IRS Aided Millimeter-Wave Sensing and Communication: Beam Scanning, Beam Splitting, and Performance Analysis

1 IRS Aided Millimeter -W a v e Sensing and Communication: Beam Scanning, Beam Splitting, and Performance Analysis Renwang Li, Xiaodan Shao, Member , IEEE , Shu Sun, Member , IEEE , Meixia T ao, F ellow , IEEE , Rui Zhang, F ellow , IEEE Abstract —Integrated sensing and communication (ISA C) has attracted growing interests for enabling the future 6G wireless networks, due to its capability of sharing spectrum and hard- ware resour ces between communication and sensing systems. Howev er , existing works on ISA C usually need to modify the communication protocol to cater f or the new sensing performance requir ement, which may be difficult to implement in practice. In this paper , we study a new intelligent reflecting surface (IRS) aided millimeter-wa ve (mmW a ve) ISA C system by exploiting the distinct beam scanning operation in mmW ave communications to achieve efficient sensing at the same time. First, we pr opose a two- phase ISA C protocol aided by a semi-passive IRS, consisting of beam scanning and data transmission. Specifically , in the beam scanning phase, the IRS finds the optimal beam for reflecting signals from the base station to a communication user via its passive elements. Meanwhile, the IRS directly estimates the angle of a nearby tar get based on echo signals fr om the target using its equipped active sensing element. Then, in the data transmission phase, the sensing accuracy is further impro ved by leveraging the data signals via possible IRS beam splitting. Next, we derive the achievable rate of the communication user as well as the Cram ´ er -Rao bound and the approximate mean square err or of the target angle estimation Finally , extensive simulation results are provided to verify our analysis as well as the effectiveness of the proposed scheme. Index T erms —Integrated sensing and communication (ISA C), intelligent reflecting surface (IRS), millimeter wave (mmW a ve), beam scanning, beam splitting, target sensing, Cram ´ er -Rao bound. I . I N T RO D U C T I O N Recently , integrated sensing and communication (ISA C) has been recognized as a ke y technology for the future 6G wireless network due to its potential to enable efficient sharing of spectrum and hardware resources between communication and sensing systems [2], [3]. Meanwhile, millimeter-wa ve Part of this work was presented at the 2023 IEEE 24th International W ork- shop on Signal Processing Advances in W ireless Communications (SP A WC) [1]. R. Li, S. Sun, and M. T ao are with the Department of Electronic Engineering and the Cooperative Medianet Inno vation Center (CMIC), Shanghai Jiao T ong University , Shanghai 200240, China (e-mails: { renwanglee, shusun, mxtao } @sjtu.edu.cn). X. Shao is with the Institute for Digital Communications, Friedrich- Alexander -University Erlangen-Nuremberg, 91054 Erlangen, Germany (e- mail: xiaodan.shao@fau.de). R. Zhang is with School of Science and Engineering, Shenzhen Research Institute of Big Data, The Chinese Univ ersity of Hong Kong, Shenzhen, Guangdong 518172, China (e-mail: rzhang@cuhk.edu.cn). He is also with the Department of Electrical and Computer Engineering, National University of Singapore, Singapore 117583 (e-mail: elezhang@nus.edu.sg). (mmW av e) technology can provide high data rate for com- munication as well as high resolution for sensing, making it promising for realizing ISA C systems. Ho wever , mmW av e signals are susceptible to obstacles, and the performance of mmW a ve ISAC systems can degrade dramatically in the absence of line-of-sight (LoS) path. T o overcome this issue, intelligent reflecting surface (IRS) has been recognized as a practically viable solution [4]–[6]. IRS is generally a digitally- controlled metasurface composed of a large number of passive reflecting elements (REs) that can reflect the incident signal with independently controlled phase shifts. By le veraging IRS, a virtual LoS link can be created between two wireless nodes when their direct link is obstructed, thus allowing for uninterrupted sensing and communication. Motiv ated by the above, significant research ef forts have been dev oted to studying IRS-aided ISA C systems [7]–[14]. In [7], the joint design of transmit beamforming at the base station (BS) and reflection coefficients at the IRS is studied to maximize the signal-to-noise ratio (SNR) of radar detection while meeting communication requirements simultaneously . The works [8] and [9] address radar beampattern design prob- lems in single-user and multi-user scenarios, respectiv ely . The Cram ´ er-Rao bound (CRB) minimization for IRS-aided sensing is considered in [10]. The authors in [11] aim to maximize the radar output signal-to-interference-plus-noise ratio (SINR) while guaranteeing the communication quality . A double-IRS- aided communication radar coexistence system is considered in [12]. In [13], a feedback-based beam training approach is proposed to design BS transmit beamforming and IRS reflec- tion coefficients for simultaneous communication and sensing. The authors in [14] propose a multi-stage hierarchical beam training codebook to achie ve the desired accuracy for IRS- aided localization while ensuring a reliable communication link with the user . Notice that all of the aforementioned works adopt passiv e IRS to assist sensing, and thus their performance is hindered by the severe path loss of the BS-IRS-tar get-IRS- BS cascaded echo link, particularly in mmW av e frequencies. It is worth mentioning that existing studies on IRS-aided ISA C [7]–[12] have mostly assumed that the target angle from the IRS is known a prior within a certain range and mainly focused on the data transmission period; howe ver , there has been very limited consideration of exploiting the channel esti- mation/training period for achieving ISAC [15]. Moreover , the aforementioned works usually require protocol modifications to accommodate the new sensing performance requirements, 2 a task that might pose challenges in practical implementation. In practice, mmW av e communication systems typically adopt the transmission protocol with two phases, namely , the beam scanning/training phase and the data transmission phase [16]. In the beam scanning phase, the BS transmits reference/pilot signals using different beams from a given codebook, while the user measures the received signal po wer for each beam and feeds back the index of the beam with the maximum power to the BS. Subsequently , the BS adopts this maximum- power beam to transmit data during the data transmission phase. The abo ve beam scanning protocol can be extended to work for IRS-aided mmW av e communication systems, by applying firstly BS beam scanning to find the maximum- power beam towards the IRS, and then IRS beam scanning to find that towards the user , for the typical scenario where the LoS channel between the BS and user is severely blocked. Howe ver , the exploration of the abov e protocol for target sensing as well as the performance tradeoff between sensing and communication under this protocol remain unaddressed yet. As such, in this paper , we in vestigate a downlink IRS-aided mmW av e ISAC system, as illustrated in Fig. 1, where a “semi- passiv e” IRS consisting of passiv e REs and acti ve sensing elements (SEs) is adopted to create virtual LoS channels for the IRS to forward information from the BS to a nearby communication user as well as detect the angle of a nearby target. In particular , the SEs are used to collect the echo signals reflected from the target for its angle estimation. Compared with a fully passiv e IRS that reflects the echoes from the target back to