Sensing Accuracy Optimization for Communication-assisted Dual-baseline UAV-InSAR ††thanks: This work was supported in part by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) GRK 2680 – Project-ID 437847244. (2024)

\DeclareAcronym

snrshort = SNR,long = signal-to-noise ratio,\DeclareAcronympdfshort = PDF,long = probability density function,\DeclareAcronymsarshort = SAR,long = synthetic aperture radar,\DeclareAcronyminsarshort = InSAR,long = interferometric synthetic aperture radar,\DeclareAcronymaoshort = AO,long = alternating optimization,long-plural-form = alternating optimizations\DeclareAcronymmimoshort = MIMO,long = multiple-input multiple-output\DeclareAcronymuavshort = UAV,long = unmanned aerial vehicle,long-plural-form = unmanned aerial vehicles\DeclareAcronymfdmashort = FDMA,long = frequency-division multiple-access,\DeclareAcronym1dshort = 1D,long = one-dimensional,\DeclareAcronymislrshort = ISLR,long = integrated sidelobe ratio,\DeclareAcronympslrshort = PSLR,long = peak sidelobe ratio,\DeclareAcronym3dshort = 3D,long = three-dimensional,\DeclareAcronympsoshort = PSO,long = particle swarm optimization,\DeclareAcronym2dshort = 2D,long = two-dimensional,\DeclareAcronymdemshort = DEM,long = digital elevation model,\DeclareAcronymgsshort = GS,long = ground station,long-plural-form = ground stations\DeclareAcronymlosshort = LOS,long = line-of-sight,\DeclareAcronymscashort = SCA,long = successive convex approximation,\DeclareAcronymneszshort = NESZ,long = noise equivalent sigma zero,\DeclareAcronymwrtshort = w.r.t.,long = with respect to ,\DeclareAcronymrhsshort = r.h.s,long = right-hand side ,\DeclareAcronymgmtishort = GMTI,long = ground moving target indication,\DeclareAcronymlhsshort = l.h.s,long = left-hand side ,\DeclareAcronymbcdshort = BCD,long = block coordinate descent,\DeclareAcronymhoashort = HoA,long = height of ambiguity,

Mohamed-AmineLahmeri1, Víctor Mustieles-Pérez12, Martin Vossiek1, Gerhard Krieger12, andRobert Schober1
1Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU), Germany
2German Aerospace Center (DLR), Microwaves and Radar Institute, Weßling, Germany

Abstract

In this paper, we study the optimization of the sensing accuracy of unmanned aerial vehicle (UAV)-based dual-baseline interferometric synthetic aperture radar (InSAR) systems. A swarm of three UAV-synthetic aperture radar (SAR) systems is deployed to image an area of interest from different angles, enabling the creation of two independent digital elevation models (DEMs). To reduce the InSAR sensing error, i.e., the height estimation error, the two DEMs are fused based on weighted average techniques into one final DEM. The heavy computations required for this process are performed on the ground. To this end, the radar data is offloaded in real time via a frequency division multiple access (FDMA) air-to-ground backhaul link. In this work, we focus on improving the sensing accuracy by minimizing the worst-case height estimation error of the final DEM. To this end, the UAV formation and the power allocated for offloading are jointly optimized based on alternating optimization (AO), while meeting practical InSAR sensing and communication constraints. Our simulation results demonstrate that the proposed solution can improve the sensing accuracy by over 39% compared to a classical single-baseline UAV-InSAR system and by more than 12% compared to other benchmark schemes.

I Introduction

The use of \acuav swarms for remote sensing has recently gained attention due to their flexibility and efficiency in data collection tasks [1]. This has led to an increased use of drones in diverse applications, such as mapping, monitoring traffic, and addressing climate change [2]. For these tasks, a variety of sensors can be deployed onboard, including cameras, LiDARs, and radars. In particular, the deployment of \acsar on \acpuav has attracted significant interest due to its ability to provide very high-resolution \acsar images over local areas, even under challenging conditions, overcoming the limitations of traditional airborne and spaceborne systems. This integration has sparked multiple recent studies focusing on system design [3], trajectory and resource allocation optimization [4, 5, 6], and experimental measurement campaigns for \acuav-\acsar systems [7].

An interesting remote sensing application of \acuav swarms is \ac3d radar imaging, which can be realized using techniques, such as \acmimo radar, tomography, and interferometry [8]. In particular, \acinsar leverages the phase differences between at least two \acsar images, captured from different angles, to extract topographic information and generate \acpdem. Key performance metrics in interferometry include \acsnr, coverage, coherence, \achoa, and height error [9, 10], which are affected by the interferometric baseline, i.e., the distance between the sensing platforms. While \acinsar has been extensively studied for spaceborne and airborne platforms [8], the optimization of \acinsar performance for \acuav-based systems remains largely unexplored. In our recent research work [11], we investigated formation and resource allocation optimization for maximizing the \acinsar coverage, but for a single-baseline \acuav-\acinsar system. In contrast, dual-baseline \acinsar systems offer advantages, such as enhanced phase unwrapping and improved sensing accuracy [12]. However, results from single-baseline systems [11] do not apply to dual-baseline systems due to the different expressions for the height error and the use of multiple acquisition geometries.

