STAR-RIS Aided Integrated Sensing and Communication over High Mobility Scenario

The above works on ISAC mainly focus on the waveform design and localization. However, velocity is also an important characteristic that should be measured in high mobility scenarios. Although [26] estimated the radial velocities along two different directions, the real velocity vector was still not able to be recovered. In this work, we propose a novel STAR-RIS aided ISAC scheme, where a STAR-RIS is equipped on the outside surface of a vehicle to improve the communication service of the in-vehicle user equipment (UE) and simultaneously reflect signals to the nearby roadside units (RSUs) for tracking the location and velocity of the vehicle. Specifically, we develop an efficient transmission structure for the ISAC scheme, where a number of training sequences with orthogonal precoders and combiners are respectively utilized at BS and RSUs for parameter extraction. We also characterize the near-field static channel model of the RIS-UE link, as well as the far-field time-frequency selective channel model of the BS-RIS-RSUs links. The cascaded channel parameters (i.e., the delays, the Doppler frequency shifts, the angles of arrivals (AOAs), and the angles of departure (AODs) of the scattering paths) of the BS-RIS-RSUs links are obtained at the RSUs by the multidimensional orthogonal matching pursuit (MOMP) algorithm. With these extracted cascaded channel parameters, the RSUs can jointly realize the vehicle localization and velocity measurement. Meanwhile, the reflection and refraction phase shifts of STAR-RIS can be designed and predicted with the sensing results. Finally, we propose a trade-off design for the performance of sensing and communication by optimizing the energy splitting factors of the STAR-RIS.

The rest of this paper is organized as follows. Section II illustrates the STAR-RIS aided mmWave ISAC system model over high mobility scenario. Then, the MOMP based channel parameter extraction is proposed in Section III. Section IV introduces the proposed vehicle sensing scheme, including the vehicle localization and velocity measurement. Finally, we propose a trade-off design for sensing and communication. Simulation results are provided in Section V, and conclusions are drawn in Section VI.

Notations: Denote lowercase (uppercase) boldface as vector (matrix). $(\cdot)^{H}$, $(\cdot)^{T}$, $(\cdot)^{*}$, and $(\cdot)^{\dagger}$ represent the Hermitian, transpose, conjugate, and pseudo-inverse, respectively. $\mathbf{I}_{N}$ is an $N\times N$ identity matrix. $\mathbb{E}\{\cdot\}$ is the expectation operator. Denote $|\cdot|$ as the amplitude of a complex value. $[\mathbf{A}]_{i,j}$ and $\mathbf{A}_{\mathcal{Q},:}$ (or $\mathbf{A}_{:,\mathcal{Q}}$) represent the $(i,j)$-th entry of $\mathbf{A}$ and the submatrix of $\mathbf{A}$ which contains the rows (or columns) with the index set $\mathcal{Q}$, respectively. $\mathbf{x}_{\mathcal{Q}}$ is the subvector of $\mathbf{x}$ built by the index set $\mathcal{Q}$. $\text{diag}(\mathbf{x})$ is a diagonal matrix whose diagonal elements are formed with the elements of $\mathbf{x}$. $(x)_{n}$ is the mod operation of $x$ with respect to $n$.

To integrate sensing and communication over downlink (DL) mobility scenario, we propose a transmission structure as shown in Figure 2. The coherence time is divided into several frames, where each frame consists of a preamble for sensing and a number of orthogonal time frequency space (OTFS) modulated blocks for communication. The preamble is divided into two parts. One is for coarse beam searching, while the other is for precise beam scanning. In the first part, the space domain with respect to BS and RSUs are divided into $L_{B}^{C}\ll N_{B}$ and $L_{R}^{C}\ll N_{R}$ parts, respectively. Thus, the number of pilot sequences with orthogonal precoders and combiners in the first part is $L_{O}^{C}=L_{B}^{C}L_{R}^{C}$. In the second part, we focus on the directions where signal is detected in coarse beam searching, and implement narrower beams for beam scanning accuracy. Thus, the second part of the preamble consists of $L_{O}^{P}=L_{B}^{P}L_{R}^{P}$ orthogonal precoded pilot sequences, where $L_{B}^{P}$ and $L_{R}^{P}$ are the number of beams transmitted by BS and that received by RSUs in this part, respectively. By contrast, in one-stage full-space precise beam scanning, the space domain with respect to BS and RSUs will usually be divided into $N_{B}$ and $N_{R}$ parts, respectively. Thus, the required number of training sequences within the preamble is $N_{B}N_{R}$. However, the required number of training sequences using the two-stage structure is $L_{B}^{C}L_{R}^{C}+L_{B}^{P}L_{R}^{P}$. Since $L_{B}^{C}$, $L_{R}^{C}$, $L_{B}^{P}$, and $L_{R}^{P}$ are usually very small values with respect to $N_{B}$ and $N_{R}$, the two-stage beam scanning strategy can significantly reduce the training overhead and computational complexity compared to the one-stage one.

