | | ′ t=1 quantity hg1(r1,t,γ1,t, θd)i represents the divergence of Y g (r ,γ ; θ ) at r . In general, g (r ,γ ; θ ) can be Using this decoupled posterior density, the BAdVAMP algo- 1 1,t 1,t d 1,t 1 1,t 1,t d interpreted as denoising of the AWGN-corrupted pseudo- rithm uses a vector AMP (VAMP) technique [35] to find the measurement r = d + f , where f ∼ CN (0, I/γ ), matrix D from (14), where each column of D is tracked inde- 1,t 1,t 1,t 1,t 1,t which corresponds to the first loop i.e., τ in Algorithm 1, pendently. However, the VAMP algorithm requires complete 1 from line 3 till line 7. On the other hand, the second loop τ information of Θ to estimate D. From (15) and (17), the ML 2 performs MMSE estimation of d from the AWGN-corrupted estimate of Θ can be computed as 2,t measurements under the pseudo-prior d2,t ∼ CN (r2,t, I/γ2,t). ˆ Lastly, the estimate of H(ψt ) is given in lines 10-16. The ΘML = arg min − ln pD(D; Θ)pY|D(Y|D; Θ) dD. (19) H Θ final updated estimate of H using Algorithm 1 is given by Z The BAdVAMP algorithm uses the expectation maximization T T Hˆ = H(ψi )= YDi (Ci + Di Di )−1, (21) (EM) approach [36] to solve the problem (19) in an iterative H 2 2 2 manner. Therefore, the BAdVAMP algorithm interleaves the where Di di di di and . On the 2 = 2,1, 2,2,..., 2,Tr i = Tmax EM and VAMP algorithms to estimate H and D from Y. other hand, the estimate of Dˆ is given by 2) The BAdVAMP Algorithm: Here, we discuss the BAd-   Dˆ = Di = di , di ,..., di , (22) VAMP algorithm [23] to estimate H and D. As discussed 2 2,1 2,2 2,Tr earlier, the BAdVAMP algorithm employs the EM approach where i = Tmax. The BAdVAMP algorithm presented in to estimate ψH and the VAMP [37] algorithm to estimate Algorithm 1 recovers Hˆ and Dˆ up-to certain phase, scalar and 3 D. In particular, the BAdVAMP method performs a joint permutation ambiguities. For a unique solution, it is important estimation of parameters using the optimization problem based to remove the ambiguities from Hˆ and Dˆ . on the Gibbs free energy [23, Eq. (38)]. We can divide the BAdVAMP quantities into two groups: VAMP and EM IV. REMOVING PERMUTATION AMBIGUITY AND ˆ quantities. The VAMP quantities are {r1,γ1, r2,γ2, d, η} and RECOVERY OF G

