A G-MUSIC Algorithm for Angle-of-Arrival Estimation in ISAC Networks

A G-MUSIC Algorithm for Angle-of-Arrival Estimation in ISAC Networks Julia Equi⋆, Gábor Fodor∗§, ⋆Ericsson Research, Finland, Email: [email protected] ∗Ericsson Research, Sweden, E-mail: [email protected] §KTH Royal Institute of Technology, Sweden. E-mail: [email protected] Abstract—One of the main challenges in integrated sensing and communications (ISAC) are interference signals caused by the communication user equipment (UE) emitting in the sensing area. It is therefore necessary to design appropriate transmitter and receive precoding schemes allowing to mitigate interference signals caused by the UEs. Subspace-based algorithms, such as the multiple signal classification (MUSIC) algorithm, are widely used in radar sensing, and are promising candidates for parameter estimation in evolving integrated sensing and communications (ISAC) networks. The MUSIC algorithm gives a consistent estimate of the angle-of-arrival (AoA) localization function when the number of time samples is large as compared to the number of receive antennas. However, when the number of time samples is of the same order of magnitude as the number of receiver antennas, the traditional MUSIC algorithm may fail to separate closely spaced targets. In this paper we propose an improved version of the MUSIC algorithm, that is particularly tailored for ISAC, where bistatic sensing is performed by multi-antenna cellular base stations for simultaneous AoA estimation of sensing targets and communication UEs. As the simulation results show, the proposed method achieves superior performances compared to the traditional MUSIC algorithm, especially when the number of available time samples is of the same size or smaller as compared to the number of receiver antennas. Moreover, the proposed method allows to separate targets and UEs located close to each other, even when the traditional MUSIC fails. Index terms— Angle-of-arrival estimation, integrated sensing and communications, large sensor array, MUSIC algorithm, random matrix theory. I. INTRODUCTION A. ISAC Technology Overview Integrated sensing and communications (ISAC) has emerged as a key enabler of future perceptive cellular networks [1], and as a promising technology component of the emerging multi-functional 6G networks [2], [3]. Indeed, ISAC networks hold the promise of providing a wide range of highly accurate localization of both connected and unconnected objects and thereby extending the range of services offered by local area and cellular networks [1], [4], [5]. Recognizing the potential of ISAC networks, several standardization and regulatory organizations – such as the Institute of Electrical and Electronics Engineers (IEEE), International Telecommunication Union (ITU), 3rd Generation Partnership Project (3GPP) and European Telecommunications This work was supported by the 6G-Multiband Wireless and Optical Signalling for Integrated Communications, Sensing and Localization (6GMUSICAL) EU Horizon 2023 project, funded by the EU, Project ID: 101139176. Standards Institute (ETSI) – as well as industry consortia – such as the 5th Generation 5G Alliance for Connected Industries and Automation (5G-ACIA) and 5th Generation Automotive Association (5GAA) – have started to study the use cases, performance indicators and viable integration levels that pave the way for introducing sensing services into communication networks, see for example [6], [7]. Regarding the levels of integration, the research community has investigated and proposed several methods that utilize the ubiquitous cellular infrastructure, take advantage of the large antenna arrays deployed at cellular base stations and leverage the large computational capacity of the cellular infrastructure [8], [9]. Time-division mode ISAC separates the sensing and communications signals in the time domain, while concurrent mode ISAC