For parameter extraction, we focus on the second part of the preamble received at both $G$ RSUs and the UE. The RSUs extract parameters of the cascaded BS-RIS-RSUs channel links for the localization and velocity measurement of the vehicle, while the UE estimates the cascaded BS-RIS-UE channel link for further transmission. With sensing results, the BS then adopts OTFS modulation, and organizes pilot symbols and data symbols over delay-Doppler-angle domain with optimized beamforming. In the meantime, the STAR-RIS will work in fully refraction mode and be delicately designed, while the UE will demodulate the received OTFS blocks for data detection with the help of the estimated channel. In this work, we focus on the sensing part of the overall ISAC system. For more detailed OTFS transmission and detection, one can refer to [33].

Figure 3 illustrates the bilayer structure of the utilized STAR-RIS. The STAR-RIS is composed of two neighboring omni-RISs, where the energy splitting factors of both omni-RISs can be adjusted for different purpose. For the DL ISAC considered in this paper, the outside one can simultaneously reflect and refract impinging signals to both sides, and the inside one is set to full penetration mode. Hence, the reflected signal is only related with the outside one, while the refraction signal is first refracted by the outside one, and then penetrate through the inside one.

Define the coefficient matrices of the two omni-RISs as $\mathbf{\Omega}_{O}$ and $\mathbf{\Omega}_{I}$, respectively. Then, the overall reflection phase shift matrix $\mathbf{\Omega}_{R}$ and transmission phase shift matrix $\mathbf{\Omega}_{T}$ can be respectively represented as $\mathbf{\Omega}_{R}=\sqrt{\epsilon_{R}^{O}}\mathbf{\Omega}_{O}$ and $\mathbf{\Omega}_{T}=\sqrt{\epsilon_{T}^{O}\epsilon_{T}^{I}}\mathbf{\Omega}_{I}\mathbf{G}\mathbf{\Omega}_{O}$, where $\epsilon_{R}^{O}$ and $\epsilon_{T}^{O}$ are respectively the reflection and refraction energy splitting factors of the outside omni-RIS, and $\epsilon_{T}^{I}=1$ is the refraction energy splitting factor of the inside omni-RIS. Note that $\epsilon_{R}^{O}+\epsilon_{T}^{O}=1$ if there is no penetration loss in the omni-RISs [34]. Besides, we define $\boldsymbol{\omega}_{O}$ and $\boldsymbol{\omega}_{I}$ as the vectors respectively composed by the diagonal elements of $\mathbf{\Omega}_{O}$ and $\mathbf{\Omega}_{I}$ for further use. Moreover, $\mathbf{G}\in\mathbb{C}^{N_{S}\times N_{S}}$ is the channel between the two omni-RISs. With reference to the channel model in [35], the channel gain from the $n_{s_{1}}$-th element of one omni-RIS to the $n_{s_{2}}$-th element of another one can be expressed as

where $a$ is the size of scattering elements, $z$ is the distance between the planes of the two omni-RISs, and $d_{n_{s_{2}},n_{s_{1}}}$ is the distance between the $n_{s_{1}}$-th element of one omni-RIS and the $n_{s_{2}}$-th element of another omni-RIS. By delicately operating $\boldsymbol{\omega}_{O}$, $\boldsymbol{\omega}_{I}$, $\epsilon_{R}^{O}$ and $\epsilon_{T}^{O}$, we can design optimal $\mathbf{\Omega}_{R}$ and $\mathbf{\Omega}_{T}$ to achieve balanced performance between sensing and communication.