3Here, we present the brief overview of the BAdVAMP algorithm. We refer The estimated channel matrices obtained via the BAd- the reader to [23] for a detailed description of the algorithm. VAMP algorithm contain ambiguities, which can be classified 6 into three main categories: permutation, scalar and phase Algorithm 2 NIHT based method to recover G ambiguities. It is not possible to avoid inherent scalar and 1: Input: rg, Dˇ , S and Xb phase ambiguities for the studied system model. However, by 2: Set: F0 = 0, j = 0 and U0 (top r left singular vectors leveraging the sparsity in D, the permutation ambiguity can be g of F0) removed from the estimated matrices Hˆ and Dˆ . Here, we also discuss a low-rank matrix completion approach to recover G 3: Repeat j j jH after removing the permutation ambiguity from Dˆ . We begin 4: Set PU = U U j ˇ j 2 ˆ ˆ kPU S⊙(D−F )kF by explaining the ambiguity issue that if H and D represent 5: j Compute α = j ˇ j 2 kS⊙(PU S⊙(D−F ))k2 the estimated channel matrices obtained via the BAdVAMP j 6: Q Fj j P S Dˇ Fj algorithm, then HΣΓˆ and ΓT Σ−1Dˆ are also valid solutions Set = + α U ⊙ − 7: Set Fj+1 = H (Q) for any arbitrary permutation matrix Γ and diagonal matrix rg
j+1 j+1 Σ, where Σ comprises of scalar and phase ambiguities. 8: Set U to top rg left singular vectors of F 9: j = j +1 A. Removing Permutation Ambiguity 10: Until j > max_iter ˇ j H −1 The design of phase shifters S during the training period is 11: Output G = F (XbXb ) Xb crucial for removing the permutation ambiguity that is induced in the estimated channels. As each column of S is a K−sparse vector, we define a state matrix for the matrix Dˆ as In Algorithm 2, the operation Hrg (·) restricts the rank of the input to the r by performing SVD on the input and keeping 1, Dˆ 6=0 g S¯ =∆ l,t (23) only r singular values in the diagonal singular value matrix, l,t ˆ g (0, Dl,t =0 while replacing the remaining singular values with zero. From (23), the relationship between S¯ and S, can be expressed Now we have estimated the LIS channels and the direct ˇ ˇ ˇ as S¯ = ΓS. This relationship shows that the nth row of channel which are {H, G} and Z, respectively. Note that the state matrix, denoted by ¯sT , represents the n′ row of S scalar and phase ambiguities are still present in the LIS es- n ′ T th timated channels. However, the precoding/combining strategy given by sn′ . This suggests that the (n,n ) element of the permutation matrix Γ must be equal to one, i.e., Γ ′ = 1. and LIS phase shift design considered in this study depend on n,n ˇ ˇ ˇ Using this fact, we can compute the nth row of Γ with a non- the composite channel given by HΦG + Z. Therefore, there zero value at the location nˆ, where is no need to eliminate scalar and phase ambiguities. To give more insight into this, let us assume that Hˇ ′ and Gˇ ′ denote the T nˆ = argmax sn′ ¯s . (24) estimated channels with perfect elimination of the phase and ′ n n scalar ambiguities, then we can write HΦˇ Gˇ +Zˇ = Hˇ ′ΦGˇ ′+Zˇ
This approach yields the permutation ambiguity matrix Γ [13]. Therefore, it is sufficient to have only permutation which helps in removing the permutation ambiguity from ambiguity removed from the estimated channels to design the the BAdVAMP estimated channels, such that the recovered LIS phase shifts and MIMO precoding/combining vectors. channel matrices are given by Dˇ = ΓT Dˆ and Hˇ = HΓˆ . V. OPTIMIZATION OF LIS PHASE SHIFTS B. Estimation of G via Matrix Completion Phase shifts of all passive elements in the LIS play an im- To extract the estimate of the channel matrix G from the portant role in determining the performance of the LIS assisted permutation ambiguity removed matrix Dˇ , similar to [13], we MIMO systems. In particular, phase shifts of passive elements use a low-rank matrix completion approach. The main idea in in the LIS can be adjusted according to the propagation the low-rank matrix completion approach is to recover a low- environment such that the reflected signals add up coherently rank matrix from the set of linear measurements. In this study, at the user with the signals arriving from other paths. This we use a well-known normalized iterative hard thresholding adjustment helps improve the total channel gain or SNR of the (NIHT) [38] algorithm to recover G from the matrix Dˇ . The received signal at the user. For this purpose, in this section, NIHT algorithm works on the principle of projection based we propose a phase shift design at the LIS that maximizes the gradient descent. gain of the estimated composite downlink channel given by In a matrix completion approach, if a recoverable matrix of ˇ T ˇ T ˇ T 2 kG ΦH + Z kF . Due to the fixed precoding/beamforming size L × N has a rank of r, then r(L + N − r) measurements strategy at the AP, we have selected the total channel gain as may be sufficient for recovery [38]. In our problem, we want an optimization objective. To find the optimal phase shifts, Φ⋆, ˇ to recover G from the estimated matrix D. Therefore, we can we formulate an optimization problem which can be expressed write the matrix completion problem as as 1 ˇ 2 ˇ T ˇ T ˇ T 2 min kD − (S ⊙ F) kF (25) max G ΦH + Z (27) F 2 Φ F s.t. rank(F)= r . (26) s. t. Φ = ejφl , l =1,...,L. (28) g l,l where rg denotes the rank of the matrix G. The NIHT based Note that the optimization of phase shifts is performed at the method to find the estimate of G is presented in Algorithm 2. AP, which is capable of managing the phase shifts at the LIS. 7

