Reconfigurable-intelligent-surface-assisted Downlink Transmission Design via Bayesian Optimization
11 Reconfigurable-intelligent-surface-assistedDownlink Transmission Design via BayesianOptimization
Dong Wang, Xiaodong Wang,
Fellow, IEEE , Fanggang Wang,
SeniorMember, IEEE
Abstract
This paper investigates the transmission design in the reconfigurable-intelligent-surface (RIS)-assisteddownlink system. The channel state information (CSI) is usually difficult to be estimated at the basestation (BS) when the RIS is not equipped with radio frequency chains. In this paper, we proposea downlink transmission framework with unknown CSI via Bayesian optimization. Since the CSI isnot available at the BS, we treat the unknown objective function as the black-box function and takethe beamformer, the phase shift, and the receiving filter as the input. Then the objective function isdecomposed as the sum of low-dimension subfunctions to reduce the complexity. By re-expressing thepower constraint of the BS in spherical coordinates, the original constraint problem is converted intoan equivalent unconstrained problem. The users estimate the sum MSE of the training symbols as theobjective value and feed it back to the BS. We assume a Gaussian prior of the feedback samples andthe next query point is updated by minimizing the constructed acquisition function. Furthermore, thisframework can also be applied to the power transfer system and fairness problems. Simulation resultsvalidate the effectiveness of the proposed transmission scheme in the downlink data transmission andpower transfer.
Index Terms
Bayesian optimization, black-box function, intelligent reflecting surface, unknown CSI.
D. Wang and F. Wang are with the State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong University,Beijing 100044, China (e-mail: { } @bjtu.edu.cn).X. Wang is with the Department of Electrical Engineering, Columbia University, New York, NY 10027 USA (e-mail:[email protected]). a r X i v : . [ c s . I T ] F e b I. I
NTRODUCTION
The deployment of the massive multiple-input multiple-output (MIMO), millimeter waveare expected to achieve the high data rate and massive device connections in the wirelessnetworks [1]. However, the dense deployments of multi-antenna base stations (BSs) and accesspoints (APs) have raised serious concerns on their energy consumption, which the wirelessnetwork operators struggle to support millions of users, especially the users in the unfavorablepropagation environments [2]. The reconfigurable intelligent surface (RIS) is regarded as apromising technology in the future sixth-generation (6G) mobile communications network due toits passive and low-cost characteristics [3]–[6]. A RIS is a meta-surface equipped with integratedelectronic circuits that can be programmed to alter an incoming electromagnetic field in acustomizable way. By suitably adjusting the phase shift of each reflecting element, the RISconsumes much less energy than a regular amplify-and-forward relay [7]. Furthermore, the RIScan be flexibly deployed at the building facades, indoor walls, and ceilings, and applied to variouscommunication scenarios such as the cellular networks, the wireless power transfer networks,and the unmanned aerial vehicle networks, etc.The existing works on the RIS-assisted communication system mainly focus on cellular scenar-ios. The authors in [8] investigated the point-to-point multi-input single-output (MISO) wirelesssystem and the total received signal power is maximized based on the semidefinite relaxation(SDR). In order to reduce the high computational complexity incurred by SDR, the authorsin [9] proposed low complexity algorithms by majorization-minimization (MM) and manifoldoptimization. In [10], the authors analyzed the practical model of the reflection coefficient andapplied it to the RIS-assisted wideband orthogonal frequency-division multiplexing (OFDM)system. The authors in [11] maximized the achievable rate in the RIS-assisted OFDM system.The authors in [12] proposed the alternating optimization algorithm and the two-stage algorithmto minimize the total transmit power at the AP in the multiple-user MISO system, while in[13], the authors developed the energy-efficient designs based on the popular gradient descentsearch and the sequential fractional programming. A simplified semidefinite programming-basedreflecting scheme and a simplified maximizing the minimum Euclidean distance precoding designwere studied in the RIS-assisted MIMO system. Moreover, the authors in [14] proposed twotransmission schemes to maximize the sum rate based on the MM in the multigroup multicastcommunication systems.
The aforementioned transmission designs assume the channel state information (CSI) is avail-able at the BS and RIS, which means that the channel estimation is required before the beamform-ing and the phase shift design. The channel estimation methods in the RIS-assisted system canbe categorized into two types. One is that the radio frequency (RF) chains are assembled in theRIS for receiving the training pilots at each element. The authors in [15] presented an alternatingoptimization approach for explicit estimation of the channel gains at the RIS elements attachedto the single RF chain. Another approach was to select the RIS reflection matrix from quantizedcodebooks via beam training, but the complexity and the overhead increase rapidly when thenumber of the elements goes larger [16]. However, the equipped RF-chains of the RIS cause hugeenergy consumption and hardware cost. The authors in [17] adopted the spatial modulation toreduce the number of RF-chains. The other type is that the cascaded channel estimation withoutequipping RF-chains at the RIS. In [18], the authors switched the RIS elements ON one-by-oneand estimated the RIS-assisted channels at the BS based on received pilot symbols from theusers. The BS-RIS-user cascaded channel was estimated based on a two-stage algorithm thatincludes a sparse matrix factorization stage and a matrix completion stage [19]. A two-timescalechannel estimation framework was proposed in [20] to exploit the property of the BS-RIS andRIS-user channels. Moreover, the authors in [21] proposed a matrix-calibration-based cascadedchannel estimation for the RIS-assisted multi-user MIMO. The authors in [22] proposed anadaptive transmission protocol in conjunction with a progressive channel estimation method forthe wideband OFDM system. It is worth noting that the cascade channel estimation is usuallyadopted in the uplink system. The channel estimation for the downlink system is difficult tooperate, which remains for future research. Although the transmission design without the CSIhas been studied in the wireless power transfer and the RF identification scenarios [23], [24],to the best our knowledge, there is no literature that investigated the transmission design in thedownlink RIS-assisted system without the CSI. Therefore, it is crucial to design a downlinktransmission scheme without the CSI and keep the RIS passive or nearly passive.In this paper, we propose a transmission framework with unknown CSI for downlink RIS-assisted system via Bayesian optimization. Bayesian optimization is a sequential design strategyfor global optimization of black-box functions, which has emerged as a powerful solution forthese varied design problems such as interactive user interface, information extraction, automaticmachine learning, and sensor network, etc [25]–[27]. The Bayesian optimization adds a