To model the near-field channel, we establish a local Cartesian coordinate system $\mathcal{C}^{S}$ with respect to the STAR-RIS, and set its origin point at one corner of the STAR-RIS. Thus, the coordinate of the $(n_{S}^{x},n_{S}^{y})$-th STAR-RIS element can be denoted as $\mathbf{p}_{n_{S}^{x},n_{S}^{y}}^{S}=((n_{S}^{x}-1)d,(n_{S}^{y}-1)d,0)^{T}$, where $d$ is the distance between two adjacent STAR-RIS elements. Denote the location of the UE within $\mathcal{C}^{S}$ as $\mathbf{p}_{U}^{S}=(x_{U}^{S},y_{U}^{S},z_{U}^{S})^{T}$. Define $n_{s}=(n^{x}_{S}-1)N^{y}_{S}+n^{y}_{S}$ as the index of vectorized RIS element, then the distance between the UE and the $n_{s}$-th STAR-RIS element is

Moreover, the quasi-static flat fading near-field channel from the $n_{s}$-th element of the STAR-RIS to the UE can be represented as

Besides, $[\mathbf{a}_{\text{SU}}]_{n_{s}}$ denotes the $n_{s}$-th element of the near-field steering vector of the STAR-RIS under a spherical wavefront, which can be expressed as

where the free space path loss model [36] is adopted. $\theta_{n_{s}}$ is the elevation AOD from the $n_{s}$-th element to the user, and $\lambda$ is the wavelength of the carrier.

It is assumed that there are a number of scatterers in the environment. Hence, there are multiple scattering paths in the channel between the BS and the STAR-RIS. In practice, signals in mmWave band fade much faster than those at lower frequency band when propagating and reflecting off a surface [4]. Therefore, it is reasonable to assume that all of the NLOS paths only experience single-bounce reflection. Thus, the time-frequency selective channel of the BS-RIS link can be expressed as

where $h^{\text{BS}}_{p}\sim\mathcal{CN}(0,\lambda^{\text{BS}}_{p})$ is the channel gain of the $p$-th path between the BS and the STAR-RIS, $\nu^{\text{BS}}_{p}$ and $\tau^{\text{BS}}_{p}$ are respectively the Doppler frequency shift and the delay of the $p$-th path. $i$ is the delay index of the channel, and $I_{T}$ is the length of the channel over delay domain. Besides, $\mathbf{a}_{\text{B}}(\theta^{\text{B}}_{p},\phi^{\text{B}}_{p})$ and $\mathbf{a}_{\text{S}}(\theta^{\text{BS}}_{p},\phi^{\text{BS}}_{p})$ are the antenna steering vector of the $p$-th path, where $\{\theta^{\text{B}}_{p},\phi^{\text{B}}_{p}\}$ and $\{\theta^{\text{BS}}_{p},\phi^{\text{BS}}_{p}\}$ are respectively the azimuth and elevation AODs of the $p$-th path at the BS and the AOAs of the $p$-th path at the STAR-RIS. Note that $\mathbf{a}_{\text{S}}(\theta,\phi)=\mathbf{a}_{\text{S,x}}(\theta,\phi)\otimes\mathbf{a}_{\text{S,y}}(\theta,\phi)$, where $\otimes$ denotes the Kronecker product, $\mathbf{a}_{\text{S,x}}(\theta,\phi)\!=\![1,e^{-\jmath 2\pi\frac{d}{\lambda}\sin(\theta)\sin(\phi)},\cdots,e^{-\jmath 2\pi(N_{S}^{x}-1)\frac{d}{\lambda}\sin(\theta)\sin(\phi)}]^{T}$ and $\mathbf{a}_{\text{S,y}}(\theta,\!\phi)\!\!=\!\![1,e^{-\!\jmath 2\pi\frac{d}{\lambda}\sin(\!\theta)\!\cos(\!\phi)},\cdots,e^{-\!\jmath 2\pi(N_{S}^{x}-\!1)\frac{d}{\lambda}\!\sin(\!\theta)\!\cos(\!\phi)}]^{T}$ are respectively the array response vectors along the elevation and the azimuth dimensions, with $d=\frac{\lambda}{2}$ representing the inter-element spacing. $\mathbf{a}_{\text{B}}$ can be defined in the same way.