enables simultaneous transmissions by utilizing the spatial domain via multibeam techniques [3], [10]. Another popular ISAC integration technique is using a unified waveform for sensing and communications, in which case the same physical layer signal is used for both sensing and communication purposes [3], [11]. Note that each integration technique is characterized by some tradeoff between sensing and communications related to the time or spatial domain or to the waveform. B. Previous Works Along a closely related research line, subspace-based angle estimation schemes based on the multiple signal classification (MUSIC) algorithm, originally described in [12], have been proposed and demonstrated for localizing connected devices in 5G networks [13] and for determining the angle-ofarrival (AoA) of sensing signals reflected by passive objects specifically in ISAC networks [14]–[16]. In those works, the angle estimates of Ksources are based on Ttimes samples of the N×1received signal vector of the form y(t) = A(θ1, . . . , θK)Γs(t) + v(t),(1) where A(θ1, . . . , θK)∈ℂN×Kis the matrix containing K steering vectors corresponding to the AoA of the sources denoted by θ1, . . . , θK,Γ∈ℂK×Kis the diagonal matrix of pathgains, s(t)∈ℂK×1is the transmit signal vector by Ksources at time instant t, and v(t)∈ℂN×1is the noise vector. The traditional MUSIC algorithm is attractive due to its ability of striking a good balance between computational complexity and AoA resolution and accuracy, see for example [17], especially for the case where the number of receiver antennas Nis fixed while the number of available time samples Tis large as compared to N. However, it typically suffers from performance degradation when N and Tare of comparable order of magnitude implying the sample (empirical) covariance matrix of the channel, on which MUSIC depends, does not match the true channel covariance matrix sufficiently well [18]. In [19], low complexity randomized MUSIC is proposed designed for the large dimensional regime requiring however a very large number of receiver antennas in the order of hundreds. Random matrix theory (RMT) is a powerful tool allowing to construct consistent estimators based on the eigenvalue analysis of the covariance matrix of received signal, when the number of receiver antennas Nare of the same order of magnitude as the number of time samples T[20]. Such estimators are particularly useful for communication models where the classical approaches often fail or give poor results due to the insufficient number of time samples available for the derivation of the parameter estimator. Surprisingly, even though the derivation of these estimators is based on the assumption of large dimensions, they outperform the classical estimators even for small values of Nand T[20]. Based on results from RMT, series of works [21]–[23] extended the MUSIC algorithm to the large dimensional RMT regime for which the number of time samples Tis of the same order of magnitude as the dimension of the received signal vector N. The large dimensional RMT was introduced by Girko in [24] allowing to derive estimators of scalar quantities that are consistent, not only when Nis fixed and Tconverges to infinity but also when both Nand Tare large and converge to infinity at the same rate. The MUSICtype of estimators obtained based on this tools are referred to as the G-MUSIC algorithms [25], borrowing the first letter of the founder’s name. Works in [21], [22], [25] derived estimators following different mathematical approaches resulting in different expressions of the localization function estimates under the assumption of the white noise. In [22], a GMUSIC approach has been proposed that is able to separate closely spaced targets showing a superior performance of the RMT-based MUSIC compared to the spatial smoothing techniques suffering from a strong performance degradation when the spacing between