Optimal Design to Dual-Scale Channel Estimation for Sensing-Assisted Communication Systems

As shown in Fig. 1, each subframe consists of $N$ blocks. The beginning of each subframe is subjected to a large-scale detection periodically, which is consistent with the design of the physical layer CSI-RS reference signals. The acquisition of large-scale slow-varying channel information is generally achieved by designing the "CSI-ResourcePeriodicityAndOffset" parameter to be periodic between 4-640 timeslots[8]. The time duration of large-scale detection is denoted as $T_{l}$, which is an optimization variable. In addition, several small-scale estimations should be arranged at the start of some blocks (denoted as $M$ in Fig. 1, which is also an optimization variable). The time duration of small-scale estimation is fixed and denoted as $T_{m}$. This corresponds to the design of the physical layer DM-RS and TRS reference signals[9]. Let $T_{s}$ and $T_{b}$ denote the time duration of a symbol and a block, respectively. $M_{b}$ and $M_{m}$ denote the number of symbols for each block and small-scale estimation, respectively. Then, the fixed time durations of each block and each small-scale channel estimation are $T_{b}=M_{b}T_{s}$ and $T_{m}=M_{m}T_{s}$, respectively.

As shown in Fig. 1, to improve the achievable rate, this system entails a compromise on sensing duration design, i.e., the duration of large-scale detection $T_{l}$, the number of small-scale updates $M$, and the number of blocks for each small-scale update $N_{m},m\in\left\{1,2,....M\right\}$. Within the total $N$ blocks of each subframe, $\lfloor\frac{T_{l}}{T_{b}}\rfloor$ blocks are used for radar sensing to achieve large-scale channel detection, while the remaining $N_{t}=N-\lfloor\frac{T_{l}}{T_{b}}\rfloor=N_{1}+...+N_{M}$ blocks are for communication transmission.

We assume the channel is quasi-static in each block but varies across different blocks. The classical "one-ring" model uses the spatial correlation matrix to capture large-scale information, along with the user surrounding effects of clutter. Accordingly, the time-varying channel for the user $k$ at the block $n$ can be written as [10]

where $\Delta\theta$ is the single-side angular spread, and $r_{k}\left(\theta,f_{d}\right)$ denotes the joint angle-Doppler channel gain function on the direction of arrival (DOA) $\theta$ and the Doppler frequency $f_{d}$. With the assumption that the joint angle-Doppler channel gain $r_{k}\left(\theta,f_{d}\right)$ for different ($\theta$, $f_{d}$) are uncorrelated, we have

where $\beta_{k}$, $S_{k}^{A}\left(\theta\right)$ and $S_{k}^{D}\left(f_{d}\right)$ are the average channel gain, power-angle-spectrum and power-Doppler-spectrum, respectively. According to (1) and (LABEL:62615), the temporal-spatial autocorrelation matrix of $\boldsymbol{h}_{k,n}$ is given by

where $\rho_{k,n}$ is the temporal correlation coefficient, and $\boldsymbol{\dot{R}}_{k}\in\mathbb{C}^{L_{t}\times L_{t}}$ is the large-scale spatial correlation matrix¹¹1Given the temporal scope of resource scheduling spanning tens of milliseconds, it is general to assume that the large-scale characteristics of the communication channel keep constant[7]. Thus, the large-scale spatial matrix $\boldsymbol{\dot{R}}_{k}$ is independent of the time index.. Based on the Clarke-Jakes model[11], the temporal correlation coefficient is given by

where $J_{0}\left(\cdot\right)$ is the zeroth-order Bessel function of the first kind, and $f_{d,k}^{\max}$ is the maximum Doppler frequency of user $k$. Considering channel aging effects, the channel of the $n$-th block can be characterized as

where $\boldsymbol{h}_{k,1}$ is the initial channel vector derived through channel estimation. It is assumed that $\boldsymbol{h}_{k,n}\sim\mathcal{C}\mathcal{N}\left(0,\beta_{k}\boldsymbol{\dot{R}}_{k}\right)$ according to the central limit theorem (CLT)[12]. $\boldsymbol{e}_{t,n,k}\sim\mathcal{C}\mathcal{N}\left(0,\beta_{k}\left(1-\rho_{k,n}^{2}\right)\boldsymbol{\dot{R}}_{k}\right)$ is the independent residual error.

It can be seen from (3) that both large-scale and small-scale channel estimation results influence communication efficiency. In the following subsection, we will model and analyze the large-scale channel detection and small-scale channel estimation in detail.

With the transmitted communication signal $s_{c,k,n}\left(t\right)$ at the block $n$, the received signals of user $k$ is given by

where $z_{k,n}\left(t\right)$ is the Gaussian noise with mean zero and variance $\sigma_{c}^{2}$. $\xi_{k,n}\left(t\right)$, and $\zeta_{k,n}\left(t\right)$ are the interference caused by beamforming gain uncertainty and other users, respectively, and given by

Accordingly, the signal-to-interference-plus-noise ratio (SINR) of ISAC user $k$ is given by

To maximize the spectral efficiency, the low-complexity maximum ratio transmission (MRT) beamforming method can be adopted as

where $\boldsymbol{C}_{h,k}=\hat{\boldsymbol{R}}_{k}\left(\hat{\boldsymbol{R}}_{k}+\frac{1}{\gamma_{e}}\mathbf{I}_{\mathrm{L}_{\mathrm{t}}}\right)^{-1}\hat{\boldsymbol{R}}_{k}$ is the covariance matrix of $\boldsymbol{\tilde{h}}_{k,1}$.

