Location-aware Beam Alignment for mmWave Communications
aa r X i v : . [ ee ss . SP ] J u l Location-aware Beam Alignment for mmWaveCommunications
Igbafe Orikumhi,
Member, IEEE,
Jeongwan Kang,
Student Member, IEEE,
HenkWymeersch,
Member, IEEE, and Sunwoo Kim,
Senior Member, IEEE
Abstract
Beam alignment is required in millimeter wave communication to ensure high data rate transmission.However, with narrow beamwidth in massive MIMO, beam alignment could be computationally intensivedue to the large number of beam pairs to be measured. In this paper, we propose an efficient beamalignment framework by exploiting the location information of the user equipment (UE) and potentialreflecting points. The proposed scheme allows the UE and the base station to perform a coordinatedbeam search from a small set of beams within the error boundary of the location information, the selectedbeams are then used to guide the search of future beams. To further reduce the number of beams to besearched, we propose an intelligent search scheme within a small window of beams to determine thedirection of the actual beam. The proposed beam alignment algorithm is verified on simulation withsome location uncertainty.
Index Terms
Beam alignment, location-aware communication, codebook, initial access, beam management.
I. I
NTRODUCTION
Millimeter-wave (mmWave) spectrum has been proposed for the fifth generation (5G) commu-nication networks due to the large bandwidth available at this frequency band. Other advantages
Igbafe Orikumhi, Jeongwan Kang, and Sunwoo Kim are with the Department of Electronics and Computer Engineering,Hanyang University, Seoul, South Korea, email: {oigbafe2,rkdwjddhks77,remero}@hayang.ac.kr. Henk Wymeersch is with theDepartment of Signal and Systems, Chalmers University of Technology, Sweden, email: [email protected]. This work wassupported by Samsung Research Funding & Incubation Center of Samsung Electronics under Project Number SRFC-IT-1601-09and by the MSIT (Ministry of Science and ICT), Korea, under the ITRC (Information Technology Research Center) supportprogram (IITP-2019-2017-0-01637) supervised by the IITP (Institute for Information & Communications Technology Planning& Evaluation). Part of this work was previously presented at the 2018 Annual Allerton Conference on Communication, Control,and Computing (Allerton) [1]. such as beamforming and spatial multiplexing are leading to an increased interest in the mmWavespectrum [2]. However, the key challenge is that the mmWave spectrum suffers from severe path-loss. On the other hand, the high-frequency spectrum allows the use of a large antenna arraywith small form factor, which provides high beamforming gains to compensate for the losses.As such, beamforming at the base station (BS) and the user equipment (UE) have become anessential part of the mmWave 5G networks [3], [4].Although as the number of array antenna increases, the array gain also increases with reducedinterference [5], a new challenge of selecting the best beam pair at the transmitter and receiverexist due to the narrow beamwidth [6], which could compromise the transmission rate especiallywhen the time taken for beam alignment is large [7]. Consequently, beam alignment especiallyin a mobile environment may become the bottleneck of the communication network. Morespecifically, in the initial access phase, beams with narrow beamwidth can complicate the initialcell search since the UE and BS have to search over a large angular directional space for suitablepath to establish communication [8], [9].When a user enters a cell, the user establishes a physical connection to the BS in the initialaccess phase. In current 4G networks, the UE regularly monitor the omnidirectional signal toestimate the downlink channel. However, in 5G mmWave networks, this is difficult to achievedue to the directionality and the rapid variations of the channel [10]. The directionality maysignificantly delay the initial access procedure, especially for beams with narrow beamwidth[11]. To reduce the beam alignment overhead, efficient beam alignment algorithms are thereforerequired. Motivated by this challenge, this paper proposes a new beam alignment algorithmwhich exploits a noisy location information.Beam alignment has been previously studied under the cell search phase [12]–[16] (i.e., thephase in which the UE search and connects to a BS with a mutual agreement on transmissionparameters). In [17], an exhaustive search algorithm to determine the optimal beam pair wasproposed. The exhaustive search is known for its high complexity. To reduce the delay in theexhaustive search algorithm, an efficient hierarchical codebook adaptive algorithm is proposed in[18] to jointly search over the channel subspace. The proposed algorithm allows the UE and BSto jointly align their beams within a constrained time. In [19], a Bayesian tree search algorithmis proposed to reduce the delay in the training of beamforming and combining vectors.Beam alignment overhead can be reduced without compromising performance if the locationside information is available at both nodes. Indeed, 5G communication devices are expected to have access to location information which can be obtained from GNSS satellites, sensors and5G radio signal [20], [21]. In [22], exploiting location information for backhaul systems wasproposed. The authors showed that the time required for beam alignment can be reduced if theposition information is shared between the nodes. For stationary backhaul systems, it is easyto assume that perfect location information is available at both nodes since this informationcan be obtained during installations. However, for a mobile device, it is likely that the locationinformation is noisy and designing beam alignment algorithms might not be straightforward asin [22]. A noisy location information is considered in [23], where the authors focused on anindependent beam pre-selection at the BS and the UE. While the pre-selection algorithm canimprove the beam selection speed, the performance of the selected beams cannot be guaranteedsince the beam selection decision is weighed on the noisy location information. In addition, thedecentralized beam selection framework may degrade system performance.This work focus on achieving fast and efficient transmit and receive beam alignment subjectto a target rate constraint by exploiting the noisy location information of the UE and potentialreflecting points. The location information of the static BS is assumed to be perfectly known,while the location information of the reflecting points and UE are considered to be noisy.The contributions of this paper are summarized as follows: Firstly, we propose a beam align-ment algorithm iteratively executed at the BS and the UE that exploits the location informationof the UE and potential reflecting points, this information is used to design a subset of beamcodebook from the BS and UE codebooks, thereby reducing the number of beam steering vectorsto be searched as compared to the exhaustive search algorithm. Secondly, we propose a searchwindow in the subset of beam codebooks to further reduce the angular space to be searched.This is achieved by determining the direction of the actual beam after obtaining the local optimalbeam within the search window. Furthermore, the local optimal beam in each search is used toguide future beam search at the UE and BS thereby reducing the beam alignment overhead.Finally, we derive the Cramér-Rao bound (CRB) of the channel parameter which an unbiasedestimator should satisfy, the CRB is then used to model the channel estimation error. We showby simulations that the proposed beam alignment scheme can speed up the beam alignmentprocess and reduce the beam alignment overhead when compared to existing schemes.The rest of the paper is organized as follows. In Section II, the signal model and the codebookused in this paper are introduced. Section III first describes the use of location informationfollowed by a detail description of the proposed beam alignment algorithm. In Section IV, we
PSfrag replacements [ x BS , y BS ] [ x , y ] [ x , y ] [ x U E , y U E ] Fig. 1. Example of network scenario with two reflectors present the channel parameter estimation and rate evaluation. Numerical results are shown insection V, and the paper is concluded in Section VI.