The received signal at the $g$-th RSU consists of two parts. One is the signals reflected by the STAR-RIS, and the other is that scattered directly by the static scatterers in the environment. The first part of the channel is cascaded by the channel between the BS and the vehicle and that between the vehicle and the $g$-th RSU. Therefore, it is reasonable to assume that $\{\theta^{\text{B}}_{p},\phi^{\text{B}}_{p}\}_{p=1}^{P}$ in (II-D) are the same with that in the first part of the channel from the BS to the $g$-th RSU. Since the RSUs are close to the vehicle, the LOS path between the vehicle and the RSUs will occupy dominant power of the channel. Thus, we assume that the channel between the STAR-RIS and any RSU only consists of the LOS path. Hence, the time-frequency selective channel from the BS to the $g$-th RSU can be represented as

where $h^{\text{B},g}_{p}\sim\mathcal{CN}(0,\lambda^{\text{B},g}_{p})$ is the channel gain of the $p$-th path of the channel, $\nu^{\text{B},g}_{p}$ and $\tau^{\text{B},g}_{p}$ are respectively the Doppler frequency shift and the delay of the $p$-th path. $\mathbf{a}_{\text{R}}(\theta^{g}_{p},\phi^{g}_{p})$ is the antenna steering vector of the $p$-th path, and $\theta^{g}_{p}$ and $\phi^{g}_{p}$ are respectively the azimuth and elevation AOAs of the $p$-th path at the $g$-th RSU. Besides, $P_{g}-P$ is the number of scattering paths from the BS to the $g$-th RSU without passing through the STAR-RIS. $\{\theta^{\text{SR}_{g}},\phi^{\text{SR}_{g}}\}$ are the elevation and azimuth AODs from the vehicle to the $g$-th RSU. For $p=1,2,\ldots,P$, it can be extracted that $\nu^{\text{B},g}_{p}=\nu^{\text{BS}}_{p}+\nu^{\text{SR}_{g}}$ and $\tau^{\text{B},g}_{p}=\tau^{\text{BS}}_{p}+\tau^{\text{SR},g}$, where $\nu^{\text{SR}_{g}}$ and $\tau^{\text{SR}_{g}}$ are the Doppler frequency shift and the delay between the vehicle and the $g$-th RSU, respectively.

Define each training sequence transmitted by the BS as $\mathbf{T}=[\mathbf{t}_{0},\mathbf{t}_{1},\ldots,\mathbf{t}_{N_{T}-1}]^{T}\in\mathbb{C}^{N_{T}\times N^{RF}_{B}}$ with $N_{T}$ represents the length of the sequence. Note that we do not adjust the phase shift matrices during the training sequence, we will omit the time index $t$ of $\mathbf{\Omega}_{R}$ and $\mathbf{\Omega}_{T}$ in the following for notational simplicity. The received signal for the $n_{t}$-th time-slot of the $l_{O}^{P}$-th training sequence ($l_{O}^{P}=L_{B}^{P}(l_{R}^{P}-1)+l_{B}^{P}$, $l_{B}^{P}=1,\ldots,L_{B}^{P}$, $l_{R}^{P}=1,\ldots,L_{R}^{P}$) at the $g$-th RSU can be represented as