Bayesianmethodology to the iterative optimizer paradigm by incorporating a prior model on the space of possible target functions. A simpler surrogate model of the objective function which is cheaperto evaluate and will be used instead to solve the optimization problem. The main contributionsof our work are summarized as follows: • To the best of our knowledge, this paper is the first attempt to focus on the downlink trans-mission scheme without the channel estimation. Motivated by the Bayesian optimization,we formulate the optimization problems to minimize the mean square error (MSE) in thedata transmission. Since the CSI of the BS-RIS and the RIS-users are not available at the BSand the RIS, this optimization cannot be solved analytically. By introducing the Bayesianoptimization, the sum MSE can be regarded as the objective function. The input of theobjective function is the beamformer at the BS, the phase shift at the RIS, and the receiverfilter at the users. Then, we reformulate the original optimization problem to an unconstraintproblem by using the spherical coordinates. It is worth noting that the dimension of theinput vector is high when the number of antennas of the BS and the number of elementsof the RIS is large. We deal with this challenge by treating the objective function additivefunction of mutually exclusive lower-dimensional components [28]. • We propose a framework of the transmission scheme in the RIS-assisted downlink system.The proposed downlink data transmission scheme can be easily extended to the MIMOsystem, the multi-user MIMO system. In addition, by considering the fairness among theusers, the maximum of individual MSE can be minimized by the Bayesian optimization.Furthermore, this setup of our work can also apply to the wireless power transfer systemby choosing the received signal strength as the objective function. • Simulation results show that the proposed downlink data transmission scheme based onthe Bayesian optimization can effectively reduce the sum MSE. For the downlink wirelesspower transfer, the proposed transmission scheme can improve the received power andachieve acceptable performance compared with the known CSI scheme. We also validatethe performance by considering the fairness of users. Moreover, the proposed transmissionscheme is robust to the slow fading channel, where the updated variable is dynamicallylearned by each feedback.The remainder of the paper is organized as follows: Section II introduce the backgroundof the Bayesian optimization. Section III introduces the system model and problem formulationin the downlink data transmission system and the wireless power transfer system, respectively.
The proposed downlink transmission scheme via Bayesian optimization is studied in Section IV.Then, Section V provides simulation results. Finally, we conclude this paper in Section VI.
Notation : Boldface lowercase and uppercase letters denote vectors and matrices, respec-tively. The operators tr {·} , vec ( · ) , ( · ) T , ( · ) ∗ , ( · ) H , ( · ) − , (cid:60) ( · ) , and (cid:61) ( · ) stand for the trace, thevectorization, the transpose, the conjugate, the Hermitian, the inverse, the real part, and theimaginary part of their arguments, respectively. For any vector x , diag { x } returns a diagonalsquare matrix whose diagonal consists of the elements of x . For any matrix X , [ X ] diag representsa diagonal matrix with the same diagonal elements of X . The operation ⊗ denotes the Kroneckerproduct. I N denotes an identity matrix of size N × N .II. B ACKGROUND OF B AYESIAN O PTIMIZATION
Consider the following problem of finding a global minimizer (or maximizer) of an unknownobjective function x ∗ = arg min x ∈X f ( x ) , (1)where X is the compact set of the domain, i.e., a hyper-rectangle X = { x ∈ R D : a i ≤ x i ≤ b i } .We can only query at some x i ∈ X and obtain the corresponding observation f ( x i ) . Note that f may be nonconvex and the gradient information is not available. Since the objective functionis unknown, the Bayesian strategy is to treat it as a random function and place a prior overit. The prior captures beliefs about the behavior of the function. After gathering the functionevaluations, which are treated as data, the prior is updated to form the posterior distribution ofthe objective function, which in turn, is used to construct an acquisition function that determinesthe next query point.In particular, the widely used Gaussian prior is adopted, i.e., f ( x t ) = [ f ( x ) , f ( x ) , . . . , f ( x t )] T ∼N ( , K t ) , where K t is the t × t kernel matrix with K t ( i, j ) = k ( (cid:107) x i − x j (cid:107) ) , i, j = 1 , ..., t. Twopopular kernels are the squared exponential (SE) kernel and the M`atern kernel, given respectivelyby k SE ( a ) = exp (cid:18) − a h (cid:19) , (2) k M`atern ( a ) = 2 − ν Γ( ν ) (cid:32) √ νah (cid:33) ν B ν (cid:32) √ νah (cid:33) , (3)where Γ( · ) and B ν ( · ) are the Gamma function and the ν -th order Bessel function, respectively; h is a hyper-parameter. In Bayesian optimization, at the t -th iteration, we have samples D t = { ( x i , f ( x i )) } ti =1 and we are interested in inferring the value of f ( x t +1 ) at the next query point x t +1 . By theGaussian prior assumption, f ( x t ) f ( x t +1 ) ∼ N , K t k t +1 k T t +1 k (0) (4)with k t +1 = [ k ( (cid:107) x t +1 − x (cid:107) ) , k ( (cid:107) x t +1 − x (cid:107) ) , . . . , k ( (cid:107) x t +1 − x t (cid:107) )] T , and the property of Gaussiandistribution, we can write f ( x t +1 ) |D t ∼ N ( µ t ( x t +1 ) , σ t ( x t +1 )) , (5)where µ t ( x t +1 ) = k T t +1 K − t f ( x t ) , (6) σ t ( x t +1 ) = k (0) − k T t +1 K − t k t +1 . (7)Note that the computation complexity of (6) and (7) is high since the dimensions of the matrixand vectors grow with t . In order to reduce the complexity, we can set a window size W and compute µ t ( x t +1 ) and σ t ( x t +1 ) based on { ( x i , f ( x i )) } ti = t − W +1 instead of { ( x i , f ( x i )) } ti =1 .Define the W × vector ¯ k t +1 = [ k ( (cid:107) x t +1 − x t − W +1 (cid:107) ) , k ( (cid:107) x t +1 − x t − W +2 (cid:107) ) , . . . , k ( (cid:107) x t +1 − x t (cid:107) )] T and the W × W kernel matrix ¯ K t such that ¯ K t ( i, j ) = k ( (cid:107) x t − W + i − x t − W + j (cid:107) ) , i, j = 1 , . . . , W. Then we can replace (6)-(7) by µ t ( x t +1 ) = ¯ k T t +1 ¯ K − t f ( x t − W +1: t ) , (8) σ t ( x t +1 ) = k (0) − ¯ k T t +1 ¯ K − t ¯ k t +1 . (9)Note that in (8)-(9) the matrix ¯ K t and vector ¯ k t +1 are both functions of x t +1 . Then, the posteriormean and variance in (8)-(9) are used to construct the following acquisition function ϕ t +1 ( x ) = µ t ( x ) − (cid:112) β t +1 σ t ( x ) , (10)where β t +1 is a hyper-parameter. The next query point x t +1 is then given by x t +1 = arg min x ∈X ϕ t +1 ( x ) . (11)In practice, (10) can be solved using various numerical methods, e.g., the grid search algorithmin [29]. The Bayesian optimization procedure for solving (1) is summarized in Algorithm . Algorithm 1
The Bayesian optimization algorithm for solving (1) for t = 1 , , . . . , T do Calculate µ t +1 and σ t +1 using (8) and (9); Find x t +1 = arg min x ∈X ϕ t +1 ( x |D t ) by performing the grid search algorithm; Evaluate f ( x t +1 ) ; end for Output: x ∗ = x T .III. S YSTEM D ESCRIPTION AND P ROBLEM F ORMULATIONS
In this section, we first introduce the system model, and then provide problem formulationsfor RIS-assisted downlink data transmission and power transfer, respectively.