the BS for detection, the semi-passi ve IRS can directly estimate the angle of the target and thus significantly reduce the path loss of the receiv ed echo signal at the BS [17]– [19]. The main contrib utions of this paper are summarized as follows: • First, we propose a two-phase ISAC protocol for the con- sidered IRS-aided mmW av e ISA C system, based on the practical tw o-phase communication protocol for mmW av e systems. Specifically , in the beam scanning phase, the training signals from the BS are used not only to identify the best IRS beam for the communication user , but also to initially estimate the target’ s angle from the IRS. Then, in the subsequent data transmission phase, the data signals from the BS are also used to improv e the angle estimation accuracy via possible IRS beam splitting, while ensuring the achiev able rate of the communication user . • Second, we analyze the achiev able rate of the commu- nication user and deri ve the CRB of angle estimation for the sensing target in the beam scanning phase. The CRB analysis reveals that more REs and SEs can achie ve more accurate target angle estimation. Additionally , we deriv e an analytical approximation of the mean square error (MSE) of the angle estimation, which leads to a closed-form expression for the minimum SNR required to achieve a desired initial angle estimation accuracy . • Third, in order to the enhance sensing accuracy in the data transmission phase, we propose two IRS beam design and sensing strategies, i.e., single-beam-based sensing and Fig. 1: System model of IRS-aided ISA C. beam-splitting-based sensing, which are applied when the difference between the estimated angles of the commu- nication user and the target from the IRS in the beam scanning phase is smaller than a giv en threshold and otherwise, respectiv ely . In the latter case, the loss in the achiev able rate of the communication user is also characterized to ensure its performance. The rest of this paper is organized as follows. Section II introduces the IRS-aided mmW ave system and presents the proposed ISAC protocol. Section III analyzes the communica- tion and sensing performance in the beam scanning phase. Sec- tion IV proposes enhanced sensing strategies during the data transmission phase, with the resulted communication/sensing performance characterized. The simulation results are pro vided in Section V , and the paper is concluded in Section VI. Notations : The imaginary unit is denoted by j = √ − 1 . V ectors and matrices are denoted by bold-face lower -case and upper-case letters, respectively . C x × y denotes the space of x × y comple x-valued matrices. x ∗ , x T , and x H denote the conjugate, transpose and conjugate transpose of vector x . I denotes an identity matrix of appropriate dimensions. diag ( x ) denotes a diagonal matrix with each diagonal element being the corresponding element in x . ˙ a ( θ ) denotes the gradient vector of a ( θ ) . v ec( · ) denotes the vectorization operator . ⌊ x ⌋ denotes the flooring operation that takes the largest inte ger no greater than x . The distribution of a circularly symmetric complex Gaussian (CSCG) random vector with zero means and cov ariance matrix Σ is denoted by C N ( 0 , Σ) ; and ∼ stands for “distributed as”. The main notations used in this paper are summarized in T able I. I I . S Y S T E M M O D E L A N D P R OT O C O L D E S I G N In this section, we introduce the IRS-aided ISA C system model and the corresponding channel model, and then propose the ISAC protocol. A. System Model W e consider a downlink mmW ave ISA C system with the aid of a semi-passiv e IRS as illustrated in Fig. 1, where an N - antenna BS aims to communicate with a single-antenna user and also to detect the angle of a sensing target. The direct links between the BS and the user , as well as the target, are assumed to be blocked due to unfav orable propagation en vironment. Thus, the IRS is deployed to create virtual links for both communication and sensing. W e consider the use of semi-passiv e IRS consisting of M passiv e REs to reflect the 3 T ABLE I: Summary of Notations. Notation Description Notation Description N Antenna number of BS M Number of IRS REs M s Number of IRS SEs M e Number of IRS REs allocated for target estimation in Phase II L Codebook size τ T ime duration of beam scanning T Channel coherence time normal- ized to number of symbol durations θ B I Spatial AoA from BS to IRS θ I U Spatial AoD from IRS to the target θ I T Spatial AoD from IRS to the user α g Path gain of BS-REs channel α h Path gain of IRS-user channel α s Path gain of REs-target-SEs link ℓ IRS’ s best beam index for the user a r ( · ) ∈ C M × 1 Array response vector of REs a s ( · ) ∈ C M s × 1 Array response vector of SEs q ( · ) ∈ C M × 1 q ( θ I T ) ≜ a r ( θ I T ) ϕ ∈ C M × 1 Reflection vector of the REs transmitted signals from the BS to the user and target, and M s activ e SEs to collect the echo signals from the tar get for its angle estimation. The complex-v alued baseband transmitted signal at the BS can be expressed as x = w s , where s denotes the training/data symbol for the communication user with unit power and w ∈ C N × 1 is the transmit beamforming vector with ∥ w ∥ 2 = 1 . Then, the recei ved signal y u at the user can be expressed as y u = p P t h H u diag( ϕ ) Gw s + n u , (1) where P t is the transmit power at the BS, G ∈ C M × N represents the channel between the BS and REs, h u ∈ C M × 1 represents the channel between the REs and the user, n u ∼ C N (0 , σ 2 ) is the receiver A WGN with σ 2 representing the noise power , and ϕ ∈ C M × 1 represents the reflection vector at the REs, which can be written as ϕ =  e j ϕ 1 , e j ϕ 2 , . . . , e j ϕ M  T , (2) with ϕ i being the phase shift by the i -th RE. The SEs can simultaneously receiv e the signals transmitted from the BS and the echo signals reflected by the target 1 . In general, the angles of the target and BS with respect to the IRS are different and can be estimated by the SEs based on the received echoes. The angle between the BS and IRS can be determined in advance by the SEs, which facilitates the estimation of the target’ s angle. The recei ved signal y s ∈ C M s × 1 at the SEs can be represented as y s = p P t ( H t diag( ϕ ) G + G s ) w s + n s , (3) where H t ∈ C M s × M denotes the channel matrix of the REs- target-SEs link, G s ∈ C M s × N denotes the channel matrix of the BS-SEs link, and n s ∼ C N (0 , σ 2 I M s ) is the receiv er A WGN. B. Channel Model W e adopt the LoS channel model to characterize the mmW av e channel. For ease of exposition, we assume that uniform linear arrays (ULAs) are equipped at the BS, REs, and SEs. Thus, the BS-REs channel can be expressed as G = α g a r ( θ B I ) a H b ( ϑ B I ) , (4) 1 The radar cross section (RCS) of the communication user (terminal) is usually significantly smaller compared to the target. Hence, the echo signal reflected by the communication user can be safely ignored in the target angle estimation. where α g = λ 4 π d BI e j 2 πd BI λ [17] denotes the complex-valued path gain of the BS-REs channel with λ being the carrier wa velength and d B I being the distance between the BS and IRS, ϑ B I = sin( ς B I ) with ς B I denoting the angle of departure (AoD) from the BS, θ B I = sin( ζ B I ) with ζ B I denoting the angle of arri val (AoA) to the IRS, and a r ( · ) ( a b ( · )) denotes the array response vector associated with the REs (BS). The array response vector for a ULA with M elements of half- wa velength spacing and the center of the ULA as the reference point can be expressed as a ( θ ) = h e − j ( M − 1) π θ 2 , e − j ( M − 3) π θ 2 , . . . , e j ( M − 1) π θ 2 i T . (5) The IRS-user channel h u can