the targets is of the order of 1/N. However, the transmission models in [21], [22], [25] considered only the AoA at the reception side similar to the model in Eq. (1) in which the transmit part of the signal with the angle-of-departure (AoD), typical to ISAC models, is lacking. Moreover, those models ignore the joint transmission framework for which both communication and sensing might occur at the same time. To address these issues, we present the problem formulation considered in this work. C. Problem Formulation We consider a typical bistatic ISAC transmission model with Mtransmit and Nreceiver antennas of the form y(t) = A(θ1, . . . , θK)ΓBH(ϕ1, . . . , ϕK)x(t) + v(t),(2) where the Kemitting sources from Eq. (1) are now considered as non emitting targets, x(t)∈ℂM×1is the transmit sensing signal vector, and, compared to Eq. (1), B(ϕ1, . . . , ϕK)∈ℂM×Kis an additional matrix containing Ksteering vectors corresponding to the AoD of the targets denoted by ϕ1, . . . , ϕK. Under the above model, an improved performance of the classical MUSIC method was obtained in [26] by deriving an optimal precoder of the transmit signal under the assumption of much higher number of time samples than the number of receiver antennas. Further, in [27] a method improving performance in terms of AoA estimation was proposed, under assumption of a small number of available time samples. However, the lack of time samples is “compensated” in [27] by the assumption on availability of a large number of independent frequency orthogonal frequency division multiplexing (OFDM) symbols which allows to achieve the consistency of the proposed method in the classical regime for which Nis fixed but the number of OFDM symbols is large, instead of T. Building on the research line of the large dimensional RMT, in this work we propose an estimator of the AoA in the ISAC bistatic scenario with a simultaneous presence of non emitting targets and, in addition compared to the model considered in Eq. (2), communication UEs causing interference to target sensing. Specifically, the proposed estimator is derived based on the usage of fixed rank perturbation models [28] for which the sum of the number of targets and communication UEs is small as compared to Nand T. The simulation results show that the proposed estimator outperforms the traditional MUSIC algorithm even for realistically small Nand T. Moreover, when the targets/UEs are close to each other, the proposed algorithm allows to separate them whereas the traditional MUSIC fails to do so. The remainder of the paper is structured as follows. In Section II, we describe the sensing scenario and establish the sensing system model. In Section III, after introducing a background on the MUSIC algorithm and some useful results from the RMT, we present the proposed estimator of the localization function and the angle estimator based on this function. Simulation results are provided in Section IV. Notations: The superscripts (·)Tand (·)Hcorrespond to the transpose and the Hermitian transpose of a matrix, respectively. Indenotes the identity matrix of dimension n×n. The symbol a.s. −→, stands for the almost sure convergence and the symbol ℑ(z)denotes the imaginary part of the complex number z. II. SYSTEM MODEL We consider a bistatic sensing scenario in a cellular ISAC network with Ktargets and Lsingle antenna user equipment (UE)s as depicted in Figure 1. The first base station (BS) is equipped with a sensing transmitter linear antenna array denoted by Tx-s, while the second BS is equipped with a sensing receiver linear antenna array denoted by Rx-s. We assume that the UEs are transmitting uplink (UL) signals during the sensing phase. 