Notations
Throughout this paper, matrices and vector symbols are represented by uppercase and low-ercase boldface respectively. A ∗ , A T and A H represent the complex conjugate, transpose andHermitian transpose of the matrix A respectively. The mathematical expectation is denoted as E [ . ] . tr ( A ) represent the trace of matrix A . The Kronecker product between two matrices A and B is denoted as A ⊗ B . II. S YSTEM M ODEL
Consider a wireless network scenario operating in the mmWave frequency band and consistingof one UE, one BS, M dominant paths with one line of sight (LOS) path and M − reflectedpaths as shown in Fig 1. The orientation of the UE is assumed to be fixed, however, the proposedscheme can also be extended to a scenario where the orientation of the UE is not known. Insuch scenario, the UE’s orientation can be estimated along with the location information [24]after which the beam alignment algorithm proposed in this paper can be applied. The Cartesiancoordinate of each node defines its position while the location of the BS is assumed to be known by the UE. The location information in this paper is discussed in detail in the Section III. Weassume that the UE and the BS are equipped with receive and transmit uniform linear array(ULA) N r and N t antennas respectively. Furthermore, we assume the communication is made inblocks (i.e., at discrete instance), where each block consists of N slots. The channel is assumedto remain constant within each block and change independently between blocks. From the N slots, N a slots are used for the control phase within which beam alignment will be achieved,while N − N a slots are used for the data transmission phase. Note that the proposed beamalignment protocol requires both the uplink and downlink communication as the protocol isiteratively executed by both the BS and the UE. We assume the BS and UE are allowed to dwellin a slot with a fixed beamformer. More specifically, within each BS-slot, the BS beamformeris kept fixed and used to transmit to the UE while the UE takes several measurements withdifferent UE beam directions. Similarly, in the UE-slot, the UE beamformer is fixed while theBS can take several measurements with different BS beam directions. We fix the number ofbeams that can be processed per slot as N b . It can be observed that while data transmission canbe improved by selecting the best beam pair, the number of slots N a taken to achieve beamalignment should be as low as possible to reduce beam alignment overhead [11], [25]. Hence,the proposed scheme aims to speed up the beam pair search subject to a rate constraint. A. Signal and Channel Model
In this section, we present the signal model. We assume beamforming vector v k where k ∈{ , , . . . , N t } is employed at the BS while the UE employs beamforming vector u j , where j ∈ { , , . . . , N r } . Furthermore, we assume the beam vectors are normalized to unity: k v k k = k u j k = 1 . The downlink received signal can be expressed as y j,k = u H j Hv k s + u H j n , (1)where u j ∈ C N r × and v k ∈ C N t × , s is the transmitted symbol with unit energy | s | = √ E s , E s is the transmit energy, n ∈ C N r × is the complex Gaussian noise vector with zero mean andcovariance σ I . The downlink channel H is expressed as [26] H = p N t N r M − X m =0 α m a r ( θ m ) a H t ( φ m ) , (2)where M is the number of multi-paths, consisting of one LOS path and M − reflected paths,specifically, m = 0 refers to the LOS path, and m = 1 , . . . , M − , refers to the m -th NLOS path passing through the m -th reflecting points. θ m and φ m are the angle of arrival (AOA) andangle of departure (AOD) of the m -th path at the receiver and transmitter in the downlink moderespectively, α m ∼ CN (0 , σ α ) denotes the instantaneous random complex gain for the m -th path.The corresponding array response vectors at the UE and BS denoted as a r ( θ m ) and a t ( φ m ) aregiven by a r ( θ m ) = 1 √ N r (cid:2) , e − jπ cos θ m , . . . , e − jπ ( N r −
1) cos θ m (cid:3) T , (3) a t ( φ m ) = 1 √ N t (cid:2) , e − jπ cos φ m , . . . , e − jπ ( N t −
1) cos φ m (cid:3) T , (4)respectively. We assume that the steering vectors are drawn from a codebook and the design ofthe codebook is presented as follows. B. Codebook Structure
The AOD and AOA are computed from the estimated location information and are associatedwith transmit and receive beamforming vectors respectively from a given codebook. The focusof this paper is not the codebook design, therefore we refer our readers to [27] for details oncodebook design. In this paper, the codebooks are designed to achieve approximately equal gainbut with narrow beams at the broadside and wide beams at the endfire [28]. The pointing anglesat the UE and BS denoted as ¯ θ j and ¯ φ k respectively are separated into grids as follows ¯ θ j = arccos (cid:18) − j − N r − (cid:19) , j = 1 , . . . , N r , (5) ¯ φ k = arccos (cid:18) − k − N t − (cid:19) , k = 1 , . . . , N t , (6)where ≤ arccos ( x ) ≤ π . The receive and transmit codebook is defined as U = [ u , . . . , u N r ] , (7) V = [ v , . . . , v N t ] , (8)where u j = a r (¯ θ j ) and v k = a t ( ¯ φ k ) are the beam steering vectors over the discrete grid angles. Itcan be observed from (5) and (6), that the codebook covers a large angular space. The exhaustivesearch requires that the BS and UE search through the entire codebook U and V respectively.The beam alignment is achieved by selecting the beam pair that maximizes the downlink rate,which can be mathematically expressed as max { u , v }∈U×V R d ( u , v ) , (9) where R d ( u , v ) = log E s (cid:12)(cid:12) u H Hv (cid:12)(cid:12) σ ! . (10)Note that in (9), we assume that the channel state information is perfectly known, which enablesus the know the SNR at the receiver and evaluate average rate under any choice of v and u . However, performing beam alignment by exhaustive search method may incur high systemoverhead. In addition, due to the dynamic nature of the channel, especially in a mobile scenariothis method may not be suitable for 5G communication.Hence, we focus on achieving fast transmit and receive beam alignment subject to a targetrate R , where R ≤ R d ( u , v ) . The proposed scheme is discussed in details in the followingsection. III. B EAM A LIGNMENT WITH L OCATION I NFORMATION
In this section, we focus on the proposed beam alignment algorithm. We assume that thelocation information of possible reflecting points can be independently estimated by the BS andthe UE [24], [29]. However, we note that the location information could be erroneous, and theuncertainty of the location information at the nodes are included in the design of the proposedalgorithm.