where $\widetilde{h}^{\text{B},g}_{p}=h^{\text{B},g}_{p}\mathbf{a}_{\text{S}}^{T}(\theta^{\text{SR}_{g}},\phi^{\text{SR}_{g}})\mathbf{\Omega}_{R}\mathbf{a}_{\text{S}}(\theta^{\text{BS}}_{p},\phi^{\text{BS}}_{p})$ and $\{\theta^{g}_{p},\phi^{g}_{p}\}=\{\theta^{g},\phi^{g}\}$ for $p=1,2,\ldots,P$. And $\widetilde{h}^{\text{B},g}_{p}=h^{\text{B},g}_{p}$ for $p=P+1,\ldots,P_{g}$. Besides, $\mathbf{F}_{\text{B}}(l_{B}^{P})$ and $\mathbf{W}_{g}^{H}(l_{R}^{P})$ are the precoder at the BS and the combiner at the $g$-th RSU of the $l_{O}^{P}$-th training sequence, respectively. $\mathbf{n}^{g}_{n_{t},l_{O}^{P}}\sim\mathcal{CN}(0,\sigma_{N}^{2})$ is the additive white Gaussian noise (AWGN) of the $g$-th RSU at the $n_{t}$-th time slot of the $l_{O}^{P}$-th training sequence. By defining $\bar{h}^{\text{B},g}_{p}=\widetilde{h}^{\text{B},g}_{p}e^{-\jmath 2\pi\nu^{\text{B},g}_{p}\tau^{\text{B},g}_{p}}$, $\mathbf{v}_{l_{O}^{P}}(\nu^{\text{B},g}_{p})=e^{\jmath 2\pi\nu^{\text{B},g}_{p}\!((l_{O}^{P}\!-\!1)(\!N_{CP}\!+\!N_{T}\!))T_{s}}\![1,\!e^{\jmath 2\pi\nu^{\text{B},g}_{p}T_{s}}\!,\ldots,e^{\jmath 2\pi\nu^{\text{B},g}_{p}(\!N_{T}\!-\!1)T_{s}}]^{T}$, and $[\mathbf{A}_{d}(\tau^{\text{B},g}_{p})]_{:,n_{t}}=[\delta(1-(n_{t}-\tau^{\text{B},g}_{p}/T_{s})_{N_{T}}),\delta(2-(n_{t}-\tau^{\text{B},g}_{p}/T_{s})_{N_{T}}),\ldots,\delta(N_{T}-(n_{t}-\tau^{\text{B},g}_{p}/T_{s})_{N_{T}})]^{T}$, (7) can be rewritten as

and the received signal $\mathbf{Y}^{g}_{l_{O}^{P}}$ at the $g$-th RSU by stacking the whole $l_{O}^{P}$-th pilot sequence can be further derived as

where $\mathbf{Y}^{g}_{l_{O}^{P}}=[\mathbf{y}^{g}_{1,l_{O}^{P}},\ldots,\mathbf{y}^{g}_{N_{T},l_{O}^{P}}]\in\mathbb{C}^{N_{R}^{RF}\times N_{T}}$ and $\mathbf{N}^{g}_{l_{O}^{P}}=[\mathbf{n}^{g}_{1,l_{O}^{P}},\ldots,\mathbf{n}^{g}_{N_{T},l_{O}^{P}}]\in\mathbb{C}^{N_{R}^{RF}\times N_{T}}$.

Meanwhile, the received signal for the $l_{O}^{P}$-th training sequence at UE can be written as

where $\bar{h}^{\text{BU}}_{p}=h_{p}^{\text{BS}}\mathbf{a}_{\text{SU}}^{T}\mathbf{\Omega}_{T}\mathbf{a}_{\text{S}}(\theta^{\text{BS}}_{p},\phi^{\text{BS}}_{p})$, and $\mathbf{n}^{\text{UE}}_{l_{O}^{P}}$ is the AWGN of the $l_{O}^{P}$-th sequence at the UE.

Since the channel is modeled as the combination of a number of scattering paths, sparsity appears in the AOA of the RSU, the AOD of the BS, the delay, and Doppler dimensions. Besides, since orthogonal precoder and combiner are implemented respectively at the BS and the RSUs for different training sequences, the received signal for each training sequence corresponds to different AODs at the BS as well as different AOAs at the RSUs. Thus, the angular parameters of each training sequence can be determined easily, and we mainly focus on the estimation of other channel parameters. Specifically, if the precoder and combiner of the relative received sequences do not align with the scattering paths, the signal power will be very low and close to the noise power. Therefore, the AOAs and AODs can be estimated by searching the index of the received sequences with highest power. Then, we can focus on estimating the delays and Doppler frequency shifts of the paths.

