Joint Beam Training and Positioning For Intelligent Reflecting Surfaces Assisted Millimeter Wave Communications
aa r X i v : . [ c s . I T ] S e p Joint Beam Training and Positioning For IntelligentReflecting Surfaces Assisted Millimeter WaveCommunications
Wei Wang,
Member, IEEE , and Wei Zhang,
Fellow, IEEE
Abstract —Intelligent reflecting surface (IRS) offers a costeffective solution to link blockage problem in mmWave commu-nications, and the prerequisite of which is the accurate estimationof (1) the optimal beams for base station/access point (BS/AP)and mobile terminal (MT), (2) the optimal reflection patternsfor IRSs, and (3) link blockage. In this paper, we carry outbeam training design for IRSs assisted mmWave communicationsto estimate the aforementioned parameters. To acquire theoptimal beams and reflection patterns, we firstly perform randombeamforming and maximum likelihood estimation to estimateangle of arrival (AoA) and angle of departure (AoD) of the lineof sight (LoS) path between BS/AP (or IRSs) and MT. Then,with the estimate of AoAs and AoDs, we propose an iterativepositioning algorithm that achieves centimeter-level positioningaccuracy. The obtained location information is not only a fringebenefit but also enables us to cross verify and enhance theestimation of AoA and AoD, and facilitates the prediction ofblockage indicator. Numerical results show the superiority ofour proposed beam training scheme and verify the performancegain brought by location information.
I. I
NTRODUCTION
Millimeter-wave (mmWave) band, ranging from 30GHz to300GHz, has attracted great interests from both academia andindustry for its abundant spectrum resources [1], [2]. TheWi-Fi standard IEEE 802.11ad runs on the 60GHz (V band)spectrum with data transfer rates of up to 7 Gbit/s [3], [4].In 3GPP Release 15, 24.25-29.5GHz and 37-43.5GHz, as themost promising frequencies for the early deployment of 5Gmillimeter wave systems, are specified based on a TDD accessscheme [5]. The millimeter scale wavelength, on one hand,renders massive antennas integratable on an antenna array withportable size [6], and, on the other hand, results in severe freespace path loss especially for non-line-of-sight (NLoS) paths.Directional transmission enabled by beamforming techniquesis an energy efficient transmission solution to compensate forthe path loss in mmWave communications [7]. By properlyadjusting the phase shifts of each antenna elements, it concen-trates the emitted energy in a narrow beam between transmitterand receiver. However, the directional link is easily blocked byobstacles like human bodies, walls, and furniture, attributed tothe millimeter scale wavelength [8]. Once LoS path is blocked,it is highly possible that the blocked link cannot be restoredno matter how the beam direction is adjuested, as the NLoSpaths are not strong enough to serve as a qualified alternative
W. Wang and W. Zhang are with the School of Electrical Engineering andTelecommunications, The University of New South Wales, Sydney, Australia(e-mail: [email protected]; [email protected]). link. Channel measurement campaigns reveal that power of theLoS component is about 13dB higher than the sum of powerof NLoS components [9]. Therefore, blockage is the biggesthindrance to the large scale applications of mmWave band inmobile communication systems.Recently, intelligent reflecting surface (IRS) [10]–[13],a.k.a. reconfigurable intelligent surface (RIS) [14], [15], largeintelligent surface (LIS) [16], passive (intelligent) reflec-tors/mirrors [17]–[19], or programmable metasurface [20]–[22], is proposed as an energy-effective and cost-effectivehardware structure for future wireless communications. IRSis essentially a new type of electromagnetic surface struc-ture which is typically designed by deliberately arranginga set of sophisticated passive scatterers or apertures in aregular array to achieve the desired ability for guiding andcontrolling the flow of electromagnetic waves [23]. Currentapplications of IRS to wireless communications can be cat-egorized into two types, namely IRS modulator and IRS“relay”. In [20]–[22], amplitude/phase modulations over IRSare investigated. Through controlling the reflection coefficientof IRS, the incident carrier wave from a feed antenna can bedigitally modulated without requiring high-performance radiofrequency (RF) chains. A more extensive application of IRSis IRS “relay”, in which the radiated power from BS/APtowards IRS is reflected to MT via intelligently managingthe phase shifters on IRS [10]–[19]. It is noteworthy thatthe rationale behind IRS “relay” and conventional amplify-and-forward (AF) relay is significantly different. AF relayfirstly receives signal and then re-generates and re-transmitssignal. In contrast, IRS only reflects the ambient RF signalsas a passive array and bypasses conventional RF modulessuch as power amplifier, filters, and ADC/DAC [11]. Hence,IRS “relay” incurs no additional power consumption and isfree from thermal noise introduced by RF modules. In thissense, IRS can be regarded as a smart “mirror” that enablesus to change the paradigm of wireless communications fromadjusting to wireless channel to changing wireless channel[14], [24]. As an active way to make wireless channel better,IRS “relay” assisted wireless communications have attractedgreat interests from researchers. In [10], IRS is applied tommWave communications to provide effective reflected pathsand thus enhance signal coverage. In [12], [15], [17], jointoptimization of the transmit beamforming by active antennaarray at the BS/AP and reflect beamforming by passive phaseshifters at the IRS is carried out. In [18], empirical studiesare performed to analyze the capability of signal coverage enhancement for IRSs assisted mmWave MIMO at 28GHz. In[19], the reconfigurable 60GHz IRS is designed, implementedand deployed to strengthen mmWave connections for indoornetworks threatened by blockage. The objective of the work isto validate IRS’s capability to address link blockage problemin mmWave communications, and beam training design is notinvestigated. Although extensive analytical and empirical stud-ies have been done on IRSs assisted wireless communicationsin the aforementioned literature, these work either assumethe availability of channel state information (CSI) or accuratemeasurement of BS/AP, MT and IRS’s position and direction.Current study of channel parameter estimation of IRSsassisted wireless communications either focuses on non-mmWave frequency band or are based on an assumption ofhardware upgrade. In [11], a practical transmission protocoland channel estimation are firstly proposed for an IRS-assistedorthogonal frequency division multiplexing (OFDM) systemunder frequency-selective channels. In [13], by exploiting thechannel correlation among different users, a channel esti-mation scheme with reduced training overhead is proposed.Specifically, with a typical user’s reflection channel vector,estimation of the other users’ reflection channel vector can besimplified as the estimation of a multiplicative coefficient. In[16], to facilitate channel estimation of IRSs assisted link overmmWave band or LoS dominated sub-6GHz band, an upgradeof IRS’s structure is proposed to add a small number ofchannel sensors to sense and process incident signal. Although[16] is intended to mmWave band, the proposed compressivesensing and deep learning algorithms are incompatible tocurrent structure of IRS which is without channel sensors.Due to the deployment of multiple IRSs, beam training ofIRSs assisted mmWave communications requires much heaviertraining overhead than traditional mmWave communications.Also, as the purpose of IRSs is to anti blockage and expandcoverage, an accurate prediction of blockage is essential tobeam selection by BS/AP. In addition, the lack of RF chainsresults in the inability of IRSs to sense signal, which furthercomplicates beam training for the paths assisted by IRSs.These three features jointly render traditional beam trainingmethods [25], [26] incompetent in IRSs assisted mmWavecommunications. Despite the aforementioned new challengesof integrating IRSs to mmWave communications, a notableadvantage is that the estimation of path parameters, e.g.,AoA/AoD and blockage indicator, can be cross verified, thanksto the relatively large number of deployed IRSs. Specifically,three accurate estimates of AoA/AoD, associated with otheressential information, e.g., direction of arrays, can yield thelocation of MT, and the location of MT will in turn reproducethe path parameters. In this way, the path parameters of IRSsassisted mmWave MIMO can be enhanced according to theirgeometric relationship. To estimate the channel parameters ofIRSs assisted mmWave communications, we have made thefollowing contributions in this paper: • We propose a simple and flexible beam training methodfor IRSs assisted mmWave MIMO by breaking it downinto several mathematically equivalent sub-problems, andwe further perform random beamforming and maximumlikelihood (ML) estimation to jointly estimate AoA and AoD of the dominant path in each sub-problem. Theproposed scheme does not require feedback from MT attraining stage, and thus can be performed in a broadcast-ing manner. Hence, the required training overhead willnot increase over MT number. • We prove the uniqueness of the AoA and AoD estimatedby beam training with random beamforming. We furtherstudy the impact of training length, and we prove thatlarger training length almost surely results in smallerpairwise error probability of AoA, AoD pair. • By sorting the reliability of the estimated AoA, AoDpairs, we propose an iterative positioning algorithm toestimate the location of MT, and, through numericalanalysis, we show that the algorithm achieves centimeter-level positioning accuracy. • With the estimated position of MT, we propose to crossverify and enhance the estimation of path parameters, i.e.,AoA and AoD, according to their geometric relationship.We further propose an accurate method of blockageprediction by comparing the ML estimate of pathloss andMT position based estimate of pathloss.Numerical results show the superiority of our proposed beamtraining scheme and verify the performance gain brought bylocation information.The rest of the paper is organized as follows. Section IIintroduces the system model. In Section III, we break downthe beam training design of IRSs assisted mmWave communi-cations. In Section IV, we propose beam training with randombeamforming. In Section V, we study the interplay betweenpositioning and beam training. In Section VI, numerical resultsare presented. Finally, in Section VII, we draw the conclusion.