also be written as h u = α h a r ( θ I U ) , (6) where α h = λ 4 π d I U e j 2 πd I U λ denotes the complex-v alued path gain of the IRS-user channel with d I U being the distance between the IRS and user , and θ I U = sin( ζ I U ) with ζ I U denoting the AoD associated with the IRS. The BS-SEs channel G s can be represented as G s = α g a s ( θ B I ) a H b ( ϑ B I ) , (7) where a s ( · ) denotes the array response vector associated with the SEs. The REs-target-SEs channel H t can be expressed as H t = α s a s ( θ I T ) a H r ( θ I T ) , (8) where θ I T = sin( ζ I T ) with ζ I T denoting AoD from the SEs, α s = q λ 2 κ 64 π 3 d 4 I T e j 4 πd I T λ refers to the complex path gain of the REs-target-SEs link [20], in which d I T denotes the distance between the IRS and target and κ denotes the RCS of the target. Considering that the locations of the BS and IRS are fixed, the BS-REs channel G and the BS-SEs channel G s are assumed to be constant for a long period, and can be estimated beforehand at the BS to achiev e the optimal transmit beamforming as w = 1 √ N a b ( ϑ B I ) . Thus, in this paper we focus on the beam training at the IRS. As a result, the received signal at the communication user in (1) can be rewritten as y u = p N P t α g h H u diag( ϕ ) a r ( θ B I ) s + n u (9) and the recei ved signal at the SEs in (3) can be rewritten as y s = p N P t α g ( H t diag( ϕ ) a r ( θ B I ) + a s ( θ B I )) s + n s . (10) Note that there exists an undetectable region, denoted as { Ω u | θ I T : | θ I T − θ B I | < 2 M s } , associated with the tar get. 4 Fig. 2: IRS-aided mmW a ve ISAC protocol. Specifically , the SEs receive strong signals from the BS and weak echo signals from the tar get. When the angles between the tar get and BS with respect to the IRS are close, the echo signals from the target cannot be effecti vely extracted from the mixed signals, and thus the target cannot be detected. This fact will be elaborated analytically in Section III-B. Therefore, our focus is on the scenarios where the target is located outside the undetectable region. C. Proposed Protocol for ISAC In this subsection, we propose a two-phase protocol for the considered IRS-aided mmW a ve ISA C system. Follo wing the existing mmW av e communication protocol, beam train- ing/scanning needs to be first conducted at the IRS, followed by data transmission. During the beam scanning phase, we adopt the widely-used discrete Fourier transformation (DFT) codebook D with L beams as follo ws, D ≜ [ a r ( η 1 ) , a r ( η 2 ) , · · · , a r ( η L )] ∈ C M × L , (11) where η i = − 1 + 2 i − 1 L , i = 1 , · · · , L , and L ≥ M . In our system model, the SEs can exploit do wnlink beam scanning for sensing by collecting the echo signals reflected from the target. In addition, the communication signals during the data transmission phase can also be utilized to refine the target sensing performance (see Section IV for details). The ISA C protocol, depicted in Fig. 2, is di vided into two phases, with T denoting the channel coherence time normalized to number of symbol durations. The first phase in volv es beam scanning with τ symbol durations, while the second phase focuses on data transmission with T − τ symbol durations. The time of beam inde x and recei ved signal power feedback as well as target angle estimation is ignored. • Phase I (beam scanning): The BS sends downlink training signals. The REs sweep the beams in the codebook D , while the SEs collect the echo signals reflected from the target. At the end of IRS beam scanning, the communica- tion user identifies the IRS’ s best beam and corresponding receiv ed power , and feeds them back to the IRS controller (directly or via the BS). Meanwhile, the SEs estimate the target angle based on the receiv ed echo signals. • Phase II (data transmission and enhanced sensing): The BS sends do wnlink data signals. If the target is detected to be located in the vicinity of the communication user , REs reflect the data signal to wards the user with the IRS’ s best beam found in Phase I, which is also reflected tow ards the target to enhance the estimation accuracy (thus termed as IRS single-beam). On the other hand, if the user and target are detected to be well separated, IRS beam splitting is adopted where a certain portion of REs are used for target sensing and the remaining REs are for communication with their corresponding optimal beam, provided that the achie vable rate of the user is ensured within an acceptable mar gin. At the end of data transmission, the SEs further estimate the target angle based on the received echo signals. I I I . P E R F O R M A N C E A NA L Y S I S F O R I N I T I A L B E A M S C A N N I N G A. Achievable Rate of Communication User In this subsection, we first analyze the maximum channel gain obtained in the beam scanning phase, and then derive the achiev able rate of the communication user . Assume the duration of one beam is equal to one symbol duration for simplicity , we hav e τ = L . The BS’ s reference signal can be set as s [ t ] = 1 , ∀ t = 1 , 2 , . . . , L . The receiv ed signal at the user in (9) can be expressed as y u [ t ] = p N P t α g h H u diag( ϕ [ t ]) a r ( θ B I ) + n u [ t ] = p N P t α g α h a H r ( θ I U ) diag( ϕ [ t ]) a r ( θ B I ) + n u [ t ] = p N P t α g α h ϕ T [ t ] diag( a H r ( θ I U )) a r ( θ B I ) + n u [ t ] = p N P t α g α h  a H r ( θ I U ) ϕ [ t ]  + n u [ t ] , (12) where θ I U = θ I U − θ B I , and ϕ [ t ] ∈ D . W e assume that ℓ is the best beam index, i.e., ℓ = arg max t,t =1 , ··· ,L | y u [ t ] | 2 . (13) Let δ u =   θ I U − η ℓ   denote the spatial direction dif ference between θ I U and its adjacent beam η ℓ  0 ≤ δ u ≤ 1 L  . Then, by denoting the best beam as ϕ ⋆ = a r ( η ℓ ) , the IRS beamforming gain can be expressed as   a H r ( θ I U ) ϕ ⋆   =      M X m =1 e j π δ u ( − M − 1 2 + m − 1 )      = sin( π M δ u 2 ) sin( π δ u 2 ) . (14) The function sin( πM x 2 ) sin( πx 2 ) exhibits beha vior similar to that of the sinc function and has a zero value at 2 M . This function decreases monotonically over x ∈ [0 , 2 M ] . Thus, when the user’ s angle θ I U is exactly aligned with the angle of the best beam, the IRS beamforming gain reaches its maximum value, i.e., δ u = 0 and   a H r ( θ I U ) ϕ ⋆   = M . When the user’ s angle θ I U lies in the middle of two adjacent beam angles, the IRS beamforming gain is the lowest, i.e., δ u = 1 L and   a T r ( θ I U ) ϕ ⋆   = sin  π M 2 L  sin − 1 ( π 2 L ) . After beam scanning, the user finds the IRS’ s best beam index ℓ and corresponding receiv ed signal power , and feeds them back to the IRS controller . In Phase II, the REs can then adopt this beam to reflect the signals from the BS to the communication user during the data transmission phase (if IRS single-beam sensing is used). The achie vable rate of 5 the user in bits per second (bps) by taking into account the beam scanning ov erhead is thus gi ven by R = T − τ T log 2 1 + N P t | α g | 2 | α h | 2 σ 2 sin 2 ( π M δ u 2 ) sin 2 ( π δ u 2 ) ! . (15) B. CRB and MSE for Initial Sensing in Phase I In this subsection, the target angle is first estimated via the maximum likelihood estimator (MLE). Then, the CRB and an approximated closed-form expression for MSE associated with the angle estimation is derived. The receiv ed echo signals at the SEs in (10) by letting s [ t ] = 1 , ∀ t can be represented as y s [ t ] = p N P t α g ( H t diag( ϕ [ t ]) a r ( θ B I ) + a s ( θ B I )) + n s [ t ] = p N P t α g  α s a s ( θ I T ) a H r ( θ I T ) diag( ϕ [ t ]) a r ( θ B I ) + a s ( θ B I )) + n s [ t ] = p N P t α g  α s a s ( θ I T ) a H r ( θ I T ) ϕ [ t ] + a s ( θ B I )) + n s [ t ] , (16) where θ I T = θ I T − θ B I . Note that the