IRS-Aided Energy Efficient UAV Communication

The channel between any two nodes in the system, i.e., UAV, IRSs, and UEs, are denoted as

where $\tilde{{\mathbf{h}}}_{\text{I}_{w}}(t)\in\mathbb{C}^{M_{w}\times 1},\tilde{{\mathbf{h}}}_{wk}(t)\in\mathbb{C}^{M_{w}\times 1}$, and $\tilde{h}_{\text{U}_{k}}(t)\in\mathbb{C}$ denote the UAV to the $w$-th IRS channel, the $w$-th IRS to the $k$-th UE channel, and the UAV to the $k$-th UE channel, respectively. We split the channel into the large scale fading factors and small scale fading channel, where for the large scale fading factors, the distances between the UAV to the $w$-th IRS, the $w$-th IRS to the $k$-th UE, and the UAV to the $k$-th UE are denoted as $d_{\text{I}_{w}}(t),d_{wk},$ and $d_{\text{U}_{k}}(t)$, respectively, where we observe that $d_{wk}$ is constant since the locations of UEs and IRSs are fixed. The pathloss exponent is denoted as $\alpha_{i},\ i\in\left\{\text{I}_{w},wk,\text{U}_{k}\right\}$, and $\beta_{0}$ is the pathloss at a reference distance. The small scale fading channels are denoted as

where $\tilde{{\mathbf{g}}}_{\text{I}_{w}}(t),\tilde{{\mathbf{g}}}_{wk}(t)$, and $\tilde{g}_{\text{U}_{k}}(t)$ denote the small scale fading channels between the UAV to the $w$-th IRS, the $w$-th IRS to the $k$-th UE, and the UAV to the $k$-th UE, respectively. The vectors ${\mathbf{e}}_{\text{I}_{w}}=\left[\exp(j\psi_{\text{I}_{w,1}}),...,\exp(j\psi_{\text{I}_{w,{M_{w}}}})\right]^{\text{T}}$ and ${\mathbf{e}}_{wk}=\left[\exp(j\psi_{w,1k}),...,\exp(j\psi_{w,{M_{w}}k})\right]^{\text{T}}$ denote the LoS path phases with $\psi_{\text{I}_{w,m}}$ and $\psi_{w,mk}$ representing the phases of the channels between the UAV to the $m_{w}$-th IRS element of the $w$-th IRS and the $m_{w}$-th IRS element of the $w$-th IRS to the $k$-th UE, respectively. The LoS path phase of the channel between the UAV to the $k$-th UE is denoted as $\psi_{\text{U}_{k}}$.The small scale channels follow the independent Rician distribution with unit power and the $K$-factor as $\kappa_{i},\ i\in\left\{\text{I}_{w},wk,\text{U}_{k}\right\}$, and with $\bar{{\mathbf{g}}}_{\text{I}_{w}}(t)\sim{\mathcal{C}}{\mathcal{N}}\left({\boldsymbol{0}}_{M_{w}},{\mathbf{I}}_{M_{w}}\right),\bar{{\mathbf{g}}}_{wk}(t)\sim{\mathcal{C}}{\mathcal{N}}\left({\boldsymbol{0}}_{M_{w}},{\mathbf{I}}_{M_{w}}\right)$, and $\bar{g}_{\text{U}_{k}}(t)\sim{\mathcal{C}}{\mathcal{N}}\left(0,1\right)$.

We assume that the LoS paths between the IRSs and UEs always exist. For the channels associated with the UAV, we adopt the well-known probabilistic LoS model [31], which models the LoS path existence probability as a variable dependent on the elevation angle between a node and a UAV. The LoS path existence probability is given as [31]