A. Exploiting Location Information
Location information can be obtained with the use of available positioning technologies. Thelocation information obtained at the BS and UE may be noisy due to latency in the positioninformation exchange or due to the use of different positioning technologies at the UE and BS.For instance, the BS may be able to estimate potential reflecting points more accurately than theUE due to interactions with multiple UEs.We define the location matrix L i ∈ R × M containing the actual location coordinate ofthe nodes, where the nodes refer to the potential reflecting points and either the BS or UE.Specifically, when the location information is estimated from the BS, the location informationmatrix contains the coordinate information of the UE and the reflecting points. Similarly, whenthe location information is measured from the UE, the location information matrix contains thecoordinate information of the BS and the reflecting points. Hence, we express L i as follows L i = [ l , l , . . . , l M − ] , (11) where l = l BS for i = UE , l UE for i = BS , (12)and l m = [ x m , y m ] T is the location information of the node along path m . The location informa-tion of the observer is denoted as l BS for the BS and l UE for the UE. Note from (11) that whenthe UE is the observer, l = l BS , and when the BS is the observer, l = l UE . Furthermore, wemodel the independent location information of the UE and M − reflecting points available atthe BS as ˆ L BS = L BS + E BS , (13)where E BS is the matrices containing the random location estimation errors of the x and y coordinates made by the BS given as E BS = [ e BS , e BS , . . . , e BS M − ] , (14)where the superscript BS and UE are used to indicate the observation at the BS and UErespectively. In this paper, we adopt a uniform bounded error model for the location estimationerror [23], [28]. We assume that all the estimates lie within a disk centered on the estimatedlocation. Let S ( r m ) be the two-dimensional disk centered at the estimated location of the nodein path m with radius r m . Here, we refer to the disk with radius r m as the uncertainty region inpath m . The random estimation error e BS m is uniformly distributed in S (cid:0) r BS m (cid:1) in (14), such that r BS m is the maximum position error of the node in path m as seen from the BS. In this paper, weassume that the location information of the BS is perfectly known. In addition, when the UE isthe observer, r UE = r UEBS , and when the BS is the observer, r BS = r BSUE , where r UEBS is the maximumposition error of the BS observed at the UE and and r BSUE is the maximum position error of theUE observed at the BS. On the other hand, we assume that the location of the UE could containsome uncertainty when observed by the UE itself, hence, we denote the maximum position errorof the UE when observed by the UE itself as r UEUE . For ease of notation, we denote the vector r i = (cid:2) r i , r i , . . . , r iM − (cid:3) containing the maximum location errors of the nodes observed at theUE and BS for i ∈ { UE , BS } .From the estimated location information at the BS (i.e., ˆ L BS ), the AOD of the m -th path canbe computed as ˆ φ m = π − arctan (cid:18) ˆ x BS m − x BS ˆ y BS m − y BS (cid:19) , m = 0 , . . . , M − , (15) Similarly, the independent location information of the BS and M − reflecting points availableat the UE can be modelled as (13) and (14), where the random estimation error e UE m from theUE is uniformly distributed in S ( r UE m ) . Following a similar procedure in (13) to (15), the AOAof the m -th path to the UE can be evaluated from the location information available at the UEas ˆ θ m = π − arctan (cid:18) ˆ x UE m − ˆ x UE ˆ y UE m − ˆ y UE (cid:19) , m = 0 , . . . , M − , (16)where ˆ x UE and ˆ y UE are the estimated x and y coordinate of the UE since its position is uncertain.The estimated distance information at the BS and UE can be obtained from ˆ L BS and ˆ L UE respectively as ˆ d BS m = q ( x BS − ˆ x BS m ) + ( y BS − ˆ y BS m ) , (17) ˆ d UE m = q (ˆ x UE − ˆ x UE m ) + (ˆ y UE − ˆ y UE m ) . (18) B. Proposed Coordinated Beam Alignment
We aim to reduce the search overhead by taking advantage of the location information whileaccounting for the location estimation error. We assume that the BS is located at the originand its location is perfectly known by the UE. Based on the estimated location information andmaximum location information error, we propose a coordinated beam alignment algorithm withbeams subsets at the BS and UE denoted as B m BS ⊂ V and B m UE ⊂ U respectively.
1) Construction of the Codebook Subset B m BS and B m UE : Define f BS ( φ ) = arg min v ∈V k v − a t ( φ ) k F , (19) f UE ( θ ) = arg min u ∈U k u − a r ( θ ) k F , (20)as the functions that return the closest beam vectors to φ and θ respectively. Then the optimalrate can be obtained by searching through a subset of beam vectors in the uncertainty regionswhich is summarized in the following proposition. Proposition . The optimal rate can be recovered from solving the following reduced searchproblem within the uncertainty region as max m max ( u , v ) ∈B m BS ×B m UE R d ( u , v ) , (21) PSfrag replacements [ x BS , y BS ] r BS (cid:2) ˆ x BS , ˆ y BS (cid:3) r BS (cid:2) ˆ x BS , ˆ y BS (cid:3) ˆ φ m − ǫ m ˆ φ m + ǫ m Uncertainty region ˆ φ m γ m x y Fig. 2. Example of two-dimensional system model showing the estimated location of the UE, a reflecting point and theuncertainty region measured by the BS. where B m BS , h f BS ( ¯ φ k ) : ˆ φ m − ǫ m ≤ ¯ φ k ≤ ˆ φ m + ǫ m i , ∀ k, (22) B m UE , h f UE (¯ θ j ) : ˆ θ m − δ m ≤ ¯ θ j ≤ ˆ θ m + δ m i , ∀ j, (23)are the subset of beam vectors that lies within the error boundary of the m -th path, ¯ θ j and ¯ φ k are given by (5) and (6) respectively, the parameters ǫ m and δ m are given by ǫ m = arctan ˆ d BS m sin ˆ φ m + r BS m ˆ d BS m cos ˆ φ m ! − ˆ φ m , (24) δ m = arctan ˆ d UE m sin ˆ θ m + r UE m ˆ d UE m cos ˆ θ m ! − ˆ θ m , (25)respectively, the derivations of ǫ m and δ m are presented in Appendix A. Remark.
From Proposition 1, it can be observed that the number of beams in the set B m UE and B m BS decreases as the maximum location error tends to zero (i.e., with high precision in the locationinformation). Furthermore, the fewer beam steering vectors in B m UE and B m BS enable fast beamalignment as compared to exhaustively searching over the entire angular space. While the optimal rate can be obtained from (21), in what follows, we present a low complexitybeam alignment procedure to speed up the beam search subject to a target rate constraint R .
