6D Radar Sensing and Tracking in Monostatic Integrated Sensing and Communications System

A massive MIMO based monostatic ISAC system operating in mmWave or Terahertz frequency bands with OFDM modulation is depicted in Fig. 1, which employs only one dual-functional BS for wireless communications and radar sensing at the same time. Generally, we consider that the BS consists of one hybrid unit (HU) and one radar unit (RU), where both the HU and the RU are configured with uniform planar arrays (UPAs). By designing the beamforming strategy, HU is responsible for transmitting downlink communications signals and receiving uplink communications signals, as well as transmitting downlink sensing signals to realize dynamic target sensing; while RU is responsible for receiving echo signals to realize dynamic target sensing.

The HU and RU are each equipped with one UPA of $N_{H}=N^{x}_{H}\times N_{H}^{z}$ and $N_{R}=N_{R}^{x}\times N_{R}^{z}$ antenna elements, named as HU-UPA and RU-UPA, respectively. Assume that both the HU-UPA and the RU-UPA are vertically mounted on the 2D plane $y=0$ at BS side as shown in Fig. 2, and the antenna spacing between the antennas distributed along x-axis and z-axis are $d_{x}=d\leq\frac{\lambda}{2}$ and $d_{z}=d\leq\frac{\lambda}{2}$, respectively, with $\lambda$ being the wavelength. Without loss of generality, we denote the position of the $n_{H}$-th antenna element in the HU-UPA as $\mathbf{p}_{n_{H}}=\mathbf{p}_{0_{H}}+[d\cdot n_{H}^{x},0,d\cdot n_{H}^{z}]^{T}$, where $\mathbf{p}_{0_{H}}$ is the position of the reference element, $n_{H}^{x}\in\{0,1,...,N_{H}^{x}-1\}$ and $n_{H}^{z}\in\{0,1,...,N_{H}^{z}-1\}$ are the antenna indices. Here we use two types of index numbers to represent the same antenna, that is, the $n_{H}$-th antenna may also be named as the $(n^{x}_{H},n^{z}_{H})$-th antenna. Similarly, we denote the position of the $n_{R}$-th antenna element in the RU-UPA as $\mathbf{p}_{n_{R}}=\mathbf{p}_{0_{R}}+[d\cdot n_{R}^{x},0,d\cdot n_{R}^{z}]^{T}$ with $n_{R}^{x}\in\{0,1,...,N^{x}_{R}-1\}$ and $n_{R}^{z}\in\{0,1,...,N^{z}_{R}-1\}$. Generally, according to the spatial consistency of the arrays, we further assume that the HU-UPA and RU-UPA are co-located and are parallel to each other, i.e., $\mathbf{p}_{0_{H}}=\mathbf{p}_{0_{R}}=[0,0,0]^{T}$, such that they may see the targets at the same propagation directions¹¹1Since the normal communications and sensing distance is much longer than the protection distance between HU-UPA and RU-UPA, it can be considered that HU-UPA and RU-UPA are located in the same position.[23]. Besides, by balancing the hardware costs and system performance, we assume that the HU-UPA employs the hardware architecture based on phase shifter (PS) structure, in which a total of $N_{H,RF}\ll N_{H}$ radio frequency (RF) chains are deployed, and each antenna is connected to a PS to realize beamforming. On the other side, we assume that the RU-UPA employs the fully-digital receiving array, with each antenna connected to one RF chain, to realize super-resolution sensing performance, which follows the setting in [10] and [23].

Suppose that the ISAC system emits OFDM signals with $M$ subcarriers, where the lowest frequency and the subcarrier interval of OFDM signals are $f_{0}$ and $\Delta f$, respectively. Then the transmission bandwidth is $W=(M-1)\Delta f$, and the frequency of the $m$-th subcarrier is $f_{m}=f_{0}+m\Delta f$, where $m=0,1,...,M-1$. Further, we consider that an OFDM frame contains $N$ consecutive OFDM symbols, where the time interval between adjacent OFDM symbols is $T_{s}=T^{\prime}_{s}+T_{g}$, with $T^{\prime}_{s}=\frac{1}{\Delta f}$ and $T_{g}$ being the OFDM symbol duration and guard interval, respectively.