In this work, we study a dual-baseline \acuav-based \acinsar system, where a swarm consisting of one master and two slave \acpuav is deployed to generate two independent \acpdem of a target area, which are then fused into a single \acdem based on weighted averaging [12]. Additionally, the radar data is offloaded to the ground in real time. We investigate the joint optimization of the \acuav formation and communication power allocation for minimization of the worst-case height error in the final \acdem under communication and sensing constraints. Our main contributions can be summarized as follows:

  • We propose an approximate bi-static \acsnr expression valid for the considered sensing application.

  • Based on the Cramér–Rao bound of the phase error, we derive a tractable upper bound for the complex expression of the height error of the final \acdem.

  • We formulate and solve a joint optimization problem for \acuav formation and communication power allocation to minimize the derived upper bound on the height error, while satisfying sensing and communication constraints.

  • Our simulation results demonstrate the effectiveness of the considered dual-baseline \acinsar system compared to single-baseline systems and other benchmark schemes.

Notations:In this paper, lower-case letters x𝑥xitalic_x refer to scalar variables, while boldface lower-case letters 𝐱𝐱\mathbf{x}bold_x denote vectors. {a,,b}𝑎𝑏\{a,...,b\}{ italic_a , … , italic_b } denotes the set of all integers between a𝑎aitalic_a and b𝑏bitalic_b. |||\cdot|| ⋅ | denotes the absolute value operator. Nsuperscript𝑁\mathbb{R}^{N}blackboard_R start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT represents the set of all N𝑁Nitalic_N-dimensional vectors with real-valued entries. For a vector 𝐱=(x1,,xN)TN𝐱superscriptsubscript𝑥1subscript𝑥𝑁𝑇superscript𝑁\mathbf{x}=(x_{1},...,x_{N})^{T}\in\mathbb{R}^{N}bold_x = ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_x start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT, 𝐱2subscriptnorm𝐱2||\mathbf{x}||_{2}| | bold_x | | start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT denotes the Euclidean norm, whereas 𝐱Tsuperscript𝐱𝑇\mathbf{x}^{T}bold_x start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT stands for the transpose of 𝐱𝐱\mathbf{x}bold_x. For real-valued multivariate functions f(𝐱)𝑓𝐱f(\mathbf{x})italic_f ( bold_x ), f𝐱(𝐚)=(fx1(𝐚),,fxN(𝐚))T𝑓𝐱𝐚superscript𝑓subscript𝑥1𝐚𝑓subscript𝑥𝑁𝐚𝑇\frac{\partial f}{\partial\mathbf{x}}(\mathbf{a})=\Big{(}\frac{\partial f}{%\partial x_{1}}(\mathbf{a}),...,\frac{\partial f}{\partial x_{N}}(\mathbf{a})%\Big{)}^{T}divide start_ARG ∂ italic_f end_ARG start_ARG ∂ bold_x end_ARG ( bold_a ) = ( divide start_ARG ∂ italic_f end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG ( bold_a ) , … , divide start_ARG ∂ italic_f end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT end_ARG ( bold_a ) ) start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT denotes the partial derivative of f𝑓fitalic_f \acwrt 𝐱𝐱\mathbf{x}bold_x evaluated for an arbitrary vector 𝐚𝐚\mathbf{a}bold_a. For any Boolean expression 𝒮𝒮\mathcal{S}caligraphic_S, 𝟙{𝒮}1𝒮\mathds{1}\{\mathcal{S}\}blackboard_1 { caligraphic_S } denotes the indicator function, which equals 1 if 𝒮𝒮\mathcal{S}caligraphic_S is true and 0 otherwise.