2) Design of the Search Window:
As discussed in the previous section, by exploiting thelocation information, we can limit the number of beam vectors in the codebook V and U , hence,we obtained the new beam subsets B m BS and B m UE for each of the m -th path. Instead of sweepingthrough the entire beam vectors in B m BS and B m UE , we can further reduce the number of beamvectors to be searched by measuring across a small window of beams W BS and W UE at the BSand UE respectively to determine the direction of beam search as shown in Fig 3. Let the sizeof the window at the BS (resp. UE) be denoted as W BS = |W BS | (resp. W UE = |W UE | ), where W BS , W UE are jointly determined at the BS and UE and correspond to the number of beams thatcan be processed in a slot N b . Furthermore, let the index of the estimated beam in B m BS (resp. B m UE ) be denoted as I m BS (resp. I m UE ), then the set of beam vectors in W BS and W UE are definedas follows W BS = B BS for W BS ≥ B BS , [ v B BS − W BS +1 , . . . , v B BS ] for W BS < B BS and B BS − I BS < W BS / , [ v , . . . , v W BS ] for W BS < B BS and I BS < W BS / , (cid:2) v I BS −⌊ W BS / ⌋ , . . . , v I BS + ⌊ W BS / ⌋ (cid:3) for W BS < B BS and I BS > W BS / , (26) W UE = B UE for W UE ≥ B UE , [ v B UE − W UE +1 , . . . , v B UE ] for W UE < B UE and B UE − I UE < W UE / , [ v , . . . , v W UE ] for W UE < B UE and I UE < W UE / , (cid:2) v I UE −⌊ W UE / ⌋ , . . . , v I UE + ⌊ W UE / ⌋ (cid:3) for W UE < B UE and I UE > W UE / , (27)where B BS = |B BS | , B UE = |B UE | and ⌊ ( a/b ) ⌋ denotes the floor of the operation. Note that sincethe design of the window is similar for each path, the index m is dropped for ease of notationin (26) and (27).When the size of the window is larger or equal to the size of the beam set, the beam vectorsin the window are given by the the beam vectors in the beam set. If the size of the beam set islarger than the size of the window (i.e., B BS > W BS ), the range of beam vectors in the searchwindow is defined by (26), where the center of the window is determined by the position of theestimated beam vector. The implementation of the window is discussed in the proposed beamalignment procedure in the following section. PSfrag replacements Optimal beam direction B m BS Direction of searchEstimated beam directionFirst searchNext search Search window W m BS Local optimal beam in search window
Fig. 3. Example of proposed scenario showing two successive beam pointing vector search in B m BS with a search window of W m BS = 5 beam vectors, where the local optimal beam determines the direction of search, hence the two pointing vectors on theleft are not searched.
3) Low Complexity Beam Alignment:
The process begins with the acquisition of the locationinformation. The BS estimates the noisy location information of the UE and the possible reflectingpoints at n = 1 as discussed in Section III-A, while the UE estimates its own location informationand the location information of the reflecting points. Thereafter, the BS and UE evaluate theAODs and AOAs for each path respectively. We assume that the BS and UE agree on the orderingof the paths based on the angle information (i.e., based on the computed AOA and AOD), where m = 0 denotes the LOS path and m = 1 , . . . , M − denote the path through the first reflectingpoint to the M − reflecting points respectively.In the beam alignment phase, (i.e., ≤ n ≤ N a ), the BS and UE jointly search the beamvectors in the uncertainty region corresponding to each of the m -th path to determine the pair ofbeam vectors that satisfies the target rate R . Specifically, in the downlink mode, the BS selectsthe beam steering vector v m from B m BS which is closest to the computed AOD ˆ φ m and transmitto the UE, such that v m = f BS ( ˆ φ m ) , (28) while the UE selects a combining vector u m which correspond to the computed AOA from B m UE such that u m = f UE (ˆ θ m ) . (29)The observed rate from path m i.e., R d,m ( u m , v m ) is compared with a threshold rate R , if thetarget R is met, the UE sends a message to the BS to move to data transmission phase. Notethat R d,m ( u , v ) is the rate obtained in path m which can be evaluated as (10). On the hand, ifthe target rate is not satisfied, the transmit beamforming vector at the BS is kept fixed while theUE takes several measurements within a small window of beam set W m UE ⊂ B m UE to determinethe direction of beam search in B m UE . At the end of the UE beam measurement, a local optimalbeam which maximizes the rate in the window is selected at the UE. If the local optimal beamsatisfies the target rate requirement, the beamforming vector from the BS and the local optimalbeam are said to be aligned and the UE sends a message to the BS to begin data transmission,we refer to this phase as the UE beam alignment phase. If the target rate is not met at the endof the UE beam alignment phase, the UE sends a message to the BS to continue with beamalignment. The objective at the UE in each UE beam alignment phase can be mathematicallyexpressed as u op = arg max u ∈W m UE R d,m ( u , v m ) . (30)In the uplink mode, the UE transmits to the BS with the local optimal beam u op obtainedfrom the UE beam alignment phase where we refer to this phase as the BS beam alignmentphase. In this phase, the local optimal beam at the UE is kept fixed, while the BS takes severalmeasurements from a small window of beam vectors W m BS ⊂ B m BS to determine the direction ofsearch in B m BS as shown in Fig. 3. At the end of the measurements, a local optimal beam isselected at the BS, this local optimal beam is used to transmit to the UE for the next UE beamalignment phase. The objective in each BS beam alignment phase can be expressed as v op = arg max v ∈W m BS R u,m ( v , u op ) , (31)where the uplink rate can be evaluated as R u,m ( v , u op ) = log E s (cid:12)(cid:12) v T H T u ∗ op (cid:12)(cid:12) σ ! , (32)where H T is the uplink channel. In the subsequent UE and BS alignment phase, the search isperformed over the local optimal beam obtained from the previous beam search as shown in Fig
3, (i.e., the window is centered on the local optimal beam direction obtained from the previoussearch). The BS and UE alternately perform beam alignment based on the procedure discussedabove for path m .The beams are said to be aligned if R d,m ( u op , v op ) ≥ R , (33)is satisfied. If the objective in (33) is not satisfied for path m , the search is performed for path m + 1 . The process is repeated for each path until (33) is satisfied or the scan over all thepaths are carried out. A summary of the proposed coordinated beam alignment is presented inAlgorithm 1. Remark.