2 Targets UE UE Sensing signals Sensing signals UL signals BS2 Rx-s Tx-s BS1 Figure 1. Considered scenario in a cellular ISAC network with closely spaced targets and communication UEs. The BSs are capable of performing sensing transmission at BS1 and sensing reception at BS2. We define the transmit and the receive steering vectors by b(ϕ)≜1 √Mh1, e−jπ cos ϕ, . . . , e−jπ(M−1) cos ϕiT ∈ℂM×1 (3) and: a(θ)≜1 √Nh1, e−jπ cos θ, . . . , e−jπ(N−1) cos θiT ∈ℂN×1 (4) respectively, where ϕ∈[0,2π]and θ∈[0,2π]. The received narrowband signal model at Rx-s at time instant t∈[0, . . . , T −1] is given by y(t) = K X k=1 γka(θk)bH(ϕk)x(t) + L X l=1 ˜γla(˜ θl)˜xl(t) + v(t) (5) where y(t)∈ℂN×1,γkand ˜γldenote the complex pathgains corresponding to the target kwith k= 1, . . . , K and to the UE lwith l= 1, . . . , L, respectively, assumed both to be deterministic, b(ϕk)and a(θk)are the steering vectors defined respectively in Eq. (3) and (4) corresponding to the target AoDs with ϕk∈(0,2π)and the target AoAs with θk∈(0,2π), respectively. The steering vector corresponding to the UE AoAs is denoted by a(˜ θl)with ˜ θl∈(0,2π). The vector x(t)∈ℂM×1denotes the transmit sensing signal vector at time tassumed to be an orthogonal pilot sequence. The communication signal of the UE lat time tis denoted by ˜xl(t)∈ℂand the vector ˜ x(t) = [˜x1(t),...,˜xL(t)]T∈ℂL×1 corresponding to LUE transmit signals is assumed to be a an orthogonal pilot sequence independent of x(t). Finally, v(t)∈ℂN×1is the noise vector with elements following the complex Gaussian distribution with variance σ2 n. Note that in this model the delay and Doppler are omitted without loss of generality. III. ANGLE ESTIMATION A. Background on the Traditional MUSIC Algorithm The covariance matrix of the received signal from Eq. (5) is given by R=AΓBHBΓHAH+e Ae Γ2e AH+σ2 nIN∈ℂN×N, where A≜[a(θ1),...,a(θK)] ∈ℂN×K(6) is the channel matrix between the Ktargets and the receiver node, B≜[b(ϕ1),...,b(ϕK)] ∈ℂM×K(7) is the channel matrix between the Ktargets and the transmitter node, Γ≜diag {γ1, . . . , γK} ∈ ℂK×K(8) is the diagonal matrix containing Ktarget pathgains, e A≜[a(˜ θ1),...,a(˜ θL)] ∈ℂN×L(9) is the channel matrix between the LUEs and the receiver node and e Γ≜diag {˜γ1,...,˜γL} ∈ ℂL×L(10) is the diagonal matrix containing LUE pathgains. We define the signal eigen subspace matrix as Us≜[u1,...,uK,uK+1,...,uK+L]∈ℂN×(K+L) containing the signal eigenvectors of Rdenoted by u1,...,uK,uK+1,...,uK+Lcorresponding to the K+L largest eigenvalues. We also define the noise eigen subspace matrix as Un≜[uK+L+1,...,uN]∈ℂN×(N−K−L) containing the noise eigenvectors of Rdenoted by uK+L+1,...,uNcorresponding to the N−K−Lsmallest eigenvalues. The basic idea of the MUSIC algorithm [12] is based on the observation that any vector belonging to the signal subspace is orthogonal to the noise subspace such that a(θk)HUnUH na(θk) = 0 (11) for k= 1, . . . , K +L, where now the index kcorresponds to a target or to a UE. With defining the true localization function [25] by η(θ)≜a(θ)HUsUH sa(θ)(12) and observing that Eq. (11) is equivalent to finding the K+L largest peaks of the localization function for θ∈[0,2π], the AoAs can be expressed as θk=argmaxθη(θ) for k= 1, . . . , K +L. In practice, instead of the true R, we have the sample covariance matrix: b R=1 TYYH∈ℂN×N(13) where Ycontains Tobservations of the received vector given in Eq. (5): Y= [y(0),...,y(T−1)] ∈ℂN×T. Let ˆ u1,...,ˆ uK+Lbe the eigenvectors of b Rcorresponding to the K+Llargest eigenvalues. We define the estimated signal eigen subspace matrix as b Us≜[ˆ u1,...,ˆ uK+L]N×(K+L). 3 The MUSIC localization function estimate, referred to as traditional MUSIC, is given by ˆηtrad(θ) = a(θ)Hb Usb UH sa(θ).