Notations:
Column vectors (matrices) are denoted bybold-face lower (upper) case letters, x ( n ) denotes the n -thelement in the vector x , ( · ) ∗ , ( · ) T and ( · ) H represent conju-gate, transpose and conjugate transpose operation, respectively, || · || denotes the Frobenius norm of a vector or a matrix, ⊙ is Hadamard product. Subtraction and addition of the cosineAoAs/AoDs are defined as θ ⊖ φ , ( θ − φ + 1) mod 2 − and θ ⊕ φ , ( θ + φ + 1) mod 2 − to guarantee the resultis within the range [ − , .II. S YSTEM M ODEL
Consider a communication link between the BS/AP andan MT operating in mmWave band, where both ends adoptuniform linear array (ULA) antenna structure. To reducewireless link blockage rate and thus guarantee the reliablelinkage between BS and MT, a number of IRSs are deployedin the cell as shown in Fig. 1, and BS/AP is able to controlIRSs via cable or lower frequency radio link.The channel response between BS/AP and MT without theassistance of IRSs is represented as [26] H BM = ζ LoS δ a M ( θ BM, ) a HB ( φ BM, )+ L X l =2 δ l a M ( θ BM,l ) a HB ( φ BM,l ) (1)where ζ LoS ∈ { , } is the indicator of blockage of theLoS path, and δ l , θ BM,l and φ BM,l are channel gain, cosine mmWaveBS/AP IRScontrollerIRS 2blockageLink ALink BIRS controllerIRS 1 Link C
Fig. 1. Illustration of IRSs assisted mmWave communications of AoA, and cosine of AoD of the l -th path, respectively.The parameters ( ζ LoS , δ , θ BM, , φ BM, ) characterize LoSpath, which are of particular interest to us in mmWavecommunications. According to [14], the path gain of LoS is δ = λe − j πdBM πd BM , where λ is the wavelength, and d BM is thedistance between BS and MT. Further, the steering vectors aregiven by a M ( θ BM,l ) = [1 , e jπ θ BM,l , · · · , e jπ ( N M − θ BM,l ] T a B ( φ BM,l ) = [1 , e jπ φ BM,l , · · · , e jπ ( N B − φ BM,l ] T where N B is the number of antennas of BS/AP, N M is thenumber of antennas of MT.We also assume that IRSs adopt ULA antenna structure.Thus, the channel response of the reflected path from BS toMT assisted (reflected) by the i -th IRS is H BR i M = ζ V LoS,i ¯ δ BR i M a M ( θ R i M ) a HR i ( φ R i M )diag { ¯ g i } a R i ( θ BR i ) a HB ( φ BR i )= ζ V LoS,i δ BR i M (¯ g i ) a M ( θ R i M ) a HB ( φ BR i ) (2)where ζ V LoS,i ∈ { , } is the indicator of blockage of the pathreflected by the i -th IRS and ¯ δ BR i M = √ ξλe − j π ( dBRi + dRiM ) π ( d BRi + d RiM ) [14], in which ξ is reflection loss, d BR i is the distance betweenBS and the i -th IRS, d R i M is the distance between the i -thIRS and MT. The equivalent path gain of the IRS reflectedpath can be written as δ BR i M (¯ g i ) , ¯ δ BR i M a HR i ( φ R i M ) diag { ¯ g i } a R i ( θ BR i )= ¯ δ BR i M a HR i ( φ R i M ⊖ θ BR i )¯ g i (3)The steering vector a R i ( φ R i M ) is given by a R i ( φ R i M ) = [1 , e jπ φ RiM , · · · , e jπ ( N Ri − φ RiM ] T (4)where N R i is the number of passive reflectors of the i -th IRS.Hence, the channel response between BS and MT with the assistance of N IRS
IRSs is represented as H = H BM + N IRS X i =1 γ i H BR i M = ζ LoS δ a M ( θ MB, ) a HB ( φ MB, ) | {z } LoS component + L X l =2 δ l a M ( θ MB,l ) a HB ( φ MB,l ) | {z } NLoS component + N IRS X i =1 γ i ζ V LoS,i δ BR i M (¯ g i ) a M ( θ R i M ) a HB ( φ BR i ) | {z } V LoS component (5)where γ i = (cid:26) , when the i th IRS is activated0 , when the i th IRS is deactivated indicates the activation status of the i -th IRS and γ i can beconfigured by BS/AP.When beam pattern of the reflection vector ¯ g i is omnidi-rectional, IRS works as a scatterer that diffuses the energyradiated from BS. When ¯ g ∗ i = a R ( φ R i M ⊖ θ BR i ) , IRS worksas a “mirror” that builds a virtual LoS (VLoS) path betweenBS and MT, and thus the energy from BS will be concentratedon MT, and φ R i M ⊖ θ BR i is termed as the optimal reflectionangle of the i -th VLoS path. We can categorize channelcomponents of H into three types as in Eq. (5), namely LoS path component , VLoS path component , and
NLoS pathcomponent . LoS path component is the direct path between BSand MT, VLoS path component consists of the paths betweenBS and MT reflected by IRSs, and NLoS path componentconsists of the paths between BS and MT reflected by scatters,e.g., walls, human bodies, and etc.III. B
REAKDOWN OF B EAM T RAINING FOR
IRS S A SSISTED M M W AVE
MIMOAs NLoS path component usually varies fast and its weightto the channel is marginal especially in mmWave band, weare more interested in LoS path and VLoS paths. Hence,beam training of IRSs assisted mmWave MIMO intends toestimate (1) the optimal reflection angle φ R i M ⊖ θ BR i of IRSsand (2) the path parameters ( ζ BM, , δ BM, , θ BM, , φ BM, ) of the LoS path and ( ζ BR i M , δ BR i M ( g ∗ i ) , θ R i M , φ BR i ) ofthe VLoS paths. For conventional mmWave communications,training overhead can be significantly reduced by exploitingthe sparse nature of mmWave channel [26], [27]. However,with the assistance of IRSs, the sparse channel of mmWaveband is artificially converted into rich scattering channel. Theincreased scattering effect, together with the unknown optimalreflection angle, jointly complicates the process of beamtraining. In this section, to make the over-complicated problemtractable, we propose to beak down beam training of IRSsassisted mmWave MIMO into two sub-problems, and we willfurther show that the two sub-problems are mathematicallyequivalent.At first, it is noteworthy that AoA/AoD of the LoS pathbetween IRSs and BS/AP can be accurately pre-measured,since both IRSs and BS/AP are pre-configured. Thus, θ BR i mmWaveBS/AP RandombeamformingMT Step 1 Step 2 mmWaveBS/APIRScontroller IRS RandombeamformingDirectionalbeamforming MT
Fig. 2. Two steps of beam training with random beamforming in IRSs assistedmmWave communications and φ BR i are used as prior knowledge hereafter. Then, beamtraining of IRSs assisted mmWave MIMO is carried out in thefollowing two steps as illustrated in Fig. 2. Step 1.
De-activate all the IRSs, and estimate the parameters ( δ BM, , θ BM, , φ BM, ) of LoS pathTo estimate the parameters, measures of channel are col-lected via Tx/Rx random beamforming in BS/AP side andMT side, i.e., y = p P T x m H H BM f s + m H ¯ w = p P T x ζ LoS δ BM, m H a M ( θ BM, ) a HB ( φ BM, ) f + L X l =2 p P T x δ BM,l m H a M ( θ BM,l ) a HB ( φ BM,l ) f | {z } ν + m H ¯ w (6)where P T x is transmit power, ¯ w ∼ CN ( , σ w I N M ) is thezero-mean complex Gaussian additive noise, s = 1 is thepilot signal sent by the user, f and m are transmit randombeamforming vector at BS/AP side and receive random beam-forming vector at MT side , respectively, and the entries of f and m are phase-only complex variables with invariableamplitude [28], i.e., f = 1 √ N B (cid:0) e jπ̺ , e jπ̺ , · · · , e jπ̺ NB (cid:1) T m = 1 √ N M (cid:0) e jπσ , e jπσ , · · · , e jπσ NM (cid:1) T ̺ n B is the phase shift value of the n B -th analog phase shifterin BS/AP side, σ n M is the phase shift value of the n M -thanalog phase shifter in MT side.As NLoS paths are much weaker than LoS path in mmWaveband, i.e., δ BM,l ( l = 2 , · · · , L ) are small compared to δ BM, ,we are very less likely to build an effective communicationlink via NLoS paths. Hence, the AoA, AoD pair that weare interested in is merely ( ζ LoS , δ
BM, , θ BM, , φ BM, ) , andthe term ν will be treated as interference. Considering thesmall scale and randomness of δ BM,l ( l = 2 , · · · , L ) , weassume that ν follows complex Gaussian distribution for the A good random beamforming codebook can be derived offline by highperformance computers, and they will be pre-configured in BS/AP, IRS andMT side. simplicity of analysis . Then, the beam training problem forIRSs assisted mmWave MIMO communications is formulatedas the estimation of ( ζ LoS , δ
BM, , θ BM, , φ BM, ) from thefollowing received signal y = p P T x ζ LoS δ BM, m H a M ( θ BM, ) a HB ( φ BM, ) f + ν + m H ¯ w (7)Adding the subscript n to y to denote the received signal inthe n -th time slot, we have y n = p P T x ζ LoS δ BM, m Hn a M ( θ BM, ) a HB ( φ BM, ) f n + ν n + m Hn ¯ w = p P T x ζ LoS δ BM, ( f Tn ⊗ m Hn ) b ( θ BM, , φ BM, )+ ν n + m Hn ¯ w where b ( θ BM, , φ BM, ) , vec ( a M ( θ BM, ) a HB ( φ BM, )) .To estimate AoA and AoD, N channel measurements are tobe collected and concatenated, and its vector form is derivedas y = p P T x ζ LoS δ BM, Db ( θ BM, , φ BM, ) + ν + w | {z } n (8)where y = [ y , y , · · · , y N ] T D = [ f ⊗ m ∗ , f ⊗ m ∗ , · · · , f N ⊗ m ∗ N ] T ν = [ ν , ν , · · · , ν N ] T ∼ CN ( , σ ν I N ) w = (cid:2) m H ¯ w , m H ¯ w , · · · , m HN ¯ w (cid:3) T Since E ( ¯ w ( ι ) ¯ w ∗ ( ι )) = E (cid:0) m Hι ¯ w ι ¯ w Hι m ι (cid:1) = σ , E ( ¯ w ( ι ) ¯ w ∗ ( κ )) = E (cid:0) m Hι ¯ w ι ¯ w Hκ m κ (cid:1) = 0 , ∀ ι = κ the covariance of the equivalent noise w is thus E ( ww H ) = σ w I N . Let n , ν + w , as ν and w are independent of eachother, we have n ∼ CN (cid:0) , (cid:0) σ w + σ ν (cid:1) I N (cid:1) .Based on the above analysis, beam training for the linkbetween BS/AP and MT is summarized as follows. Sub-problem 1:
How to accurately estimate the parameterset ( ζ LoS , δ
BM, , θ BM, , φ BM, ) from y . Step 2.