where $a_{i}$ and $b_{i}$ are the design parameters dependent on the environment, and $\theta_{i}(t)$ is the elevation angle, which can be expressed as $\theta_{i}(t)=\frac{180}{\pi}\sin^{-1}\left(\frac{H_{\text{A}}-H_{i}}{d_{i}(t)}\right)$. Note that, due to the height of the installed IRSs, we consider the design parameters $a_{i}$ and $b_{i}$ as variables. Defined in [31], the design parameters are dependent on the average building height and building density with respect to the ground. Therefore, by taking the height of the installed IRSs on buildings as the new effective ground, the average building height and density which the IRSs see will decrease, resulting in a relatively rural environment with respect to the original environment. This new consideration can effectively capture the fact that highly located IRSs are favorable in obtaining the LoS path. Note that, the design parameters $a_{i}$ and $b_{i}$ can adapt to each IRS or UE, thus, this model can effectively consider the different heights of the IRSs or even consider the advantages of highly located UEs. Accordingly, the effective channel between the nodes can be expressed as

where $\nu_{i}<1,\ i\in\left\{\text{I}_{w},\text{U}_{k}\right\},$ is the additional attenuation factor due to the loss of the LoS path. From hereafter, we take $\nu_{i}=0$ for equational conciseness, which will be explained in Section III. Note that, the proposed methods can still be applied when $\nu_{i}\neq 0$. In conclusion, the overall channel for the $k$-th UE is given as

where ${\boldsymbol{\Phi}}_{w}$ is the phase control matrix of the $w$-th IRS. The phase control matrix can be represented as ${\boldsymbol{\Phi}}_{w}=\text{diag}\left({\boldsymbol{\phi}}_{w}\right)$, with ${\boldsymbol{\phi}}_{w}=\left[\exp(j\phi_{w,1}),...,\exp(j\phi_{w,{M_{w}}})\right]^{\text{T}}$, thus, the $m$-th IRS element of the $w$-th IRS will shift the phase of the impinging signal with the factor $\phi_{w,m}$.

We adopt the time-division multiple access (TDMA) technique at the UAV to serve the $K$ UEs to efficiently prevent inter-user interference. The achievable rate for the $k$-th UE can be denoted as

where $B$ is the bandwidth, $P_{c}$ is the transmit power from the UAV, and $\sigma^{2}$ denotes the noise spectral density.

Since we assumed to use the rotary-wing UAV, we adopt the rotary-wing UAV flight power consumption model from [23], which is given as

where $V$ is the UAV speed and the other variables are listed in Table I. The total energy consumption of the UAV can be expressed as

where $\tau$ is the communication time and $T$ is the total flight time of the UAV. We observe that the power consumption is dependent on the UAV speed, and that the model itself is quite complicated. This leads us to believe that the UAV speed should also be jointly optimized to effectively minimize the UAV power consumption. Also, we can confirm that while the power consumption for communication usually has the magnitude of $1$ W or smaller, the power consumption from UAV hovering has the magnitude of $100$ W. Thus, it is obvious that we must consider the UAV flight power to successively achieve energy efficient UAV communication systems.

In the SISU case, we omit the indexes $w$ and $k$ for brevity. To minimize the UAV energy consumption while sufficing a specific data constraint, it is trivial to operate the IRS phase shifts to maximize the achievable rate. Since the UAV and UE are both single antenna devices, the optimal phase shifts for the IRS are given in a simple closed form. Note that, we need the instantaneous channel state information to derive the optimal phase shifts. To focus on the achievable theoretical performance, we assume to have perfect channel state information (CSI) throughout this paper. The reflection coefficient of the $m$-th IRS element is then given as

where $h_{\text{I},m}(t)$ and $h_{m}(t)$ denote the $m$-th element of the channel vectors ${\mathbf{h}}_{\text{I}}(t)$ and ${\mathbf{h}}(t)$, respectively. This will coherently combine the UAV-UE channel with the UAV-IRS-UE channel, optimally maximizing the achievable rate.

To gain tractability, we adopt two frequently used assumptions as in [23]. First, to capture the randomness of the channel, we use the expected achievable rate instead of the instantaneous achievable rate. Note that, this might seem as if perfect CSI is unnecessary. However, to obtain the expected achievable rate, we need the IRS to constantly adapt to the instantaneous channels for successful coherent combination, thus, the assumption of perfect CSI is necessary. Second, due to the involved form of (6), we assume that the LoS path existence probabilities are constant with respect to time, e.g., fix the probabilities with the average elevation angles. Nevertheless, the simplified model still captures the advantage of the highly located IRSs.