The choice for the dictionaries of the sets about the channel parameters can be defined as $\mathbf{\Psi}_{\theta,\phi}^{\text{R}}\!=\![\mathbf{a}_{\text{R}}(\{\bar{\theta},\bar{\phi}\}_{1}),\ldots,\mathbf{a}_{\text{R}}(\{\bar{\theta},\bar{\phi}\}_{N_{\theta,\phi}^{\text{R}}})]$, $\mathbf{\Psi}_{\theta,\phi}^{\text{B}}\!=\![\mathbf{a}_{\text{B}}(\{\bar{\theta},\bar{\phi}\}_{1}),\ldots,\mathbf{a}_{\text{B}}(\{\bar{\theta},\bar{\phi}\}_{N_{\theta,\phi}^{\text{B}}})]$, $\mathbf{\Psi}_{\nu}(l_{O}^{P})\!=\![\mathbf{v}_{l_{O}^{P}}(\bar{\nu}_{1}),\ldots,\mathbf{v}_{l_{O}^{P}}(\bar{\nu}_{N_{\nu}})]$, and $\mathbf{\Psi}_{\tau}\!=\![\mathbf{a}_{d}(\bar{\tau}_{1}),\ldots,\mathbf{a}_{d}(\bar{\tau}_{N_{\tau}})]$, where $[\mathbf{a}_{d}(\bar{\tau})]_{n_{T}}=\delta(n_{T}-\bar{\tau}/T_{s})$ . Thus, the number of channel parameters to be searched is $K=4$. For notational simplicity, we define $N_{1}^{s}=N_{\theta,\phi}^{\text{R}}$, $N_{2}^{s}=N_{\theta,\phi}^{\text{B}}$, $N_{3}^{s}=N_{\nu}$, $N_{4}^{s}=N_{\tau}$ as the dimensions of the above dictionaries, respectively. Besides, we define the dimension of the elements within each dictionary as $N_{1}^{a}=N_{R}$, $N_{2}^{a}=N_{B}$, $N_{3}^{a}=N_{T}$, $N_{4}^{a}=N_{T}$.

With this multi-dimensional dictionary configuration, and ignoring quantization effects caused by the finite resolution of the dictionaries, we can define $\mathcal{C}\in\mathbb{C}^{N_{1}^{s}\times N_{2}^{s}\times N_{3}^{s}\times N_{4}^{s}}$ as

where $\mathbf{j}=(j_{1},j_{2},j_{3},j_{4})$ is the set of dictionary indexes. And we define $\mathcal{J}=\{\mathbf{j}\in\mathbb{N}^{K}\ \text{s.t.}\ j_{k}\leq N^{s}_{k},\forall k\leq K\}$ for further use. Then, the equivalent channel gain representation $\mathcal{H}_{u}\in\mathbb{C}^{N_{R}\times N_{B}\times N_{V}}$ of the $u$-th delay tap over angle-delay-Doppler domain in tensor form can be finally written as

Since our purpose is to transform the channel estimation problem into the multi-dimensional sparse estimation problem, we can relabel the sub-indexes by their dictionary counterparts for cleaner formulation by substituting the entry indexes $(n_{r},n_{b},n_{\nu},u)$ with $\mathbf{i}=(i_{1},i_{2},i_{3},i_{4})$ as $[\mathcal{H}_{i_{4}}(l_{O}^{P})]_{i_{1},i_{2},i_{3}}=\sum\limits_{\mathbf{j}\in\mathcal{J}}\prod\limits_{k=1}^{K}[\mathbf{\Psi}_{k}(l_{O}^{P})]_{i_{k},j_{k}}[\mathcal{C}]_{\mathbf{j}}$ for every combination of $i_{k}\leq N_{k}^{a},\forall k\leq K$, where $\mathbf{\Psi}_{1}(l_{O}^{P})=\mathbf{\Psi}_{\theta,\phi}^{\text{R}}$, $\mathbf{\Psi}_{2}(l_{O}^{P})=\mathbf{\Psi}_{\theta,\phi}^{\text{B}}$, $\mathbf{\Psi}_{3}(l_{O}^{P})=\mathbf{\Psi}_{\nu}(l_{O}^{P})$, $\mathbf{\Psi}_{4}(l_{O}^{P})=\mathbf{\Psi}_{\tau}$. And we define $\mathcal{I}=\{\mathbf{i}\in\mathbb{N}^{K}\ \text{s.t.}\ i_{k}\leq N^{a}_{k},\forall k\leq K\}$ for further use. This allows us to rewrite the received signal at the $n_{r}^{RF}$-th RF chain of the $g$-th RSU for the $l_{O}^{P}$-th training sequence as

where $\mathbf{A}_{D}(i_{3})$ is an all-zero matrix with only $[\mathbf{A}_{D}(i_{3})]_{i_{3},i_{3}}=1$. Then, the weight of the contribution of $\sum\limits_{\mathbf{j}\in\mathcal{J}}\prod_{k=1}^{K}[\mathbf{\Psi}_{k}(l_{O}^{P})]_{i_{k},j_{k}}[\mathcal{C}]_{\mathbf{j}}$ to $[\mathbf{Y}^{g}_{l_{O}^{P}}]_{n_{r}^{RF},n_{t}}$ is

which is denoted as the measurement matrix. Taking into account that the noise is white, the maximum likelihood estimator is given by the minimum mean square estimator as (21) at the top of the next page.

and it can be solved by the MOMP algorithm for a sparse solution.