Activate the i -th IRS, de-activate the rest IRSs, andestimate the parameters ( δ BR i M , θ R i M , φ R i M ⊖ θ BR i ) of the i -th VLoS path. Repeat the above process for the rest IRSs.As φ BR i is known, with the transmit beamforming vector f = a B ( φ BRi ) √ N B , BS/AP is able to concentrate its power towardsIRSs via transmit beamforming. Simultaneously, IRS performs Although we assume that ν follows Gaussian distribution in theoreticalanalysis, the channel model to be applied in numerical simulation is still thecluster based model as in (1). passive random reflection and MT performs receive randombeamforming , the received signal at MT side is written as y = p P T x m H ( H BM + H BR i M ) a B ( φ BR i ) √ N B + m H ¯ w = p N B P T x ζ V LoS,i δ BR i M m H a M ( θ R i M ) a HR i ( φ R i M ⊖ θ BR i )¯ g i + p P T x ζ LoS δ m H a M ( θ MB, ) a HB ( φ MB, ) a B ( φ BR i ) √ N B | {z } ν + L X l =2 p P T x δ l m H a M ( θ MB,l ) a HB ( φ MB,l ) a B ( φ BR i ) √ N B | {z } ν + m H ¯ w | {z } w (9)The interference term ν and ν are insignificant due to(1) the small NLoS path coefficients δ l ( l = 2 , · · · , L ) in mmWave band, (2) the spatial filtering impact, i.e., a HB ( θ MB,l ) a B ( φ BR i ) ≈ , ( l = 1 , , · · · , L ) for | φ BR i − θ MB,l | > N B .Similar to (8), by concatenating N channel measurements,we have y = p N B P T x ζ V LoS,i δ BR i M Db ( θ R i M , φ R i M ⊖ θ BR i )+ ν + ν + w | {z } n (10)where D = [ g ⊗ m ∗ , g ⊗ m ∗ , · · · , g N ⊗ m ∗ N ] T Based on the above analysis, beam training for the reflectedpath between BS/AP and MT assisted by the i -th IRS issummarized as follows. Sub-problem 2:
How to accurately estimate the parameterset ( ζ V LoS,i , δ BR i M , θ R i M , φ R i M ⊖ θ BR i ) from y . Remark 1.
We can find that Sub-problem 1 and Sub-problem2 are mathematically equivalent. Owing to the flexible controlover IRS, we are capable to decompose the complicated non-sparse channel estimation problem of IRSs assisted mmWaveMIMO into a set of equivalent sub-problems of beam trainingdesign.
IV. B
EAM T RAINING W ITH R ANDOM B EAMFORMING
In this section, ML estimation method is applied to estimatethe path parameters ( δ, θ, φ ) of LoS/VLoS paths from channelmeasurements sampled by random Rx/Tx beamforming. Fur-thermore, the feasibility of random beamforming based beamtraining is verified. A. Maximum Log-likelihood Estimation of ( δ, θ, φ ) For conciseness of expression, we write the unified modelof sub-problem 1 and sub-problem 2 as y = ζδ Db ( θ, φ ) + n (11)where ζ is the indicator of blockage, δ is equivalent path gain ( δ = √ P T x δ BM, or δ = √ P T x N B δ BR i M ), θ is cosine AoA, φ is equivalent cosine AoD ( φ = φ BM, or φ = φ R i M ⊖ θ BR i ),and b ( θ, φ ) , vec ( a Rx ( θ ) a HT x ( φ )) .It is noteworthy that estimation of ( δ, θ, φ ) should be per-formed merely when ζ = 1 , as the measurement vector y giventhat ζ = 0 contains no information about ( δ, θ, φ ) . Therefore,we estimate the parameters ( δ, θ, φ ) through maximizing log-likelihood function under the assumption that ζ = 1 , i.e., (ˆ δ, ˆ θ, ˆ φ ) = arg max δ,θ,φ L ( δ, θ, φ ) (12)where L ( δ, θ, φ ) = log P ( y | ζ = 1 , δ, θ, φ )= − N log π − N log σ − k y − δ Db ( θ, φ ) k σ (13)and the conditional probability is P ( y | ζ, δ, θ, φ ) = 1 π N det( σ I N ) e − ( y − ζδ Db ( θ,φ )) H ( y − ζδ Db ( θ,φ )) σ (14)
1) Estimation of δ : Before the derivation of ˆ θ, ˆ φ , weshould find the expression of ˆ δ . To this end, we ignore termsindependent thereof and set ∂ L ( δ, θ, φ ) ∂δ = 0 (15)Expanding Eq. (15), we have Re n ( Db ( θ, φ )) H ( y − δ Db ( θ, φ )) o = 0 (16)From Eq. (16), the optimal ˆ δ is derived as ˆ δ = b H ( θ, φ ) D H y k Db ( θ, φ ) k (17)
2) Estimation of θ and φ : Next, we will jointly estimate θ and φ . Substituting Eq. (17) into Eq. (13), we have L ( δ, θ, φ )= − N log π − N log σ − (cid:13)(cid:13)(cid:13) y − Db ( θ,φ ) b H ( θ,φ ) D H k Db ( θ,φ ) k y (cid:13)(cid:13)(cid:13) σ (18)Since (cid:13)(cid:13)(cid:13)(cid:13) y − Db ( θ, φ ) b H ( θ, φ ) D H k Db ( θ, φ ) k y (cid:13)(cid:13)(cid:13)(cid:13) = y H ( I − Db ( θ, φ ) b H ( θ, φ ) D H k Db ( θ, φ ) k ) y , (19)the beam training problem is formulated as P θ,φ (cid:13)(cid:13)(cid:13)(cid:13) b H ( θ, φ ) D H k Db ( θ, φ ) k y (cid:13)(cid:13)(cid:13)(cid:13) s.t. − ≤ θ < − ≤ φ < P1 is a non-convex problem. However, as there are only tworeal-valued variables to be estimated, a simple but efficienttwo-step algorithm can be readily applied to solve P1. To facilitate the development of the two-step algorithm, we firstlyderive the partial derivatives of the objective function asfollows.Let g ( θ, φ ) , (cid:13)(cid:13)(cid:13) b H ( θ,φ ) D H k Db ( θ,φ ) k y (cid:13)(cid:13)(cid:13) , the derivative of g ( θ, φ ) with respect to θ is ∂g ( θ, φ ) ∂θ = ∂ b H ( θ,φ ) D H yy H Db ( θ,φ ) ∂θ b H ( θ, φ ) D H Db ( θ, φ ) − b H ( θ, φ ) D H yy H Db ( θ, φ )( b H ( θ, φ ) D H Db ( θ, φ )) ∂ b H ( θ, φ ) D H Db ( θ, φ ) ∂θ =2 Re b H ( θ, φ ) D H yy H D ∂ b ( θ,φ ) ∂θ b H ( θ, φ ) D H Db ( θ, φ ) − b H ( θ, φ ) D H yy H Db ( θ, φ )( b H ( θ, φ ) D H Db ( θ, φ )) b H ( θ, φ ) D H D ∂ b ( θ, φ ) ∂θ ! (20)where ∂ b ( θ,φ ) ∂θ = vec (cid:0) ( a Rx ( θ ) ⊙ ϑ Rx ) a HT x ( φ ) (cid:1) and ϑ Rx =[0 , jπ, · · · , jπ ( N r − T . Similarly, the derivative of g ( θ, φ ) with respect to φ is ∂g ( θ, φ ) ∂φ = 2 Re b H ( θ, φ ) D H yy H D ∂ b ( θ,φ ) ∂φ b H ( θ, φ ) D H Db ( θ, φ ) − b H ( θ, φ ) D H yy H Db ( θ, φ )( b H ( θ, φ ) D H Db ( θ, φ )) b H ( θ, φ ) D H D ∂ b ( θ, φ ) ∂φ ! (21)where ∂ b ( θ,φ ) ∂φ = vec (cid:0) a Rx ( θ )( a T x ( φ ) ⊙ ϑ T x ) H (cid:1) and ϑ T x =[0 , jπ, · · · , jπ ( N t − T .Further, the two-step algorithm is explained as follows. Step 1. Joint AoA and AoD Coarse Search
Set quantization level Z θ and Z φ , and then exhaustivelysearch for the N pk largest maxima that satisfy g ( θ ˆ ι , φ ˆ κ ) > g ( θ ˆ ι − , φ ˆ κ ) g ( θ ˆ ι , φ ˆ κ ) > g ( θ ˆ ι +1 , φ ˆ κ ) g ( θ ˆ ι , φ ˆ κ ) > g ( θ ˆ ι , φ ˆ κ − ) g ( θ ˆ ι , φ ˆ κ ) > g ( θ ˆ ι , φ ˆ κ +1 ) over the discrete grid D , (cid:26) ( θ ι , φ κ ) (cid:12)(cid:12)(cid:12) θ ι = − ι − Z θ , ι = 1 , , · · · , Z θ ,φ κ = − κ − Z φ , κ = 1 , , · · · , Z φ (cid:27) (22) Step 2. Joint AoA and AoD Fine Search
For a given discrete maximum ( θ ˆ ι , φ ˆ κ ) T , run gradientdescent search starting from ( θ (1) , φ (1) ) T = ( θ ˆ ι , φ ˆ κ ) T asfollows (cid:18) θ ( i +1) φ ( i +1) (cid:19) = (cid:18) θ ( i ) φ ( i ) (cid:19) ⊕ λ ∂g ( θ,φ ) ∂θ (cid:12)(cid:12) θ = θ ( i ) ∂g ( θ,φ ) ∂φ (cid:12)(cid:12) φ = φ ( i ) ! (23)where λ is the preset step size. The iteration stops when ( θ ( i +1) ⊖ θ ( i ) ) + ( φ ( i +1) ⊖ φ ( i ) ) ≤ ǫ , where ǫ is a presetparameter. Repeat the above operations over the rest N pk − maximaderived in Step 1, and select the best one as (ˆ θ, ˆ φ ) . Then, theexact value of the estimated path gain ˆ δ can be subsequentlyobtained by substituting (ˆ θ, ˆ φ ) into (17). Remark 2.