(14) The angle estimates are obtained from the local maxima of the localization function for θ∈[0,2π]: ˆ θk,trad =argmaxθˆηtrad(θ) for k= 1, . . . , K +L. B. Background on RMT Before presenting the main result of the paper, we need to introduce some background from RMT necessary for derivation of our estimator. The results are based on the limiting spectrum analysis of the received signal sample covariance matrix which is a random matrix, i.e., whose entries are random variables. The concatenation of the received signals from Eq. (5) for Ttime samples can be written in the matrix form: Y=AΓBHX+e Ae Γe X+V=P+e P+V∈ℂN×T,(15) where P=AΓBHXwith A,B, and Γdefined respectively in Eq. (6), (7), and (8), e P=e Ae Γe Xwith e Aand e Γdefined respectively in Eq. (9) and (10), X= [x(0),...,x(T−1)] ∈ ℂN×Tis the sensing transmit signal matrix, e X= [˜ x(0),..., ˜ x(T−1)] ∈ℂL×Tis the UE transmit signal matrix with UE transmit vectors ˜ x(t)∈ℂL×1as columns, and V= [v(0),...,v(T−1)] ∈ℂN×Tis the noise matrix. We note that the received matrix is written as a sum of three matrices with two matrices Pand e P, corresponding to the signal part of rank K+L, and of the matrix V, corresponding to the noise part of rank equal to min(N, T). We now state the general assumptions considered in this paper. Assumption 1. The dimension ratio defined by cN≜N/T satisfies: cN→c∈ℝ+as N→ ∞ and T→ ∞. Assumption 2. The dimension ratio defined by ¯cM≜M/T satisfies: ¯cM→¯c∈ℝ+as M→ ∞ and T→ ∞. Assumption 3. The sum of the number of distinct targets and UEs K+Lis fixed as N→ ∞,M→ ∞, and T→ ∞. Under the above assumptions, the model in Eq. (15) corresponds to the so-called fixed rank perturbation model [28] in RMT literature where the signal part P+e Pof small rank K+Lcan be viewed as a perturbation of the noise part Vof full rank. We first recall the characterization of the limiting spectrum of the noise part, namely, the limiting eigenvalue distribution of the sample covariance matrix of the noise denoted by 1 TVVH. Let µbe the probability measure of the limiting eigenvalue distribution of 1 TVVH. We recall the definition of the Stieltjes transform [29]: Definition 1. The Stieltjes transform of a probability measure µis the complex function m(z) = Z1 t−zµ(dt) defined on ℂ+={z:ℑ(z)>0}. As the entries of the matrix Vfollow the complex Gaussian distribution of variance σ2 n,µis the Marcenko– Pastur distribution with the support given by the interval [σ2 n(1 −√c)2, σ2 n(1 + √c)2], and m(x) = σ2 n(1 −c)−x+q(σ2 n(1 −c)−x)2−4cσ4 nx 2cσ2 nx(16) for x∈(σ2 n(1 + √c)2,∞)with the limiting cratio defined in Assumption 1. C. The G-MUSIC Algorithm for Bistatic ISAC We present the main result in the following theorem based on the application of the results from [21] to our transmission model in Eq. (15). Theorem 1. Let Assumptions 1, 2, and 3 hold true. Let ˆ u1,...,ˆ uK+Lbe the signal eigenvectors of b Rdefined in Eq. (13) with respective K+Llargest eigenvalues ˆ λ1> . . . > ˆ λK+L. For θ∈[0,2π], the estimate of the G-MUSIC localization function is given by ˆη(θ) = K+L X k=1 f(ˆ λk)a(θ)Hˆ ukˆ uH ka(θ)(17) where, for x∈[σ2 n(1 + √c)2,∞],f(x)is defined as f(x)≜g′(x) m(x)g(x) with m(x)defined in Eq. (16),g(x)defined as g(x)≜m(x)(xcm(x) + c−1) and g′(x)the derivative of g(x). As N→ ∞,M→ ∞, and T→ ∞, for θ∈[0,2π], we have the almost sure convergence of the proposed estimator η(θ)−ˆη(θ)a.s. −→ 0 where η(θ)is the true localization function defined in Eq. (12). Proof. The elements of the proof are provided in Appendix A. It should be noted that compared to the traditional MUSIC algorithm, the proposed algorithm has the same complexity, since the additional term f(ˆ λk)in Eq. (17) is a constant. IV. SIMULATION RESULTS In this section, we present the simulation results showing the performance of the proposed G-MUSIC localization function estimator and compare its performance with the traditional MUSIC method for closely spaced targets and UEs. We consider a scenario with a limited number of time samples available that is smaller or equal to the number of receiver antennas. We consider two targets with equal pathgains γ1=γ2= 1 with the AoAs and AoDs equal to [20◦,22◦]and [40◦,42◦], respectively. We consider also two communication UEs with pathgains ˜γ1= ˜γ2= 1 and true AoAs equal to 4 [35◦,37◦]. The AoA search grid is set to [θmin :θstep :θmax] = [10◦: 