We employ the spherical coordinates $(r,\theta,\phi)$ to represent one position in 3D space. As shown in Fig. 2, $r$ represents the polar distance with mathematical range of $r\geq 0$, $\theta$ represents the horizontal angle with mathematical range of $0^{\circ}\leq\theta\leq 180^{\circ}$, and $\phi$ represents the pitch angle with mathematical range of $-90^{\circ}\leq\phi\leq 90^{\circ}$. Moreover, the spherical coordinate $(r,\theta,\phi)$ may be translated to its Cartesian counterpart $(x,y,z)$ through

Since BS is located at the origin of coordinate system, we denote the service area of BS as $\{(r,\theta,\phi)|r_{min}\leq r\leq r_{max},\theta_{min}\leq\theta\leq\theta_{max},\phi_{min}\leq\phi\leq\phi_{max}\}$. Suppose that there are $P$ single-antenna communications users, $K$ dynamic targets, as well as widely distributed static environment within this service area. We assume that the 6D motion parameters of the $k$-th dynamic target are $\{r_{k},\theta_{k},\phi_{k},v_{r,k},\omega_{\theta,k},\omega_{\phi,k}\}$, in which $(r_{k},\theta_{k},\phi_{k})$ represents the position of the $k$-th target, $v_{r,k}$, $\omega_{\theta,k}$ and $\omega_{\phi,k}$ represent the radial velocity, horizontal angular velocity, and pitch angular velocity, respectively. Besides, we assume that the position of the $p$-th user is $(R_{p},\vartheta_{p},\varphi_{p})$, which are known and stationary to BS, due to the fact that they can be easily obtained through user reporting, or other techniques[24, 25, 26].

The task of ISAC system is to sense all $K$ dynamic targets while serving the communications of all $P$ users. As described in Fig. 3, the proposed ISAC framework consists of two stages: sensing beam scanning (SBS) stage and sensing beam tracking (SBT) stage. For the aspect of communications, BS continuously generates $P$ communications beams towards $P$ users to maintain communications service during both SBS stage and SBT stage. For the aspect of sensing, BS generates one sensing beam that can scan the service area during SBS stage, during which the BS may detect the targets and estimate their parameters. To proceed, BS generates a single sensing beam to track all $K$ dynamic targets in a time division manner during SBT stage, that is, the sensing beam may sequentially illuminate each target and continuously track them. In this work, we mainly focuses on the SBT stage, and refer readers to our previous work [27] for more details on the SBS stage.

Assume that $K$ dynamic targets possess different physical directions. At each direction, the BS may adopt one OFDM frame, i.e., $N$ consecutive OFDM symbols, to realize dynamic target sensing. As shown in Fig. 3, we divide the SBT stage into $L$ tracking time slots (TTSs), and each TTS lasts for a time duration of $T_{TTS}=K\cdot N\cdot T_{s}$. Besides, each TTS is further divided into $K$ unit time slots (UTSs), and each UTS lasts for a time duration of $T_{UTS}=N\cdot T_{s}$. Clearly, there is $T_{TTS}=K\cdot T_{UTS}$. During $L$ consecutive TTSs, BS should track all $K$ dynamic targets using the methods such as Kalman filtering with $T_{TTS}$ as time step, which is named as multiple-shots tracking. To realize continuous target tracking and reliable users communications, in the $(l,k^{\prime})$-th UTS, BS needs to generate $P$ communications beams pointing to $P$ users and generate one sensing beam pointing to the sensing tracking direction $(\xi_{lk^{\prime}},\eta_{lk^{\prime}})$ from HU-UPA, where $k^{\prime}=1,2,...,K$. The BS needs to update the motion parameter sensing results of the $k^{\prime}$-th dynamic target within this UTS, known as single-shot sensing. Ideally, $(\xi_{lk^{\prime}},\eta_{lk^{\prime}})$ should be equal to the physical direction of the $k^{\prime}$-th dynamic target in the $(l,k^{\prime})$-th UTS.

Here we assume that the transmission power of BS is $P_{t}$, the energy of sensing beam is $\rho_{lk^{\prime}}P_{t}$, and the remaining energy $(1-\rho_{lk^{\prime}})P_{t}$ is evenly distributed among communications beams, where $\rho_{lk^{\prime}}\in[0,1]$ is the power distribution coefficient used in the $(l,k^{\prime})$-th UTS. Due to the short duration of one UTS, we keep the directions of transmitting beams unchanged within one UTS. Then the transmission signals from HU-UPA on the $m$-th subcarrier of the $n$-th symbol in the $(l,k^{\prime})$-th UTS should be represented as