first part of y s [ t ] is due to the BS-REs-target-SEs link, while the second part is due to the BS-SEs link. In order to estimate the target angle, the BS-SEs link signal should be first canceled out. Fortunately , the angle θ B I can be estimated in advance since the positions of the BS and IRS are fixed. By exploiting the asymptotic orthogonality of the array steering v ectors [21], the BS-REs- target-SEs link can be extracted from y s [ t ] . Specifically , the echo signals b y s [ t ] from the target can be extracted as b y s [ t ] =  I M s − a s ( θ B I ) a H s ( θ B I ) M s  y s [ t ] . (17) Note that by ignoring the noise, we have y s [ t ] ≜ a s ( θ B I ) a H s ( θ B I ) M s y s [ t ] = p N P t α g  α s M s a s ( θ B I ) a H s ( θ B I ) a s ( θ I T ) a H r ( θ I T ) ϕ [ t ] + a s ( θ B I )  , (18) due to 1 M s a H s ( θ B I ) a s ( θ B I ) = 1 . Note that f ( θ I T ) ≜     a H s ( θ B I ) a s ( θ I T ) M s     2 = sin 2 ( π M s θ I T 2 ) M 2 s sin 2 ( π θ I T 2 ) , (19) whose mainlobe is located within [0 , 2 M s ] . When θ I T = 0 , we hav e f ( θ I T ) = 1 . When θ I T = 2 k M s , k = 1 , 2 , · · · , M s − 1 , we hav e f ( θ I T ) = 0 . When 2 M s ≤ | θ I T | ≤ 2 − 2 M s , f ( θ I T ) ≤ 6 . 25% in the case of M s ≥ 5 . Thus, when | θ I T | ≥ 2 M s , we hav e y s [ t ] ≈ √ N P t α g a s ( θ B I ) . Therefore, when 0 ≤ | θ I T | < 2 M s , which corresponds to the undetectable region Ω u of the target, the echo signal from the target cannot be extracted from the BS-SEs direct signal, and thus the target angle cannot be estimated. Otherwise, the extracted signal can be further expressed as b y s [ t ] = p N P t α g α s a s ( θ I T ) a H r ( θ I T ) ϕ [ t ] + n s [ t ] . (20) No L nformation Region ߩ ௡௜ ߩ ௧௛ Threshold Region SNR (dB) MSE (dB) Fig. 3: MSE versus SNR. By collecting all echo signals from the target during the L symbol durations for beam scanning, we ha ve Y = [ b y s [1] , b y s [2] , · · · , b y s [ L ]] = p N P t α g α s a s ( θ I T ) a H r ( θ I T ) [ ϕ [1] , ϕ [2] , · · · , ϕ [ L ]] + N ≜ p N P t α g α s a s ( θ I T ) q H ( θ I T ) X + N , (21) where X ≜ [ ϕ [1] , ϕ [2] , · · · , ϕ [ L ]] , q ( θ I T ) ≜ a r ( θ I T ) , and N ≜ [ n s [1] , n s [2] , · · · , n s [ L ]] . Let R x ≜ 1 L XX H represent the covariance matrix of X . W ith the codebook designed as in (11), we have X = D and R x = I M . For ease of notation, we simply re-denote θ I T by θ . Let ξ = [ θ , Re { α s } , Im { α s } ] T ∈ R 3 × 1 denote the vector of the unknown parameters to be estimated, which includes the target’ s angle and the complex channel coefficients. Particularly , we are interested in charac- terizing the MSE for estimating the angle. By vectorizing (21), we have v ec( Y ) = α s v ec( U ( θ )) + v ec( N ) , (22) where U ( θ ) = √ N P t α g a s ( θ ) q H ( θ ) X . Then, the target angle can be estimated according to the follo wing theorem. Theor em 1: The angle estimated via the MLE is given by θ MLE = arg max θ   a H s ( θ ) YX H q ( θ )   2 , (23) which can be solved by exhausti ve search ov er [ − 1 , 1] . Pr oof: See Appendix A. Next, we derive the CRB and MSE of the angle estimation. The MSE curve, characterizing the performance of MLE, can typically be di vided into three regions: asymptotic, threshold, and no-information regions [22], as depicted in Fig. 3. In the high SNR regime, the MSE is identical to the CRB, which is known as the asymptotic region. As SNR decreases to a threshold ρ th , the MSE starts to de viate from the CRB due to the presence of outliers, giving rise to the threshold region. Upon further decrease in SNR to a threshold ρ ni , the desired signals become indistinguishable from the noise, resulting in a uniformly distrib uted estimation result across the entire parameter space, referred to as the no-information region. Therefore, a useful method to predict the MSE is the method of interval errors (MIE), which divides MSE into two parts: a local error term close to the true value (i.e., the CRB), and an outlier term accounting for global errors. The MIE 6 approach is widely employed for MSE prediction under MLE [23]–[26]. Thus, we employ MIE to predict the MSE of angle estimation in this paper . Define ˆ θ as the estimation of θ , the MSE can be expressed as MSE = E [( ˆ θ − θ ) 2 ] = Pr( no outlier ) E [( ˆ θ − θ ) 2   no outlier ] + Pr( outlier ) E [( ˆ θ − θ ) 2   outlier ] , (24) where “outlier” denotes the event that ˆ θ falls outside the mainlobe of the objectiv e function. When MSE is located in the no-information region, we have ˆ θ ∼ U ( − 1 , 1) and E [( ˆ θ − θ ) 2   outlier ] = 1 3 . When MSE is located in the asymptotic region, we hav e E [( ˆ θ − θ ) 2   no outlier ] = CRB I where CRB I denotes the CRB of angle estimation in Phase I. Let p = Pr( no outlier ) , the MSE can be re written as MSE = 1 − p 3 + p · CRB I . (25) In the sequel, we first deriv e CRB I , and then find the prob- ability of the e vent of “no outlier”. Let F ∈ R 3 × 3 denote the Fisher information matrix (FIM) for estimating ξ . Since N ∼ C N ( 0 , σ 2 I LM s ) , each entry of F is gi ven by [27] F i,j = 2 σ 2 Re ( ∂ U H ∂ ξ i ∂ U ∂ ξ j ) , i, j ∈ { 1 , 2 , 3 } , (26) where U = α s v ec( U ( θ )) . The FIM can be partitioned as F =  F θθ F θ α F T θ α F αα  , (27) where α = [Re { α s } , Im { α s } ] T . The CRB for estimating the angle θ is defined as CRB ( θ ) = [ F − 1 ] 1 , 1 = [ F θθ − F θ α F − 1 αα F T θ α ] − 1 . (28) Then, the CRB for target sensing is gi ven by the following theorem. Theor em 2: The CRB for estimating the angle θ is giv en by CRB( θ ) = σ 2 2 | α s | 2  tr  ˙ U ( θ ) ˙ U H ( θ )  − | tr ( U ( θ ) ˙ U H ( θ ) ) | 2 tr( U ( θ ) U H ( θ ))  . (29) Pr oof: See Appendix B. W ith the array response vector defined as in (5), we obtain a H s ( θ ) ˙ a s ( θ ) = 0 , ˙ a H s ( θ ) a s ( θ ) = 0 , (30) q H ( θ ) ˙ q ( θ ) = 0 , ˙ q H ( θ ) q ( θ ) = 0 , ∀ θ . (31) Consequently , the CRB for estimating the angle θ can be simplified as CRB I = σ 2 2 LN P t | α s | 2 | α g | 2 ( M ∥ ˙ a s ( θ ) ∥ 2 + M s ∥ ˙ q ( θ ) ∥ 2 ) = 6 σ 2 LN P t | α g | 2 | α s | 2 π 2 M M s ( M 2 + M 2 s − 2) ≜ 1 ρ t π 2 N L 6 M M s ( M 2 + M 2 s − 2) , (32) where ρ t ≜ P t | α g | 2 | α s | 2 σ 2 represents the SNR of the target. For a giv en set of SNR and the number of BS antennas, we can improve the sensing accuracy by increasing the codebook size L , the number of REs M , and the number of SEs M s , according to (32). It is worth noting that the estimation of the spatial direction θ I T exhibits an interesting characteristic: the resulting CRB remains unaf fected by the spatial direction itself, as the employed codebook uniformly di vides the entire spatial direction space. Howe ver , when estimating the physical angle ζ I T of the target, the corresponding CRB becomes dependent on the specific physical angle. In addition, it can be observed from (32) that the number of REs M and the number of SEs M s play equal roles in the CRB. Giv en that SEs hav e higher hardware cost, it would be more fav orable to deploy more REs than SEs ( M > M s ) if the total number of elements are fixed (i.e, M + M s is fixed). Next, we find the probability of the event of “no outlier”, which can be determined according to the following theorem. Theor em 3: The probability of the ev ent of “no outlier” under the MLE can be approximated by p ≈  1 − 1 2 exp  − Lρ t N M M s 2  M s + M − 2 . (33) Pr oof: See Appendix C. By substituting (32) and (33) back into (25), we can obtain an approximate closed-form expression of MSE. W ith the obtained MSE, we have the follo wing corollary . Cor ollary 1: The no-information threshold ρ ni of the MSE under the MLE can be approximated by ρ ni ≈ − 2 LN M M s ln h 2  1 − (1 − β ni ) 1 M s + M − 2 i , (34) where β ni ∈ (0 , 1) is an empirical parameter , and is usually set as 8 9 [23]. The breakdown threshold ρ th of the MSE under the MLE can be approximated by ρ th ≈ 2 LN M M s [ln ( ρ 0 ) + ln (ln ( ρ 0 ))] , (35) where ρ 0 = π 2 M 2 M 2 s ( M s + M − 2) 2 . Pr oof: The proof is similar to that in [23] and thus omitted here due to space limitation. Based on Corollary 1, it is observed that when the SNR of the target surpasses ρ th , the MSE coincides with the CRB. If the SNR lies within the range of ρ ni and ρ th , the MSE de viates from the CRB. Con versely , when the SNR falls below ρ ni , the MSE becomes independent of the SNR. Consequently , in order to achieve accurate estimation of the target angle, it is imperativ e for the SNR to e xceed ρ th . I V . E N H A N C E D S E N S I N G D U R I N G D A TA T R A N S M I S S I O N At the end of IRS beam scanning process, the communi- cation user identifies the IRS’ s best beam index ℓ and the corresponding recei ved signal po wer, and feeds them back to the IRS controller where the corresponding user’ s SNR γ ℓ can be obtained. The SEs perform an initial estimation of the target angle according to Theorem 1. Subsequently , two cases arise in the data transmission phase. If the estimated angle is close to the communication user, all REs should 7 reflect the data signal to wards the user with the IRS’ s best beam, which also helps achiev e high SNR at the target to enhance the estimation accuracy (i.e., IRS single-beam). On the other hand, if the angles of the user and the target with respect to the IRS are well separated, beam splitting could be conducted such that some of REs adopt a different beam to provide beamforming gain to improve the sensing accuracy , provided that the achiev able rate of the user with the remaining REs is still satisfactory . In this section, we in vestigate the enhancement of target angle estimation when the user’ s achiev able rate has sufficient margin, considering the above two IRS beamforming strategies, respecti vely . A. Single Beam Based Sensing A straightforward strategy for REs is to adopt the best beam ϕ ⋆ = a r ( η ℓ ) for the communication user . In this case, the echo signals from the target in (20) can be rewritten as b y s [ t ] = p N P t α g α s a s ( θ I T ) a H r ( θ I T ) ϕ ⋆ s [ t ] + n s [ t ] . (36) The IRS beamforming gain for the target can be expressed as   a H r ( θ I T ) ϕ ⋆   =      M X m =1 e j π δ t ( − M − 1 2 + m − 1 )      = sin( π M δ t 2 ) sin( π δ t 2 ) , (37) where δ t ≜ | θ I T − η ℓ | denotes the spatial direction difference between θ I T and the best beam η ℓ . As described in Section III-B, this IRS beamforming gain is small when δ t ≥ 2 M . Therefore, the region where the target estimation can benefit from this strategy is gi ven by { Ω t | θ I T : | θ I T − θ B I − η ℓ | < 2 M , θ I T / ∈ Ω u } . In the following, we first estimate the target angle via the MLE, and then deri ve its CRB. By collecting the echo signals from the target in Phase II, we have Y 2 = [ b y s [ τ + 1] , b y s [ τ + 2] , · · · , b y s [ τ + τ 2 ]] = p N P t α g α s a s ( θ I T ) q H ( θ I T ) ϕ ⋆ × [ s [ τ + 1] , s [ τ + 2] , · · · , s [ τ + τ 2 ]] + N 2 ≜ p N P t α g α s a s ( θ I T ) q H ( θ I T ) X 2 + N 2 , (38) where X 2 ≜ ϕ ⋆ [ s [ τ + 1] , s [ τ + 2] , · · · , s [ τ + τ 2 ]] ∈ C M × τ 2 , τ 2 = T − τ , and a r ( θ I T ) is replaced by q ( θ I T ) as in (21). The data signals are assumed to be independent of each other and satisfy E ( | s [ τ i ] | 2 ) = 1 for τ i ∈ ( τ , τ + τ 2 ] . Thereby , X 2 X H 2 = τ 2 ϕ ⋆ ( ϕ ⋆ ) H . Then, we collect the echo signals in Phase I and Phase II together to estimate the angle of the target. By collecting (21) and (38), we hav e Y = p N P t α g α s a s ( θ I T )  q H ( θ I T ) X , q H ( θ I T ) X 2  +[ N , N 2 ] . (39) Accordingly , the target angle can be estimated based on the following theorem. Theor em 4: The angle estimated via the MLE by combining the echo signals in Phase I and Phase II together is giv en by θ MLE = arg max θ   a H s ( θ ) YX H q ( θ ) + a H s ( θ ) Y 2 X H 2 q ( θ )   2 LM + ( T − τ ) | q H ( θ ) ϕ ⋆ | 2 , (40) which can be found by exhaustiv e search over [ − 1 , 1] . Pr oof: The proof is similar to that in Appendix A. Thus, it is omitted here due to space limitation. Note that the MLE needs to know the transmitted signals X 2 during Phase II, which is not av ailable due to random data signals. Ho wever , there are alternati ve estimation methods av ailable that do not rely on the kno wledge of transmitted signals in practice. One such method is the Multiple Signal Classification (MUSIC) algorithm, which can approach MLE performance in high SNR scenarios without the need for information about the transmitted signals [28]. Next, we consider the CRB of angle estimation via the MLE. W e can adopt the MIE employed in Section III-B to predict the MSE. Howe ver , due to the existence of the denominator term in (40), the chi-square distribution cannot be adopted for the MLE. Consequently , the method utilized in Theorem 3 to calculate the probability of the “no out- lier” ev ent cannot be applied in this scenario. Hence, we only explore the CRB (instead fo MSE) of angle estimation. For notational conv enience, we omit θ I T here. Let U 2 ≜ √ N P t α g a s  q H X , q H X 2  . W e thus ha ve ˙ U 2 = p N P t α g ˙ a s  q H X , q H X 2  + p N P t α g a s  ˙ q H X , ˙ q H X 2  . (41) According to Theorem 2, the CRB for angle estimation in the whole phase (i.e., by combining the echo signals in Phase I and Phase II together) can be obtained as follo ws, CRB w = σ 2 2 | α s | 2  tr( ˙ U 2 ˙ U H 2 ) − | tr( U 2 ˙ U H 2 ) | 2 tr( U 2 U H 2 )  = σ 2 2 | α s | 2  β 1 M ∥ ˙ a s ∥ 2 2 + β 2 | q H ϕ ⋆ | 2 ∥ ˙ a s ∥ 2 2 + β 1 M s ∥ ˙ q ∥ 2 2 + β 2 M s | ˙ q H ϕ ⋆ | 2 − | β 2 | 2 M 2 s | q H ϕ ⋆ ( ϕ ⋆ ) H ˙ q | 2 β 1 M s ∥ q ∥ 2 2 + β 2 M s | q H ϕ ⋆ | 2  − 1 ( a ) ≤ σ 2 2 | α s | 2 1 β 1 M ∥ ˙ a s ∥ 2 2 + β 1 M s ∥ ˙ q ∥ 2 2 + β 2 | q H ϕ ⋆ | 2 ∥ ˙ a s ∥ 2 2 = 1 ρ t π 2 N × 6 LM M s ( M 2 + M 2 s − 2) + τ 2 | q H ϕ ⋆ | 2 M s ( M 2 s − 1) ≜ CRB up , (42) where CRB w denotes the CRB for angle estimation in the whole phase, β 1 ≜ N P t | α g | 2 L , β 2 ≜ N P t | α g | 2 τ 2 , ( a ) holds by ignoring the term β 1 M s ∥ q ∥ 2 2 , the equality holds when q = ϕ ⋆ , and | q H ϕ ⋆ | 2 =    sin( π M δ t / 2) sin( π δ t / 2)    2 . Compared with the CRB in (32), it is observed that the first component of CRB up (i.e., LM M s ( M 2 + M 2 s − 2) ) represents the contrib ution of Phase I estimation, and the second component of CRB up (i.e., τ 2 | q H ϕ ⋆ | 2 M s ( M 2 s − 1) ) represents the contrib ution of Phase II estimation. The inequal- ity CRB w ≤ CRB up < CRB I indicates that the estimation accuracy is improv ed by combining Phase I and Phase II together as compared to Phase I. Notably , there are distinct behaviors exhibited by the REs in Phase II compared to Phase I. On one hand, the REs conduct beam scanning in Phase I, thus providing more flexibility to sense the target. 