To effectively express the probabilistic LoS model, we define a state variable as

for $i\in\left\{\text{U},\text{I}\right\}$, where $s_{\text{U}}$ and $s_{\text{I}}$ denote the states of the UAV-UE path and UAV-IRS path, respectively, and $s_{i}=1$ represents the existence of the selected path. With this variable, we can upper bound the expected achievable rate using the Jensen’s inequality as

where $\bar{h}_{\mathbf{s}}(t)$ is the overall channel between the UAV and the UE. Due to the channel states, the overall channel is now a variable dependent on ${\mathbf{s}}$. We observe that the closed form of (16) can be derived by computing $\mathbb{E}\left[|\bar{h}_{{\mathbf{s}}}(t)|^{2}\right]$. However, $\mathbb{E}\left[|\bar{h}_{{\mathbf{s}}}(t)|^{2}\right]$ is difficult to derive due to the joint consideration of the UAV-UE channel and the UAV-IRS-UE channel.

To compute the expected channel gain, we first assume that the IRS phases are always coherently combining the UAV-UE channel and the UAV-IRS-UE channel. In result, the overall channel norm value is given as

By defining a new variable $C(t)=\sum^{M}_{m=1}\lvert h_{\text{I},m}(t)\rvert\lvert h_{m}(t)\rvert$, we can expand the expected channel gain as

where $\lvert h_{\text{U}}(t)\rvert C(t)$ can be separated due to independence. Note that $C(t)$ is the sum of $M$ i.i.d. variables. Considering that the number of IRS elements, i.e., $M$, is predicted to be installed in large quantities, we employ the central limit theorem and assume that $C(t)$ follows the Gaussian distribution. Overall, (16) can be shown as

with the overall signal-to-noise ratio (SNR) in the closed form expression as

where $s_{i}^{2}$ is denoted as $s_{i}$ since $s_{i}^{2}=s_{i}$, and the time independent variables $\gamma_{i}$ and $\gamma_{i}^{\prime}$ are defined as

where $\mu\left(\kappa_{i}\right)$ denotes the mean of the Rician distribution with the $K$-factor as $\kappa_{i}$, and $\mu_{\text{I}}=M\mu\left(\kappa_{\text{I}}\right)\mu\left(\kappa\right)$. Note that in (17), the channel terms without LoS paths are zero due to the assumption $\nu_{i}=0$, where the case $\nu_{i}\neq 0$ gives only constant differences in (21)-(24).

In Fig. 2, we plot the approximated achievable rate and the simulated achievable rate for the SISU case with the same parameters in Section V. We observe that the approximated achievable rate is tightly bound with the simulated achievable rate. Thus, hereinafter, we consider $\hat{R}(t)$ as the achievable rate of the UE.

In this subsection, we formulate the optimization problem for the SISU case. Since optimizing with respect to continuous time will result in an infinite number of variables, we adopt the path discretization technique as in [23], where the trajectory is discretized into small segments. By segmenting the trajectory into $N$ pieces, the UAV energy consumption minimization problem can be formulated as

where $T_{n},\tau_{n},V_{n},{\mathbf{q}}\left[n\right]$ and $\hat{R}_{n}$ are the UAV flight time, UAV transmission time, UAV speed, UAV horizontal location, and achievable rate at the $n$-th segment, respectively. We assume the UAV has fixed initial and final locations ${\mathbf{q}}_{0}$ and ${\mathbf{q}}_{\text{F}}$, respectively, as in (25). The transmission time is trivially bounded by the flight time as in (26), and we assume that the UE has a fixed data constraint $Q$ as in (28). One characteristic of the path discretization technique is it assumes that the channels are constant within a segment. To uphold this assumption, for the segment length $\Delta_{n}=\lVert{\mathbf{q}}[n+1]-{\mathbf{q}}[n]\rVert$, we bound the length with $\min\{\Delta_{\max},V_{\max}T_{n}\}$, where $\Delta_{\max}$ is the maximum length of a segment to conserve this assumption, and $V_{\max}$ is the maximum speed of the UAV.

