Intelligent Reflecting Surface-Aided Maneuvering Target Sensing: True Velocity Estimation

Assume that the directions of the target with respect to the BS and IRS, denoted by ${\theta}_{tb}$ and ${\theta}_{it}$ in Fig. 2(b), are well estimated in advance. To estimate the true velocity, the BS transmits a probing beam

where $d$ and $\lambda$ represent the inner spacing of the ULA and the wavelength, respectively. Here, $\mathbf{a}(\cdot)$ represents the steering vector of the antenna array of the BS towards the target. Note that due to the use of the narrow beam, the effect of the non-light-of-sight (NLoS) path is considered negligible.

The echo of the $k$-th pilot symbol $s_{k}$ received by the BS is given by

where

represents the response vector of the IRS, $\mathbf{G}$ denotes the channel between BS and IRS, which is modeled as a Rician channel comprising a light-of-sight (LoS) path and a number of NLoS paths, $\mathbf{\Psi}$ denotes the phase shifter which is designed to align toward the target³³3The optimized design of phase shifter will be left as future work., $\alpha_{d}$ and $\alpha_{r}$ denote the complex channel gain, accounting for the path loss and target reflectivity⁴⁴4The fluctuations of target reflectivity are typically modeled using the standard Swerling classes. In this paper, we adopt the Swerling I model where the reflectivity is assumed to be constant within a symbol [17].. $\mu_{d}$ and $\mu_{r}$ represent the Doppler frequencies corresponding to the direct link and IRS link, respectively, $T_{s}$ denotes the symbol period, and $\{\mathbf{p}_{r,k}\}_{k=1}^{N_{r}}$ denote the noise vectors whose elements are drawn independently from a complex Gaussian distribution with zero mean and covariance matrix $\frac{\sigma_{r}^{2}}{N}\mathbf{I}$ [13].

Since the Doppler effect depends on the mobility of the target, we propose to estimate the true velocity $\mathbf{v}$ by exploiting the Doppler frequencies $\{\mu_{d},\mu_{r}\}$.In the following, we first discuss the geometrical insights of the Doppler effect to reveal the relation between true velocity $\mathbf{v}$ and the Doppler frequencies $\{\mu_{d},\mu_{r}\}$.

The Doppler frequency of the relative motion between the BS and the maneuvering target is given by

where $v$ denotes the amplitude of the true velocity, and $\theta_{v}$ represents the angle between the velocity direction and the $x$-axis, which is marked in red in Fig. 2. Note that $v\cos\left(\theta_{v}-\theta_{tb}\right)$ is the projection of velocity on the line across the BS and target, which is marked in dark green in Fig. 2(a). Then, the Doppler frequency corresponding to the direct link is given by

The deployment of the IRS creates another path, which allows the BS to observe the velocity from an additional perspective. The Doppler frequency of the relative motion between the IRS and the maneuvering target is given by

Then, the Doppler frequency corresponding to the IRS link is given by

where $\theta_{1}={\frac{\theta_{tb}+\theta_{it}}{2}-\theta_{v}}$ and $\theta_{2}=\frac{\theta_{it}-\theta_{tb}}{2}$ which are marked in dark green and purple in Fig. 2(b), respectively.

From (7), $\mu_{r}$ comes from two projections:

1. 1.

   $\cos\theta_{1}$: This term projects the velocity to the angular bisector of the angle $\angle_{\text{BS-Target-IRS}}$. Note that the angular bisector is perpendicular to the tangent line of the ellipse with foci at the BS and IRS.

2. 2.

   $\cos\theta_{2}$: This term projects the projected speed $v\cos\theta_{1}$ to the line connecting the BS and the target (or the line connecting the IRS and the target). Since the direction of the projected speed $v\cos\theta_{1}$ is fixed, there will be no ambiguity on the velocity at the projection corresponding to $\cos\theta_{2}$.