8 Consequently , even if the number of SEs is one (i.e., M s = 1 ), we can still estimate the target angle as in (32). On the other hand, the REs fix their beamforming direction during Phase II, instead of perusing further beam scanning. As such, if the number of SEs is one in Phase II, the second component of CRB up becomes zero, and thus the sensing performance cannot be improv ed ef fectively . B. Beam Splitting Based Sensing The single-beam-based sensing is effecti ve only when the target locates in the vicinity of the communication user . T o fur- ther enhance target estimation, we propose IRS beam splitting when the communication user’ s achiev able rate has sufficient margin. Specifically , REs can be divided into two groups: one for communication and the other for target sensing, with different beams, respectiv ely . Howe ver , beam splitting can potentially degrade the communication SNR and introduce interference between the two beams. Therefore, it becomes important to determine the conditions and the portion of REs for applying the beam splitting. W e first examine the impact of beam splitting on data transmission. Suppose M e REs are allocated for target angle estimation, while the remaining M − M e REs are dedicated to data transmission in Phase II. Since the initial target angle is estimated in Phase I, the M e REs are specifically designed to align with the estimated angle in order to maximize the signal power at that target. The remaining M − M e elements are set to be the last M − M e entries of the IRS’ s best beam to serve the communication user . For the ease of analysis, we assume that the user exactly locates at its best beam direction (i.e., δ u = 0 ), and the following analysis can be e xtended to the case where δ u  = 0 . When all REs are used for communication, the IRS’ s best beam is given by ϕ ⋆ = e − j π ( M − 1) θ I U 2 h 1 e j π θ I U . . . e j π ( M − 1) θ I U i T . (43) Then, we have | a H r ( θ I U ) ϕ ⋆ | = M . When M e REs are allocated for target angle estimation, the reflection coefficients of REs are given by ϕ e = e − j π ( M − 1) θ I U 2 h 1 , e j π θ I T , . . . , e j π ( M e − 1) θ I T , e j π M e θ I U , . . . , e j π ( M − 1) θ I U i T . (44) Then, the IRS beamforming gain for the communication user can be obtained as G IRS ≜ | a H r ( θ I U ) ϕ e | =      M − M e + ϱ sin  π M e δ U T 2  sin  π δ U T 2       , (45) where δ U T = | θ I U − θ I T | denotes the spatial direc- tion difference between the user and target, and ϱ = exp  j π δ U T ( M e − 1) 2  . Due to the existence of ϱ and sin( π M e δ U T / 2) , G IRS may be smaller than M − M e , which means that the M e REs allocated for target estimation may negati vely impact the communication rate performance. Thus, it is crucial to determine the condition under which G IRS does 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 UT 0 10 20 30 40 50 60 IRS beamforming gain G IRS G IRS y = M - M e x = 11/M Fig. 4: IRS beamforming gain G IRS versus δ U T when M = 60 . not deviate much from M − M e . Then, we have G 2 IRS =( M − M e ) 2 + sin 2  π M e δ U T 2  sin 2  π δ U T 2  + 2( M − M e ) sin  π M e δ U T 2  cos  π ( M e − 1) δ U T 2  sin  π δ U T 2  ( a ) ≈ ( M − M e ) 2 + sin 2  π M e δ U T 2  sin 2  π δ U T 2  + 2( M − M e ) π δ U T sin( π M e δ U T ) , (46) where ( a ) holds due to 2 sin  π M e δ U T 2  cos  π ( M e − 1) δ U T 2  ≈ sin( π M e δ U T ) for M e ≫ 1 , and sin  π δ U T 2  ≈ π δ U T 2 for δ U T ≪ 1 . W e find that the effect on G IRS mainly stems from sin( π M e δ U T ) . When sin( π M e δ U T ) = − 1 , i.e., M e δ U T = 3 2 + 2 k , k = 0 , 1 , 2 , . . . , G IRS can be much smaller than M − M e , and thus de grade the communication SNR dramat- ically . It is also observed that when M e δ U T > 2 , the value of sin( π M e δ U T / 2) sin( π δ U T / 2) in (45) will be very small. Consequently , the threshold of δ U T is selected as 11 2 M e . When δ U T > 11 2 M e , the effect of beam splitting on the communication can be neglected. Note that the effect of beam splitting on the target is similar to (45). Therefore, the threshold of δ U T is set to 11 M . When δ U T > 11 M , it can be reg arded that the M e REs and the other M − M e REs cause only negligible interference to each other . Overall, the condition for choosing beam splitting is when the target angle falls in the region { Ω e | θ I T : | θ I T − θ B I − η ℓ | > 11 M , θ I T / ∈ Ω u } 2 . Fig. 4 illustrates G IRS versus δ U T when M e is set to different values. It is found that 11 M is a good threshold. When δ U T > 11 M , G IRS fluctuates only slightly around M − M e . Next, we determine the element number for beam splitting. The user feeds the IRS’ s best beam index ℓ and corresponding receiv ed signal power back to the IRS controller where the 2 Note that the above analysis assumes that the user exactly locates at the optimal beam direction, i.e., δ u = 0 . When δ u  = 0 , we have δ U T ∈ {| θ I T − η ℓ | − δ u , | θ I T − η ℓ | + δ u } . In the worse case, we ha ve δ U T = | θ I T − η ℓ | − 1 L > 10 M , where the user and target are also well separated. This is the reason why we do not choose δ U T as 7 2 M e , which is not suf ficient in general to ensure the user and target to be well separated. 9 -5 0 5 10 15 20 25 Transmit Power (dBm) 10 -8 10 -6 10 -4 10 -2 10 0 MSE or CRB MSE Analytical Approximation in (25) CRB I in (32) No-information threshold in (34) Breakdown threshold in (35) Fig. 5: MSE versus transmit power in Phase I when M = 64 and M s = 12 . 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 Ratio of Beam Scanning Time and Coherence Time, /T 3.8 3.9 4 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 Achievable Rate (bps/Hz) 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 MSE 10 -6 Achievable Rate in (15) with u =0 Achievable Rate in (15) Averaged over u Achievable Rate in (15) with u =1/L MSE Fig. 6: Achiev able rate and MSE v ersus ratio of beam scanning time and coherence time with P t = 20 dBm. corresponding user’ s SNR γ ℓ can be obtained. If the SNR γ ℓ of the communication user is lar ger than the predefined target γ , the beam splitting can be performed in the region Ω e . Howe ver , the exact angle of the user is unav ailable with only the knowledge of the best beam η ℓ . Thus, we consider the worst case to decide the element number for beam splitting. Specifically , the user’ s SNR is gi ven by γ ℓ = N P t | α g | 2 | α h | 2 σ 2 sin 2 ( π M δ u 2 ) sin 2 ( π δ u 2 ) ≜ G ch sin 2 ( π M δ u 2 ) sin 2 ( π δ u 2 ) , (47) where G ch ≜ N P t | α g | 2 | α h | 2 σ 2 is the unkno wn channel gain. Since the expression sin 2 ( π M δ u 2 ) / sin 2 ( π δ u 2 ) decreases as δ u increases within the interval [0 , 1 /L ] , we first assume that the user exactly locates at the optimal beam direction (i.e., δ u = 0 ) to obtain the worst channel gain G ch,min = γ ℓ M 2 under a giv en γ ℓ . Then, we set δ u in (48) to 1 /L to obtain the worst IRS beamforming gain. Therefore, when M e REs are split for target sensing, the worst channel gain multiplies the worst IRS beamforming gain, leading to the worst SNR γ e of the user as follows, γ e = G ch,min sin 2 ( π ( M − M e ) δ u 2 ) sin 2 ( π δ u 2 ) = γ ℓ M 2 sin 2 ( π ( M − M e ) 2 L ) sin 2 ( π 2 L ) ≥ γ . (48) Therefore, the element number for target sensing is given by M e ≤  M − 2 L π arcsin  M r γ γ ℓ sin  π 2 L   (49a) ( a ) ≈  M  1 − r γ γ ℓ  , (49b) where ( a ) holds due to sin( x ) ≈ x and arcsin( x ) ≈ x for x ≪ 1 . In order to improv e the sensing accuracy , M e can be set as the right-hand side of (49a). It is noted that when the user actually locates at the optimal beam direction, we can obtain M e = j M  1 − q γ γ ℓ k . Finally , we consider the performance of tar get estimation after beam splitting. Following the similar procedure in Sec- tion IV -A, we can obtain the corresponding CRB of angle estimation. Consequently , the CRB is similar to that in (42), with q H ϕ ⋆ replaced by q H ϕ e . V . S I M U L AT I O N R E S U LT S In this section, numerical examples are provided to validate our analysis and e valuate the performance of our proposed protocol. The carrier frequency is f c = 28 GHz. Other system parameters are set as follo ws unless specified otherwise later: N = 64 , M = 64 , M s = 12 , L = M , P t = 30 dBm, T = 1000 , d B I = 30 m, d I U = 10 m, d I T = 5 m, ζ B I = − 60 ◦ , ζ I U = 0 ◦ , ζ I T = 30 ◦ , σ 2 = − 120 dBm, and κ = 7 