A notable advantage of the proposed scheme isthat it does not need feedback at random beamforming stage,which enables BS/AP and IRSs to broadcast its pilot signal.Therefore, its training overhead does not increase over thenumber of MTs.B. Uniqueness of The Estimated AoA and AoD Pair
To delve into the effectiveness of beam training with randombeamforming, conditions under which ( θ, φ ) can be accuratelyestimated from the measurement signal y are studied in theideal scenario without noise or interference.Firstly, two definitions of uniqueness are introduced asfollows.(1) Uniqueness of measurement signal representation ,namely y = δ Db ( θ, φ ) = e δ Db ( e θ, e φ ) , ∀ e δ ∈ C , ∀ ( e θ, e φ ) = ( θ, φ ) (24)(2) Uniqueness of estimated AoA and AoD pair , namely (cid:13)(cid:13)(cid:13)(cid:13) b H ( θ, φ ) D H k Db ( θ, φ ) k y (cid:13)(cid:13)(cid:13)(cid:13) > (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) b H ( e θ, e φ ) D H k Db ( e θ, e φ ) k y (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) , ∀ ( e θ, e φ ) = ( θ, φ ) (25)Uniqueness of measurement signal representation means thatany AoA, AoD pair ( e θ, e φ ) that differs from ( θ, φ ) cannotconstruct the measurement signal y . It is an inherent propertyof the sampling method , which is primarily determined by D . By contrast, uniqueness of the estimated AoA and AoDdepends on both sampling method and estimation method . Itindicates that AoA, AoD pair can be accurately estimated fromthe measurement signal y using a specific estimation method.In the following Theorem, we will study the relationshipbetween the above two types of uniqueness. Theorem 1.
As long as uniqueness of measurement signalrepresentation is satisfied, ML method is capable to accuratelyestimate the AoA, AoD pair.Proof.
See Appendix A.According to Theorem 1, the uniqueness of AoA and AoDestimation is equivalent to the uniqueness of measurementsignal representation, which means we just need to investigatethe conditions on which uniqueness of measurement signalrepresentation can be achieved.Before studying the sensing matrix D , we will observe thesignal space of channel response. The vectorized response ofLoS path, namely h = δ b ( θ, φ ) , is a high dimensional ( N r N t -dimensional) variable that is characterized by ( δ, θ, φ ) , and wedefine the signal space of h as S , { δ b ( θ, φ ) | δ ∈ C , − ≤ θ, φ < } (26) S is a nonlinear k -dimensional ( k = 3 ) submanifold of C N r N t with the parameters ( δ, θ, φ ) [29], [30]. As b ( θ, φ ) is the Kronecker product of two array steering vectors, S isindeed the so-called array manifold [31]. Thus, one channelrealization ˇ h with the parameters (ˇ δ, ˇ θ, ˇ φ ) can be seen asa point in the array manifold. The dimensionality k can beinterpreted as an “information level” of the signal, analogousto the sparsity level in compressive sensing problems [29],[32], [33]. In [29], it is proved that signals obeying manifoldmodels can also be recovered from only a few measurements,simply by replacing the traditional compressive sensing modelof sparsity with a manifold model for h . The above statementis supported by Lemma 1. Lemma 1.
For a random orthoprojector Φ ∈ C M × N , thefollowing statement (1 − ǫ ) r MN ≤ k Φh − Φh k k h − h k ≤ (1 + ǫ ) r MN , ∀ h , h ∈ S , h = h (27) holds with high probability, when dimensionality M of theprojected low-dimensional space is sufficient , where h ∈S , h ∈ S , h = h , < ǫ < is the isometry constant [29]. Remark 3. k h − h k is the Euclidean distance betweentwo points h , h on the manifold, and k Φh − Φh k is theEuclidean distance between the projected points Φh , Φh onthe image of S (namely Φ S ). The isometry constant ǫ measuresthe degree that the pairwise Euclidean distance between pointson S is preserved under the mapping Φ . Apparently, Lemma1 indicates that k Φh − Φh k > is satisfied with highprobability, as it is a weaker condition than Lemma 1. Although the sensing matrix D is not necessarily anorthoprojector, via singular value decomposition, it can bedecomposed as D = e Ψ e Λ e Φ , where e Ψ ∈ C M × M , e Λ ∈ C M × M ,and e Φ ∈ C M × N . Then, we have k Dh − Dh k = k e Λ e Φh − e Λ e Φh k , where e Φ is indeed the orthoprojector, and e Λ isa diagonal matrix with non-zero elements that scales thecomponent in each dimension. k e Φh − e Φh k > implicates k Dh − Dh k > , which is equivalent to Dh = Dh ,namely, δ Db ( θ , φ ) = δ Db ( θ , φ ) , ∀ ( δ , θ , φ ) =( δ , θ , φ ) . Thus, it is easy to find that Db ( θ , φ ) = µ Db ( θ , φ ) , ∀ ( θ , φ ) = ( θ , φ ) , ∀ µ ∈ C , where µ , δ δ .To conclude, the randomly generated sensing matrix D hasa large probability to guarantee the uniqueness of ML basedjoint AoA and AoD estimation. C. On The Impact of Training Length N Theorem 1 indicates that, with random beamforming, Eq.(25) holds with high probability. In other words, in noiselessscenario, the distance gap between the highest peak (globaloptimum) and other peaks (other local optimums) exist withhigh probability. However, in practice, corrupted by noise and The sufficient number of M is related to ǫ and several manifold-relatedfactors, e.g., condition number, volume, and geodesic covering regularity.Detailed analysis can be referred to [29], [30]. In practice, the exact rela-tionship between the sufficient number and its dependent factors is of limitedsignificance due to the following two reasons, (1) the received measurementsignal is corrupted by noise, (2) M can be online adjusted according to channelconditions. -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8-1-0.8-0.6-0.4-0.200.20.40.60.8 (a) N = 4 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8-1-0.8-0.6-0.4-0.200.20.40.60.8 (b) N = 8 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8-1-0.8-0.6-0.4-0.200.20.40.60.8 (c) N = 12 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8-1-0.8-0.6-0.4-0.200.20.40.60.8 (d) N = 16 Fig. 3. Contour plots of g ( θ, φ ) with different training lengths (Red crossrepresents the position of the first peak, purple asterisk represents the positionof the second peak, and black square is the position of the actual AoA, AoDpair) TABLE I. Peak values of g ( θ, φ ) over training lengthTraining Length Peak 1 Peak 2 Peak 1 − Peak 2 N = 4 N = 8 N = 12 N = 16 interference, the highest peak may (1) shift to its adjacentpoints, or (2) be transcended and replaced by other peaks.Error Type 1 incurs mild AoA, AoD estimation error followedby power loss of an acceptable level; Error Type 2 incurssignificant AoA, AoD estimation error followed by beammisalignment. Apparently, we would like to avoid Error Type2. To study the estimation error, the pairwise error probability(PEP) of any two parameter sets ( θ, φ ) and ( e θ, e φ ) is derivedin the following theorem. Theorem 2.