The problem (27) is non-convex optimization problem since VI. NUMERICAL RESULTS ˇ T ˇ T ˇ T 2 kG ΦH +Z kF can be shown to be a non-concave function In this section, we evaluate the performance of our pro- over Φ [39]. Therefore, in order to solve the optimization prob- posed two-stage channel estimation technique for the LIS lem, we rely on its approximation. For notational simplicity, assisted MIMO system. First, the performance of the proposed ˜ ˇ T ˜ ˇ T ˜ ˇ T let us assume that G = G , H = H and Z = Z . Let BAdVAMP based channel estimation algorithm is compared ˜ ˜ ˜ ˜ ˜ ˜ 2 A = GΦH + Z, now we can write kGΦH + ZkF in (27) as with its counterpart the BiGAMP algorithm [13], in terms of N M normalized mean squared error (NMSE). Later, we present 2 (l) 2 kAkF = |ai,j | , (29) the achievable rate performance of the LIS assisted MIMO i=1 j=1 communication system using the optimized LIS phase shift X X where design. L 2 2 MIMO channel parameters: We use narrowband geomet- l ( ) jφl ˜ ai,j = g˜i,le hl,j +˜zi,j . (30) ric channel model (1) to obtain the MIMO channels H, G

l=1 and Z. The number of propagation paths in Z and H are set X We can rewrite (30) as to LZ = LH = 64. Due to highly correlated propagation conditions, the number of propagation paths in G is set to L ∗ L 2 l LG =8 with rg = rank(G)=8. Similar to [23], the singular a( ) = g˜ ejφl h˜ +˜z g˜ ejφl h˜ +˜z i,j i,l l,j i,j i,l l,j i,j values of H are selected to obtain a desired condition number l ! l ! X=1 X=1 κ(H) while ensuring that E kHk2 = LM. In (1), angles L ∗ L F of departure and arrival in both azimuth and elevation are = g˜ ejφl h˜ g˜ ejφl h˜ + |z˜ |2+ i,l l,j i,l l,j i,j distributed with the Laplacian distribution [26]. For simplicity, l=1 ! l=1 ! X L X the angular spreads (standard deviations) of these angles are 0 ∗ jφl assumed equal and set to 10 . Similar to H, we also ensure 2 Re z˜ g˜i,le h˜l,j . (31) i,j E G 2 E Z 2 ( l=1 !) that k kF = NL and k kF = NM. Throughout X this section, the inter-element spacing at the user is assumed The first term in (31) can be further expressed as to be half-wavelength,  whereas, we consider two cases of inter- L ∗ L element spacing at the AP with d = λ/2 and d =4λ. jφl ˜ jφl ˜ g˜i,le hl,j g˜i,le hl,j = (32) BAdVAMP initialization: For initialization of the BAd- l=1 ! l=1 ! 0 0 VAMP algorithm, we draw r1,tr and ψH from the i.i.d. LX X L L 2 2 CN (0, 10) and CN (0, 1) distributions, respectively. The initial 2 jφl ˜ ∗ −jφl ˜∗ jφq ˜ |g˜i,l| e hl,j + g˜i,le hl,j g˜i,qe hl,j 0 −3 value of γ1,tr is set to 10 and it is assumed that the statistical l=1 l=1 q6=l   distribution of the matrix D, i.e., θd, is known at the AP. X X X We note that the first term in (32) has no phase informa- Inner EM iterations are set to τ1,max = 1 and τ2,max = 0. jφ 2 tion because of the fact that |e l | = 1. On the other The maximum number of iterations, Tmax, for both BAdVAMP hand, the second term which represents the cross terms in and BiGAMP schemes is fixed to 300, where the maximum L ∗ jφl ˜ (32) has negligible value as 2Re{z˜i,j( l=1 g˜i,le hl,j )} >> number of restarts in BAdVAMP is set to 10. L L (˜g∗ e−jφl h˜∗ g˜ ejφq h˜ ). Thus, we ignore the l=1 q6=l i,l l,j i,q l,j P first term in (31) and other terms which are independent of A. BAdVAMP versus BiGAMP channel estimation phaseP P shifts and we can rewrite our approximated total φ In this subsection, various experiments are carried out to channel gain maximization optimization problem as compare the performance of the proposed BAdVAMP based N M L channel estimation scheme with the BiGAMP channel es- ∗ jφl ˜ max 2Re z˜i,j g˜i,le hl,j . (33) timation scheme [13]. The NMSE performance comparison φl i=1 j=1 ( l !) X X X=1 presented here is for the LIS channels H and G only. For the ˆ From (33), we can obtain the phase of the lth element in the estimated matrix H (before removing the ambiguities), the ⋆ ⋆ NMSE is given by LIS, given by φl = Φl,l, that maximizes the approximate version of the total channel gain as kH − HPˆ k2 NMSE Hˇ = F , (35) N M ˜ kHk2 Im i j z˜i,j g˜i,lhl,j F ⋆ −1 =1 =1
φl = tan . (34) where P denotes the generalized permutation matrix which  nPN PM ˜ o  Re i=1 j=1 z˜i,j g˜i,lhl,j perfectly removes the permutation, scalar and phase ambigui-   Since the fixed eigenmodenP transmitP precodingo and receive ties. This perfect elimination of ambiguities is only considered combining scheme is considered in this study, after obtaining in this subsection for comparison purposes. Φ⋆ using (34), we compute the precoding/combining vectors In Fig. 2, we plot NMSE results of the LIS estimated using the composite channel matrix given by GΦ˜ ⋆H˜ + Z˜. It channels H and G against various values of SNR for both is also possible to perform a joint optimization of downlink BAdVAMP and BiGAMP based channel estimation schemes. precoding and LIS phase shift, however, the investigation of Here, we set the total number of transmit antennas and receive the joint optimization is out of the scope of this study. antennas to M = N = 64. The number of passive elements in LIS is L = 64 and the length of the training sequences is 8