0.1◦: 50◦]. The noise variance σ2 nis assumed to be equal to 1. The results are averaged over 1000 runs. In Figure 2 A. (top part), the magnitude of the normalized localization function given in Eq. (17) is depicted (in red) versus angles (in degrees) for different values of the sensing samples T= 16,24,32 for a fixed number of sensing receiver antennas equal to N= 32, with a zoomed view around the true target AoAs and UE AoAs depicted in sub-figures B. and C., respectively. The sensing and communication Signalto-Noise Ratios are assumed to be both equal to 15 dB. The results are compared to the performance of the traditional MUSIC localization function (in blue) provided in Eq. (14) as well as to the ground truth localization function (in black) given in Eq. (12). We observe a superior performance of the proposed algorithm with the higher number of time samples, the closer the magnitude of the localization function to the ground truth which presents four peaks at true target/UE AoAs [20◦,22◦,35◦,37◦]. For the given SNR, it could be seen that the traditional MUSIC results in only two peaks and therefore fails to accurately estimate the four AoAs, for all considered values of T. The proposed estimator allows to differentiate well closely spaced targets and UEs for T= 32 and T= 24 as four distinct peaks are observed. For the smallest number of time samples T= 16, we can see that the G-MUSIC fails to separate the AoAs of the targets whereas two peaks are obtained at the UE estimated AoAs. This can be explained by the fact that the UE signals are less correlated compared to the target signals which present additional propagation path corresponding to the transmittertarget link. V. CONCLUSIONS In this paper, we proposed a G-MUSIC estimator based on the results from RMT on fixed rank perturbation models. The simulation results show that the proposed estimator allows to achieve a better angle estimation performance and is particularly useful for closely spaced target/UEs separation for a limited number of time samples available for which the traditional MUSIC fails. The proposed algorithm is applicable to ISAC scenarios in which it can be useful to estimate the UE parameters such as AoAs and can subsequently be used for interference mitigation caused by the UEs. Even though the proposed estimator derivation requires assumptions on infinite values of the number of antennas and time samples, we have demonstrated the applicability of the G-MUSIC approaches to realistic system parameter values compatible with currently deployed communication systems. It should be noted that the simplified narrowband signal model considered in this paper can be trivially extended to wideband frequency signals which are more realistic for ISAC scenarios. Moreover, in practice, due to the existence of the communication link between the UE and the BS(s), it will be possible to differentiate the target AoAs from the UE AoAs and hence to design appropriate transmit-receive precoding schemes based on these estimates. 10 15 20 25 30 35 40 45 50 Angles (degree) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Normalized localization function Ground truth G-MUSIC, T=32 MUSIC, T=32 G-MUSIC, T=24 MUSIC, T=24 G-MUSIC, T=16 MUSIC, T=16 19 19.5 20 20.5 21 21.5 22 22.5 23 0.95 0.96 0.97 0.98 0.99 1 34 34.5 35 35.5 36 36.5 37 37.5 0.95 0.96 0.97 0.98 0.99 1 True target AoAs True UE AoAs B. A. C. Figure 2. Top Figure A.: Normalized localization function versus angles (in degrees) of the proposed G-MUSIC method (in red) compared to the traditional MUSIC (in blue) and the ground truth (in black) for different values of sensing samples available, for equal sensing and communication SNR of 15 dB. Bottom Figure B.: Zoomed view around the true target AoAs. Bottom Figure C.: Zoomed view around the true UE AoAs. APPENDIX A. Appendix: Elements of the Proof of Theorem 1 We now consider a general fixed rank perturbation model from RMT as considered in [21] satisfying Assumption 1: YN=QN+WN(18) where WN∈ℂN×Thas complex Gaussian entries with variance 1/T and QN∈ℂN×Tis a matrix of fixed rank ras N, T → ∞. Assumption 4. The matrix QNhas a fixed rank ras N, T → ∞. Denoting by QN=UNΩNVH Nthe singular value decomposition of QNwith ΩN≜diag {ω1,N , . . . , ωr,N }, ΩNconverges to the diagonal matrix O=   ω1Ij10 ... 