For the received signal of the $l_{O}^{P}$-th sequence at the $n_{r}^{RF}$-th RF chain which has higher signal power, only the signal of one scattering path is involved. Thus, we can only search for one set of $\mathbf{j}$ with each received sequence. Note that the AODs at the BS and the AOAs at the $g$-th RSU can be derived by

respectively. Therefore, the dictionary indexes $j_{1}$ and $j_{2}$ for the $l_{O}^{P}$-th sequence at the $n_{r}^{RF}$-th RF chain can be directly determined. Note that the estimated AODs at the BS are coarse estimates, and can be refined by further beam scanning within more narrow ranges. Besides, consider the case where the angles are not on the grids, we can gather the received sequences of several neighbor grids with higher signal strength, and select the central one of them to determine the objective received sequence and its angle dictionary indexes. Then, the remained dictionary indexes $j_{3}$ and $j_{4}$ will be both searched to maximize the projection with $[\mathbf{Y}^{g}_{l_{O}^{P}}]_{n_{r}^{RF},:}$. In this problem, the projection and matching step is equivalent to the expression

By defining $\mathbf{y}_{\mathbf{\Phi}}({l_{O}^{P},n_{r}^{RF}})\in\mathbb{C}^{1\times N_{1}^{s}\times N_{2}^{s}\times N_{3}^{s}\times N_{4}^{s}}$ as $[\mathbf{y}_{\mathbf{\Phi}}({l_{O}^{P},n_{r}^{RF}})]_{1,\mathbf{i}}=[\mathbf{Y}^{g}_{l_{O}^{P}}]_{n_{r}^{RF},:}^{*}[\mathbf{\Phi}_{n_{r}^{RF}}(l_{O}^{P})]_{:,\mathbf{i}}$, the problem (24) can be rewritten as

Since searching all the parameters of one scattering path simultaneously will incur huge computational complexity, we turn to alternatively search the parameters, and refine them by means of a number of iterations. For example, in the $l$-th iteration, the refinement of $j_{k}$ can be implemented by (26) at the top of the next page.

For the first iteration, the parameters are also alternatively initialized. For the search of initial ${j_{k},k=\{3,4\}}$, define $\mathcal{E}$ as the set of dictionary indexes that we have an initial estimation, and define $\overline{\mathcal{E}}$ as the set of dictionary indexes that we do not have an initial estimation excluding $k$. Note that $k\notin\mathcal{E}$, $k\notin\overline{\mathcal{E}}$, and $\mathcal{E}\cup\{k\}\cup\overline{\mathcal{E}}=\{1,2,\ldots,K\}$. By splitting the indexes among different sets, and implementing some relaxations [37], (26) can be further derived as (27) at the top of the next page.

After searching the dictionary indexes for the parameters of the path, the corresponding equivalent channel gain $[\mathcal{C}]_{\widehat{\mathbf{j}}}$ should be calculated. By utilizing the measurement matrix and the dictionary matrices, we have

After the implementation of the MOMP algorithm for all the received sequences with highest signal power, the equivalent channel gain as well as the channel parameters of all the scattering paths can be obtained. Note that the UE can also acquire its related channel parameters by resorting to the MOMP algorithm in a similar way.

For user localization and velocity measurement, we should firstly separate the STAR-RIS aided channel from the whole channel between the BS and each RSU. From (7), one can notice that the cascaded scattering paths reflected by the STAR-RIS contain a certain degree of Doppler frequency shift. Besides, the paths to be recognized have the same AOA at the $g$-th RSU, while others usually have different AOAs. With those observations, we develop a path recognition algorithm for the RSUs as illustrated in Algorithm 1. Firstly, the paths with non-zero Doppler frequency shift are picked out. Then, the picked paths are divided into different sets, where each set consists of the paths that have the same AOA at the $g$-th RSU. Finally, due to the reflection of the STAR-RIS, the set which contains maximal number of paths will be chosen.