The PEP
P e (cid:16) ( θ, φ ) → ( e θ, e φ ) (cid:17) that ( θ, φ ) ismistaken as ( e θ, e φ ) in relatively high SNR regime can beapproximated as P e (cid:16) ( θ, φ ) → ( e θ, e φ ) (cid:17) ≈ Q (cid:18) | δ | σ d ( D , θ, φ, e θ, e φ ) (cid:19) (28) where d ( D , θ, φ, e θ, e φ ) , k Db ( θ, φ ) k − | b H ( e θ, e φ ) D H Db ( θ, φ ) | k Db ( e θ, e φ ) k Proof.
See Appendix B.Theorem 2 indicates that PEP is inversely proportional to d ( D , θ, φ, e θ, e φ ) . To build the connection between PEP andtraining length N , Proposition 1 is derived. Proposition 1. d ( D N , θ, φ, e θ, e φ ) is monotonically increasingover training length N , where D N = (cid:2) D HN − d N (cid:3) H , i.e., d ( D N , θ, φ, e θ, e φ ) ≥ d ( D N − , θ, φ, e θ, e φ ) (29) and the equality holds only if b H ( e θ, e φ ) d N d HN b ( e θ, e φ ) b H ( e θ, e φ ) d N d HN b ( θ, φ ) = b H ( e θ, e φ ) D HN − D N − b ( e θ, e φ ) b H ( e θ, e φ ) D HN − D N − b ( θ, φ ) (30) Proof.
See Appendix C.To verify Proposition 1, we plot the contour of g ( θ, φ ) withdifferent training lengths in noiseless scenario in Fig. 3. We set δ = 1 , θ = 0 , φ = 0 . As can be seen that the gap between thefirst and the second peaks increases over training length, andthe value of which is given in Table I. In addition, we can findthat position of the first peak is invariant to training length andremains the same as the actual AoA, AoD pair, while positionof the second peak varies. This verifies the uniqueness of MLbased joint AoA, AoD estimation. Remark 4.
According to Proposition 1, with random beam-forming, the PEP probability of an erroneous estimate ( e θ, e φ ) being mistaken as the authentic parameters ( θ, φ ) decreasesalmost surely over training length N . Therefore, an appro-priate N can guarantee a satisfying accuracy of parameterestimation in scenarios with different SNR and interferencelevels. V. I
NTERPLAY B ETWEEN P OSITIONING AND B EAM T RAINING
In IRSs assisted mmWave MIMO system, BS/AP and IRSs,with their positions and array directions being known by allthe MTs, can be seen as anchor nodes or beacons. The AoDsderived at beam training stage enable MT to estimate its ownposition. Hence, IRSs assisted mmWave MIMO system isendowed with the capability of high-accuracy localization. Theacquired position information is not only a fringe benefit, butalso in turn facilitates beam training. The interplay betweenbeam training and indoor positioning is explained as follows.AoD estimate of the unblocked reliable links can yield theposition of MT, and the position of MT, associated with anchorpositions and anchor directions, can improve the precision ofAoD/AoA estimation and assist in the decision of blockageindicator ζ . A. Reliability of The Estimated AoA, AoD Pair (ˆ θ, ˆ φ ) To be concise, we treat BS/AP and IRSs as identical anchornodes. The η = 1 -st anchor is BS/AP and the rest N IRS anchors ( η = 2 , , · · · , N IRS +1 ) are IRSs. Although we havealready obtained N IRS +1 sets of path parameters (ˆ δ η , ˆ θ η , ˆ φ η ) ,we should be aware that the estimation is performed under theassumption that ζ η = 1 . In practice, LoS and VLoS paths maysuffer from blockage (namely ζ η = 0 ) by moving obstacles,which will jeopardize the estimation of (ˆ δ η , ˆ θ η , ˆ φ η ) . Otherthan blockage, insufficient training length or low SNR may incur Error Type 2 of joint AoA and AoD estimation, whichis defined in Section IV. C.Therefore, it is essential to select the trustworthy parametersas the input of positioning algorithm. To this end, we introducethe metric – residual signal power ratio ̟ η , to measure thereliability of (ˆ δ η , ˆ θ η , ˆ φ η ) , i.e., ̟ η = k y η − ˆ δ η Db (ˆ θ η , ˆ φ η ) k k y η k (31)Recall that (ˆ δ η , ˆ θ η , ˆ φ η ) are obtained by minimizing k y η − δ η Db ( θ η , φ η ) k , the yielded estimate (ˆ δ η , ˆ θ η , ˆ φ η ) will thusalways result in k y η − δ η Db ( θ η , φ η ) k ≤ k y η k . Therefore,the range of ̟ η is ̟ η ∈ [0 , .Since the dominant component of mmWave channel is LoSpath, the reconstructed signal ˆ δ η Db (ˆ θ η , ˆ φ η ) should accountfor the majority of the received signal y given that theparameters (ˆ δ η , ˆ θ η , ˆ φ η ) are accurate and residual signal powerratio ̟ η should be smaller. Conversely, when blockage orError Type 2 occurs, the parameters (ˆ δ η , ˆ θ η , ˆ φ η ) are heavilybiased, and thus ̟ η should be larger. Following the aboveheuristics, anchors’ reliability can be sorted. B. AoD Based Positioning1) Geometric Relationship Between AoDs and MT Position:
We denote the index set of the reliable links as N , positioncoordinates of the η -th anchor as p η , ULA direction of the η -th anchor as e η . Note that p η , e η are known by MTs. Thedirection vector of the LoS path between MT and the η -thanchor is p − p η k p − p η k , where p is the position of MT. Thus,the geometric relationship between AoDs and MT positionis expressed as ˆ φ η = ( p − p η ) T e η k p − p η k | {z } φ η ( p ) + ε η , η ∈ N (32)where ˆ φ η is the estimate of cosine AoD of the η -th linkderived in beam training stage, φ η ( p ) is the actual cosine AoDthat is dependent on position p , and ε η is estimation error.For illustrative purposes, a typical scenario of IRSs assistedmmWave communications is shown in Fig. 4.
2) Taylor Series Method for AoD Based Positioning:
In theideal case, when ε η = 0 , we have ˆ φ η = φ η ( p ) . The equation φ η ( p ) = ( p − p η ) T e η k p − p η k corresponds to a right circular cone.There are unknown variables of MT’s position coordinates,thus the minimum sufficient number of unblocked links toestimate the 3-D position of MT is |N | = 3 , which is theintersection of the three right circular cones. As IRSs are cost-effective compared with conventional mmWave devices, theycan be massively installed with minimal effort. We can expectthat IRSs assisted mmWave with a large number of delicatelyplaced IRSs is capable to guarantee |N | ≥ unblocked linkswith high probability. In practice, estimation error ε η cannot be zero. To estimatethe 3-D position p = ( x, y, z ) T , least square criterion isadopted, i.e., min p ξ φ ( p ) , X η ∈N (cid:16) ˆ φ η − φ η ( p ) (cid:17) s.t. p ∈ S (33)where S is the position range of indoor MT, e.g., the 3-Dspace of lecture hall. As the objective function ξ φ ( p ) is non-convex, it is non-trivial to derive the analytical solution tothe problem. Fortunately, Taylor-series estimation method iscapable to effectively solve a large class of position-locationproblems [34]. Starting with a rough initial guess, the Taylor-series estimation method iteratively improves its guess at eachstep by determining the local linear least-sum-squared-errorcorrection [34]. In AoD based positioning, with the initialposition guess ˆ p , the following approximation can be obtainedthrough Taylor series expansion by neglecting m -th orderterms ( m ≥ ), i.e., φ η ( p ) ≈ φ η (ˆ p ) + ( p − ˆ p ) T ∂φ η ( p ) ∂ p (cid:12)(cid:12)(cid:12)(cid:12) p =ˆ p (34)where the first order derivative is denoted as ∂φ η ( p ) ∂ p = k p − p η k e η − ( p − p η ) T e η p − p η k p − p η k k p − p η k (35)Substituting (37) into (32), we have ˆ φ η − φ η (ˆ p ) ≈ ∂φ η ( p ) ∂ p T (cid:12)(cid:12) p =ˆ p ( p − ˆ p ) + ε η , η ∈ N (36)Its matrix form is written as ∆ φ ≈ A T ∆ p + ε (37)where ∆ p = p − ˆ p , ε = [ ε , · · · , ε |N | ] T , and ∆ φ = [ ˆ φ − φ (ˆ p ) , · · · , ˆ φ |N | − φ |N | (ˆ p )] T (38a) A = (cid:20) ∂φ ( p ) ∂ p (cid:12)(cid:12) p =ˆ p , · · · , ∂φ |N | ( p ) ∂ p (cid:12)(cid:12) p =ˆ p (cid:21) (38b)On the basis of (37), the Taylor series method for AoD basedpositioning is summarized in Algorithm 1.