A. System Model
The system model is shown in Fig. 1. We consider an RIS-assisted multi-user MISO system,where the BS equipped with M antennas serves K single-antenna users with the help of an N -element RIS. There is no direct link between the BS and users. The RIS is usually installed on asurrounding wall to assist the transmissions between the BS and the users. Each element of theRIS is configurable and programmable via an RIS controller. Moreover, the RIS is connected toa smart controller, which can be used to exchange the information with the BS and the users.The transmitted signal at the BS is given by x = W s , (12)where W = [ w , w , . . . , w M ] T ∈ C M × K is the linear precoding matrix at the BS; s =[ s , s , . . . , s K ] T contains independent and identically distributed (i.i.d.) transmitted data symbolsto users each with zero mean and unit variance. The transmission power constraint is given by tr { W H W } ≤ P . Then, the received signal at the RIS is written as y = Hx , (13)where H ∈ C N × M is the channel matrix from the BS to the RIS. The RIS is a reflection deviceand it effectively applies a phase shift to each element of the received signal y and then forwardsit to the users. The received signal at users is then given by r = [ r , r , . . . , r K ] T = F ΦHW s + u , (14) RIS
User 1BS User K Fig. 1. The RIS-assisted multiuser MISO system. where F = [ f , f , . . . , f K ] T ∈ C K × N denotes the channel matrix between the RIS and theusers; Φ = diag { φ } with φ = [ φ , φ , . . . , φ N ] T and φ n = e jθ n is the phase shift at the n -thRIS antenna element, n = 1 , , . . . , N ; u = [ u , u , . . . , u K ] T contains i.i.d. zero-mean circularlysymmetric complex Gaussian noise at the user, i.e., u k ∼ CN (0 , γ ) . Finally, each user k scalesits received signal r k by c k ∈ C , which is represented by ˆ s = Cr = CF ΦHW s + Cu , (15)where C = diag { c , c , . . . , c K } . B. Problem Formulations
The above RIS-assisted system can be employed for downlink data transmission or powertransfer. For either case, we need to design the matrices { W , Φ , C } , which depend on thechannels { H , F } . The key challenge is then to obtain the optimal design without knowing thechannels; and for that we resort to the tool of Bayesian optimization. In what follows, we firstformulate the optimization problems for data transmission and power transfer.
1) Downlink Data Transmission:
For downlink data transmission, one objective is to min-imize the sum MSE, given by f MSE ( W , Φ , C ) (cid:44) E {(cid:107) ˆ s − s (cid:107) } (16) = E {(cid:107) ( CF ΦHW − I ) s + Cu (cid:107) } (17) = tr { ( CF ΦHW − I ) H ( CF ΦHW − I ) } + γ tr { C H C } (18) = tr { W H H H Φ H F H C H CF ΦHW + I } − tr { W H H H Φ H F H C H }− tr { CF ΦHW } + γ tr { C H C } . (19)Then the optimization problem becomes min { W , Φ , C } f MSE ( W , Φ , C ) (20a) s . t . tr { W H W } ≤ P (20b) ≤ θ n ≤ π, ∀ n = 1 , , . . . , N. (20c)Note that since the channel state information (CSI) { H , F } is not available, the above opti-mization problem cannot be solved analytically. Remark 1:
To evaluate the expectation in (26a), at each iteration t , we can transmit κ pilotsymbol vectors s (1) , . . . , s ( κ ) , and evaluate an approximate objective value f MSE ( W , Φ , C ) ≈ κ κ (cid:88) n =1 (cid:107) ˆ s ( n ) − s ( n ) (cid:107) . (21)Simulation results in Sec.V indicate that one pilot symbol vector, e.g., κ = 1 suffices.To take into account the fairness among users, we can also minimize the maximum ofindividual MSE. In particular, the MSE of user k is given by f MSE k ( w k , Φ , c k ) (cid:44) E {| ˆ s k − s k | } (22) = E {| ( c k f k ΦH M (cid:88) k =1 w k s k + c k u k − s k | } (23) = ( c k f k ΦHw k − H ( c k f k ΦHw k −
1) + γ c H k c k (24) = w H k H H Φ H f H k c H k c k f k ΦHw k − (cid:60){ w H k H H Φ H f H k c H k } + γ c H k c k + 1 . (25)Then we can consider the following min-max formulation: min { w k , Φ ,c k } max k f MSE k ( w k , Φ , c k ) (26a) s . t . (20b) , (20c) . (26b) Note that the objective function in (26a) is not smooth which is hard to be optimized via Bayesianoptimization. Therefore we employ the following smooth approximation to the maximizationoperator [30]: max { x , x , . . . , x K } ≈ η ln K (cid:88) k =1 exp( f MSE k ( w k , Φ , c k ) η ) , (27)where η > is a parameter. Large η leads to high accuracy of the approximation, but it alsocauses the problem to be nearly ill-conditioned. When η is chosen appropriately, we can use thefollowing smooth reformulation of (26a)-(26b): min { w k , Φ ,c k } η ln K (cid:88) k =1 exp( f MSE k ( w k , Φ , c k ) η ) (28a) s . t . (20b) , (20c) . (28b)
2) Downlink Power Transfer:
For downlink wireless power transfer, the receiving filter C is not needed at the users. The total harvested power at the users is defined as f p ( W , Φ ) (cid:44) E { rr H } (29) = tr { F ΦHW W H H H Φ H F H } + γ K. (30)The optimization problem is then max { W , Φ } f p ( W , Φ ) (31a) s . t . (20b) , (20c) . (31b)We can also maximize the minimum of individual harvested power. In particular, theharvested power at user k is defined as f p k ( W , Φ ) (cid:44) E { r k r H k } (32) = f k ΦHW W H H H Φ H f H k + γ . (33)Then we consider the following max-min formulation: max { W , Φ } min k f p k ( W , Φ ) (34a) s . t . (20b) , (20c) . (34b) Similarly as before, a smooth reformulation of (34a)-(34b) is given by max { W , Φ } η ln K (cid:88) k =1 exp( f p k ( W , Φ ) η ) (35a) s . t . (20b) , (20c) . (35b)IV. D OWNLINK T RANSMISSION D ESIGNS W ITH U NKNOWN
CSIIn this section, we propose a Bayesian optimization based approach to solve the total MSEminimization problem in (20) based on observing the sample MSE. Note that the same approachcan be used to solve the other problems formulated in Sec. III.