where $(\Gamma_{p},\Upsilon_{p})\!=\!(\cos\varphi_{p}\cos\vartheta_{p},\sin\varphi_{p})$ is the spatial-domain direction corresponding to the physical direction $(\vartheta_{p},\varphi_{p})$ of the $p$-th user, and $(\Xi_{lk^{\prime}},\Theta_{lk^{\prime}})=(\cos\eta_{lk^{\prime}}\cos\xi_{lk^{\prime}},\sin\eta_{lk^{\prime}})$ is the spatial-domain direction corresponding to the sensing tracking direction $(\xi_{lk^{\prime}},\eta_{lk^{\prime}})$. Besides, $\mathbf{w}_{c,p,lk^{\prime}}=\sqrt{\frac{(1-\rho_{lk^{\prime}})P_{t}}{PN_{H}}}\mathbf{a}_{H}(\Gamma_{p},\Upsilon_{p})$ and $\mathbf{w}_{s,lk^{\prime}}=\sqrt{\frac{\rho_{lk^{\prime}}P_{t}}{N_{H}}}\mathbf{a}_{H}(\Xi_{lk^{\prime}},\Theta_{lk^{\prime}})$ are the communications beamforming vector for the $p$-th user and the sensing beamforming vector for the sensing tracking direction, respectively, and $\mathbf{a}_{H}(\Psi,\Omega)$ is the array steering vector of HU-UPA with the form

where $\otimes$ denotes the Kronecker product, and

Moreover, $s^{c,p,lk^{\prime}}_{n,m}$ and $s^{s,lk^{\prime}}_{n,m}$ are communications signals for the $p$-th user and sensing detection signals, respectively.

Based on (2), the BS may realize both communications function and sensing function by optimizing $\rho_{lk^{\prime}}$ during each UTS, which may be conceived through the power allocation strategy proposed in [27], via maximizing the sensing performance while ensuring users communications performance. To that end, we only consider dynamic target sensing problem while omitting the design of communications function in this work. Consequently, there is always one beam towards the sensing tracking direction $(\xi_{lk^{\prime}},\eta_{lk^{\prime}})$, and we can rewrite the transmission signals from the HU-UPA on the $m$-th subcarrier of the $n$-th OFDM symbol in the $(l,k^{\prime})$-th UTS as

where $\grave{\rho}_{lk^{\prime}}P_{t}$ is the power allocated to $(\xi_{lk^{\prime}},\eta_{lk^{\prime}})$ direction, and $s^{t,lk^{\prime}}_{n,m}$ is the signal transmitted to $(\xi_{lk^{\prime}},\eta_{lk^{\prime}})$ direction.

The motion of dynamic target can be described from two levels: 1) long-term motion, and 2) short-term motion. Specifically, the dynamic target undergoes long-term motion within $L$ TTSs. Given the long tracking time of the target, the radial velocity and angular velocities of the target are susceptible to various disturbances. In long-term motion, the time interval for describing the target motion is $T_{TTS}$. We refer to the 6D motion parameters of the $k$-th target at the $l$-th TTS as the state of this target, denoted as

where $l=0,1,...,L-1$. Without loss of generality, during the motion process, the direction in which the polar distance decreases, the direction in which the horizontal angle value decreases, and the direction in which the pitch angle value decreases are taken as the positive directions for radial velocity, horizontal angular velocity, and pitch angular velocity, respectively. Then the 6D motion model of the $k$-th dynamic target for long-term motion can be represented as

where $\mathbf{u}^{Long}_{k,l}=[u^{Long}_{r,k,l},u^{Long}_{\theta,k,l},u^{Long}_{\phi,k,l}]^{T}$ represents the random disturbance during the long-term motion.

In addition to sense the long-term motion, the BS needs to observe the $k^{\prime}$-th dynamic target within the $(l,k^{\prime})$-th UTS. The dynamic target also undergoes short-term motion within this UTS. Due to the short duration of the short-term motion, one usually assumes that the velocities of the dynamic target remain constant within one UTS time, i.e., $N$ OFDM symbol times[10]. Then the 6D motion parameters of the $k$-th dynamic target within the $n$-th OFDM symbol time of the $(l,k^{\prime})$-th UTS can be expressed as

with $n=0,1,...,N-1$, which satisfies

Naturally, the long-term motion model corresponds to the multiple-shots tracking of dynamic target, while short-term motion model corresponds to single-shot sensing of dynamic target. The ISAC system needs to utilize $\mathbf{S}^{Short}_{k,lk^{\prime},n}$ as much as possible to sense and track $\mathbf{S}^{Long}_{k,l}$ of the $k$-th dynamic target, that is, the ISAC system needs to utilize single-shot sensing to realize multiple-shots tracking.