3) Reliable Link Set N : An intuitive method to constructthe set of reliable links is to select |N | links with the |N | smallest ̟ η to avoid unreliable AoDs resulted from blockageand Error Type 2 of joint AoA, AoD estimation. However, itis non-trivial to determine the exact value of |N | . Although |N | = 3 anchors are theoretically sufficient to yield theposition of MT in the ideal noiseless case, more anchors aredesirable in practice for positioning algorithm to enhance theaccuracy of position estimation.To utilize as many reliable anchors as possible, the followingstrategy is proposed to iteratively construct the reliable link set N . Firstly, we sort the anchors in ascending order accordingto ̟ η . Then, starting from |N | = 3 anchors, we iterativelyincrease the number of anchors used for positioning in Algo-rithm 1, and by the end of each iteration, we calculate thecost ξ φ ( e MT ) |N | . Finally, we select the output corresponding to MT Z YX
RISBS Reference antenna of a ULADirection vector of then-th anchor Position of the n-th anchorPosition of MT Obstacle Cosine AoD of the n-thanchor
Fig. 4. A typical scenario of RIS assisted mmWave communications
Algorithm 1:
Taylor Series Method For AoD BasedPositioning
Initialization : Generate an initial guess of MT position ˆ p Input : The estimate of cosine AoDs of a selected set ofreliable links, i.e., ˆ φ η , ( η ∈ N ) , positions of anchors p η , ( η ∈ N ) , directions of ULA on the anchors e η , ( η ∈ N ) , and iteration stopping parameter ǫ . Repeat With the given ˆ p , generate φ η (ˆ p ) , ( η ∈ N ) according to Eq. (38a) and A according to Eq.(38b). Find the least square estimate of ∆ p , i.e., ˆ∆ p = ( AA T ) − A ∆ φ Update ˆ p , i.e., ˆ p ← ˆ p + ˆ∆ p . Until k ˆ∆ p k < ǫ .the largest |N | that satisfies ξ φ ( e MT ) |N | ≤ ξ th as the estimatedposition of MT, where ξ th is a preset threshold . C. Parameter Estimation With The Aid of MT Position
With the estimated position ˆ p , channel parameters can berefined according to the geometric relationship.
1) AoD Estimation:
With ˆ p , AoD estimation is updated by ¯ φ η = (ˆ p − p η ) T e η k ˆ p − p η k , η ∈ { , , · · · , N IRS + 1 } (39)
2) AoA Estimation:
To estimate AoA, the direction of ULAin MT’s side is essential. Therefore, we firstly find the least An appropriate ξ th can be obtained by carrying out a great number ofMonte Carlo experiments offline. In our numerical experiment, we find that √ ξ th = 0 . results in a good performance. square estimate of e MT by solving the following optimizationproblem. min e MT ξ θ ( e MT ) , X η ∈N (cid:18) (ˆ p − p η ) T e MT k ˆ p − p η k ⊖ ˆ θ η (cid:19) s.t. k e MT k = 1 (40)Note that N can be derived in the iterative process accordingto Section V. A. 3.The objective function of (40) can be rewritten in matrixform as ξ θ ( e MT ) = k P T e MT ⊖ ˆ θ k (41)where P = (cid:20) (ˆ p − p η ) k ˆ p − p η k · · · (ˆ p − p η |N| ) k ˆ p − p η |N| k (cid:21) , ˆ θ =[ˆ θ η , · · · , ˆ θ η |N| ] T and N = { η , · · · , η |N | } . The optimizationproblem can be solved via projected gradient descent method[35], in which we iteratively update e MT as follows. d MT,i +1 = e MT,i − λ ∂ξ θ ( e MT ) ∂ e MT (cid:12)(cid:12)(cid:12) e MT = e MT,i e MT,i +1 = d MT,i +1 k d MT,i +1 k (42)where λ is step size and ∂ξ θ ( e MT ) ∂ e MT = P (cid:16) P T e MT ⊖ ˆ θ (cid:17) .Finally, with ˆ e MT yielded by projected gradient descentmethod, AoA estimation is updated by ¯ θ η = (ˆ p − p η ) T ˆ e MT k ˆ p − p η k (43)
3) Prediction of Blockage:
As a prerequisite of our pro-posed blockage prediction method, we firstly introduce theestimation of δ η , which is dependent on the values of ( θ η , φ η ) .Note that the parameter estimate obtained in Section IV byML estimation is under the assumption that ζ η = 1 , whileit is probable that ζ η = 0 in fact. It would be misleading inthe estimation of δ η by directly substituting (ˆ θ η , ˆ φ η ) into (17).Therefore, we will use the estimates of AoA and AoD refinedby position to assist the estimation of δ η and ζ η , as they arecross verified by multiple anchors and are thus more reliable.Substituting (¯ θ η , ¯ φ η ) into (17), we have ¯ δ η = b H (¯ θ η , ¯ φ η ) D H y k Db (¯ θ η , ¯ φ η ) k = ζ η δ η b H (¯ θ η , ¯ φ η ) D H Db ( θ η , φ η ) + b H (¯ θ η , ¯ φ η ) D H n k Db (¯ θ, ¯ φ ) k = ζ η δ η f (¯ θ η , ¯ φ η ) + ¯ n (44)where f (¯ θ η , ¯ φ η ) , b H (¯ θ η , ¯ φ η ) D H Db ( θ η ,φ η ) k Db (¯ θ η , ¯ φ η ) k , ¯ n ∼ CN (0 , σ n ) ,and σ n = σ w + σ ν k Db (¯ θ η , ¯ φ η ) k (or σ n = σ w + σ ν + σ ν k Db (¯ θ η , ¯ φ η ) k ). Thus, wehave ¯ δ η = (cid:26) δ η f (¯ θ η , ¯ φ η ) + ¯ n, ζ η = 1¯ n, ζ η = 0 (45)Theoretically, with the knowledge of δ η , f (¯ θ η , ¯ φ η ) and σ n ,the decision of ζ η can be made by comparing the probabilitiesof ¯ δ η conditioned on ζ η = 0 and ζ η = 1 . However, accurateestimation of f (¯ θ η , ¯ φ η ) and σ n is challenging in practice. With TABLE II. Simulation Parameters
Parameter Value
Operating frequency GHzTx power of BS/AP [0 , dBmNoise power − dBmPosition of IRSs (5 , − , . , (5 , , . , (0 , − , . , (0 , , . , ( − , − , . , ( − , , . , ( − , , . , (10 , , . , ( − , , . , (10 , , . , ( − , − , . , (10 , − , . Position of BS/AP (0 , , Direction of IRSs’ ULA (0 , , , (1 , , , (0 , , , (0 , , , (0 , , , (1 , , , (0 , , , (0 , , , (0 , , , (1 , , , (0 , , , (0 , , Direction of BS/AP’s ULA ( √ , √ , Reflection loss −
10 log ξ dBSize of obstacles . × . × . metersAltitude of MT [1 . , . metersNumber of users , , Number of NLoS paths , respect to δ η , its amplitude | δ η | is estimable from the distanceof MT, while its phase cannot be accurately estimated fromthe distance, as it is very sensitive to distance estimation errorand may be affected by random initial phase of local oscillatorin transmitter side.Alternatively, a heuristic method is proposed to decideblockage indicator by comparing the pathloss estimated from (¯ θ, ¯ φ ) and pathloss estimated from ˆ p , i.e., (cid:12)(cid:12)(cid:12)(cid:12)
10 log | ¯ δ η | −
10 log | δ η (ˆ p ) | (cid:12)(cid:12)(cid:12)(cid:12) ζ η = 1 ⋚ ζ η = 0 P L th (46)where | δ η (ˆ p ) | = (cid:12)(cid:12)(cid:12) √ P Tx λe − j πdBM πd BM (cid:12)(cid:12)(cid:12) , η = 1 (cid:12)(cid:12)(cid:12)(cid:12) √ ξP Tx N B λe − j π ( dBRη + dRηM ) π ( d BRη + d RηM ) (cid:12)(cid:12)(cid:12)(cid:12) , η = 2 , · · · , N IRS +1 BS/AP to MT distance d BM and IRS to MT distance d R η M areattainable from ˆ p , and P L th is the preset threshold of pathlossdistance (In numerical simulations, we set P L th = 6 dB ).VI. N UMERICAL R ESULTS
In this section, we numerically study the performance ofthe proposed joint beam training and positioning scheme forIRSs assisted mmWave MIMO.