Similar to (P4) in [23], we transform (P1) into an equivalent problem as

where $\delta_{b}=3U_{\text{tip}}^{-2}$, and $\delta_{p}=\frac{1}{2}d_{0}\rho sA_{\text{p}}$. The transformation with regard to slack variables $A_{n}$ and $y_{n}$, which are the $n$-th elements of ${\mathbf{A}}$ and ${\mathbf{y}}$, are equivalent as in [23], thus, we omit the transformation details. The slack variables $u_{n}$ and $v_{n}$ are the $n$-th elements of the vectors ${\mathbf{u}}$ and ${\mathbf{v}}$, where they upper bound the UAV-IRS distance and the UAV-UE distance, respectively, which are given as

Also, $R_{n}$ is the achievable rate with the slack variables $u_{n}$ and $v_{n}$ instead of $d_{\text{I},n}$ and $d_{\text{U},n}$, respectively, which can be expressed as

We observe that by considering the IRS, the achievable rate has an involved form compared to the achievable rate with no IRSs, which makes the analysis difficult. To the best of our knowledge, this is the first paper to acknowledge the achievable rate for a general number of IRSs.

While the objective function is a convex function, the constraints (33)-(35) do not follow the standard convex optimization problem form. To resolve this issue, we adopt the well known SCA technique. By deriving a global bound of a constraint that meets specific conditions, the SCA technique iteratively solves the optimization problem and guarantees to converge to a KKT condition point. One way to obtain the global lower bound satisfying the specific conditions is to apply the first-order Taylor expansion to a convex function on a local point. The right hand side of the constraints (33) and (35) have been shown to be convex in [23], while $R_{n}$ from (34), which is defined in (38), needs to be analyzed. Therefore, we propose the following lemma.

Since $p_{i}$ and $(1-p_{i})$ are non-negative constants, $R_{n}$ is a weighted linear sum of the function $f(u_{1},...,u_{Z})$ with different values of $Z$, thus, through Lemma 1, we confirm that $R_{n}$ is indeed a convex function. Due to the convexity of $R_{n}$, we can successfully use the first-order Taylor expansion of $R_{n}$ as the lower bound.

Adopting the lower bound of the constraints (33)-(35), the overall optimization problem in the $\ell$-th iteration can be expressed as

where the lower bound functions are defined as

The values $A_{n}^{(\ell)},y_{n}^{(\ell)},\Delta_{n}^{(\ell)},{\mathbf{q}}^{(\ell)}[n],u_{n}^{(\ell)},v_{n}^{(\ell)},$ and $R_{n}^{(\ell)}$ are the local points to obtain the lower bound, which are the results of the $\left(\ell-1\right)$-th iteration, and the gradient $\nabla R_{n}$ can be calculated through the result of Appendix B. Note that, for the first iteration, we must define an initial case to compute the problem (P3). We will specify the initial case in Section V.

Since the lower bounded constraints are all affine functions, the problem (P3) is now in the standard convex optimization problem form, thus, can be solved effectively via problem solving tools such as CVX [32]. By iteratively solving the optimization problem and updating the local points, the solutions of (P3) will be in the feasible region of the original problem (P2), which converges to a KKT condition point. The proposed optimization technique is summarized in Algorithm 1. Note that, since the optimization problem is from the upper bound in (16), the resulting solution may not satisfy the data constraint in reality. To settle this issue, we can give a margin to the UE constraint $Q$, e.g., $Q+\chi$ for some positive constant $\chi$. Thus, by restraining the constraint, it will have a larger probability to satisfy the constraint in practice.