II System Model

We consider three rotary-wing \acpuav, denoted by Uk,k{0,1,2}subscript𝑈𝑘𝑘012U_{k},k\in\{0,1,2\}italic_U start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , italic_k ∈ { 0 , 1 , 2 }, performing \acinsar sensing over a target area. U0subscript𝑈0U_{0}italic_U start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, the master drone, transmits and receives radar echoes, while U1subscript𝑈1U_{1}italic_U start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and U2subscript𝑈2U_{2}italic_U start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT, the slave drones, only receive. We use a \ac3d coordinate system, where the x𝑥xitalic_x-, y𝑦yitalic_y-, and z𝑧zitalic_z-axes represent the range direction, the azimuth direction, and the altitude, respectively. The mission time T𝑇Titalic_T is divided into N𝑁Nitalic_N slots of duration δtsubscript𝛿𝑡\delta_{t}italic_δ start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT, with T=Nδt𝑇𝑁subscript𝛿𝑡T=N\cdot\delta_{t}italic_T = italic_N ⋅ italic_δ start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT. The drone swarm forms a dual-baseline interferometer with two independent observations acquired by (U0,U1)subscript𝑈0subscript𝑈1(U_{0},U_{1})( italic_U start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_U start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) and (U0,U2)subscript𝑈0subscript𝑈2(U_{0},U_{2})( italic_U start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_U start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ), respectively. The considered \acuav-\acsar systems operate in stripmap mode [13] and fly at a constant velocity, vysubscript𝑣𝑦v_{y}italic_v start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT, following a linear trajectory that is parallel to a line, denoted by ltsubscript𝑙𝑡l_{t}italic_l start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT, which is parallel to the y𝑦yitalic_y-axis and passes in time slot n𝑛nitalic_n through reference point 𝐩t[n]=(xt,y[n],0)T3subscript𝐩𝑡delimited-[]𝑛superscriptsubscript𝑥𝑡𝑦delimited-[]𝑛0𝑇superscript3\mathbf{p}_{t}[n]=(x_{t},y[n],0)^{T}\in\mathbb{R}^{3}bold_p start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT [ italic_n ] = ( italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT , italic_y [ italic_n ] , 0 ) start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT, see Figure 1. The position of Uksubscript𝑈𝑘U_{k}italic_U start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT in time slot n{1,,N}𝑛1𝑁n\in\{1,...,N\}italic_n ∈ { 1 , … , italic_N } is 𝐪k[n]=(xk,y[n],zk)Tsubscript𝐪𝑘delimited-[]𝑛superscriptsubscript𝑥𝑘𝑦delimited-[]𝑛subscript𝑧𝑘𝑇\mathbf{q}_{k}[n]=(x_{k},y[n],z_{k})^{T}bold_q start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT [ italic_n ] = ( italic_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , italic_y [ italic_n ] , italic_z start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT, with the y𝑦yitalic_y-axis position vector 𝐲=(y[1]=0,y[2],,y[N])TN𝐲superscript𝑦delimited-[]10𝑦delimited-[]2𝑦delimited-[]𝑁𝑇superscript𝑁\mathbf{y}=(y[1]=0,y[2],...,y[N])^{T}\in\mathbb{R}^{N}bold_y = ( italic_y [ 1 ] = 0 , italic_y [ 2 ] , … , italic_y [ italic_N ] ) start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT given by:

y[n+1]=y[n]+vyδt,n{1,N1}.formulae-sequence𝑦delimited-[]𝑛1𝑦delimited-[]𝑛subscript𝑣𝑦subscript𝛿𝑡for-all𝑛1𝑁1\displaystyle y[n+1]=y[n]+v_{y}\delta_{t},\forall n\in\{1,N-1\}.italic_y [ italic_n + 1 ] = italic_y [ italic_n ] + italic_v start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT italic_δ start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT , ∀ italic_n ∈ { 1 , italic_N - 1 } .(1)

For simplicity, we denote the position of Uksubscript𝑈𝑘U_{k}italic_U start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT in the across-track plane (i.e., xzlimit-from𝑥𝑧xz-italic_x italic_z -plane) by 𝐪k=(xk,zk)T2,k{0,1,2}formulae-sequencesubscript𝐪𝑘superscriptsubscript𝑥𝑘subscript𝑧𝑘𝑇superscript2for-all𝑘012\mathbf{q}_{k}=(x_{k},z_{k})^{T}\in\mathbb{R}^{2},\forall k\in\{0,1,2\}bold_q start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = ( italic_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , italic_z start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT , ∀ italic_k ∈ { 0 , 1 , 2 }. The interferometric baseline, bksubscript𝑏𝑘b_{k}italic_b start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT, which refers to the distance between sensors U0subscript𝑈0U_{0}italic_U start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT and Uksubscript𝑈𝑘U_{k}italic_U start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT, is given by:

bk(𝐪0,𝐪k)=𝐪k𝐪02,k{1,2}.formulae-sequencesubscript𝑏𝑘subscript𝐪0subscript𝐪𝑘subscriptnormsubscript𝐪𝑘subscript𝐪02for-all𝑘12\displaystyle b_{k}(\mathbf{q}_{0},\mathbf{q}_{k})=||\mathbf{q}_{k}-\mathbf{q}%_{0}||_{2},\forall k\in\{1,2\}.italic_b start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( bold_q start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , bold_q start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) = | | bold_q start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT - bold_q start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT | | start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , ∀ italic_k ∈ { 1 , 2 } .(2)