A. Settings of Numerical Experiment
We assume that IRSs-assisted mmWave MIMO system isdeployed in an indoor scenario, e.g., lecture hall, and thelength, width and height of which are meters, metersand meters, respectively. The rest system parameters arelisted in Table II. For simplicity, we assume that AoA, AoD ofNLoS paths follow uniform distribution, i.e., θ BM,l , φ
BM,l ∼ U (0 , π ) , l = 2 , ..., L , and path coefficient follows complexGaussian distribution, i.e., δ l ∼ CN (0 , σ l ) , l = 2 , ..., L and
10 log δ σ l = 20 dB. We model user (MT holder) as a cubewith its length, width and height being . m, . m and . m,respectively. We denote position of the MT held by user Number of blocked links P e r c en t age Number of MTs: 20Number of MTs: 50Number of MTs: 100
Fig. 5. Blockage rate with different user densities as ( x, y, z ) , where x, y, z follow uniform distribution, i.e., x, y ∼ U ( − , and z ∼ U (1 . , . . Users are uniformlydistributed in the lecture hall under the non-overlapping con-straint. For a typical MT, the other MT holders are its potentialobstacles, and thus the blockage probability increases with userdensity.To gain insights into the relationship between user densityand blockage probability, Fig. 5 is presented where there are IRSs deployed, which means a total of LoS/VLoS links areavailable. From the Fig. 5, we can see that when the number ofMTs is , more than of channel realizations experienceno link blockage, the largest number of blocked links is , andthe percentage of which is less than ; when the number ofMTs is , more than of channel realizations experienceless than blocked links, the largest number of blocked linksis , and the percentage of which is less than ; when thenumber of MTs is , more than of channel realizationsexperience less than blocked links, the largest number ofblocked links is , and the percentage of which is almostnegligible. Note that when there exists at least unblockedlink, uninterrupted communication over mmWave band can beguaranteed, and when there exist at least unblocked links,positioning algorithm can be performed to locate MT andmeanwhile enhance parameter estimation. B. Performance of Beam Training With Random Beamforming
In Fig. 6, we study the mean squared error (MSE) perfor-mance of the estimated AoA and AoD, which fundamentallydetermines the accuracy of beam alignment and positioning.Cram´er-Rao bound (CRB) is adopted as the benchmark. Sincethe estimation of ( θ, φ ) is part of the joint estimation of ( δ, θ, φ ) , CRBs of θ and φ are obtained as the last two diagonalelements of the inverse of Fisher information matrix w.r.t. ( δ, θ, φ ) . The detailed derivation of CRB is omitted, as itfollows the standard procedure. When the training length is , to study the performance limit of joint AoA and AoDestimation, we firstly measure the MSE of AoA and AoDwhen the channel is with merely LoS path. As can be seenfrom Fig. 6(a) that, from dBm to dBm the empirical MSE ofboth AoA and AoD is significantly higher than CRB, and theperformance gap gradually turns to be constant from dBmto above. It indicates that, from dBm to dBm the jointestimation of ( θ, φ ) experiences Error Type 2 as mentionedin Section IV. C, in which the estimated AoA and AoD pairis far apart from their authentic values, and from dBm toabove only Error Type 1 happens, in which the estimationerror is mild and tightly lower bounded by CR bound. In Tx Power/dBm -8 -6 -4 -2 M SE (a) Training length N=16 MSE of AoD φ in LoS Channel ModelMSE of AoA θ in LoS Channel ModelCR Bound of AoD φ in LoS Channel ModelCR Bound of AoA θ in LoS Channel ModelMSE of AoD φ in LoS + NLoS Channel ModelMSE of AoA θ in LoS + NLoS Channel Model Tx Power/dBm -8 -6 -4 -2 M SE (b) Training length N=8 MSE of AoD φ in LoS Channel ModelMSE of AoA θ in LoS Channel ModelCR Bound of AoD φ in LoS Channel ModelCR Bound of AoA θ in LoS Channel ModelMSE of AoD φ in LoS + NLoS Channel ModelMSE of AoA θ in LoS + NLoS Channel Model Fig. 6. MSE performance of AoA/AoD estimated by random beamformingbased beam training in LoS channel model and LoS + NLoS channel model,where NLoS path number is , training length is N = 8 , . practice, NLoS path’s impacts on beam training cannot beoverlooked. Therefore, we further carry out the simulation ofrandom beamforming based beam training in LoS + NLoSchannel (the number of NLoS paths is ). As can be seen fromFig. 6(a) that, from dBm to dBm the MSE performancein LoS + NLoS channel is slightly worse than that in LoSchannel, which indicates that noise is the main detrimentalfactor. From dBm to above, the MSE curves turn to be flat,and this is because the impact of NLoS path, namely ν inEq. (8) (or ν , ν in Eq. (10)), does not diminish over SNR.The good point is that MSE from dBm to above is around − and the performance is satisfactorily accurate for bothbeam alignment and positioning over short distance. To studythe impact of training length, MSE performance comparisonis reproduced when N = 8 in Fig. 6(b). A notable differencefrom N = 16 case is that the flat curves of empirical MSEstart from dBm, and the values of which are around − ,which verifies the benefits of increasing training length.In Fig. 7, we study the performance of beam alignmentbased on (ˆ θ, ˆ φ ) estimated from random beamforming. Notethat, although the high precision of AoA and AoD estimationis desirable, it is indeed not essential if the beamformingcodebook is a pre-configured finite set, e.g., discrete FourierTransform (DFT) codebook. Traditional beam sweeping basedbeam alignment methods [26] estimate the beam gain (receivedpower level) of different beams (or beam pairs) throughexhaustive/hierachical beam sweeping and select the strongestbeam (or beam pair) from the pre-configured finite set. It isapparently unfair to compare their performance using MSEof AoA/AoD, as beam sweeping based methods estimateAoA/AoD from finite angle bins. To this end, we adoptdata rate after beam alignment as the performance metric,and we select the beam pair that has the minimum angledifference to (ˆ θ, ˆ φ ) to align the beam. From Fig. 7, we cansee that random beamforming with training length N = 16 has almost the same performance as exhaustive beam sweep- Tx Power/dBm D a t a R a t e / bp s / H z Random Beamforming Based Beam Training (Training length N = 8)Random Beamforming BasedBeam Training (Training length N = 16)Exhaustive Beam Sweeping(Training length N = 256)Hierarchical Beam Sweeping(Training length N = 16)
Fig. 7. Data rate comparison in LoS + NLoS channel model, where NLoSpath number is . Tx Power/dBm R M SE Indoor positioning with training length N=8Indoor positioning with training length N=16
Fig. 8. Accuracy of Indoor Positioning ing, while training length of the latter is as extravagant as N = N r N t = 256 ( N r =16 is the antenna number of MT, and N t = 16 is the antenna (reflector) number of IRS or BS/AP).Hierarchical beam sweeping reduces the training overhead to N = 4 log min( N r , N t ) + 2 log N r ,N t )min( N r ,N t ) = 16 , but itsperformance is severely degraded within the transmit powerrange from dBm to dBm. To study the impact of traininglength N , we also plot the rate curve of random beamformingwith N = 8 . From dBm to dBm, there exists a performancegap between random beamforming with N = 8 and randombeamforming with N = 16 , and the performance gap becomesnegligible from dBm to above. C. Performance of Positioning Algorithm and Location Infor-mation Aided Parameter Estimation
In Fig. 8, the accuracy of indoor positioning of IRSs assistedmmWave MIMO is studied in terms of root mean squarederror (RMSE). When the training length is N = 16 , RMSEis . meter at dBm, and converges to . meter from dBm to dBm, which indicates that, with the aid ofIRSs, mmWave MIMO achieves centimeter accuracy in indoorscenario. When the training length is N = 8 , RMSE is . meter at dBm, and converges to . meter from dBm to dBm. Considering the reduced training length, the accuracylimit of . meter for N = 8 case in high SNR regimes isacceptable. However, the positioning accuracy of N = 8 caseis not satisfying in low SNR regimes. Through case analysis, Tx Power/dBm B l o ck age P r ed i c t i on A cc u r a cy (a) Training length N=16 Position Aided Blockage PreditionBlockage Predition Based OnK-means Clustering (Residual Ratio)Blockage Predition Based On K-meansClustering (Received Signal Power)
Tx Power/dBm B l o ck age P r ed i c t i on A cc u r a cy (b) Training length N=8 Position Aided Blockage PreditionBlockage Predition Based OnK-means Clustering (Residual Ratio)Blockage Predition Based On K-meansClustering (Received Signal Power)