In the $(l,k^{\prime})$-th UTS, the BS transmits the detection signals through HU-UPA at the beginning of sensing, which will be reflected by dynamic targets and cause echoes. Then, the RU-UPA will receive the sensing echo signals. Let us define the path from the $n_{H}$-th antenna of HU-UPA to the $k$-th dynamic target and then back to the $n_{R}$-th antenna of RU-UPA as the $(n_{H},k,n_{R})$-th propagation path. Then we denote $\tau^{lk^{\prime}\!,n}_{k,n_{H}\!,n_{R}}\!=\!(D^{lk^{\prime},n}_{k,n_{H}}+D^{lk^{\prime},n}_{k,n_{R}})/c$ as the time delay of the $(n_{H},k,n_{R})$-th propagation path in the $n$-th OFDM symbol time during the $(l,k^{\prime})$-th UTS, where $c$ represents the speed of light, $D^{lk^{\prime},n}_{k,n_{H}}$ is the distance between the $n_{H}$-th antenna of HU-UPA and the $k$-th dynamic target, and $D^{lk^{\prime},n}_{k,n_{R}}$ is the distance between the $n_{R}$-th antenna of RU-UPA and the $k$-th dynamic target.

Suppose that the signal transmitted by the $n_{H}$-th antenna of HU-UPA is $s(t)$, and the corresponding passband signal is $\mathcal{R}\{s(t)e^{j2\pi f_{0}t}\}$. Then the echo signal will be a delayed version of the transmitting signal with amplitude attenuation. Specifically, the passband echo signal received by the $n_{R}$-th antenna through the $(n_{H},k,n_{R})$-th propagation path at the $n$-th OFDM symbol during the $(l,k^{\prime})$-th UTS is $\mathcal{R}\{\alpha^{lk^{\prime}}_{k}s(t-\tau^{lk^{\prime},n}_{k,n_{H},n_{R}})e^{j2\pi f_{0}(t-\tau^{lk^{\prime},n}_{k,n_{H},n_{R}})}\}$. The corresponding baseband echo signal is $\alpha^{lk^{\prime}}_{k}s(t-\tau^{lk^{\prime},n}_{k,n_{H},n_{R}})e^{-j2\pi f_{0}\tau^{lk^{\prime},n}_{k,n_{H},n_{R}}}$, and the baseband equivalent channel is

where $\alpha^{lk^{\prime}}_{k}$ is the channel fading factor and $\delta(\cdot)$ denotes the Dirac delta function. Taking the Fourier transform of (21), the baseband frequency-domain channel response can be obtain as

Thus the frequency-domain sensing echo channel of the $(n_{H},k,n_{R})$-th propagation path on the $m$-th subcarrier of the $n$-th OFDM symbol during the $(l,k^{\prime})$-th UTS is

Based on (1), the Taylor expansion approximation of $D^{lk^{\prime},n}_{k,n_{H}}$ can be expressed as $D^{lk^{\prime},n}_{k,n_{H}}\approx r^{Short}_{k,lk^{\prime},n}-(n^{x}_{H}d\cos\phi^{Short}_{k,lk^{\prime},n}\cos\theta^{Short}_{k,lk^{\prime},n}+n^{z}_{H}d\sin\phi^{Short}_{k,lk^{\prime},n})$. Similarly, $D^{lk^{\prime},n}_{k,n_{R}}$ can be approximated as $D^{lk^{\prime},n}_{k,n_{R}}\approx r^{Short}_{k,lk^{\prime},n}-(n^{x}_{R}d\cos\phi^{Short}_{k,lk^{\prime},n}\cos\theta^{Short}_{k,lk^{\prime},n}+n^{z}_{R}d\sin\phi^{Short}_{k,lk^{\prime},n})$. Let us denote

and then there is

Then (23) can be rewritten as

In narrowband OFDM systems, the Doppler squint effect and beam squint effect are typically negligible [28], and thus (26) can be further represented as

We denote $\mathbf{H}^{lk^{\prime}\!,n,m}_{k}\in\mathbb{C}^{N_{R}\times N_{H}}$ as the overall frequency-domain sensing echo channel matrix on the $m$-th subcarrier at the $n$-th OFDM symbol within the $(l,k^{\prime})$-th UTS from the HU-UPA to the $k$-th dynamic target and then back to the RU-UPA, whose $(n_{R},n_{H})$-th element is $[\mathbf{H}^{lk^{\prime}\!,n,m}_{k}]_{n_{R},n_{H}}=h^{lk^{\prime},n,m}_{k,n_{H},n_{R}}$. Moreover, the matrix $\mathbf{H}^{lk^{\prime}\!,n,m}_{k}$ can be decomposed as

where $\mathbf{a}_{R}(\Psi,\Omega)$ is the array steering vector for spatial-domain direction $(\Psi,\Omega)$ of RU-UPA with the form

Here $\otimes$ denotes the Kronecker product, and

Besides, $\alpha^{lk^{\prime}}_{k}$ is usually modeled as $\alpha^{lk^{\prime}}_{k}=\sqrt{\frac{\lambda^{2}}{(4\pi)^{3}(r^{Long}_{k,l})^{4}}}\sigma_{k}$, and $\sigma_{k}$ is the radar cross section (RCS) of the $k$-th dynamic target. Without loss of generality, we assume that the RCS follows the Swerling $\rm I$ model.