From the proposed algorithm, it can be observed that for a given BS beamformingvector in each downlink mode, a local optimal receive beam vector can be obtained at the UE.The local optimal beam vector can be used to transmit to the BS in the uplink mode to guide theselection of beamforming vector during the BS beam alignment phase. In addition, determiningthe search direction from the search window can further reduce the number of beam vectors tobe searched and thereby speed up the beam alignment process.At the end of the beam alignment process, the UE and BS move to data transmission phasewhere we evaluate the effective rate as follows R eff = (cid:20)(cid:18) − N a N (cid:19) R d,m ( u op , v op ) (cid:21) + , (34)where ( N − N a ) /N is the fraction of the slots allocated to data transmission and [ a ] + denotes max ( a, . For the exhaustive search algorithm, the parameter N a can be evaluated as N a = ⌈ N t N r /N b ⌉ , where N b is the number of beams processed in each slot, N r and N t correspond tothe number of beams at the UE and BS respectively as given by (5) and (6).IV. C RAMÉR -R AO B OUNDS FOR C HANNEL P ARAMETERS E STIMATION
In the previous sections, it is assumed that the channel matrices and the channel state in-formation are perfectly known. Unfortunately, this assumption may not be true in real-worldtransmission but have to be obtained by channel estimation algorithms [26]. As a consequence,a more or less accurate estimate of the channel is available for the rate computations. Hence,we estimate the AOA, AOD and α m using the Cramér-Rao Bound [30] and the estimates areused to reconstruct the channel. We focus on the downlink channel estimation where the UE Algorithm 1
Coordinated Beam Alignment in one block
Input: L i , N , W BS , W UE , r UE , r BS Output: u op , v op , N a for n = 1 , , . . . , N do if n = 1 then Estimate ˆ L BS and ˆ L UE end if Evaluate ˆ φ m and ˆ θ m ∀ m from (15) and (16) n ← n + 1 for m = 0 , , . . . , M − do Select v m and u m from (28) and (29) respectively. if R d,m ( u m , v m ) ≥ R then Begin data transmission phase (line 27) else
Construct a finite beam vectors set B m UE according to (23) Set the window according to (26)
Determine u op from (30) and transmit to BS n ← n + 1 end if if R u,m ( v , u op ) ≥ R then Go to line 27 else
Construct a finite beam vectors set B m BS according to (22) Set the window according to (26)
Determine v op according to (31) and transmit to UE n ← n + 1 Return to line 9 end if end for
Output v op , u op and N a = n Evaluate performance using (34) end for takes measurements in different spatial direction and sends feedback messages to the BS usingthe local optimal beam u op as discussed in the previous section.In general, the channel matrix estimate ˜ H can be considered as the sum of the true channelmatrix H and the channel estimation error matrix E which can be expressed as ˜ H = H + E . (35)Since the channel estimation error matrix E is not known, we treat the elements of E as randomvariables such that E ∼ CN ( , Σ H ) , where Σ H is the covariance matrix obtained from theFisher information matrix which will be discussed subsequently. In what follows, we derive theFisher information matrix. A. Likelihood Function
Let P and Q denote the sum of the number of beam steering vectors in the uncertaintyregions at the BS and UE respectively, where P = P m B m BS and Q = P m B m UE . If each of the P transmit beamforming vectors are measured against each of the Q receive beamforming vectors,the observation Y ∈ C P × Q can be expressed in the form of a matrix given by Y = [ y , y , . . . , y Q ] , (36)where y q = [ y ,q , y ,q , . . . , y P,q ] T .Let y = vec( Y ) = (cid:2) y T , y T , . . . , y T Q (cid:3) T , (37)then y = vec (cid:0) U H HV (cid:1) + vec ( N )= M − X m =0 V T ⊗ U H | {z } A vec ( H ) + n = M − X m =0 A (cid:0) a ∗ t ( φ m ) ⊗ a H r ( θ m ) (cid:1) + n , (38)where U , [ u , u , . . . , u P ] , V , [ v , v , . . . , v Q ] , N denotes a complex matrix whose entriesare assumed to be uncorrelated, each with zero mean and variance σ and n ∼ CN ( , σ I P Q ) . The mean vector can be expressed as m = M − X m =0 A (cid:0) a ∗ t ( φ m ) ⊗ a H r ( θ m ) (cid:1) . (39) Let θ = [ φ , . . . , φ M − , θ , . . . , θ M − , α , . . . , α M − ] T , then the likelihood function of the randomvector in (38) conditioned on θ is obtained from the PDF which can be expressed as [31] f ( y | θ ) ∝ πσ ) P Q exp (cid:18) − σ ( y − m ) H ( y − m ) (cid:19) , (40)and the log-likelihood function can be expressed as L ( y ) = ln f ( y | θ )= − P Q ln πσ − σ ( y − m ) H ( y − m ) . (41) B. Cramér-Rao Bound
In this section, we derive the CRB of the channel parameters. Let ˜ θ be the unbiased estimatorof θ , the mean squared error (MSE) is bounded by [32] E (cid:20)(cid:16) ˜ θ − θ (cid:17) (cid:16) ˜ θ − θ (cid:17) T (cid:21) (cid:23) J − θ , (42)where J θ is the Fisher information matrix (FIM) defined as J θ , − E y | θ (cid:20) ∂ L ( y ) ∂ θ ∂ θ T (cid:21) . (43)The FIM can be structured as J θ = J ( φ m , φ m ) J ( φ m , θ m ) J ( φ m , α m ) J ( θ m , φ m ) J ( θ m , θ m ) J ( θ m , α m ) J ( α m , φ m ) J ( α m , θ m ) J ( α m , α m ) , (44)in which J ( x t , x r ) is defined as J ( x t , x r ) , − E y | θ (cid:20) ∂ L ( y ) ∂x t ∂x r (cid:21) . (45)To evaluate the parameters of J θ , we consider the following Lemma 1. Lemma . The trace and Kronecker product operations have the following respective properties[33], [34] x H Ay = tr (cid:0) Ax H y (cid:1) = tr (cid:0) Ayx H (cid:1) , (46) ( A ⊗ B ) ( C ⊗ D ) = ( AC ) ⊗ ( BD ) . (47) The entries of the matrix J θ are given as follows J ( φ m , φ m ) = 2 α m π E s σ tr (cid:18) A H A (cid:0) sin φ m ˜ a t ( φ m )˜ a H t ( φ m ) (cid:1) ⊗ (cid:0) a r ( θ m ) a H r ( θ m ) (cid:1)(cid:19) , (48) J ( φ m , θ m ) = − α m π E s σ Re (cid:26) tr (cid:18) A H A (cid:0) sin φ m ˜ a t ( φ m ) a T r ( θ m ) (cid:1) ⊗ (cid:0) sin θ m a t ( φ m )˜ a T r ( θ m ) (cid:1)(cid:19)(cid:27) , (49) J ( φ m , α m ) = − α m πE s σ Im (cid:26) tr (cid:18) A H A (cid:0) sin φ m a ∗ t ( φ m )˜ a H t ( φ m ) (cid:1) ⊗ (cid:0) a r ( θ m ) a H r ( θ m ) (cid:1)(cid:19)(cid:27) , (50) J ( θ m , θ m ) = 2 α m π E s σ tr (cid:18) A H A (cid:0) a ∗ t ( φ m ) a T t ( φ m ) (cid:1) ⊗ (cid:0) ˜ a ∗ r ( θ m )˜ a T r ( θ m ) sin θ m (cid:1)(cid:19) , (51) J ( θ m , α m ) = 2 α m πE s σ Im (cid:26) tr (cid:18) A H A (cid:0) a ∗ t ( φ m ) a T t ( φ m ) (cid:1) ⊗ (cid:0) sin θ m a r ( θ m )˜ a T r ( θ m ) (cid:1)(cid:19)(cid:27) , (52) J ( α m , α m ) = 2 E s σ tr (cid:18) A H A (cid:0) a ∗ t ( φ m ) a T t ( φ m ) (cid:1) ⊗ (cid:0) a r ( θ m ) a H r ( θ m ) (cid:1)(cid:19) . (53)The detail derivation of the entries is relegated to Appendix B. Remark.