0ωsIjs   , where ω1> . . . > ωswith j1+. . . +js=r. Let ˆ u1,...,ˆ urbe the signal eigenvectors of YNYH N with the respective rhighest eigenvalues ˆ λ1> . . . > ˆ λr. Under Assumption 4 and the system model Eq. (18) with QN=A(θ1, . . . , θr)Xwhere A(θ1, . . . , θr)∈ℂN×ris the matrix of steering vectors defined by Eq. (4) and X∈ℂr×T 5 is a deterministic matrix, it was shown in [21] that the localization function can be written as ˆη(θ) = r X k=1 ζ(ˆ λk)a(θ)Hˆ ukˆ uH ka(θ)(19) where, for x∈[σ2 n(1 + √c)2,∞],ζ(x)is defined as ζ(x)≜xm(x)cm(x)−1−c x′ xm(x)2cm(x)−1−c x with m(x)defined in Eq. (16) and ()′is the derivative of a function of x. We need to show that our system model in Eq. (15) satisfies Assumption 4 with r=K+L. The matrix P+e P= AΓBHX+e Ae Γe X(of fixed rank K+Lfrom Assumption 3 as N, M, T → ∞) can be written as P+e P=¯ ASHwhere ¯ A= [Ae A]∈ℂN×(K+L)with Aand e Adefined as in Eq. (6) and (9) respectively, and SHis defined as follows: SH≜ΓBHX e Γe X∈ℂ(K+L)×T. We define the following diagonal matrix: D=Γ 0K×K 0L×Le Γ∈ℂ(K+L)×(K+L), as N,M,T→ ∞. We can show that SHS→D2using the assumption that Xand e Xare independent orthogonal unitary matrices. We have AHA→IKas N→ ∞and e AHe A→ILobserving that nondiagonal terms of AHAand e AHe Aare sums of geometric series and by using Assumption 3 from which Kand Lare fixed while N→ ∞. Similarly BHB→IKas M→ ∞. Hence, we show that ¯ AH¯ A→IK+L. From this we have (P+e P)H(P+e P)→D2, as N, M, T → ∞. Hence, P+e P satisfies Assumption 4 and the diagonal matrix of its singular value decomposition converges to the diagonal matrix Das N,M,T→ ∞, where Dhas K+Ldiagonal elements. This allows us to prove that the expression of the localization function in Eq. (19) (with g(x) = m(x) (xcm(x) + c−1)) is applicable to our model with the signal subspace of dimension K+L. This concludes the proof of Theorem 1. REFERENCES [1] J. A. Zhang, X. Huang, Y. J. Guo, J. Yuan, and R. W. Heath, “Multibeam for joint communication and radar sensing using steerable analog antenna arrays,” IEEE Trans. on Veh. Techn., vol. 68, no. 1, pp. 671–685, 2019. [2] F. Liu, C. Masouros, A. P. Petropulu, H. Griffiths, and L. Hanzo, “Joint radar and communication design: Applications, state-of-the-art, and the road ahead,” IEEE Trans. on Comm., vol. 68, no. 6, pp. 3834–3862, 2020. [3] A. Zhang, M. L. Rahman, X. Huang, Y. J. Guo, S. Chen, and R. W. Heath, “Perceptive mobile networks: Cellular networks with radio vision via joint communication and radar sensing,” IEEE Veh. Technol. Mag., vol. 16, no. 2, pp. 20–30, 2021. [4] C. Mollén, G. Fodor, R. Baldemair, J. Huschke, and J. Vinogradova, “Joint multistatic sensing of transmitter and target in OFDM-based JCAS system,” in IEEE EUCNC& 6G SUMMIT, 2023, pp. 144–149. [5] T. Ropitault, C. R. C. M. da Silva, S. Blandino, A. Sahoo, N. Golmie, K. Yoon, C. Aldana, and C. Hu, “IEEE 802.11bf WLAN sensing procedure: Enabling the widespread adoption of WiFi sensing,” IEEE Comm. Stand. Mag., vol. 8, no. 1, pp. 58–64, 2024. [6] 3GPP, “Feasibility study on integrated sensing and communications,” 3rd Generation Partnership Project, Technical Report (TR) 22.837, Jun. 2024, version 19.4.0. [7] ITU-R, “Future technology trends of terrestrial International Mobile Telecommunications (IMT) systems towards 2030 and beyond,” Internatinal Telecommunication Union, Technical Report (TR) M.2516-0, Nov. 2022, version 11/2022. [8] X. Chen, Z. Feng, Z. Wei, J. A. Zhang, X. Yuan, and P. Zhang, “Concurrent downlink and uplink joint communication and sensing for 6G networks,” IEEE Trans. on Veh. Techn., pp. 1–6, 2023. [9] C.