Proposition 1: With the MRT transmit beamforming method in (21) and the predicted results in (17), the SINR $\gamma_{k,n}$ in the n-th block is given by

It can be seen from (22) that both small and large-scale estimations influence the communication rate. Large-scale characteristics are reflected in $\mathrm{tr}\left(\boldsymbol{C}_{h,k}\right)$, while small-scale characteristics are evident in both $\rho_{k,n}$ and $\mathrm{tr}\left(\boldsymbol{C}_{h,k}\right)$. The spectral efficiency (bps/Hz) in the $n$-th block for user $k$ can be written as

Thus, based on the frame structure shown in Fig. 1, the system achievable rate is given by

where the second term at the right-hand side of (24) is the rate discount caused by the dual-scale channel estimation, and $(MT_{m}+\text{mod}(T_{l},T_{b}))$ is the time duration occupied by dual-scale channel estimation.

In this paper, we aim to maximize communication efficiency given in (24) while satisfying sensing requirements by jointly optimizing the large-scale sensing duration, the number of small-scale updates, and the number of blocks in each small-scale update. Thus, the optimization problem can be formulated as

where $\varGamma_{k}$ is the allowed maximum detection error for user $k$; the constraint (25b) gives the sensing requirements for each user; the constraint (25c) limits the number of communication transmission blocks to $N-\lfloor\frac{T_{l}}{T_{b}}\rfloor$; (25d) constraints the times of small-scale and large-scale within scheduling cycle, while (25e) constraints the number of blocks in each update not less than 1.

When $M$ and $T_{l}$ are fixed, the second row of (24) is constant and $\mathbf{P}1$ can be equivalently reformulated as

Since this discrete problem is intractable, we leverage the following proposition to obtain closed-form solutions.

Proposition 2: The closed-form solutions of $N_{m}$ in $\mathbf{P}2$ are given by

where $N_{t}=\sum_{m=1}^{M}{N_{m}}$. That is, after an even distribution equal to $\lfloor\frac{N_{t}}{M}\rfloor$ for each update, the remaining $\text{mod}(N_{t},M)$ blocks need to spread evenly in random $\text{mod}(N_{t},M)$ updates, which will have $\lfloor\frac{N_{t}}{M}\rfloor+1$ blocks, and the other $M-\text{mod}(N_{t},M)$ updates keep to have $\lfloor\frac{N_{t}}{M}\rfloor$ blocks.

Based on the results in (27), the achievable rate in (24) can be summarized as

With the well-known inequality $\mathrm{tr}\left(AB\right)\leqslant\left\|A\right\|_{2}\mathrm{tr}\left(B\right)$, the lower bound of $\mathrm{SE}_{k,n}$ is given by

where $P_{t}$ is total transmit power of all users[11]. The channel estimation quality is reflected in $\boldsymbol{C}_{h,k}$, for which we have the simpler form

Then, the following proposition is motivated.

Proposition 3: For $\left(h-1\right)T_{b}<T_{l}<hT_{b},\forall h\in\left\{1,...,N\right\}$, when $\lfloor\frac{T_{l}}{T_{b}}\rfloor$ is fixed and radar’s large-scale sensing accuracy is reasonable, i.e., $\hat{\boldsymbol{R}}_{k}=\beta_{k}\boldsymbol{\dot{R}}_{k}-\boldsymbol{R}_{r,k}\succeq 0$, the derivative of achieved throughput monotonically decreases and then it is segmented concave on $T_{l}$.

According to Proposition 3, time-related resource optimization is a convex problem with fixed $M$ and blocks occupied by $T_{l}$, i.e., $\lfloor\frac{T_{l}}{T_{b}}\rfloor$. Thus, the optimal solution of this subproblem can be derived via a two-dimension search over $M$ and $\lfloor\frac{T_{l}}{T_{b}}\rfloor$. For these two discrete variables, $\lfloor\frac{T_{l}}{T_{b}}\rfloor$ is between $\lfloor\frac{T_{l}^{min}}{T_{b}}\rfloor$ and $N$, where $T_{l}^{\min}$ is determined by the sensing requirements in (25b), and $M$ is between 1 and $N-\lfloor\frac{T_{l}}{T_{b}}\rfloor$. Accordingly, there are $\sum_{T_{l}=\lfloor\frac{T_{l}^{min}}{T_{b}}\rfloor}^{N}{\left(N-\lfloor\frac{T_{l}}{T_{b}}\rfloor\right)}$ loops and each loop optimizes a single variable via the interior-point method with complexity $\mathcal{O}\left(1^{3.5}\right)$. The total computational complexity of the two-dimensional search-based optimization algorithm is $\sum_{T_{l}=\lfloor\frac{T_{l}^{min}}{T_{b}}\rfloor}^{N}{\left(N-\lfloor\frac{T_{l}}{T_{b}}\rfloor\right)}$.