As the essence of beam alignment is to select the beam pair that meets a target rate,when the location information is accurately known at the BS and the UE, the results convergeto the optimal beam with the actual values of θ to realize a lossless transmission. C. Downlink Channel Estimate and Rate Evaluation
From the FIM, we express the covariance matrix of the estimation error of θ as Σ θ = J − θ .The error covariance matrix of H can be obtained by a Jacobian transformation matrix T givenas Σ H = TΣ θ T T , (54)where T = ∂ h /∂ θ , h = vec( H ) is obtained by the column vectorization of the channel matrix H . The vector h can be expressed as h = vec ( H )= M − X m =0 α m (cid:0) a ∗ t ( φ m ) ⊗ a H r ( θ m ) (cid:1) . (55)Consequently, we obtain the entries of T as T = (cid:20) ∂ h ∂φ m , ∂ h ∂θ m , ∂ h ∂α m (cid:21) , (56) where ∂ h ∂φ m = − αjπ sin φ m (˜ a t ( φ m ) ⊗ a r ( θ m )) , (57) ∂ h ∂θ m = αjπ sin θ m ( a ∗ t ( φ m ) ⊗ ˜ a ∗ r ( θ m )) , (58) ∂ h ∂α m = ( a ∗ t ( φ m ) ⊗ a r ( θ m )) . (59)The rate can then be computed with the estimated channel ˆ H = H + E as follows R d ( u op , v op ) = log E s (cid:12)(cid:12)(cid:12) u H op ˆHv op (cid:12)(cid:12)(cid:12) σ + E s E (cid:26)(cid:12)(cid:12)(cid:12) u H op (cid:16) ˆH − H (cid:17) v op (cid:12)(cid:12)(cid:12) (cid:27) = log E s (cid:12)(cid:12)(cid:12) u H op ˆHv op (cid:12)(cid:12)(cid:12) σ + E s E n(cid:12)(cid:12) u H op Ev op (cid:12)(cid:12) o , (60)where E ∼ CN ( , Σ H ) . V. N UMERICAL R ESULTS
In this section, we present the simulation setup and show the performance of the proposedbeam alignment algorithm.
A. Simulation Setup
We consider a scenario with two potential reflecting points as shown in Fig. 1. The actuallocation coordinates of the BS, two reflecting points and the UE are given as l BS = [0 , , l = [50 , , l = [50 , − and l UE = [100 , respectively. Furthermore, the maximum locationerror of the BS, two reflecting points and the UE when observed from the BS are given by r BSBS = 0 , r BS = 11 , r BS = 15 and r BSUE = 13 respectively, while the maximum location error ofthe BS and the reflecting points when observed from the UE are given as r UEBS = 0 , r UE = 18 , r UE = 17 and r UEUE = 7 respectively, where all measurements are in meters. Furthermore, themaximum location estimation errors are only used to validate the proposed algorithm and alsoallow us to compare the proposed beam alignment scheme with the two-step beam alignment in[23].As discussed in Section II, we focus on the initial access phase, and the simulations are per-formed over independent blocks. The location estimation and beam alignment are performed -20 -15 -10 -5 0 5 1002468101214 Fig. 4. Rate vs SNR for varying number of antennas in the uncertainty region. at the beginning of each block as summarized in Algorithm I. The metric used to select a beamis the rate loss of the beam compared with the optimal beam pair, where the optimal beamspair are achieved with the precise location information and perfect channel state information.We set the target rate to of the optimal rate, i.e., R = 0 .
95 max u , v R d ( u , v ) except statedotherwise, the size of the window is set to the number of beams that can be processed in eachslot (i.e., W BS = N b ). B. Results and Discussion
Fig. 4 shows the rate as a function of the SNR obtained from the beam vectors correspondingto the initial estimate of ˆ φ m and ˆ θ m in the uncertainty region. The results are shown with varyingnumber of antennas. A performance loss can be observed when compared with the rates obtainedfrom selecting the optimal beams. From (5) and (6), as the number of antennas increases, thebeamwidth becomes narrow, and the severity of beam misalignment increases. As observed fromthe figure, the use of a low number of array antenna (lower array gain) may outperform a highnumber of antenna array depending on the severity of the location estimation error. In addition,the result shows that although location information with some degree of precision is expected tobe available to 5G systems, beam alignment is required in the uncertainty region of the locationinformation especially for a large number of antenna arrays with narrow beamwidth. -20 -15 -10 -5 0 5 10 15 20024681012141618 Fig. 5. Comparison of rate vs SNR, with varying number of antennas. Fixed parameters: N b = 5 , N = 100 . In Fig. 5, we compare the proposed beam alignment scheme with the optimal beam selectionscheme. When Fig. 5 is compared with Fig. 4, the effectiveness of the proposed beam alignmentcan be observed from the rates achieved. When the proposed beam alignment is compared withthe optimal aligned beams plots, a performance gap is observed which is due to the beamalignment overhead given by (34). However, to achieve the optimal beam pair, we assume thatthe channel information of each node is perfectly known which is difficult to achieve in practice.Furthermore, the result shows that exploiting location information can further reduce the beamalignment overhead with a slight penalty on the performance. The beam alignment overheadcan be reduced with high precision in the location information. In addition, it can be observedthat the gap between the proposed beam alignment scheme and the optimal aligned beam plotsincreases with an increasing number of antenna which is mainly due to the fact that as thenumber of antenna increases, the beamwidth decreases. Hence, as the number of beams in theuncertainty region increases, the number of slots N a required for beam alignment also increases.The proposed algorithm is compared with the results obtained from the Two-Step algorithmin [23], with N t = N r = 64 antennas in Fig 6. From Fig 6, it can be observed that the proposedscheme outperforms the two step algorithm proposed in [23], even though the penalty of beamalignment