-X. Wang, X. You, X. Gao, X. Zhu, Z. Li, C. Zhang, H. Wang, Y. Huang, Y. Chen, H. Haas, J. S. Thompson, E. G. Larsson, M. D. Renzo, W. Tong, P. Zhu, X. Shen, H. V. Poor, and L. Hanzo, “On the road to 6G: Visions, requirements, key technologies, and testbeds,” IEEE Comm. Surveys Tutorials, vol. 25, no. 2, pp. 905–974, 2023. [10] M. U. Baig, J. Vinogradova, G. Fodor, and C. Mollén, “Joint communication and sensing beamforming for passive object localization,” in ITG WSA&SCC 2023, pp. 1–6. [11] M. L. Rahman, J. A. Zhang, X. Huang, Y. J. Guo, and R. W. Heath, “Framework for a perceptive mobile network using joint communication and radar sensing,” IEEE Trans. Aerosp. Electron. Syst., vol. 56, no. 3, pp. 1926–1941, Jun. 2020. [12] R. Schmidt, “Multiple emitter location and signal parameter estimation,” IEEE Trans. on Antennas and Propagation, vol. 34, no. 3, pp. 276–280, 1986. [13] T. Ma, Y. Xiao, and X. Lei, “Channel reconstruction-aided MUSIC algorithms for joint AoA&AoD estimation in MIMO systems,” IEEE Wireless Comm. Letters, vol. 12, no. 2, pp. 322–326, 2023. [14] L. Leyva, D. Castanheira, A. Silva, and A. Gameiro, “Two-stage estimation algorithm based on interleaved OFDM for a cooperative bistatic ISAC scenario,” in IEEE VTC2022-Spring, pp. 1–6. [15] X. Cheng, X. Yuan, W. Jiang, L. Zhu, and Y. Zhang, “Deep active learning for multi-source AoA tracking in mmwave-based ISAC systems,” in 2023 IEEE Globecom Workshops, pp. 632–637. [16] D. Luo, H. Wu, H. Luo, B. Lin, and F. Gao, “Moving target sensing for ISAC systems in clutter environment,” in IEEE WCNC2024, pp. 1–6. [17] F. Wen, J. Shi, and Z. Zhang, “Joint 2D-DOD, 2D-DOA, and polarization angles estimation for Bistatic EMVS-MIMO radar via PARAFAC analysis,” IEEE Trans. on Veh. Techn., vol. 69, no. 2, pp. 1626–1638, 2020. [18] P. Vallet, X. Mestre, and P. Loubaton, “Performance analysis of an improved MUSIC DoA estimator,” IEEE Trans. on Sig. Processing, vol. 63, no. 23, pp. 6407–6422, 2015. [19] B. Li, S. Wang, J. Zhang, X. Cao, and C. Zhao, “Fast randomizedMUSIC for mm-wave massive MIMO radars,” IEEE Trans. on Veh. Techn., vol. 70, no. 2, pp. 1952–1956, 2021. [20] R. Couillet and M. Debbah, Random Matrix Methods for Wireless Communications. Cambridge University Press, Nov. 2011. [21] W. Hachem, P. Loubaton, X. Mestre, J. Najim, and P. Vallet, “A subspace estimator for fixed rank perturbations of large random matrices,” Journ. of Multiv. Analysis, vol. 114, pp. 427–447, 2013. [22] P. Vallet, X. Mestre, and P. Loubaton, “Performance analysis of an improved MUSIC DoA estimator,” IEEE Trans. on Sig. Processing, vol. 63, no. 23, pp. 6407–6422, 2015. [23] D. Schenck, X. Mestre, and M. Pesavento, “Probability of resolution of MUSIC and g-MUSIC: An asymptotic approach,” IEEE Trans. on Sig. Processing, vol. 70, pp. 3566–3581, 2022. [24] V. L. Girko, An Introduction to Statistical Analysis of Random Arrays. Berlin, Boston: De Gruyter, 1998. [25] X. Mestre, “Improved estimation of eigenvalues and eigenvectors of covariance matrices using their sample estimates,” IEEE Trans. on Information Theory, vol. 54, no. 11, pp. 5113–5129, 2008. [26] L. Zhang, Y. H. Chew, and W.-C. Wong, “Transmitter precoder design to improve the performance of the MUSIC algorithm,” IEEE Trans. on Sig. Processing, pp. 1–1, 2012. [27] L. Lin, H. Zheng, C. Zhou, and Z. Shi, “Channel covariance matrix construction for DOA estimation with limited communication symbols,” in IEEE RadarConf2023, pp. 1–5. [28] F. Benaych-Georges and R. R. Nadakuditi, “The eigenvalues and eigenvectors of finite, low rank perturbations of large random matrices,” Adv. in Math., vol. 227, no. 1, pp. 494–521, 2011. [29] J. Silverstein, “Strong convergence of the empirical distribution of eigenvalues of large dimensional random matrices,” Journal of Multivariate Analysis, vol. 55, no. 2, pp. 331–339, 1995. 6