A. Spherical Representation of Variables
We treat the objective in (20) as a black-box function with input variables { W , Φ , C } .In order to employ Bayesian optimization, the input variables should be real-valued and theoptimization should be unconstrained. We first transform { W , Φ , C } into real-valued vectors.The phase shift Φ consists of N real variables θ = [ θ , θ , . . . , θ N ] T ∈ [0 , π ] N .For the beamforming matrix W ∈ C M × K , we can first vectorize it as w = vec ( W ) ∈ C MK and then convert it to real-valued as ˜ w = [ (cid:60){ w T } , (cid:61){ w T } ] T ∈ R = [ ˜ w , ˜ w , . . . , ˜ w MK ] MK .Then the power constraint (20b) becomes ˜ w + ˜ w + · · · + ˜ w MK ≤ P. (36)In order to remove this constraint, we parameterize the vector ˜ w by spherical coordinates ψ =[ ψ , ψ , . . . , ψ MK − ] T ∈ [0 , π ] MK − as follows ˜ w m = √ P cos ψ m m − (cid:89) n =1 sin ψ n , m = 1 , , . . . , M K − , (37) ˜ w MK = √ P sin ψ MK − MK − (cid:89) n =1 sin ψ n , m = 2 M K. (38)It can be easily checked that ˜ w defined by (37)-(38) always satisfies (36). Moreover, the filteringmatrix C consists of K complex variables c = [ c , c , . . . , c K ] T ∈ C K , which can be representedas K real variables ˜ c = [ (cid:60){ c T } , (cid:61){ c T } ] T ∈ R K . Note that we observe from simulations that itselements ˜ c k ∈ [ − , under the Rayleigh fading channel model. Therefore, we can also represent ˜ c using the spherical coordinate, i.e., ˜ c k = cos γ k , with γ k ∈ [0 , π ] . Hence the constrained optimization problem in (20) with complex-valued variables is converted to an unconstrainedoptimization with D = 2( M + 1) K + N − real-valued variables min x f MSE ( x ) , (39)with x = [ θ T , ψ T , γ T ] T . B. Optimization via Coordinate Decomposition
When the dimension D is high, the optimization in (11) is hard to solve. A typical ap-proach is to partition the D -dimensional decision variable x into L non-overlapping segments: x (1) , ..., x ( L ) . We postulate that the objective function can be decomposed as the following: f ( x ) ≈ f (1) ( x (1) ) + f (2) ( x (2) ) + · · · + f ( L ) ( x ( L ) ) . (40)Moreover, given { ( x i , f ( x i )) } ti = t − W +1 , we further assume that f ( (cid:96) ) ( x ( (cid:96) ) t +1 ) and f ( x t − W +1: t ) arejointly Gaussian. Similar to (8)-(9), we obtain the posterior mean and the posterior variance of f ( (cid:96) ) ( x ( (cid:96) ) t +1 ) as µ ( (cid:96) ) t ( x ( (cid:96) ) t +1 ) = (¯ k ( (cid:96) ) t +1 ) T ¯ K − t f ( x t − W +1: t ) , (41) ( σ ( (cid:96) ) t ( x ( (cid:96) ) t +1 )) = k (0) − (¯ k ( (cid:96) ) t +1 ) T ¯ K − t ¯ k ( (cid:96) ) t +1 , (42)where ¯ k ( (cid:96) ) t +1 = [ k ( (cid:107) x ( (cid:96) ) t +1 − x ( (cid:96) ) t − W +1 (cid:107) ) , k ( (cid:107) x ( (cid:96) ) t +1 − x ( (cid:96) ) t − W +2 (cid:107) ) , . . . , k ( (cid:107) x ( (cid:96) ) t +1 − x ( (cid:96) ) t (cid:107) )] T . The acquisitionfunction ϕ ( (cid:96) ) t +1 of the segment x ( (cid:96) ) is then given by ϕ ( (cid:96) ) t +1 ( x ( (cid:96) ) ) = µ ( (cid:96) ) t ( x ( (cid:96) ) ) − (cid:112) β t +1 σ ( (cid:96) ) t ( x ( (cid:96) ) ) . (43)And the next query segment x ( (cid:96) ) t +1 is given by x ( (cid:96) ) t +1 = arg min x ( (cid:96) ) ϕ ( (cid:96) ) t +1 ( x ( (cid:96) ) ) , (cid:96) = 1 , . . . , L. (44)Finally we combine all segments to obtain the query point x t +1 and the corresponding objectivefunction value f ( x t +1 ) .Moreover, we can further consider multiple, say Q , random partitions of x . For eachpartition, we compute the query point x t +1 according to the above procedure and the objectivevalue f ( x t +1 ) . Then we choose the partition that results in the minimum value of objectivevalue.In summary, the practical implementation of the proposed transmission scheme is shownin Fig. 2. The total training procedure consists of two stages: the initial stage and the updating BS RIS
Transmitting phase
User 1 ...