is considered in our proposed scheme. This is because in the proposed algorithm, the -20 -15 -10 -5 0 5 102468101214 Fig. 6. Comparison of proposed scheme and the two step algorithm in [23]. Fixed parameters: N t = N r = 64 , N b = 5 , N = 100 . beam alignment is jointly coordinated by the BS and UE, while in [23], the beam alignment iscarried out independently at each node.Fig. 7 compares the effective rates obtained with the channel estimate as a function of SNR.The plots without location information are obtained by transmitting with the beam set ( V and U ). With the knowledge of the location uncertainty region, beam alignment is performed usingthe beam set ( B m BS and B m UE ), where m correspond to the LOS path in this plot. In this result,we evaluate the FIM given by (44) and the rates are computed from (60), where the expectationis taken over the channel estimation error matrix. In the plot of the exhaustive search withoutlocation information, the channel is estimated by transmitting symbols with each of the 16-dimensional beams vectors in V and U since N t = N r = 16 . A performance gap can be observedwhen the proposed beam alignment is compared with the exhaustive search without locationinformation. The improvement in the proposed algorithm is due to the fast beam alignmentprocedure discussed in Section III-B3, leading to a small beam alignment penalty when comparedto the exhaustive search. With the location information and maximum location estimation errorknown a priori , the channel parameters can be estimated by transmitting with the few beamsin B m BS and B m UE which correspond to the set of beams vectors in the uncertainty region of the -20 -15 -10 -5 0 5 10 15 20024681012141618 Fig. 7. Effective rates with channel estimate as a function of SNR. Fixed parameters: N t = N r = 16 , N b = 5 , N = 100 , R = max u , v R d ( u , v ) . m -th path. We plot the effective rate of the exhaustive search and the proposed beam alignmentalgorithm in the location uncertainty region. It is observed that for a fixed N b and N , theperformance of both schemes is quite close to the optimal rate plot. This is because there arefew beams in the location uncertainty region and hence, the beam alignment overhead is low.As the number of beams in the uncertainty region increases due to narrow beams at the BSand UE (increasing N r and N t ), the effect of the search window can be observed, and theefficiency of the proposed scheme with search window become more noticeable as shown inFig. 8. In the figure, the effective rate of the proposed scheme is plotted showing the effectof the proposed window for varying number of antennas. As the number of antennas increasefrom N t = N r = 16 to N t = N r = 64 , the number of beams in the uncertainty region alsoincrease, thereby requiring more slots for beam alignment. However, with the implementation ofthe search window, the proposed scheme is able to achieve fast beam alignment compared to theplot of the proposed beam alignment scheme without the implementation of the search window,hence, a better effective rate performance is achieved as shown in the result. The improvementobtained from the use of the search window in the proposed scheme is as a result the reductionin the number of beams to be searched after determining the direction of search in the window. -20 -15 -10 -5 0 5 10 15 20024681012141618 Fig. 8. Comparison of the proposed beam alignment scheme with and without search window in the uncertainty region. Fixedparameters: N b = 5 , N = 100 , R = max u , v R d ( u , v ) . From Fig. 7 and Fig. 8, it can be deduced that with high precision in location information,there is no need to estimate the channel with a large number of beam vectors from V and U since similar performance can be achieved with much fewer beam set in the uncertainty region.Intuitively, with some degree of precision of the location information at the BS and UE andthe knowledge of the maximum location error, the number of beams required to achieve thebest estimate of θ and φ can be reduced. Hence, exploiting location information in the proposedscheme reduces the beam alignment overhead.VI. C ONCLUSION
In this paper, we propose a coordinated beam alignment algorithm which exploits the noisylocation information of the UE, and potential reflecting points. The proposed beam alignmentenables the UE and BS to jointly coordinate the beam alignment to combat the path-loss in themmWave range. The scheme speeds up the beam alignment procedure by focusing the searchin a specific region bounded by the location error margin. Furthermore, an intelligent search isperformed in this specific region by searching across a small window of beams to determinethe direction of the actual beam. The numerical results show that the proposed algorithm can significantly improve the beam alignment speed when compared to the scenario with perfectchannel information and other existing schemes.A PPENDIX AP ROOF OF P ROPOSITION I Proof:
Without loss of generality, we define ǫ m = γ m − ˆ φ m as the half angle of the uncertaintyregion in path m from the BS as shown in Fig. 2, where γ m is the angle between the antennanormal and the maximum location error boundary of the node in path m . Then we can express y m = ˆ d BS m sin ˆ φ m , (61) x m = ˆ d BS m cos ˆ φ m , (62)where ˆ d BS m can be obtained from equation (17) for the BS, and (18) for the UE. From simplegeometry, we can obtain tan γ m = y m + r BS m x m = ˆ d BS m sin ˆ φ m + r BS m ˆ d BS m cos ˆ φ m , m = 1 , . . . , M, (63)by solving for γ m , the following can be obtained γ m = arctan ˆ d BS m sin ˆ φ m + r BS m ˆ d BS m cos ˆ φ m ! . (64)By solving for ǫ m = γ m − ˆ φ m , equation (64) can be expressed as ǫ m + ˆ φ m = arctan ˆ d BS m sin ˆ φ m + r BS m ˆ d BS m cos ˆ φ m ! . (65)Following similar steps from (61) to (65) we obtain δ m + ˆ θ m = arctan ˆ d UE m sin ˆ θ m + r UE m ˆ d UE m cos ˆ θ m ! , (66)which concludes the proof. A PPENDIX BP ROOF OF THE ENTRIES OF THE M ATRIX J θ In this section, we present the derivation of the entries of the J θ . Proof:
The proof begins with the calculations of J ( φ m , φ m ) , − E y | θ (cid:20) ∂ L ( y ) ∂φ m (cid:21) . (67)Note that L ( y ) in (45) is given by the log-likelihood function in (41) and by differentiating L ( y ) w.r.t φ m , we obtain ∂L ( y ) ∂φ m = E s σ (cid:20) ( y − m ) H ∂ m ∂φ m + ∂ m H ∂φ m ( y − m ) (cid:21) , (68)taking the second differential of (68) w.r.t φ m we have ∂ L ( y ) ∂φ m = E s σ (cid:20) − ∂ m H ∂φ m ∂ m ∂φ m + ( y − m ) H ∂ m ∂φ m + ∂ m H ∂φ m ( y − m ) − ∂ m H ∂φ m ∂ m ∂φ m (cid:21) . (69)By taking the expectation of (69) w.r.t. L ( y ) we obtain J ( φ m , φ m ) = 2 E s σ (cid:20) ∂ m H ∂φ m ∂ m ∂φ m (cid:21) . (70)Next, we will determine ∂ m H /∂φ m , where m is given by (39) ∂ m H ∂φ m = ∂∂φ m α m (cid:18) a T t ( φ m ) ⊗ a H r ( θ m ) (cid:19) A H = α m jπ sin φ m (cid:18) ˜ a H t ( φ m ) ⊗ a H r ( θ m ) (cid:19) A H , (71)where ˜ a t ( φ m ) = 1 √ N t (cid:20) , e − jπ cos φ m , . . . , ( N t − e − jπ ( N t −
1) cos φ m (cid:21) T . (72)Similarly, ∂ m /∂φ m can be obtained as ∂ m ∂φ m = − A α m jπ sin φ m (cid:18) ˜ a t ( φ m ) ⊗ a r ( θ m ) (cid:19) . (73)By substituting (71) and (73) in (70), we obtain J ( φ m , φ m ) = 2 α m π E s σ tr (cid:18) A H A (cid:0) sin φ m ˜ a t ( φ m )˜ a H t ( φ m ) (cid:1) ⊗ (cid:0) a r ( θ m ) a H r ( θ m ) (cid:1)(cid:19) , (74)where ˜ a r ( θ m ) = 1 √ N r (cid:20) , e − jπ cos θ m , . . . , ( N r − e − jπ ( N r −
1) cos θ m (cid:21) T . (75) Next, we compute J ( φ m , θ m ) , − E y | θ (cid:20) ∂ L ( y ) ∂φ m ∂θ m (cid:21) , (76)where we take the second derivative of (68) with respect to φ m and θ m given as ∂ L ( y ) ∂φ m ∂θ m = E s σ (cid:20) − ∂ m H ∂φ m ∂ m ∂θ m + ∂ m H ∂θ m ∂ m ∂φ m (cid:21) . (77)The partial derivative of m w.r.t θ m can be obtained as ∂ m ∂θ m = α m jπ sin θ m A (cid:0) a ∗ t ( φ m ) ⊗ ˜ a ∗ r ( θ m ) (cid:1) , (78)where the partial derivative of m H w.r.t θ m can be obtained similarly as ∂ m H ∂θ m = − α m jπ sin θ m (cid:0) a T t ( φ m ) ⊗ ˜ a T r ( θ m ) (cid:1) A H . (79)By substituting (71), (73), (78) and (79) into (77), and using the properties in Lemma 1, (76)can be simplified as J ( φ m , θ m ) = − α m π E s σ Re (cid:26) tr (cid:18) A H A (cid:0) sin φ m ˜ a t ( φ m ) a T r ( θ m ) (cid:1) ⊗ (cid:0) sin θ m a t ( φ m )˜ a T r ( θ m ) (cid:1)(cid:19)(cid:27) , (80)where Re { x } denotes the real part of the complex variable x .Next, we compute J ( φ m , α m ) , − E y | θ (cid:20) ∂ L ( y ) ∂φ m ∂α m (cid:21) , (81)which can be obtained by taking the second derivative of (68) with respect to φ m and α m givenas, ∂ L ( y ) ∂φ m ∂α m = E s σ (cid:20) ∂ m H ∂φ m ∂ m ∂α m + ∂ m H ∂α m ∂ m ∂φ m (cid:21) , (82)where ∂ m ∂α m = A (cid:0) a ∗ t ( φ m ) ⊗ a r ( θ m ) (cid:1) , (83)and ∂ m H ∂α m = (cid:0) a T t ( φ m ) ⊗ a H r ( θ m ) (cid:1) A H . (84)By using the properties defined in Lemma 1 and substituting (71), (73), (83) and (84) into (82)we obtain J ( φ m , α m ) = − α m πE s σ Im (cid:26) tr (cid:18) A H A (cid:0) sin φ m a ∗ t ( φ m )˜ a H t ( φ m ) (cid:1) ⊗ (cid:0) a r ( θ m ) a H r ( θ m ) (cid:1)(cid:19)(cid:27) , (85) where Im { x } is the imaginary part of the complex variable x .Next, we determine J ( θ m , θ m ) from J ( θ m , θ m ) , − E y | θ (cid:20) ∂ L ( y ) ∂θ m (cid:21) , (86)similar to (70) as follows J ( θ m , θ m ) = 2 E s σ (cid:20) ∂ m H ∂θ m ∂ m ∂θ m (cid:21) , (87)which can be evaluated by substituting (78) and (79) into (87) as J ( θ m , θ m ) = 2 α m π E s sin θ m σ (cid:20)(cid:0) a T t ( φ m ) ⊗ ˜ a T r ( θ m ) (cid:1) × A H A ( a ∗ t ( φ m ) ⊗ ˜ a ∗ r ( θ m )) (cid:21) = 2 α m π E s σ tr (cid:18) A H A (cid:0) a ∗ t ( φ m ) a T t ( φ m ) (cid:1) ⊗ (cid:0) ˜ a r ( θ m )˜ a H r ( θ m ) sin θ m (cid:1)(cid:19) . (88)Next, we evaluate the entries of J ( θ m , α m ) defined as J ( θ m , α m ) , − E y | θ (cid:20) ∂ L ( y ) ∂θ m ∂α m (cid:21) , (89)where the second derivative of (68) is taken with respect to respect to θ m and α m as follows J ( θ m , α m ) = E s σ (cid:20) ∂ m H ∂θ m ∂ m ∂α m + ∂ m H ∂α m ∂ m ∂θ m (cid:21) , (90)which can be evaluated by substituting (78), (79), (83) and (84) into (90) as follows J ( θ m , α m ) = 2 α m πE s σ sin θ m Im (cid:26)(cid:0) a T t ( φ m ) ⊗ ˜ a T r ( θ m ) (cid:1) × A H A (cid:0) a ∗ t ( φ m ) ⊗ a r ( θ m ) (cid:1)(cid:27) = 2 α m jπE s σ Im (cid:26) tr (cid:18) A H A (cid:0) a ∗ t ( φ m ) a T t ( φ m ) (cid:1) ⊗ (cid:0) sin θ m a r ( θ m )˜ a T r ( θ m ) (cid:1)(cid:19)(cid:27) . (91)Finally, the factor J ( α m , α m ) , − E y | θ (cid:20) ∂ L ( y ) ∂α m (cid:21) (92)can be evaluated as: J ( α m , α m ) = 2 E s σ (cid:20) ∂ m H ∂α m ∂ m ∂α m (cid:21) , (93)where the parameters ∂ m H /∂α m and ∂ m /∂α m can be obtained by substituting (83) and (84)into (93), from which we obtain J ( α m , α m ) = 2 E s σ tr (cid:18)(cid:0) a T t ( φ m ) ⊗ a H r ( θ m ) A H A (cid:0) a ∗ t ( φ m ) ⊗ a r ( θ m ) (cid:1)(cid:19) = 2 E s σ tr (cid:18) A H A (cid:0) a ∗ t ( φ m ) ⊗ a r ( θ m ) (cid:1)(cid:0) a T t ( φ m ) ⊗ a H r ( θ m ) (cid:1)(cid:19) = 2 E s σ tr (cid:18) A H A (cid:0) a ∗ t ( φ m ) a T t ( φ m ) (cid:1) ⊗ (cid:0) a r ( θ m ) a H r ( θ m ) (cid:1)(cid:19) . (94)This concludes the proof of the entries of J θ . R EFERENCES [1] I. Orikumhi, J. Kang, C. Park, J. Yang, and S. Kim, “Location-aware coordinated beam alignment in mmwavecommunication,” in , Oct.2018, pp. 386–390.[2] T. S. Rappaport, S. Sun, R. Mayzus, H. Zhao, Y. Azar, K. Wang, G. N. Wong, J. K. Schulz, M. Samimi, and F. Gutierrez,“Millimeter wave mobile communications for 5G cellular: It will work!”
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