Fig. 9. Accuracy of Blockage Predication we find that the correlation between residual ratio ̟ η and theaccuracy of (ˆ θ η , ˆ φ η ) is weakened by the increased level ofnoise and the reduced training length. In other words, a small ̟ η may misleadingly correspond to an unreliable anchor node,and thus results in inaccurate estimate of position. To improvethe accuracy, a more sophisticated positioning algorithm thatiteratively sorts the reliability will be developed in the future.In Fig. 9, the accuracy of blockage prediction is studied. Asthe major benefit of location information, accurate blockagepredication is essential for smooth and uninterrupted mmWavecommunications. For the purpose of comparison, two methodsare adopted as benchmarks, which are (1) received powerbased blockage prediction and (2) residual ratio based block-age prediction. For (1), it is straightforward that unblockedlinks have significantly higher received signal level than thatof blocked links. However, as power level is an absolutequantity, without the prior knowledge such as the likely rangeof received power, it is possible to mistake the unblockedlink between MT and faraway anchor as a blocked link. Incontrast, residual ratio in (2) is a relative quantity, which is notdependant on the likely range of received power. However, theoptimal threshold that is essential for blockage predication isunavailable either. Therefore, we adopt the K-means clusteringmethod to partition the observations into clusters, i.e.,blocked links and unblocked links. When the training lengthis N = 16 , we can see from the figure that position aidedblockage prediction is slightly erroneous merely at dBm andbecomes errorless when transmit power increases. With respectto the benchmark methods, although the prediction accuracy ofresidual ratio based K-means clustering method is worse thanposition aided blockage prediction, its error rate is below . ,which is acceptable. By contrast, the predication error rate ofreceived power based K-means clustering method is nearly . , which indicates that the prediction is almost random.When the training length reduces to N = 8 , the superiorityof position aided blockage prediction is more remarkable, andthis is owing to the cross-validation mechanism enabled bylocation information.In Fig. 10, MSE performance evaluation of AoA/AoDrefined by location information is performed. To this end,we intentionally filter out the blocked links, and reserveAoA/AoD estimate of the unblocked links. As can be seen thatAoA/AoD refined by location information is more accuratethan AoA/AoD estimated by beam training with random Tx Power/dBm -4 -3 -2 M SE (a) Training length N=16 AoD estimated by beam training with random beamformingAoA estimated by beam training with random beamformingAoD refined by location informaitonAoA refined by location informaiton
Tx Power/dBm -4 -3 -2 -1 M SE (b) Training length N=8 AoD estimated by beam training with random beamformingAoA estimated by beam training with random beamformingAoD refined by location informationAoA refined by location informaiton
Fig. 10. MSE performance of AoA/AoD refined by location information beamforming. This is because location information is derivedby multiple anchors, and AoA/AoD refinement accordingto geometric relationship means that the estimation is crossverified. It is noteworthy that the performance enhancementis more significant when the training length is N = 8 , fromwhich we find the potential to reduce training length of beamtraining with the aid of location information. Another notablepoint is that AoA refined by location information is alwaysworse than AoD refined by location information. This isbecause the direction vector ˆ e MT is derived from estimationin (40), while the direction vectors of anchors e η are wellknown. VII. C ONCLUSION
In this paper, beam training for IRSs assisted mmWavecommunications is studied. By breaking down beam trainingfor IRSs assisted mmWave MIMO into several mathematicallyequivalent sub-problems, we perform random beamformingand maximum likelihood estimation to derive the optimalbeam of BS/AP and MT and the optimal reflection pattern ofIRSs. Then, by sorting the reliability of the estimated AoA,AoD paris, we propose an iterative positioning algorithm toacquire the position of MT, and with which we are able tocross verify and enhance the estimation of AoA and AoD, andaccurately predict link blockage. Numerical results show thesuperiority of our proposed beam training scheme and verifythe performance gain brought by location information. A
PPENDIX AP ROOF OF T HEOREM y = Db ( θ, φ ) , accordingto Cauchy-Schwarz inequality, we have (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) b H ( e θ, e φ ) D H k Db ( e θ, e φ ) k Db ( θ, φ ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ k Db ( θ, φ ) k (47)Then, the proof of Eq. (25) is reduced to prove that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) b H ( e θ, e φ ) D H k Db ( e θ, e φ ) k Db ( θ, φ ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = k Db ( θ, φ ) k (48)namely Db ( θ, φ ) = µ Db ( e θ, e φ ) , ∀ µ ∈ C , ∀ ( θ, φ ) = ( e θ, e φ ) ,which is mathematically equivalent to Eq. (24).A PPENDIX BP ROOF OF T HEOREM
P e (cid:16) ( θ, φ ) → ( e θ, e φ ) (cid:17) = P r (cid:13)(cid:13)(cid:13)(cid:13) b H ( θ, φ ) D H k Db ( θ, φ ) k y (cid:13)(cid:13)(cid:13)(cid:13) < (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) b H ( e θ, e φ ) D H k Db ( e θ, e φ ) k y (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = P r − | b H ( θ, φ ) D H n | k Db ( θ, φ ) k + | b H ( e θ, e φ ) D H n | k Db ( e θ, e φ ) k − ℜ (cid:8) δ n H Db ( θ, φ ) (cid:9) + 2 ℜ ( δ n H Db ( e θ, e φ ) b H ( e θ, e φ ) D H Db ( θ, φ ) k Db ( e θ, e φ ) k ) > k δ Db ( θ, φ ) k − | δ b H ( e θ, e φ ) D H Db ( θ, φ ) | k Db ( e θ, e φ ) k ! ≈ P r N > k δ Db ( θ, φ ) k − | δ b H ( e θ, e φ ) D H Db ( θ, φ ) | k Db ( e θ, e φ ) k ! (49)where N =2 ℜ ( − δ n H Db ( θ, φ ) + δ n H Db ( e θ, e φ ) b H ( e θ, e φ ) D H Db ( θ, φ ) k Db ( e θ, e φ ) k ) and ℜ{·} is the real part of a complex number. Eq. (49)is obtained by neglecting the component − | b H ( θ,φ ) D H n | k Db ( θ,φ ) k + | b H ( e θ, e φ ) D H n | k Db ( e θ, e φ ) k in high SNR regime. Since N is a Gaussianrandom variable, we have N ∼N , σ | δ | k Db ( θ, φ ) k − | b H ( e θ, e φ ) D H Db ( θ, φ ) | k Db ( e θ, e φ ) k !! According to the definition of Q function, (28) is obtained. A PPENDIX CP ROOF OF P ROPOSITION d ( D n , θ, φ, e θ, e φ ) as d ( D n , θ, φ, e θ, e φ )= k D n b ( θ, φ ) k − | b H ( e θ, e φ ) D Hn D n b ( θ, φ ) | k D n b ( e θ, e φ ) k = b H ( θ, φ ) D Hn − D n − b ( θ, φ ) + b H ( θ, φ ) d n d Hn b ( θ, φ ) − (cid:12)(cid:12)(cid:12) b H ( e θ, e φ ) D Hn − D n − b ( θ, φ ) + b H ( e θ, e φ ) d n d Hn b ( θ, φ ) (cid:12)(cid:12)(cid:12) b H ( e θ, e φ ) D Hn − D n − b ( e θ, e φ ) + b H ( e θ, e φ ) d n d Hn b ( e θ, e φ ) Thus d ( D n , θ, φ, e θ, e φ ) − d ( D n − , θ, φ, e θ, e φ )= b H ( θ, φ ) d n d Hn b ( θ, φ ) + | b H ( e θ, e φ ) D Hn − D n − b ( θ, φ ) | b H ( e θ, e φ ) D Hn − D n − b ( e θ, e φ ) − | b H ( e θ, e φ ) D Hn − D n − b ( θ, φ ) + b H ( e θ, e φ ) d n d Hn b ( θ, φ ) | b H ( e θ, e φ ) D Hn − D n − b ( e θ, e φ ) + b H ( e θ, e φ ) d n d Hn b ( e θ, e φ ) For the purpose of conciseness, let ˇ a = b H ( e θ, e φ ) d n d Hn b ( e θ, e φ );ˇ b = b H ( e θ, e φ ) d n d Hn b ( θ, φ );ˇ c = b H ( e θ, e φ ) D Hn − D n − b ( e θ, e φ );ˇ d = b H ( e θ, e φ ) D Hn − D n − b ( θ, φ ) . As b H ( e θ, e φ ) d n and d Hn b ( θ, φ ) are numbers, rather thanvectors, we have b H ( θ, φ ) d n d Hn b ( θ, φ ) = | b H ( e θ, e φ ) d n d Hn b ( θ, φ ) | b H ( e θ, e φ ) d n d Hn b ( e θ, e φ ) = | ˇ b | ˇ a Then, d ( D n , θ, φ, e θ, e φ ) − d ( D n − , θ, φ, e θ, e φ )= | ˇ b | ˇ a + | ˇ d | ˇ c − | ˇ b + ˇ d | ˇ a + ˇ c = | ˇ b | ˇ c (ˇ a + ˇ c ) + | ˇ d | ˇ a (ˇ a + ˇ c ) − ˇ a ˇ c | ˇ b + ˇ d | ˇ a ˇ c (ˇ a + ˇ c )= | ˇ b | ˇ c (ˇ a + ˇ c ) + | ˇ d | ˇ a (ˇ a + ˇ c ) − ˇ a ˇ c | ˇ b | − ˇ a ˇ c | ˇ d | − a ˇ cRe { ˇ b ∗ ˇ d } ˇ a ˇ c (ˇ a + ˇ c )= | ˇ b | ˇ c + | ˇ d | ˇ a − a ˇ cRe { ˇ b ∗ ˇ d } ˇ a ˇ c (ˇ a + ˇ c )= | ˇ a ˇ d − ˇ b ˇ c | ˇ a ˇ c (ˇ a + ˇ c ) ≥ and equality holds when ˇ a ˇ d − ˇ b ˇ c = 0 , namely, b H ( e θ, e φ ) d n d Hn b ( e θ, e φ ) b H ( e θ, e φ ) d n d Hn b ( θ, φ ) = b H ( e θ, e φ ) D Hn − D n − b ( e θ, e φ ) b H ( e θ, e φ ) D Hn − D n − b ( θ, φ ) (50)R EFERENCES[1] Z. Pi and F. Khan, “An introduction to millimeter-wave mobile broad-band systems,”
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