User KRIS controller
Feedback phase
Bayesian optimization
Update phase
Fig. 2. The proposed scheme via Bayesian optimization in the downlink data transmission. stage. First, in the initialization process, the BS, the RIS, and the users randomly initialize inputvector x t for W times. After each initialization, the BS transmits the pilot symbol s ( t ) to theusers. Then the users estimate the objective value in (21) and feed it back to the BS. Basedon the W initial sample points and the corresponding objective value, we randomly generate Q possible partitions of x . Then the BS computes x in (44) and transmits pilot s (1) . The usersestimate the objective value f ( x ) and feed it back to the BS. Then the BS chooses the partitioncorresponds to a minimum of f ( x ) .In the updating process, the BS transmits the pilot symbol s ( t ) to the users. Then the usersestimate the objective value f ( x t ) and feedback to the RIS and the BS. The main difference withthe initial stage is that the Bayesian optimization is operated at each time to obtain the next querypoint x t +1 . Then the BS informs the RIS and the users of the updated Φ and C , respectively.Finally, by repeating this updating process, the sum MSE is minimized. The proposed Bayesianoptimization based training scheme is shown in Algorithm .V. S IMULATION R ESULTS
In this section, we present numerical results to verify the performance of the proposedtransmission scheme based on Bayesian optimization. The BS is equipped with M = 2 antennas.The number of elements in RIS is N = 2 if there is no specific introduction. The parameter β t +1 in (10) is set to . t +2) . The window size is W = 20 . The SE kernel is chosen in Bayesianoptimization and the hyper-parameter h is learned by the maximum likelihood estimation in [26]. Algorithm 2
The proposed training scheme based on Bayesian optimization Initialization process: for t = − W + 1 , . . . , do The BS, the RIS, and the users randomly initialize x t . The BS transmits pilot s ( t ) to users. The users estimate the objective value in (21) and feed it back to the BS. end for t = 1 for each one of the Q possible partitions of x do The BS computes x in (44). The BS transmits pilot s (1) to users. The users estimate the objective value f ( x ) and feed it back to the BS. end for The BS chooses the partition corresponds to minimum of f ( x ) . Updating process: for t = 2 , . . . , T do The BS transmits pilot s ( t ) to users. The users estimate the objective value f ( x t ) and feed it back to the BS. for (cid:96) = 1 , , . . . , L do Compute µ ( (cid:96) ) t and σ ( (cid:96) ) t using (41) and (42), respectively; Update x ( (cid:96) ) t +1 in (44) using the grid search algorithm; end for x t +1 = ∪ L(cid:96) =1 x ( (cid:96) ) t +1 ; The BS transmits the updated Φ and C to the RIS and the users; end for Output: { W , Φ , C } .The number of partition Q is equal to the input dimension D . The smooth parameter is η = 50 .The simulation results are obtained by taking an average over 1000 random realizations and themaximum iteration time T = 350 .Furthermore, for the initialization of the proposed algorithm, we set the initial phase shift of SNR(dB) S u m M SE Bayesian optimizationTransmission with known CSI
Fig. 3. The sum MSE performance of the proposed Bayesian optimization method and the known CSI scheme is evaluated inthe downlink data transmission. The proposed “Bayesian optimization” has an acceptable performance with the “Transmissionwith known CSI” in the Appendix. each reflecting element follows a uniform distribution, i.e., θ i ∈ U (0 , π ) . Similarly, the filteringelement is γ k ∈ U (0 , π ) . For the initialization of ψ , we first randomly generate the vector ˜ w satisfying the power constraint (36), and then compute the corresponding ψ by inverting thespherical coordinate transformation (37), (38) as follows: ψ = arccos (cid:18) ˜ w √ P (cid:19) , (45) ψ m = arccos (cid:18) ˜ w m √ P Π m − n =1 sin ψ n (cid:19) , for m = 2 , . . . , M K − . (46) A. Downlink Data Transmission
In the downlink data transmission, the Rayleigh fading is assumed for each link. The sumMSE performance of the proposed Bayesian optimization scheme and the known CSI scheme isshown in Fig. 3. The derivation of the benchmark “Transmission with known CSI” is shown in The number of iterations S u m M SE SNR = 0 dBSNR = 10 dBSNR = 20 dB
Fig. 4. The convergence time for the proposed transmission scheme is evaluated in the downlink data transmission. It showsthat the proposed scheme can achieve convergence through iterations.
Appendix. As we can see, the proposed Bayesian optimization scheme is acceptable comparedwith the scheme with known CSI, especially at the medium SNR ( < SNR < ). It is worthnoting that the sum MSE performance degrades in the low SNR regime since the noise variesdramatically in each feedback. Moreover, the sum MSE is convergent at 0.7 and does not reduceany more. This is because the objective function is approximated as the sum of low-dimensionsubfunctions, which makes the performance loss.The convergence time for the proposed scheme is evaluated in Fig. . The SNR is set to { dB , dB , dB } . Although the sum MSE is high at the initial samples, the sum MSEgradually convergent after 150 times in both the high or the low SNR regimes. This showsthat the number of iterations of the proposed algorithm is acceptable in practice. Note that themore iterations guarantee better performance since the objective value of the new sample pointmay be stable at the local optimal. Moreover, it can be seen that slight fluctuations appear atthe convergent point. This is because the noise can affect the objective value at each feedback,which makes the updated { W , Φ , C } fluctuate at the convergent point. Number of reflection elements S u m M SE Known CSI with SNR=10dBKnown CSI with SNR=20dBBayesian Optimization with SNR=10dBBayesian Optimization with SNR=20dB
Fig. 5. The sum MSE performance of different reflection elements for the proposed scheme and the transmission scheme withknown CSI is evaluated in the downlink data transmission. When
SNR = 20 dB, the sum MSE of the proposed “Bayesianoptimization” with ten reflecting elements is nearly same as the “Known CSI” with two reflecting elements in Appendix.