The perpendicular baseline, denoted by b,ksubscript𝑏bottom𝑘b_{\bot,k}italic_b start_POSTSUBSCRIPT ⊥ , italic_k end_POSTSUBSCRIPT, is the magnitude of the projection of Uksubscript𝑈𝑘U_{k}italic_U start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT’s baseline vector perpendicular to U0subscript𝑈0U_{0}italic_U start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT’s \aclos to 𝐩t[n]subscript𝐩𝑡delimited-[]𝑛\mathbf{p}_{t}[n]bold_p start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT [ italic_n ] and is given by:

b,k(𝐪0,𝐪k)=bk(𝐪0,𝐪k)cos(θ0αk(𝐪0,𝐪k)),k{1,2},formulae-sequencesubscript𝑏bottom𝑘subscript𝐪0subscript𝐪𝑘subscript𝑏𝑘subscript𝐪0subscript𝐪𝑘subscript𝜃0subscript𝛼𝑘subscript𝐪0subscript𝐪𝑘for-all𝑘12b_{\bot,k}(\mathbf{q}_{0},\mathbf{q}_{k})=b_{k}(\mathbf{q}_{0},\mathbf{q}_{k})%\cos\Big{(}\theta_{0}-\alpha_{k}(\mathbf{q}_{0},\mathbf{q}_{k})\Big{)},\forallk%\in\{1,2\},italic_b start_POSTSUBSCRIPT ⊥ , italic_k end_POSTSUBSCRIPT ( bold_q start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , bold_q start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) = italic_b start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( bold_q start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , bold_q start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) roman_cos ( italic_θ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT - italic_α start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( bold_q start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , bold_q start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) ) , ∀ italic_k ∈ { 1 , 2 } ,(3)

where θ0subscript𝜃0\theta_{0}italic_θ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT is the fixed look angle that U0subscript𝑈0U_{0}italic_U start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT’s \aclos has with the vertical, and αksubscript𝛼𝑘\alpha_{k}italic_α start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT is the angle between the interferometric baseline bksubscript𝑏𝑘b_{k}italic_b start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT and the horizontal plane.

II-A \acinsar Performance

Next, we introduce the relevant \acinsar sensing performance metrics.

II-A1 \acinsar Coverage

Let rk,k{0,1,2}subscript𝑟𝑘𝑘012r_{k},k\in\{0,1,2\}italic_r start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , italic_k ∈ { 0 , 1 , 2 }, denote Uksubscript𝑈𝑘U_{k}italic_U start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT’s slant range \acwrt 𝐩t[n]subscript𝐩𝑡delimited-[]𝑛\mathbf{p}_{t}[n]bold_p start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT [ italic_n ]. The slant range is independent of time and is given by:

rk(𝐪k)=(xkxt)2+zk2,k{0,1,2}.formulae-sequencesubscript𝑟𝑘subscript𝐪𝑘superscriptsubscript𝑥𝑘subscript𝑥𝑡2superscriptsubscript𝑧𝑘2for-all𝑘012\displaystyle r_{k}(\mathbf{q}_{k})=\sqrt{(x_{k}-x_{t})^{2}+z_{k}^{2}},\forallk%\in\{0,1,2\}.italic_r start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( bold_q start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) = square-root start_ARG ( italic_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT - italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_z start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG , ∀ italic_k ∈ { 0 , 1 , 2 } .(4)

The radar swath is designed to be centered \acwrt ltsubscript𝑙𝑡l_{t}italic_l start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT. To this end, the look angle of the slave \acpuav, denoted by θk(𝐪k),k{1,2}subscript𝜃𝑘subscript𝐪𝑘𝑘12\theta_{k}(\mathbf{q}_{k}),k\in\{1,2\}italic_θ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( bold_q start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) , italic_k ∈ { 1 , 2 }, is adjusted such that the beam footprint is centered around 𝐩tsubscript𝐩𝑡\mathbf{p}_{t}bold_p start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT, i.e., θk(𝐪k)=arctan(xkxtzk)subscript𝜃𝑘subscript𝐪𝑘subscript𝑥𝑘subscript𝑥𝑡subscript𝑧𝑘\theta_{k}(\mathbf{q}_{k})=\arctan\left(\frac{x_{k}-x_{t}}{z_{k}}\right)italic_θ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( bold_q start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) = roman_arctan ( divide start_ARG italic_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT - italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT end_ARG start_ARG italic_z start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_ARG ). The swath width of Uksubscript𝑈𝑘U_{k}italic_U start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT can be approximated as follows [13]:

Sk(𝐪k)=Θ3dBrk(𝐪k)cos(θk(𝐪k)),k{0,1,2},formulae-sequencesubscript𝑆𝑘subscript𝐪𝑘subscriptΘ3dBsubscript𝑟𝑘subscript𝐪𝑘subscript𝜃𝑘subscript𝐪𝑘for-all𝑘012\displaystyle S_{k}(\mathbf{q}_{k})=\frac{\Theta_{\rm 3dB}r_{k}(\mathbf{q}_{k}%)}{\cos(\theta_{k}(\mathbf{q}_{k}))},\forall k\in\{0,1,2\},italic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( bold_q start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) = divide start_ARG roman_Θ start_POSTSUBSCRIPT 3 roman_d roman_B end_POSTSUBSCRIPT italic_r start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( bold_q start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) end_ARG start_ARG roman_cos ( italic_θ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( bold_q start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) ) end_ARG , ∀ italic_k ∈ { 0 , 1 , 2 } ,(5)

where Θ3dBsubscriptΘ3dB\Theta_{\mathrm{3dB}}roman_Θ start_POSTSUBSCRIPT 3 roman_d roman_B end_POSTSUBSCRIPT is the -3 dB beamwidth in elevation.