In this subsection, we formulate the optimization problem to minimize the UAV energy consumption in a SIMU scenario. However, this can be performed very easily with the similar expression as in [23]. The resulting optimization problem is given as

where $\tau_{nk}$ and $\hat{R}_{nk}$ are the transmit time with the $k$-th UE at the $n$-th segment and the achievable rate for the $k$-th UE at the $n$-th segment, respectively, and $Q_{k}$ is the data constraint for the $k$-th UE. Since we assume that the channels are approximately constant within a segment, the TDMA assumption is legitimate with the constraint (52). With the multiple UE expansion, we observe that the transmission for all UEs must be considered in each flight time as in (52), and the data constraint for each UE must be sufficed independently as in (53). We neglect the transformation process of (P4) due to redundancy. However, (P4) will be a subset within the general MIMU case, thus, can be effectively solved with the latter solutions in Section IV.

In the MIMU scenario, the most straightforward approach to exploit the multiple IRSs is to use all of them simultaneously. We denote this approach as the general IRS usage case. In order to realize this concept, we must define the UE achievable rate with regard to multiple IRSs. Fortunately, due to TDMA, only one UE is considered in a time instance, thus, the phase shifts of the IRSs can be easily obtained with the same approach as (13). Assuming that the IRSs concentrate on the $k$-th UE, the achievable rate is obtained as

with the state vector as ${\mathbf{s}}=\left[s_{\text{U}_{k}},s_{\text{I}_{1}},...,s_{\text{I}_{W}}\right]^{\text{T}}$, and the SNR as

where $\gamma_{i}$ and $\gamma_{i}^{\prime}$ follow the same definitions as (21)-(24).

In result, the optimization problem will have the same expression as (P4), and the problem can be transformed in the equivalent form as

where we observe that the slack variables $v_{nk},A_{nk}$, and $u_{nw}$ have an additional index due to the MIMU generalization. Using Lemma 1, we have shown that $R_{nk}$ is a convex function with respect to $u_{nw}$ and $v_{nk}$ for a general number of IRSs. Thus, we can effectively solve (P5) with the same method as (P2).

While exploiting all the IRSs simultaneously might be the most straightforward method, it may not be the most probable solution. Considering that the IRS is a communication assisting object, the IRSs are likely to be located far apart. Thus, due to the double propagation characteristic of the IRS, the power from the IRSs that are far apart from a UE is likely to be negligible. Also, the assumption that every IRS is controlled simultaneously for a single UE may be implausible. Thus, the resulting solutions may be less practical.

Accordingly, we consider a more practical case that each UE is supported by only one IRS, where we denote it as the IRS matching case. The formulated optimization problem will effectively evaluate and decide which IRS to use per instance. The overall problem can be expressed as

where $\hat{R}_{nwk}$ represents the achievable rate of the $k$-th UE at the $n$-th segment when the UE is matched with the $w$-th IRS. Hence, with the maximum function in (67), we pick the best IRS per instance for communication. Since we exploit only a single IRS, $\hat{R}_{nwk}$ is equivalent to (19) by considering the signals from only the $w$-th IRS, where we assume the signals from other IRSs are negligible. While we can still adopt the SCA technique due to the convexity of $\max_{w}\hat{R}_{nwk}$, the gradient is quite difficult to compute. Instead, we introduce a matching variable $\eta_{nwk}$, where it represents the communication time between the $k$-th UE in the $n$-th segment using the $w$-th IRS. The overall problem is given as

where with the additional constraints (72) and (73), the solution will effectively force only one $\eta_{nwk}$, which is associated with the largest $\hat{R}_{nwk}$, to $\tau_{nk}$, and the other $\eta_{nw^{\prime}k}$ values to zero. By adopting the matching variable $\eta_{nwk}$, we successfully removed the maximum operation in (67), and the gradient of $\hat{R}_{nwk}$ can be calculated through the results from the previous section since $\hat{R}_{nwk}$ has the same expression as (19). As we can observe, (P7) is the same as (P4) with the additional linear constraint (72), thus, we can solve the problem equivalently through the SCA technique. Note that, by choosing a single IRS for each instance, the IRS matching case can also compensate for the rather strong assumption that the LoS path between the IRSs and UEs always exist, since IRSs closely located to a UE will have a high probability to have an LoS path. Thus, the IRS matching case is more practical than the general IRS usage case.