5 0

0 -5

-5 -10

-10

-15

-15 NMSE (dB) NMSE (dB)

-20 -20

-25 -25

-30 -30 -5 0 5 10 15 -5 0 5 10 15 SNR (dB) SNR (dB)

Fig. 2: NMSE performance of estimated H and G against Fig. 3: NMSE performance of estimated H and G against SNR values with d = {λ/2, 4λ} and M = N = L = 64. SNR values with d = {λ/2, 4λ} and M = N = L = 36.

-2 2 2 Tr = 500. The SNR is defined as (1/σn), where σn is the noise variance. The entries of the training matrix Xb follow -4 the CN (0, 1) distribution and then it is normalized, such that -6 2 kXbkF = pu. The sparsity rate K/N of the matrix S during the training stage is set to 0.1, meaning that each column of -8 S consists of K non-zero entries with K = ⌈0.1N⌉, where -10 ⌈·⌉ represents the ceiling function. The condition number of NMSE (dB) the LIS channel matrix H is set to κ(H) = 100. In Fig. 2, -12 two cases of inter-element spacing at the AP is considered: -14 d = λ/2 and d = 4λ. Fig. 2 indicates that the proposed BAdVAMP based channel estimation technique estimates the -16 LIS channels H and G with a higher quality as compared to -18 the BiGAMP algorithm [13], especially at moderate to high 40 80 120 160 200 240 SNR regimes. For the channel matrix H, it can be seen that the NMSE performance gap between the two algorithms widens Fig. 4: NMSE performance of estimated H and G against as the SNR increases. This higher quality estimation of the κ(H) values with M = N = L = 64. LIS channels is due to the fact that the VAMP part of the BAdVAMP algorithm is robust to ill-conditioned matrices as compared to the AMP based BiGAMP algorithm [23]. Further, more ill-conditioned. Fig. 4 indicates the robustness of the we note that the performance of both the schemes is similar proposed BAdVAMP scheme over the BiGAMP scheme when for the estimated LIS channel G. learning ill-conditioned channels. Since both the schemes suf- A similar trend can be seen in Fig. 3 which demonstrates fer from higher ill-conditioning, we note that the performance the NMSE performance against various SNR values with M = degradation of the BAdVAMP scheme is with a slower rate N = L = 36. Here, the rest of the simulation parameters are than the BiGAMP scheme. the same as used in Fig. 2. The spatial correlation in H with d = λ/2 is higher than the case when d = 4λ. Both Fig. 2 B. Achievable rate with BAdVAMP and optimized Φ and Fig. 3 indicate that the NMSE performance improves when Here, we evaluate the achievable rate performance of the the channel matrix is spatially less correlated i.e., for d =4λ LIS assisted MIMO system using the proposed LIS phase shift case. However, in both the cases, BAdVAMP is preferable as design presented in Section V. For the rest of the results pre- it yields the estimated channels with higher quality than its sented in this section, we fix Ns =2, inter-element spacing at counterpart BiGAMP. the AP to d = λ/2 and pu =1. The value of the training length To examine the performance of the proposed scheme against in the first stage of the estimation process is set to 64, i.e., 2 2 2 different conditioning levels of the channel matrix H, we plot Td = 64 and σe = σnM/(pu +σnM). We do not assume that NMSE performance against various κ(H) values in Fig. 4. the ambiguities are eliminated perfectly but it is considered In this figure, we set M = N = L = 64, SNR= 10dB and here that only the permutation ambiguity is removed from the d = λ/2. To induce conditioning in H, we vary the condition estimated channels using the framework provided in Section number of the channel matrix H, i.e., κ(H), from 40 to 240. IV. Achievable rates of two different configurations are plotted Note that as κ(H) increases, the channel matrix H becomes in in Fig. 5, where the cases considered are: i) M = N = 36 9