dBsm. The curv es of MSE are obtained by a veraging ov er 1000 independent realizations of the noise. A. Communication and Sensing T r ade-off in Phase I W e first consider the MSE performance of MLE in Phase I. Fig. 5 illustrates a comparison of the analytical approx- imation of MSE in (25) and the actual MSE with respect to the transmit power when M = 64 and M s = 12 . There are some interesting findings. Firstly , the deri ved analytical approximation closely matches the actual MSE in both low- SNR and high-SNR regimes. When the transmit power falls in the threshold region, the deriv ed analytical approximation slightly overestimates the actual MSE. Overall, the deriv ed analytical approximation performs well. Secondly , Corollary 1 can well predict the no-information threshold and breakdown threshold. Thus, in order to ensure reliable angle estimation in Phase I, the transmit power should exceed the breakdo wn threshold. Next, we in vestig ate the communication and sensing trade- off in Phase I. Fig. 6 sho ws the achie vable rate and MSE v ersus the ratio of beam scanning time and coherence time, when all REs are adopted for communication during data transmission period. The curve labeled “ Achie vable Rate in (15) A veraged ov er δ u ” is generated by taking the expectation of (15) over δ u , where δ u ∼ U (0 , 1 L ) . Since exhausting beam scanning is adopted, we hav e τ = L as in Section III-A. The achiev able rate and MSE exhibit different variations versus the IRS beam scanning time. As the time of beam scanning increases, the IRS beamforming gain (14) in the case of δ u = 1 L increases at the expense of reduced data transmission time, leading to an initial increase and subsequent decrease in the achie vable 10 -5 0 5 10 15 20 25 Transmit Power (dBm) 10 -8 10 -6 10 -4 10 -2 10 0 MSE or CRB MSE in Phase I CRB I MSE in the whole phase CRB w CRB up Fig. 7: MSE versus transmit po wer in Phase II when ζ I U = 0 ◦ and ζ I T = 0 ◦ . -80 -60 -40 -20 0 20 40 60 80 IT (degree) 10 -8 10 -6 10 -4 10 -2 10 0 MSE Fig. 8: MSE versus the target angle in beam-splitting- based sensing when M = 64 and M e = 36 . rate. In the case of δ u = 0 , the IRS beamforming gain already reaches its peak when L = M . Thus, increasing the beam scanning time further only leads to a reduction in the achie v- able rate. Howe ver , when considering the average effect, the achiev able rate initially rises and subsequently declines. On the other hand, the MSE monotonically decreases as the time of beam scanning increases. Thus, a proper beam scanning time that balances achiev able rate and MSE is desired. Howe ver , with our proposed enhanced sensing approaches in Section IV, we can first ensure the communication quality in Phase I, and then further improve sensing accurac y in Phase II. B. Communication and Sensing T r ade-off in Phase II W e first consider the MSE performance in Phase II as depicted in Fig. 7, where the target angle ( ζ I T = 0 ◦ ) is close to the communication user’ s angle. Several interesting findings are inferred from the results. Firstly , the concise form of CRB up closely approximates CRB w , making it con venient for analysis. Secondly , when the transmit power is below 5 dBm, the curve of CRB up is nearly the same as that of CRB I . This occurs because the MSE falls within the no-information region, resulting in significant estimation errors in Phase I and | q H ϕ ⋆ | 2 ≈ 0 . As a result, the sensing accuracy cannot be improv ed in Phase II. Thirdly , the no-information threshold and breakdown threshold in Phase II closely resemble those in Phase I since the REs allocated for sensing in Phase II point their beam to wards the angle estimated in Phase I. Consequently , if the angle estimation in Phase I is inaccurate, the REs for sensing cannot point their beam towards the target effecti vely in Phase II, thus resulting in limited performance improv ement. Next, the influence of the undetectable region on the sensing performance is illustrated in Fig. 8, which depicts the MSE versus the target angle in beam-splitting-based sensing when M = 64 and M e = 36 . It is observed that the MSE is high when θ I T < − 44 ◦ or θ I T > 75 ◦ , primarily due to the pres- ence of the undetectable region. Specifically , the undetectable region is defined as { Ω u | − 90 ◦ < ζ I T < − 44 . 4 ◦ or 75 . 3 ◦ < ζ I T < 90 ◦ } , where 180 π arcsin  sin  − 60 π 180  + 2 12  ≈ − 44 . 4 ◦ and 180 π arcsin  sin  − 60 π 180  − 2 12 + 2  ≈ 75 . 3 ◦ . In addition, the MSE increases for | ζ I T | < 10 ◦ due to the beam splitting only conducted at arcsin(11 / M ) = 9 . 90 ◦ . T o better present the MSE performance, we focus on the region { ζ I T | − 40 ◦ ≤ ζ I T ≤ 70 ◦ } in the subsequent in vestigation. W e compare the achie vable rate and MSE versus the number of REs used for sensing in Fig. 9 and Fig. 10 where ζ I T is set to 30 ◦ and 3 ◦ , respectively . The curve labeled “Refer- ence Achiev able Rate” represents the achiev able rate obtained when M − M e REs are dedicated to communication without any interference. The curve labeled “Reference MSE in the whole phase” represents the MSE obtained when M e REs are dedicated to tar get sensing without any interference in Phase II. The leftmost point on the curve of “Reference MSE in the whole phase” represents the MSE in Phase I. It is firstly observed that the achiev able rate sho ws a decreasing trend, while the estimation accurac y exhibits an increasing trend as the number of REs allocated for sensing increases. When the user and target are well separated (Fig. 9), the achiev able rate and MSE remain relativ ely close to their references, respec- tiv ely . It is worth noting that the slight fluctuation in the curve labeled “ Achie vable Rate” is due to the minimal impact of the REs split for sensing on the communication performance. This observation is consistent with the results shown in Fig. 4, where slight fluctuation occurs when δ U T > 11 M . Ho wev er, when the user and target are close (Fig. 10), beam splitting results in a significant drop in the achie vable rate compared to its reference around M e = 25 . The MSE in the whole phase fluctuates more sev erely around the reference MSE, but it is always smaller than the MSE in Phase I. Therefore, beam- splitting-based sensing can only be adopted when the user and target are well separated to ensure the communication quality , as discussed in Section IV -B. Next, we proceed to e valuate the MSE versus the tar get angle in the single-beam-based sensing and beam-splitting- based sensing as displayed in Fig. 11 when the transmit power is P t = 30 dBm. The corresponding achiev able rate is shown in Fig. 12, with a threshold achiev able rate of the communication user set at 5.0 bps/Hz. The number of REs split for target sensing is determined as M e = 36 according 11 0 10 20 30 40 50 60 Number of IRS REs Used for Sensing, M e 1 2 3 4 5 6 7 Achievable Rate (bps/Hz) 0 0.2 0.4 0.6 0.8 1 1.2 1.4 MSE 10 -7 Achievable Rate Reference Achievable Rate MSE in the whole phase Reference MSE in the whole phase MSE in Phase I Fig. 9: Achiev able rate and MSE versus the number of REs used for sensing when P t = 30 dBm, ζ I U = 0 ◦ and ζ I T = 30 ◦ . 