The sum MSE performance of different reflection elements for the proposed scheme andthe transmission scheme with known CSI is evaluated in Fig. . We can see that the sum MSEdecreases with the increase of the number reflecting elements on the RIS. Although there is agap between the proposed scheme and the scheme with perfect CSI, the channel estimation is notrequired at the RIS and the BS, which makes our proposed scheme more practical. Moreover, itis shown that performance gain by increasing the number of reflecting elements is not obviouscompared with the benchmark. This is because the increased dimension of the input variablesmakes the optimization more difficult with the limited number of feedback. Furthermore, we cansee that the sum MSE of the proposed scheme with ten reflecting elements is nearly the same asthe sum MSE of the known CSI scheme with two elements. This reveals that the implementationof more reflecting elements without the RF-chains can achieve the same performance with theknown CSI scheme.The sum MSE performance with the different numbers of pilot symbol vectors is evaluated The number of iterations S u m M SE = 1 = 10 = 20 Fig. 6. The sum MSE performance for the proposed transmission scheme with different number of pilot symbol vectors isevaluated in the downlink data transmission. We set
SNR = 20 dB. It is shown that the one pilot symbol vector is enough tothe training process. in Fig. 6. The SNR is set to dB. We can see that the number of pilot symbol vectorsdoes not significantly affect performance. Moreover, the sum MSE of the transmission with onepilot symbol vector outperforms the transmission with 10 and 20 pilot symbol vectors. This isbecause suitable noise can make the objective value avoid the local minimum and accelerate theconvergence speed. Therefore, one pilot symbol vector is enough in the training process.Next, we plot the MSE performance of the proposed scheme with the slow fading channel inFig. 7. We assume that H and F varies slowly in each feedback. The varying channel coefficientsof each feedback are models as ˜ H = H +∆ H and ˜ F = F +∆ F , where the entries of ∆ H , ∆ F are ∈ CN (0 , ν ) . Here we set ν = 0 . . We can see that the proposed bayesian optimizationalgorithm is robust to the slow fading channel. This is because the proposed scheme adopts theprevious sample points and is not sensitive to the trivial parameter change. As a comparison,the frequent channel estimation is required for the known CSI scheme to avoid the performanceloss with the channel varying. SNR(dB) S u m M SE Bayesian optimizationTransmission with known CSI
Fig. 7. The sum MSE performance for the proposed transmission scheme with the slow fading channel is evaluated in thedownlink data transmission. It is shown that the proposed “Bayesian optimization” is robust to the slow varying channel.
In Fig. 8, The MSE performance for the proposed scheme and the transmission scheme withknown CSI is evaluated in the fairness downlink data transmission. The known CSI scheme isthe benchmark for the fairness problem, and the derivation of the known CSI scheme is similarto [31]. As we can see, the min-max MSE of the proposed scheme achieves the acceptableperformance compared with the known CSI scheme. Although the sum MSE problem and themin-max MSE problem are different in the formulation, the proposed scheme can solve theseproblems in a similar way.
B. Downlink Power Transfer
We further consider the large scale fading in the downlink power transfer. The large scalefading is modeled as κ = ςd − α , where d is the distance between the transmitter and the receiver; α = 2 . is the path loss exponent, and ς is the path loss at the reference distance m which is setto dB. Moreover, the background noise variance at each user is set to − dBm. Moreover, SNR(dB) T he m i n - m a x M SE Bayesian optimizationTransmission with known CSI
Fig. 8. The MSE performance for the proposed scheme and the transmission scheme with known CSI is evaluated in the fairnessdownlink data transmission. The proposed “Bayesian optimization” achieves an acceptable performance with “Transmission withknown CSI”. the distance between the BS and the users is fixed to m, and the distance between the usersand the RIS is set to m.The received power performance for the proposed scheme and the transmission scheme withknown CSI is evaluated in Fig. . The known CSI scheme in [5] is set to the benchmark. Thereceived power of the users improves as the increase of the transmit power. The performancegap between the known CSI scheme and the proposed transmission scheme is acceptable.In Fig. 10, the received power performance with different transmit power for the proposedscheme and the transmission scheme with known CSI is evaluated in the downlink fairnesswireless power transfer. The derivation of the benchmark “Known CSI” is similar to [31]. Ourproposed Bayesian optimization based scheme achieves a satisfactory performance comparedwith the known CSI scheme. It validates the effectiveness of the proposed scheme in both thesum power maximization and the min-max problem.
16 18 20 22 24 26 28 30
Transmit power(dBm) R e c e i v ed s u m po w e r( W ) -8 Bayesian optimizationKnown CSI [5]
Fig. 9. The received power performance with different transmit power for the proposed scheme and the transmission schemewith known CSI is evaluated in the downlink wireless power transfer. It is shown that the proposed “Bayesian optimization”achieves an acceptable performance with “Known CSI” in [5].
VI. C
ONCLUSION
In this paper, we studied the transmission scheme in the downlink RIS-assisted system withunknown CSI. The training scheme based on Bayesian optimization was proposed to minimizethe sum MSE. Moreover, the proposed scheme can be extended to the fairness problem and thewireless power transfer system. The superiority of the proposed scheme over the existing schemeis that the RF chains of the RIS and the channel estimation are not required for the transmission,which makes the implementation of the RIS more flexible and energy effective. The simulationresults has demonstrated that the proposed scheme achieved an acceptable performance comparedwith the known CSI scheme. Furthermore, it was also shown that the proposed scheme can resistthe impact of the slow fading channel. The ramification of this paper is that it provides a newperspective to design the RIS-assisted downlink transmission scheme.