Remark: The conventional mono-static radar can only access $\mu_{d}$, which corresponds to the radial component of the velocity. As a result, the tangent projection will cause ambiguity, e.g., different velocities may yield the same observation at the BS. One example is illustrated in Fig. 2(a) where two velocity vectors, i.e., $\mathbf{v}_{amb}$ (marked in gray) and $\mathbf{v}$ (marked in red), give the same projection. Similarly, for the IRS link, the velocity ambiguity occurs at the tangent line of the ellipse, as illustrated in Fig. 2(b). Thus, either of the two links, by itself, will cause velocity ambiguity. Nevertheless, by jointly considering the two links, the true velocity vector can be uniquely determined. Note that we assume that the distance between the BS and the IRS should be comparable to the distance between the target and the BS/IRS.

The following proposition provides the true velocity estimation in the IRS-aided system by jointly leveraging $\mu_{d}$ and $\mu_{r}$.

Proof: See Appendix. $\blacksquare$

We first turn off the IRS to obtain a coarse estimation of $\mu_{d}$ only based on the direct link. This can be implemented through a receiving beam with zero gain on the direction of IRS. In this case, the system degenerates to a conventional mono-static one. Assume the BS transmits $N_{d}$ pilot symbols in Stage I. Given the pilot symbol $s_{k}=1$, the received echo of the $k$-th symbol at the BS is given by

where $\{\mathbf{p}_{d,k}\}_{k=1}^{N_{d}}$ denotes the noise vector whose elements are drawn independently from a complex Gaussian distribution with zero mean and covariance matrix $\frac{\sigma_{d}^{2}}{N}\mathbf{I}$

Define $\mathbf{w}_{d}=\mathbf{a}(\theta_{tb})$ as the combiner corresponding to the received beam which points to the estimated target position. Then the output for the $k$-th symbol is given by

where $\beta_{d}=\alpha_{d}\mathbf{w}_{d}^{\mathrm{H}}\mathbf{a}(\theta_{tb})\mathbf{a}^{\mathrm{H}}(\theta_{tb})\mathbf{f}$ and $n_{d,k}=\mathbf{w}_{d}^{\mathrm{H}}\mathbf{p}_{d,k}$ denotes the combined noise following the Gaussian distribution, i.e., $n_{d,k}\sim\mathcal{CN}(0,\sigma_{d}^{2})$. Stacking $z_{d,k}$ into a vector yields

Define the steering vector in the Doppler domain as

The coarse estimation of $\mu_{d}$ is obtained as

At this stage, we turn on the IRS to obtain a refined estimation of $\mu_{d}$ and $\mu_{f}$. The BS transmits $N_{r}$ pilot symbols at this stage. Define $\mathbf{w}_{r}=\mathbf{a}(\theta_{tb})+\mathbf{a}({\theta}_{bi})$ as the combiner illuminated at the estimated target position and IRS, simultaneously, where $\theta_{bi}$ denotes the direction of the IRS with respect to the BS. Then, the output for the $k$-th symbol is given by

where $\beta_{r}=\alpha_{r}\mathbf{w}_{r}^{\mathrm{H}}\mathbf{G}\mathbf{\Psi}\mathbf{b}(\theta_{it})\mathbf{a}^{\mathrm{H}}(\theta_{tb})\mathbf{f}$ and $n_{r,k}=\mathbf{w}_{r}^{\mathrm{H}}\mathbf{p}_{r,k}$ denotes the combined noise, which follows the Gaussian distributions, i.e., $n_{r,k}\sim\mathcal{CN}(0,\sigma_{r}^{2})$.

Before proceeding, we first construct a set of $P\times 1$ signal vectors by rearranging $\{z_{r,k}\}_{k=1}^{N_{r}}$. By doing that, the Doppler frequency estimation problem can be solved by the MODE. Given a positive integer $P$, the signal vector is defined as

From (15), (16) can be rewritten as

where

The covariance matrix of $\mathbf{z}_{r,k}$ is

where $\mathbb{E}{\{\cdot\}}$ denotes the expectation over $k$ and $\mathbf{\Phi}=\mathbb{E}{\{\mathbf{x}_{r,k}\mathbf{x}_{r,k}^{\mathrm{H}}\}}$. The eigenvalue decomposition of $\mathbf{R}_{r}$ gives

where $\lambda_{1}\geq\lambda_{2}\geq\lambda_{3}=\cdots=\lambda_{P}=\sigma_{r}^{2}$ are the eigenvalues, $\mathbf{G}_{s}=\left[\mathbf{g}_{1},\mathbf{g}_{2}\right]\in\mathbb{C}^{P\times 2}$ and $\mathbf{G}_{N}=\left[\mathbf{g}_{3},\mathbf{g}_{4},\cdots,\mathbf{g}_{P}\right]\in\mathbb{C}^{P\times(P-2)}$ correspond to the signal and noise subspace, respectively. Note that the linear span of $\mathbf{G}_{s}$ is the same as that of $\mathbf{A}$, i.e.,