0 10 20 30 40 50 60 Number of IRS REs Used for Sensing, M e 1 2 3 4 5 6 7 Achievable Rate (bps/Hz) 0 0.2 0.4 0.6 0.8 1 1.2 1.4 MSE 10 -7 Achievable Rate Reference Achievable Rate MSE in the whole phase Reference MSE in the whole phase MSE in Phase I Fig. 10: Achiev able rate and MSE v ersus the number of REs used for sensing when P t = 30 dBm, ζ I U = 0 ◦ and ζ I T = 3 ◦ . -40 -30 -20 -10 0 10 20 30 40 50 60 70 IT (degree) 10 -8 10 -7 MSE Single-beam-based Sensing Beam-splitting-based Sensing Fig. 11: MSE versus the tar get angle in single-beam- based sensing and beam-splitting-based sensing when P t = 30 dBm. -40 -30 -20 -10 0 10 20 30 40 50 60 70 IT (degree) 5 5.2 5.4 5.6 5.8 6 6.2 6.4 6.6 6.8 Achievable Rate (bps/Hz) Single-beam-based Sensing Beam-splitting-based Sensing Fig. 12: Achiev able rate versus the target angle in single- beam-based sensing and beam-splitting-based sensing when P t = 30 dBm. to (49a). Sev eral interesting findings are observed. Firstly , in Fig. 11, the MSE decreases significantly around ζ I T = 0 ◦ in the single-beam-based sensing. This behavior can be attributed to the region { Ω t   | ζ I T | < arcsin(2 / M ) ≈ 1 . 80 ◦ } . Secondly , the MSE in the single-beam-based sensing also decreases for | ζ I T | < 10 ◦ due to the sidelobes of the function sin( π M x/ 2) sin( π x/ 2) . Howe ver , the MSE exhibits little improvement since the target is outside the region Ω t , and the signals cannot effecti vely reach the target. Thirdly , in the beam-splitting-based sensing, the MSE decreases for | ζ I T | > 10 ◦ due to the beam splitting conducted at arcsin(11 / M ) ≈ 9 . 90 ◦ . Ho wever , since only a portion of REs is allocated for target sensing during Phase II, the sensing performance in the beam-splitting-based sensing is inferior to that achiev ed at the re gion Ω t in the single-beam- based sensing. As for the achiev able rates sho wn in Fig. 12, the single- beam-based sensing maintains a constant achiev able rate since all REs reflect the signals to wards the user’ s direction in Phase II. Con versely , the achiev able rate in the beam-splitting- based sensing decreases from 6.7 bps/Hz to around 5.3 bps/Hz when | ζ I T | > 10 ◦ . Nevertheless, this rate remains above the 5.0 bps/Hz threshold. Consequently , the single-beam-based sensing benefits the target sensing when the angles of the user and target are close to each other . On the other hand, the beam-splitting-based sensing improves the sensing accuracy at the cost of reducing the achiev able rate of the communication user , which is advantageous when the user and target are well separated in angle and the achiev able rate of the user has sufficient margin. Furthermore, we explore the achiev able rate and MSE with respect to transmit power and target angle in Fig. 13 and Fig. 14, respectiv ely . It is notew orthy that when the transmit power is below 22 dBm, the achiev able rate remains below 5 bps/Hz, indicating insufficient rate mar gin for target sensing during Phase II by beam splitting based sensing. Hence, the sensing performance can only be improv ed in the region Ω t with single-beam based sensing. As the transmit power increases, beam splitting can be employed, resulting in further enhanced sensing performance in the region Ω e . 12 Fig. 13: Achie vable rate versus the target angle and transmit power . Fig. 14: MSE v ersus the target angle and transmit po wer . V I . C O N C L U S I O N In this paper, we hav e proposed a ne w ISA C protocol for an IRS-aided mmW ave system that utilizes downlink beam scanning/data signals for achie ving simultaneous beam training and target sensing. W e deri ve the achiev able rate of the communication user and the CRB/MSE of the target angle estimation in the beam scanning and data transmission phases, respectively . In particular, two IRS beam design and sensing strategies, namely , single-beam-based sensing and beam-splitting-based sensing, are proposed to enhance the sensing accuracy during the data transmission phase while ensuring the communication quality . Numerical results have verified the effecti veness of the proposed protocol and design. A P P E N D I X A. Proof of Theor em 1 Denoting v ec( Y ) as e y , the likelihood function of vec( Y ) giv en ξ is L ( e y ; ξ ) = 1 ( π σ 2 ) LM s exp  − 1 σ 2 ∥ e y − α s v ec( U ( θ )) ∥ 2  . (50) Then, maximizing the likelihood function is equiv alent to minimizing ∥ e y − α s v ec( U ( θ )) ∥ 2 . Therefore, the MLE can be written as ( θ MLE , α MLE ) = arg min θ,α ∥ e y − α s v ec( U ( θ )) ∥ 2 . (51) W ith any given θ , the optimal α is given by α MLE = (vec( U ( θ ))) H e y ∥ vec( U ( θ )) ∥ 2 . By substituting α MLE back into (51), yielding ∥ e y − α MLE v ec( U ( θ )) ∥ 2 = ∥ e y ∥ 2 −   (v ec( U ( θ ))) H v ec( Y )   2 ∥ v ec( U ( θ )) ∥ 2 = ∥ e y ∥ 2 −   a H s ( θ ) YX H q ( θ )   2 LM M s . (52) Thereby , the MLE of θ is giv en by (23) . B. Proof of Theor em 2 Since ∂ U ∂ θ = α s v ec( ˙ u ( θ )) and ∂ U ∂ α = [1 , j ] ⊗ vec( u ( θ )) , we hav e F θθ = 2 σ 2 Re  α s  v ec( ˙ U )  H α s v ec( ˙ U )  = 2 | α s | 2 σ 2 tr  ˙ U ˙ U H  . (53) F θ α = 2 σ 2 Re n α ∗ s v ec( ˙ U H )[1 , j ] ⊗ vec( U ) o = 2 σ 2 Re n α ∗ s [1 , j ] ⊗  v ec( ˙ U H ) v ec( U ) o = 2 σ 2 Re n α ∗ s tr  U ˙ U H  [1 , j ] o . (54) F αα = 2 σ 2 Re n ([1 , j ] ⊗ vec( U )) H [1 , j ] ⊗ vec( U ) o = 2 σ 2 Re n [1 , j ] H [1 , j ]  ⊗  v ec( U H ) v ec( U ) o = 2 σ 2 Re n [1 , j ] H [1 , j ]  ⊗  tr( U H U ) o = 2 σ 2 tr  UU H  I 2 . (55) Thus, the FIM can be obtained as in (29). C. Proof of Theor em 3 Based on the MLE in Theorem 1, we ha ve I ( θ − θ 0 ) ≜    a H s ( θ ) YX H q ( θ )    2 =    a H s ( θ )  √ N P t α g α s a s ( θ 0 ) q ( θ 0 ) H X + N  X H q ( θ )    2 =    √ N P t Lα g α s a H s ( θ ) a s ( θ 0 ) q ( θ 0 ) H q ( θ ) + a H s ( θ ) NX H q ( θ )    2 ≜    √ N P t Lα g α s f M s ( θ − θ 0 ) f M ( θ − θ 0 ) + w    2 , (56) where θ 0 is the actual angle to be estimated, f M ( θ ) ≜ sin( π M θ/ 2) sin( π θ/ 2) , and w ≜ a H s ( θ ) NX H q ( θ ) ∼ C N (0 , LM M s σ 2 ) . Therefore, 2 LM M s σ 2 I ( θ − θ 0 ) is a non-central chi-square distributed random variable with two degrees of freedom with non-centrality parameter gi ven by Υ = 2 N P t L 2 | α g | 2 | α s | 2 LM M s σ 2 f 2 M s ( θ − θ 0 ) f 2 M ( θ − θ 0 ) = 2 Lρ t N M M s f 2 M s ( θ − θ 0 ) M 2 s f 2 M ( θ − θ 0 ) M 2 . (57) Therefore, the periodogram sampled at discrete point { 2 k M s   k = 0 , 1 , . . . , M s − 1 } and { 2 i M   i = 0 , 1 , . . . , M − 1 } 13 is distributed according to 2 I ( θ − θ 0 ) LM M s σ 2 ∼    χ 2 2 (2 Lρ t N M M s ) , θ − θ 0 = 0 , χ 2 2 , θ − θ 0 = 2 M s k , k  = 0 , χ 2 2 , θ − θ 0 = 2 M i, i  = 0 , (58) where χ 2 2 and χ 2 2 ( · ) represent the central and non-central chi- square distributions with two degrees of freedoms, respec- tiv ely . Defining e I ( k ) ≜ 2 I ( 2 k M s ) LM M s σ 2 , we have e I ( k ) is χ 2 2 for k  = 0 and χ 2 2 (2 Lρ t N M M s ) for k = 0 . Then, the cdf of e I ( k ) is given by F e I ( k ) ( x ) =  1 − Q 1 ( √ 2 Lρ t N M M s , √ x ) , k = 0 , 1 − exp( − x/ 2) , k  = 0 , (59) where Q 1 ( α, β ) denotes the first-order Marcum-Q function with parameter α and β . Similarly , defining I ( i ) ≜ 2 I ( 2 i M ) LM M s σ 2 , we have I ( i ) is χ 2 2 for i  = 0 and χ 2 2 (2 Lρ t N M M s ) for i = 0 . Then, the cdf of I ( i ) is giv en by F I ( i ) ( y ) =  1 − Q 1 ( √ 2 Lρ t N M M s , √ y ) , i = 0 , 1 − exp( − y / 2) , i  = 0 . (60) Thereby , the probability of the event of “no outlier” can be expressed as p = Pr n e I (0) > max  e I ( k ) o Pr  I (0) > max  I ( i )  = M s − 1 Y k =1 Pr n e I (0) > e I ( k ) o M − 1 Y i =1 Pr  I (0) > I ( i )  , k  = 0 , i  = 0 . (61) W e consider the first part of the abo ve probability . Since e I ( k ) , k  = 0 , are i.i.d. random variables, the first part of the probability can be simplified to p 1 =  Pr n e I (0) > e I ( k ) o M s − 1 =  Z ∞ 0 p e I ( k ) ( x )  1 − F e I (0) ( x )  dx  M s − 1 , k  = 0 , (62) where p e I ( k ) ( x ) is the pdf of the exponentially distrib ution. Then, we hav e p 1 =  Z ∞ 0 1 2 e − x/ 2 Q 1  p 2 Lρ t N M M s , √ x  dx  M s − 1 =  Z ∞ 0 xe − x 2 / 2 Q 1  p 2 Lρ t N M M s , x  dx  M s − 1 =  1 − 1 2 exp  − Lρ t N M M s 2  M s − 1 . 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