Based on (29), the sensing echo channel of all $K$ dynamic targets on the $m$-th subcarrier of the $n$-th OFDM symbol in the $(l,k^{\prime})$-th UTS can be represented as

In addition, since the real physical world is composed of dynamic targets and static environment, the RU-UPA will receive both the effective echoes caused by interested dynamic targets (dynamic target echoes) and the undesired echoes caused by uninterested background environment (clutter). Following our previous work [27], we model the static environmental clutter channel on the $m$-th subcarrier of the $n$-th OFDM symbol in the $(l,k^{\prime})$-th UTS as

where $I^{\prime}$ is the total number of static environmental clutter scattering units, $\Psi^{\mathfrak{c}}_{i^{\prime}\!,lk^{\prime}}=\cos\phi^{\mathfrak{c}}_{i^{\prime}\!,lk^{\prime}}\cos\theta^{\mathfrak{c}}_{i^{\prime}\!,lk^{\prime}}$, $\Omega^{\mathfrak{c}}_{i^{\prime}\!,lk^{\prime}}=\sin\phi^{\mathfrak{c}}_{i^{\prime}\!,lk^{\prime}}$, $(r^{\mathfrak{c}}_{i^{\prime}\!,lk^{\prime}},\theta^{\mathfrak{c}}_{i^{\prime}\!,lk^{\prime}},\phi^{\mathfrak{c}}_{i^{\prime}\!,lk^{\prime}})$ is the position of the $i^{\prime}$-th clutter scattering unit, $\beta^{\mathfrak{c},lk^{\prime}}_{i^{\prime}}=\sqrt{\frac{\lambda^{2}}{(4\pi)^{3}(r^{\mathfrak{c}}_{i^{\prime}\!,lk^{\prime}})^{4}}}\sigma^{\mathfrak{c}}_{i^{\prime}}$ is the channel fading factor, and $\sigma^{\mathfrak{c}}_{i^{\prime}}$ is the RCS of the $i^{\prime}$-th clutter scattering unit that also follows the Swerling $\rm I$ model. Due to the random distribution of clutter scattering units in various directions and distances, when the number of clutter scattering units $I^{\prime}$ is large enough, $\mathbf{H}^{background}_{lk^{\prime},n,m}$ can be considered as a random channel. Therefore, a low complexity clutter channel generation method is to directly approximate $\mathbf{H}^{background}_{lk^{\prime},n,m}$ as

where $\mathbf{H}^{\mathfrak{c}}_{lk^{\prime}\!,m}$ is a complex Gaussian matrix with the dimension of $N_{R}\times N_{H}$, and $\beta^{\mathfrak{c}}_{lk^{\prime}}$ is the clutter power regulation factor. Formulas (33) and (34) indicate that since the static clutter scattering unit remains stationary for $N$ OFDM symbol times, $\mathbf{H}^{background}_{lk^{\prime},n,m}$ would remain unchanged for $N$ symbols.

Based on (32) and (34), the overall sensing echo channel of dynamic targets and static environment on the $m$-th subcarrier of the $n$-th OFDM symbol in the $(l,k^{\prime})$-th UTS is

We design that the BS employs the Kalman filtering algorithm to track the dynamic targets within $L$ TTSs. Assume that the predicted value of the 6D parameters for the $k^{\prime}$-th dynamic target within the $l$-th TTS is $\widetilde{\mathbf{S}}^{Long}_{k^{\prime},l}=[\widetilde{r}^{Long}_{k^{\prime},l},\widetilde{\theta}^{Long}_{k^{\prime},l},\widetilde{\phi}^{Long}_{k^{\prime},l},\widetilde{v}^{Long}_{r,k^{\prime},l},\widetilde{\omega}^{Long}_{\theta,k^{\prime},l},\widetilde{\omega}^{Long}_{\phi,k^{\prime},l}]^{T}$. Then, the HUUPA of the BS needs to generate one sensing beam towards $(\widetilde{\theta}^{Long}_{k^{\prime},l},\widetilde{\phi}^{Long}_{k^{\prime},l})$ direction during the $(l,k^{\prime})$-th UTS to re-sense the $k^{\prime}$-th target. Hence based on (6), the transmission signals of HU-UPA during the $(l,k^{\prime})$-th UTS can be expressed as