16 18 20 22 24 26 28 30
Transmit power(dBm) R e c e i v ed s u m po w e r( W ) -9 Bayesian optimizationKnown CSI
Fig. 10. The received power performance with different transmit power for the proposed scheme and the transmission schemewith known CSI is evaluated in the downlink fairness wireless power transfer. The proposed “Bayesian optimization” has anacceptable performance with “Known CSI”. A PPENDIX T HE T RANSMISSION S CHEME W ITH K NOWN
CSIIn this section, we iteratively solve the original problem when CSI is available at the BSand the RIS. The closed-form solution is obtained to minimize the sum MSE. Lastly, the overallalgorithm is provided.We can expand the objection function (26a) as J ( W , Φ , C ) = E {(cid:107) ( CF ΦHW − I ) s + Cu (cid:107) } (47) = tr { ( CF ΦHW − I ) H ( CF ΦHW − I ) } + σ tr { C H C } (48) = tr { W H H H Φ H F H C H CF ΦHW } − tr { W H H H Φ H F H C H }− tr { CF ΦHW } + σ tr { C H C } . (49) In the following, we focus on the optimization of the transmit beamformer W and the phaseshift matrix Φ . A. Optimize { W , C } for fixed Φ We refer to the lagrangian method to solve this problem. We introduce a introduce anauxiliary variable α . Then, let W = α ¯ W , (50) C = α − ¯ C , (51)and the lagrangian function can be written as L ( ¯ W , α ) = tr { ¯ W H H H Φ H F H ¯ C H ¯ CF ΦH ¯ W } − tr { ¯ W H H H Φ H F H ¯ C H }− tr { ¯ CF ΦH ¯ W } + α − σ tr { ¯ C H ¯ C } + λ ( α tr { ¯ W ¯ W H } − P ) , (52)where λ is the Lagrangian multiplier. Then, By setting ∂ L ∂λ = 0 and ∂ L ∂α = 0 . We can obtain α = P tr { ¯ W ¯ W H } (53) α = σ tr { ¯ C H ¯ C } λ tr { ¯ W ¯ W H } . (54)Hence, we have λα = σ P tr { ¯ C H ¯ C } . (55)By setting ∂ L ∂ W ∗ = 0 , we can get the closed-form solution of ¯ W and α as ¯ W = 2 (cid:18) H H Φ H F H ¯ C H ¯ CF ΦH + σ P tr { ¯ C H ¯ C } I (cid:19) − ¯ CF ΦH , (56) α = P tr { ¯ W ¯ W H } − , (57)Then, the optimal W can be expressed as W opt = α ¯ W . (58)By setting ∂ L ∂ C ∗ = 0 , we can obtain ¯ C = (cid:16) F ΦH ¯ W ¯ W H H H Φ H F H + σ I (cid:17) − ¯ W H H H Φ H F H , (59)Then, we can obtain the closed form solution of C as C opt = α − ¯ C . (60) B. Optimize Φ for fixed { W , C } For fixed { W , C } , the optimization problem (20) can be rewritten as min Φ tr { Φ H AΦB } − tr { Φ H D H } − tr { ΦD } (61a) s . t . ≤ θ n ≤ π, ∀ n = 1 , , . . . , N (61b)where A = F H C H CF , B = HW W H H H and D = HW CF . Since Φ is the diagonalmatrix, we have tr { Φ H AΦB } = φ H ( A (cid:12) B ) φ , (62) tr { ΦD } = d T φ , (63) tr { Φ H D H } = φ H d ∗ . (64)where d = diag { D } . Then, the problem (61) can be rewritten as min φ φ H Ξφ − d T φ − φ H d ∗ (65a) s . t . | φ n | = 1 , ∀ n = 1 , , . . . , N, (65b)where Ξ = A (cid:12) B . Then, the problem (65) can be rewritten as min φ f ( φ ) (66a) s . t . | φ n | = 1 , ∀ n = 1 , , . . . , N, (66b)where f ( φ ) = φ H Ξφ − (cid:60){ φ H d } . Since the unit modulus constraint in (66b), the problem (66)is a non-convex optimization problem and hard to solve. Here, we adopt the MM algorithm tosolve this problem [14]. The main idea is to construct a series of more tractable approximatesubproblems. Similarly to the lemma in [32], we introduce the lemma in the following. Lemma 1:
For any given solution φ t at the t -th iteration and for any feasible φ , we have φ H Ξφ ≤ y ( φ | φ t ) (cid:44) φ H Xφ − (cid:60){ φ H ( X − Ξ ) φ } + ( φ t ) H ( X − Ξ ) φ t , (67)where Ξ = λ max I N and λ max is the maximum eigenvalue of Ξ . (cid:3) Then, the surrogate objective function y ( φ | φ t ) can be constructed as g ( φ | φ t ) = y ( φ | φ t ) + 2 (cid:60){ φ H d } . (68) Algorithm 3
The transmission scheme with known CSI Initial: t = 1 , φ = φ , given H , F ; while the | f ( φ t +1 ) − f ( φ t ) | is reduced by more than ε do Calculate { W , C } using (58) and (60), respectively; Calculate q t using (71); Calculate φ t +1 using (72); Calculate f ( φ t +1 ) using (66a); end while It can be proofed that g ( φ | φ t ) satisfies the three conditions in [33]. Furthermore, the objectivefunction g ( φ | φ t ) is more tractable than the original f ( φ | φ t ) . Then, the subproblem can be solvedat the t -th iteration is given by min φ g ( φ | φ t ) (69a) s . t . | φ n | = 1 , ∀ n = 1 , , . . . , N. (69b)Since φ H φ = M , we have φ H Ξφ = M λ max . By substituting (67) into (70a), the problem (69)can be reformulated as max φ (cid:60){ φ H q t } (70a) s . t . | φ n | = 1 , ∀ n = 1 , , . . . , N. (70b)where q = ( λ max I N − Ξ ) φ t − d ∗ . (71)It can be easily seen that the optimal solution of problem (70) is given by φ t +1 = e j arg( q t ) . (72)Finally, we provide the transmission scheme with known CSI in Algorithm .R EFERENCES [1] G. C. Alexandropoulos, P. Ferrand, J. Gorce, and C. B. Papadias, “Advanced coordinated beamforming for the downlinkof future LTE cellular networks,”
IEEE Commun. Mag. , vol. 54, no. 7, pp. 54–60, Jul. 2016. [2] M. D. Renzo, M. Debbah, D. T. P. Huy, A. Zappone, M. Alouini, C. Yuen, V. Sciancalepore, G. C. Alexandropoulos,J. Hoydis, H. Gacanin, J. de Rosny, A. Bounceur, G. Lerosey, and M. Fink, “Smart radio environments empowered byreconfigurable AI meta-surfaces: an idea whose time has come,” EURASIP J. Wireless Comm. and Netw. , vol. 2019, p.129, May 2019.[3] Q. Wu and R. Zhang, “Towards smart and reconfigurable environment: Intelligent reflecting surface aided wireless network,”
IEEE Commun. Mag. , vol. 58, no. 1, pp. 106–112, May 2020.[4] M. D. Renzo, A. Zappone, M. Debbah, M. Alouini, C. Yuen, J. D. Rosny, and S. Tretyakov, “Smart radio environmentsempowered by reconfigurable intelligent surfaces: How it works, state of research, and road ahead,”