Sensing Accuracy Optimization for Communication-assisted Dual-baseline UAV-InSAR ††thanks: This work was supported in part by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) GRK 2680 – Project-ID 437847244. (1)

II-A2 \Acinsar Coherence

A key performance metric for \acinsar is coherence, representing the cross-correlation between two \acsar images. For the images acquired by (U0,Uk),k{1,2}subscript𝑈0subscript𝑈𝑘𝑘12(U_{0},U_{k}),k\in\{1,2\}( italic_U start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_U start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) , italic_k ∈ { 1 , 2 }, the total coherence can be decomposed into several decorrelation sources as follows:

γk(𝐪0,𝐪k)=γRg,k(𝐪k)γSNR,k(𝐪0,𝐪k)γother,k{1,2},formulae-sequencesubscript𝛾𝑘subscript𝐪0subscript𝐪𝑘subscript𝛾Rg𝑘subscript𝐪𝑘subscript𝛾SNR𝑘subscript𝐪0subscript𝐪𝑘subscript𝛾otherfor-all𝑘12\gamma_{k}(\mathbf{q}_{0},\mathbf{q}_{k})=\gamma_{\mathrm{Rg},k}(\mathbf{q}_{k%})\gamma_{\mathrm{SNR},k}(\mathbf{q}_{0},\mathbf{q}_{k})\gamma_{\rm other},%\forall k\in\{1,2\},italic_γ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( bold_q start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , bold_q start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) = italic_γ start_POSTSUBSCRIPT roman_Rg , italic_k end_POSTSUBSCRIPT ( bold_q start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) italic_γ start_POSTSUBSCRIPT roman_SNR , italic_k end_POSTSUBSCRIPT ( bold_q start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , bold_q start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) italic_γ start_POSTSUBSCRIPT roman_other end_POSTSUBSCRIPT , ∀ italic_k ∈ { 1 , 2 } ,(6)

where γRg,ksubscript𝛾Rg𝑘\gamma_{\mathrm{Rg},k}italic_γ start_POSTSUBSCRIPT roman_Rg , italic_k end_POSTSUBSCRIPT is the baseline decorrelation, γSNR,ksubscript𝛾SNR𝑘\gamma_{\mathrm{SNR},k}italic_γ start_POSTSUBSCRIPT roman_SNR , italic_k end_POSTSUBSCRIPT is the \acsnr decorrelation, and γothersubscript𝛾other\gamma_{\rm other}italic_γ start_POSTSUBSCRIPT roman_other end_POSTSUBSCRIPT represents the contribution from all other decorrelation sources. The \acsnr decorrelation of pair (U0,Uk)subscript𝑈0subscript𝑈𝑘(U_{0},U_{k})( italic_U start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_U start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) is affected by the \acpsnr of both \acpuav and is given by [10]:

γSNR,k(𝐪0,𝐪k)=11+SNR01(𝐪0)11+SNRk1(𝐪0,𝐪k),subscript𝛾SNR𝑘subscript𝐪0subscript𝐪𝑘11subscriptsuperscriptSNR10subscript𝐪011subscriptsuperscriptSNR1𝑘subscript𝐪0subscript𝐪𝑘\displaystyle\gamma_{\mathrm{SNR},k}(\mathbf{q}_{0},\mathbf{q}_{k})=\frac{1}{%\sqrt{1+\mathrm{SNR}^{-1}_{0}(\mathbf{q}_{0})}}\frac{1}{\sqrt{1+\mathrm{SNR}^{%-1}_{k}(\mathbf{q}_{0},\mathbf{q}_{k})}},italic_γ start_POSTSUBSCRIPT roman_SNR , italic_k end_POSTSUBSCRIPT ( bold_q start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , bold_q start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) = divide start_ARG 1 end_ARG start_ARG square-root start_ARG 1 + roman_SNR start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( bold_q start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) end_ARG end_ARG divide start_ARG 1 end_ARG start_ARG square-root start_ARG 1 + roman_SNR start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( bold_q start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , bold_q start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) end_ARG end_ARG ,(7)

where SNR0subscriptSNR0\mathrm{SNR}_{0}roman_SNR start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT denotes the \acsnr of the mono-static acquisition by U0subscript𝑈0U_{0}italic_U start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT given by [10]:

SNR0(𝐪0)=γmr03(𝐪0),subscriptSNR0subscript𝐪0subscript𝛾𝑚superscriptsubscript𝑟03subscript𝐪0\mathrm{SNR}_{0}(\mathbf{q}_{0})=\frac{\gamma_{m}}{r_{0}^{3}(\mathbf{q}_{0})},roman_SNR start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( bold_q start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) = divide start_ARG italic_γ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT end_ARG start_ARG italic_r start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ( bold_q start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) end_ARG ,(8)

where γm=σ0PtGtGrλ3cτpPRF44π3vysin(θ0)kbTsysBRgFLsubscript𝛾𝑚subscript𝜎0subscript𝑃𝑡subscript𝐺𝑡subscript𝐺𝑟superscript𝜆3𝑐subscript𝜏𝑝PRFsuperscript44superscript𝜋3subscript𝑣𝑦subscript𝜃0subscript𝑘𝑏subscript𝑇syssubscript𝐵Rg𝐹𝐿\gamma_{m}=\frac{\sigma_{0}P_{t}\;G_{t}\;G_{r}\lambda^{3}c\tau_{p}\mathrm{PRF}%}{4^{4}\pi^{3}v_{y}\sin(\theta_{0})k_{b}T_{\mathrm{sys}}\;B_{\mathrm{Rg}}\;F\;L}italic_γ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT = divide start_ARG italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT italic_G start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT italic_G start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT italic_λ start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT italic_c italic_τ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT roman_PRF end_ARG start_ARG 4 start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT italic_π start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT italic_v start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT roman_sin ( italic_θ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) italic_k start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT italic_T start_POSTSUBSCRIPT roman_sys end_POSTSUBSCRIPT italic_B start_POSTSUBSCRIPT roman_Rg end_POSTSUBSCRIPT italic_F italic_L end_ARG. Here, σ0subscript𝜎0\sigma_{0}italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT is the normalized backscatter coefficient, Ptsubscript𝑃𝑡P_{t}italic_P start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT is the radar transmit power, Gtsubscript𝐺𝑡G_{t}italic_G start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT and Grsubscript𝐺𝑟G_{r}italic_G start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT are the transmit and receive antenna gains, respectively, λ𝜆\lambdaitalic_λ is the radar wavelength, c𝑐citalic_c is the speed of light, τpsubscript𝜏𝑝\tau_{p}italic_τ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT is the pulse duration, PRFPRF\mathrm{PRF}roman_PRF is the pulse repetition frequency, kbsubscript𝑘𝑏k_{b}italic_k start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT is the Boltzmann constant, Tsyssubscript𝑇sysT_{\mathrm{sys}}italic_T start_POSTSUBSCRIPT roman_sys end_POSTSUBSCRIPT is the receiver temperature, BRgsubscript𝐵RgB_{\mathrm{Rg}}italic_B start_POSTSUBSCRIPT roman_Rg end_POSTSUBSCRIPT is the bandwidth of the radar pulse, F𝐹Fitalic_F is the noise figure, and L𝐿Litalic_L represents the total radar losses. The derivation of the bi-static \acsnr for Uksubscript𝑈𝑘U_{k}italic_U start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT, denoted by SNRksubscriptSNR𝑘\mathrm{SNR}_{k}roman_SNR start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT, is more complicated. Here, assuming a small bi-static angle |θ0θk|subscript𝜃0subscript𝜃𝑘|\theta_{0}-\theta_{k}|| italic_θ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT - italic_θ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT |, which holds for \acinsar applications [8], we propose the following approximation111Please find a detailed derivation of the approximated bi-static \acsnr expression in Appendix A.:

SNRk(𝐪0,𝐪k)γmr02(𝐪0)rk(𝐪k),k{1,2}.formulae-sequencesubscriptSNR𝑘subscript𝐪0subscript𝐪𝑘subscript𝛾𝑚superscriptsubscript𝑟02subscript𝐪0subscript𝑟𝑘subscript𝐪𝑘for-all𝑘12\mathrm{SNR}_{k}(\mathbf{q}_{0},\mathbf{q}_{k})\approx\frac{\gamma_{m}}{r_{0}^%{2}(\mathbf{q}_{0})r_{k}(\mathbf{q}_{k})},\forall k\in\{1,2\}.roman_SNR start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( bold_q start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , bold_q start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) ≈ divide start_ARG italic_γ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT end_ARG start_ARG italic_r start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( bold_q start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) italic_r start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( bold_q start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) end_ARG , ∀ italic_k ∈ { 1 , 2 } .(9)

Furthermore, the baseline decorrelation reflects the loss of coherence caused by the different angles used for the acquisition of both \acinsar images [14]:

γRg,k(𝐪k)=1Bp[2+Bp1+𝒳(𝐪k)2Bp1+𝒳1(𝐪k)],subscript𝛾Rg𝑘subscript𝐪𝑘1subscript𝐵𝑝delimited-[]2subscript𝐵𝑝1𝒳subscript𝐪𝑘2subscript𝐵𝑝1superscript𝒳1subscript𝐪𝑘\gamma_{\mathrm{Rg},k}(\mathbf{q}_{k})=\frac{1}{B_{p}}\left[\frac{2+B_{p}}{1+%\mathcal{X}(\mathbf{q}_{k})}-\frac{2-B_{p}}{1+\mathcal{X}^{-1}(\mathbf{q}_{k})%}\right],italic_γ start_POSTSUBSCRIPT roman_Rg , italic_k end_POSTSUBSCRIPT ( bold_q start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) = divide start_ARG 1 end_ARG start_ARG italic_B start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT end_ARG [ divide start_ARG 2 + italic_B start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT end_ARG start_ARG 1 + caligraphic_X ( bold_q start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) end_ARG - divide start_ARG 2 - italic_B start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT end_ARG start_ARG 1 + caligraphic_X start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( bold_q start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) end_ARG ] ,(10)