Note that the maximum likelihood estimate (MLE) of $\bm{\mu}=\{\mu_{d},\mu_{r}\}$ depends on $\mathbf{R}_{r}$, which is unknown in practice. To address this, we first obtain the estimated eigenvectors $\{\widehat{\mathbf{g}}_{i}\}_{i=1}^{P}$ and the estimated eigenvalues $\{\widehat{\lambda}_{i}\}_{i=1}^{P}$ by performing the eigenvalue decomposition on the sample covariance matrix of $\{\mathbf{z}_{r,k}\}_{k=P}^{N_{r}}$, i.e.,

Then, an approximated maximum likelihood estimate (MLE) of $\bm{\mu}=\{\mu_{d},\mu_{r}\}$ is given by [14]

where $\mathrm{tr}(\cdot)$ denotes the trace, $\Phi_{\mathbf{A}}(\bm{\mu})=\mathbf{A}\left({\mathbf{A}}^{\mathrm{H}}\mathbf{A}\right)^{-1}{\mathbf{A}}^{\mathrm{H}}$ represents the orthogonal projector onto the range space of $\mathbf{A}$, and $\mathbf{\widehat{G}}_{s}=[\widehat{\mathbf{g}}_{1},\widehat{\mathbf{g}}_{2}]$, while

and $\widehat{\sigma}_{r}^{2}=\sum_{i=3}^{P}\widehat{\lambda}_{i}/(P-2)$.

However, the problem in (25) does not have an analytical solution and is thus hard to solve directly. To solve (25), we then consider introducing some tractable intermediate variables, based on which $\mu_{d}$ and $\mu_{r}$ can be estimated. Construct a polynomial

which has $2$ roots in the complex plane. It can be validated that $c_{1}=-\left(e^{j2\pi\mu_{d}T_{s}}+e^{j2\pi\mu_{r}T_{s}}\right)$ and $c_{2}=e^{j2\pi(\mu_{d}+\mu_{r})T_{s}}$. Clearly, $\omega_{d}=e^{-j2\pi\mu_{d}T_{s}}$ and $\omega_{r}=e^{-j2\pi\mu_{r}T_{s}}$ are the roots of this polynomial. Then, the strategy of solving (25) is to obtain the intermediate variables $c_{1}$ and $c_{2}$, based on which $\mu_{d}$ and $\mu_{r}$ can be estimated, instead of estimating $\mu_{d}$ and $\mu_{r}$ directly.

Define a matrix

From the definition of $\mathbf{A}$ in (18), we have $\mathbf{C}\mathbf{A}=\mathbf{0}$, which indicates

Substituting (29) into (25) produces

where $\mathrm{vec}(\cdot)$ denotes the vectorization operator and

Recall from (27) that $\mathbf{c}\triangleq\left[c_{1},c_{2}\right]^{\mathrm{T}}$ is related to $\bm{\mu}$. Then, the MLE of $\mathbf{c}$ is given by

Define matrix

and vector

where $\widehat{g}_{i,j}$ denotes the $(i,j)$-th entry of $\widehat{\mathbf{G}}_{s}$. We have

where

Substituting (35) into (32) leads to

Note that (38) is a weighted least squares problem whose solution is given by

However, $\bf{W}$ also depends on $\mathbf{C}$ through (31) and (28). In order to tackle this challenge, we propose an iterative algorithm to find the solution, which is summarized in Algorithm 1.

Once $\mathbf{\widehat{c}}$ is obtained, we can solve the equation

to obtain two roots

The corresponding $\mu_{1}$ and $\mu_{2}$ are obtained as

where $\mathcal{P}(\cdot)$ denotes the phase of a complex number.

We can obtain the refined estimates of $\mu_{d}$ and $\mu_{r}$, i.e.,

By exploiting Proposition 1, the direction and absolute value of true velocity can be estimated.