35 40 Perfect CSI, M = 36 Perfect CSI BAdVAMP, M = 36 BAdVAMP BiGAMP, M = 36 35 30 Random Random, M = 36 No LIS Perfect CSI, M = 16 BAdVAMP, M = 16 30 25 BiGAMP, M = 16 Random, M = 16 25

20

20

15 Achievable rate (bps/Hz) Achievable rate (bps/Hz) 15

10 10

5 5 -5 0 5 10 15 20 64 100 144 196 256 SNR (dB) Number of Elements in LIS, L

Fig. 5: Achievable rate versus SNR performance with M = Fig. 7: Achievable rate versus the number of LIS elements N = {36, 16}, L = 36 and κ(H) = 160. performance with M = N = 36 and SNR= 10 dB.

35 at M = 36 and M = 16 are comparable. The reason for Perfect CSI BAdVAMP this trend is that we only use Ns =2 data streams, whereas, BiGAMP increasing the number of data streams will make the rate gap Random 30 No LIS more apparent for these two antenna configurations. Next, we evaluate the achievable rate performance against the training length, Tr in Fig. 6, where we set M = N = 25 L = 64, SNR = 10 dB and κ(H) = 160. Similar to the previous figures, we compare the achievable rate results with perfect CSI, BiGAMP and random schemes. However Achievable rate (bps/Hz) LIS Gain 20 to gain more insight into the usefulness of LIS, here we also include a case where no LIS is present in the surrounding area. It can be seen in Fig. 6, that initially the achievable

15 rate increases with the increase in the training length for both 200 400 600 800 1000 Training Length, T r BAdVAMP and BiGAMP schemes, however, at Tr = 600, the rate performance converges and does not increases with Tr. Fig. 6: Achievable rate versus SNR performance with M = This suggests that the training length of 600 is the optimal N = L = 64, SNR= 10 dB and κ(H) = 160. choice for the given network settings. Fig. 6 also indicates that the proposed scheme performs better than the case when no LIS terminal is available to assist the MIMO communication. and ii) M = N = 16. Here, we use L = 36, Tr = 800 In fact, the rate performance with no LIS is even lower than and κ(H) = 160. Fig. 5 demonstrates the achievable rate, the random (with LIS) case. This trend supports the idea given by (9), for the proposed channel estimation and optimal of deploying the LIS terminal. Finally in Fig. 7, we show phase shift design, labeled as ‘BAdVAMP’. We compare it the achievable rate performance by varying the number of with the scenario ‘Perfect CSI’, where perfect CSI is assumed passive elements, L, in the LIS from 64 to 256. Here, we to be available at the AP and the associated LIS phase shift set M = N = 36, SNR= 10 dB and κ(H) = 100. It can be design is based on exhaustive search in problem (27). For observed from Fig. 7 that the rate improves by increasing the a comparison, we also include a scenario ‘Random’, where number of passive elements in the LIS. random precoding/combining scheme and random LIS phase shifts are used. Finally, we also plot the achievable rate results with ‘BiGAMP’ based channel estimation [13] using our VII. CONCLUSIONS proposed LIS phase shift design as the phase shift optimization In this paper, we have proposed a two stage channel was not investigated in [13]. Fig. 5 indicates that the proposed estimation method for the LIS assisted MIMO communication BAdVAMP scheme achieves better rate performance than the system, where the direct channel between the AP and user is BiGAMP and random schemes. For example, at SNR= 20 estimated in the first stage and the BAdVAMP algorithm is dB, the achievable rate gap between the proposed scheme and used to estimate LIS channels in the second stage. We also the BiGAMP scheme is nearly equal to 1.65 bps/Hz. More presented a phase shift design for the LIS which approximately importantly, it can be seen that the rate performance with maximizes the channel gain at the user. Through numerical the proposed phase shift design is close to the perfect CSI simulations, we show the effectiveness of the proposed channel case. Further, we note that the achievable rate performance estimation method and phase shift design. 10

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