IEEE J. Sel. AreasCommun. , early access, 2020.[5] Q. Wu and R. Zhang, “Weighted sum power maximization for intelligent reflecting surface aided SWIPT,”
IEEE WirelessCommun. Letters , vol. 9, no. 5, pp. 586–590, May 2020.[6] M. A. ElMossallamy, H. Zhang, L. Song, K. G. Seddik, Z. Han, and G. Y. Li, “Reconfigurable intelligent surfaces forwireless communications: Principles, challenges, and opportunities,”
IEEE Trans. Cogn. Commun. Netw. , vol. 6, no. 3, pp.990–1002, 2020.[7] S. Gong, X. Lu, H. D. Thai, D. Niyato, L. Shu, D. I. Kim, and Y.-C. Liang, “Towards smart wireless communications viaintelligent reflecting surfaces: A contemporary survey,”
IEEE Commun. Surv. Tut. , early access, 2020.[8] Q. Wu and R. Zhang, “Intelligent reflecting surface enhanced wireless network: Joint active and passive beamformingdesign,” in
Proc. IEEE Global Commun. Conf. (GLOBECOM) , Dec. 2018, pp. 1–6.[9] X. Yu, D. Xu, and R. Schober, “MISO wireless communication systems via intelligent reflecting surfaces,” in
Proc.IEEE/CIC Int. Conf. Commun. China (ICCC) , Aug. 2019, pp. 735–740.[10] W. Cai, H. Li, M. Li, and Q. Liu, “Practical modeling and beamforming for intelligent reflecting surface aided widebandsystems,”
IEEE Commun. Lett. , pp. 1568–1571, Jul. 2020.[11] Y. Yang, S. Zhang, and R. Zhang, “IRS-enhanced OFDM: Power allocation and passive array optimization,” in
Proc. IEEEGlobal Commun. Conf. (GLOBECOM) , Dec. 2019, pp. 1–6.[12] Q. Wu and R. Zhang, “Intelligent reflecting surface enhanced wireless network via joint active and passive beamforming,”
IEEE Trans. Wireless Commun. , vol. 18, no. 11, pp. 5394–5409, Nov. 2019.[13] C. Huang, A. Zappone, G. C. Alexandropoulos, M. Debbah, and C. Yuen, “Reconfigurable intelligent surfaces for energyefficiency in wireless communication,”
IEEE Trans. Wireless Commun. , vol. 18, no. 8, pp. 4157–4170, Aug. 2019.[14] G. Zhou, C. Pan, H. Ren, K. Wang, and A. Nallanathan, “Intelligent reflecting surface aided multigroup multicast MISOcommunication systems,”
IEEE Trans. Signal Process. , pp. 3236 – 3251, Apr. 2020.[15] G. C. Alexandropoulos and E. Vlachos, “A hardware architecture for reconfigurable intelligent surfaces with minimal activeelements for explicit channel estimation,” in
Proc. IEEE Int. Conf. Acoust., Speech Signal Process. (ICASSP) , May 2020,pp. 9175–9179.[16] A. Taha, M. Alrabeiah, and A. Alkhateeb, “Enabling large intelligent surfaces with compressive sensing and deeplearning.” [Online]. Available: https://arxiv.org/abs/1904.10136.[17] J. Luo, S. Wang, and F. Wang, “Joint transmitter-receiver spatial modulation design via minimum euclidean distancemaximization,”
IEEE J. Sel. Areas Commun. , vol. 37, no. 9, pp. 1986–2000, Sept. 2019.[18] D. Mishra and H. Johansson, “Channel estimation and low-complexity beamforming design for passive intelligent surfaceassisted MISO wireless energy transfer,” in
Proc. IEEE Int. Conf. on Acoust. Speech Signal Process. (ICASSP) , May 2019,pp. 4659–4663.[19] Z.-Q. He and X. Yuan, “Cascaded channel estimation for large intelligent metasurface assisted massive MIMO,”
IEEEWireless Commun. Lett. , vol. 9, pp. 210–214, Feb. 2020. [20] C. Hu and L. Dai, “Two-timescale channel estimation for reconfigurable intelligent surface aided wireless communications.” arXiv , 2020. [Online]. Available: https://arxiv.org/abs/1912.07990.[21] H. Liu, X. Yuan, and Y. A. Zhang, “Matrix-calibration-based cascaded channel estimation for reconfigurable intelligentsurface assisted multiuser MIMO,” IEEE J. Sel. Areas in Commun. , early access, 2020.[22] S. Lin, B. Zheng, G. C. Alexandropoulos, M. Wen, F. Chen, and S. Mumtaz, “Adaptive transmission for reconfigurableintelligent surface-assisted OFDM wireless communications,”
IEEE J. Sel. Areas Commun. , early access, 2020.[23] S. Abeywickrama, T. Samarasinghe, C. K. Ho, and C. Yuen, “Wireless energy beamforming using received signal strengthindicator feedback,”
IEEE Trans. Signal Process. , vol. 66, no. 1, pp. 224–235, Jan. 2018.[24] S. Chen, S. Zhong, S. Yang, and X. Wang, “A multiantenna rfid reader with blind adaptive beamforming,”
IEEE InternetThings J. , vol. 3, no. 6, pp. 986–996, Dec. 2016.[25] B. Shahriari, K. Swersky, Z. Wang, R. P. Adams, and N. de Freitas, “Taking the human out of the loop: A review ofbayesian optimization,”
Proc. IEEE , vol. 104, no. 1, pp. 148–175, Jan. 2016.[26] E. Brochu, V. M. Cora, and N. de Freitas, “A tutorial on bayesian optimization of expensive cost functions, with applicationto active user modeling and hierarchical reinforcement learning,” 2010. [Online]. Available: https://arxiv.org/abs/1012.2599[27] R. Garnett, M. Osborne, and S. Roberts, “Bayesian optimization for sensor set selection,” in
Proc. ACM/IEEE Int. Conf.Inf. Process.Sensor Netw. , Jan. 2010, pp. 209–219.[28] K. Kandasamy, J. Schneider, and B. Poczos, “High dimensional bayesian optimisation and bandits via additive models,”in
Proc. Int. Conf. Mach. Learn. , Jul. 2015.[29] D. R. Jones, C. D. Perttunen, and B. E. Stuckman, “Lipschitzian optimization without the lipschitz constant,”
J. Optim.Theory Appl. , vol. 79, no. 1, pp. 157–181, Jan. 1993.[30] S. Xu, “Smoothing method for minimax problems,”
Comput. Optim. Appl. , vol. 20, no. 3, pp. 267–279, 2001.[31] H. Xie, J. Xu, and Y. Liu, “Max-min fairness in IRS-aided multi-cell MISO systems via joint transmit and reflectivebeamforming,” in
Proc. IEEE Int. Conf. Commun. (ICC) , Jun. 2020, pp. 1–6.[32] J. Song, P. Babu, and D. P. Palomar, “Sequence design to minimize the weighted integrated and peak sidelobe levels,”
IEEE Trans. Signal Process. , vol. 64, no. 8, pp. 2051–2064, Apr. 2016.[33] Y. Sun, P. Babu, and D. P. Palomar, “Majorization-minimization algorithms in signal processing, communications, andmachine learning,”