where Bp=BRgf0subscript𝐵𝑝subscript𝐵Rgsubscript𝑓0B_{p}=\frac{B_{\mathrm{Rg}}}{f_{0}}italic_B start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT = divide start_ARG italic_B start_POSTSUBSCRIPT roman_Rg end_POSTSUBSCRIPT end_ARG start_ARG italic_f start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG is the fractional bandwidth, f0subscript𝑓0f_{0}italic_f start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT is the radar center frequency, and function 𝒳𝒳\mathcal{X}caligraphic_X is given by [14]:

𝒳(𝐪k)=2(sin(θ0)𝟙{θ0>θk(𝐪k)}+sin(θk(𝐪k))𝟙{θ0θk(𝐪k)})sin(θ0)+sin(θk(𝐪k)).𝒳subscript𝐪𝑘2subscript𝜃01subscript𝜃0subscript𝜃𝑘subscript𝐪𝑘subscript𝜃𝑘subscript𝐪𝑘1subscript𝜃0subscript𝜃𝑘subscript𝐪𝑘subscript𝜃0subscript𝜃𝑘subscript𝐪𝑘\mathcal{X}(\mathbf{q}_{k})=\frac{2\Big{(}\sin(\theta_{0})\mathds{1}\{\theta_{%0}>\theta_{k}(\mathbf{q}_{k})\}+\sin(\theta_{k}(\mathbf{q}_{k}))\mathds{1}\{%\theta_{0}\leq\theta_{k}(\mathbf{q}_{k})\}\Big{)}}{\sin(\theta_{0})+\sin(%\theta_{k}(\mathbf{q}_{k}))}.caligraphic_X ( bold_q start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) = divide start_ARG 2 ( roman_sin ( italic_θ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) blackboard_1 { italic_θ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT > italic_θ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( bold_q start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) } + roman_sin ( italic_θ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( bold_q start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) ) blackboard_1 { italic_θ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ≤ italic_θ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( bold_q start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) } ) end_ARG start_ARG roman_sin ( italic_θ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) + roman_sin ( italic_θ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( bold_q start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) ) end_ARG .(11)

II-A3 Height of Ambiguity (HoA)

The \achoa is related to the sensitivity of the radar system to topographic height variations [10]. The \achoa of pair (U0,Uk)subscript𝑈0subscript𝑈𝑘(U_{0},U_{k})( italic_U start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_U start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) is given by [10]:

hamb,k(𝐪0,𝐪k)=λr0(𝐪0)sin(θ0)b,k(𝐪0,𝐪k),k{1,2}.formulae-sequencesubscriptamb𝑘subscript𝐪0subscript𝐪𝑘𝜆subscript𝑟0subscript𝐪0subscript𝜃0subscript𝑏perpendicular-to𝑘subscript𝐪0subscript𝐪𝑘for-all𝑘12\displaystyle h_{\mathrm{amb},k}(\mathbf{q}_{0},\mathbf{q}_{k})=\frac{\lambda r%_{0}(\mathbf{q}_{0})\sin(\theta_{0})}{b_{\perp,k}(\mathbf{q}_{0},\mathbf{q}_{k%})},\forall k\in\{1,2\}.italic_h start_POSTSUBSCRIPT roman_amb , italic_k end_POSTSUBSCRIPT ( bold_q start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , bold_q start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) = divide start_ARG italic_λ italic_r start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( bold_q start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) roman_sin ( italic_θ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) end_ARG start_ARG italic_b start_POSTSUBSCRIPT ⟂ , italic_k end_POSTSUBSCRIPT ( bold_q start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , bold_q start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) end_ARG , ∀ italic_k ∈ { 1 , 2 } .(12)

II-A4 \acdem Height Accuracy

The height error of the \acdem acquired by the \acinsar pair (U0,Uk)subscript𝑈0subscript𝑈𝑘(U_{0},U_{k})( italic_U start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_U start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) is given by [10]:

σhk(𝐪0,𝐪k)=hamb,k(𝐪0,𝐪k)σΦk(𝐪0,𝐪k)2π,k{1,2},formulae-sequencesubscript𝜎subscript𝑘subscript𝐪0subscript𝐪𝑘subscriptamb𝑘subscript𝐪0subscript𝐪𝑘subscript𝜎subscriptΦ𝑘subscript𝐪0subscript𝐪𝑘2𝜋for-all𝑘12\sigma_{h_{k}}(\mathbf{q}_{0},\mathbf{q}_{k})=h_{\mathrm{amb},k}(\mathbf{q}_{0%},\mathbf{q}_{k})\frac{\sigma_{\Phi_{k}}(\mathbf{q}_{0},\mathbf{q}_{k})}{2\pi}%,\forall k\in\{1,2\},italic_σ start_POSTSUBSCRIPT italic_h start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( bold_q start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , bold_q start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) = italic_h start_POSTSUBSCRIPT roman_amb , italic_k end_POSTSUBSCRIPT ( bold_q start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , bold_q start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) divide start_ARG italic_σ start_POSTSUBSCRIPT roman_Φ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( bold_q start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , bold_q start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) end_ARG start_ARG 2 italic_π end_ARG , ∀ italic_k ∈ { 1 , 2 } ,(13)

where σΦksubscript𝜎subscriptΦ𝑘\sigma_{\Phi_{k}}italic_σ start_POSTSUBSCRIPT roman_Φ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT is the random error in the interferometric phase and can be approximated in the case of high interferometric coherences by the Cramér–Rao bound [8]:

σΦk(𝐪0,𝐪k)=1γk(𝐪0,𝐪k)1γk2(𝐪0,𝐪k)2nL,k{1,2},formulae-sequencesubscript𝜎subscriptΦ𝑘subscript𝐪0subscript𝐪𝑘1subscript𝛾𝑘subscript𝐪0subscript𝐪𝑘1superscriptsubscript𝛾𝑘2subscript𝐪0subscript𝐪𝑘2subscript𝑛𝐿for-all𝑘12\sigma_{\Phi_{k}}(\mathbf{q}_{0},\mathbf{q}_{k})=\frac{1}{\gamma_{k}(\mathbf{q%}_{0},\mathbf{q}_{k})}\sqrt{\frac{1-\gamma_{k}^{2}(\mathbf{q}_{0},\mathbf{q}_{%k})}{2n_{L}}},\forall k\in\{1,2\},italic_σ start_POSTSUBSCRIPT roman_Φ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( bold_q start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , bold_q start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) = divide start_ARG 1 end_ARG start_ARG italic_γ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( bold_q start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , bold_q start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) end_ARG square-root start_ARG divide start_ARG 1 - italic_γ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( bold_q start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , bold_q start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) end_ARG start_ARG 2 italic_n start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT end_ARG end_ARG , ∀ italic_k ∈ { 1 , 2 } ,(14)

where nLsubscript𝑛𝐿n_{L}italic_n start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT is the number of independent looks employed, i.e., nLsubscript𝑛𝐿n_{L}italic_n start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT adjacent pixels of the interferogram are averaged to improve phase estimation [8]. The fusion of the two independent \acinsar \acpdem is performed based on inverse-variance weighting, such that the height of an arbitrary target estimated by \acinsar pair (U0,Uk)subscript𝑈0subscript𝑈𝑘(U_{0},U_{k})( italic_U start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_U start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ), denoted by hksubscript𝑘h_{k}italic_h start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT, is weighted by wk(𝐪0,𝐪k)=1σhk2(𝐪0,𝐪k)subscript𝑤𝑘subscript𝐪0subscript𝐪𝑘1subscriptsuperscript𝜎2subscript𝑘subscript𝐪0subscript𝐪𝑘w_{k}(\mathbf{q}_{0},\mathbf{q}_{k})=\frac{1}{\sigma^{2}_{h_{k}}(\mathbf{q}_{0%},\mathbf{q}_{k})}italic_w start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( bold_q start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , bold_q start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) = divide start_ARG 1 end_ARG start_ARG italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_h start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( bold_q start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , bold_q start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) end_ARG, k{1,2}𝑘12k\in\{1,2\}italic_k ∈ { 1 , 2 }, and averaged as h1w1+h2w2w1+w2subscript1subscript𝑤1subscript2subscript𝑤2subscript𝑤1subscript𝑤2\frac{h_{1}w_{1}+h_{2}w_{2}}{w_{1}+w_{2}}divide start_ARG italic_h start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_w start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_h start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_w start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_ARG start_ARG italic_w start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_w start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_ARG. The final height error of the fused \acdem is characterized by [12]:

σh(𝐪0,𝐪1,𝐪2)=k{1,2}wk2(𝐪0,𝐪k)σhk2(𝐪0,𝐪k)(k{1,2}wk(𝐪0,𝐪k))2.subscript𝜎subscript𝐪0subscript𝐪1subscript𝐪2subscript𝑘12superscriptsubscript𝑤𝑘2subscript𝐪0subscript𝐪𝑘subscriptsuperscript𝜎2subscript𝑘subscript𝐪0subscript𝐪𝑘superscriptsubscript𝑘12subscript𝑤𝑘subscript𝐪0subscript𝐪𝑘2\displaystyle\sigma_{h}(\mathbf{q}_{0},\mathbf{q}_{1},\mathbf{q}_{2})=\sqrt{%\frac{\sum\limits_{k\in\{1,2\}}w_{k}^{2}(\mathbf{q}_{0},\mathbf{q}_{k})\sigma^%{2}_{h_{k}}(\mathbf{q}_{0},\mathbf{q}_{k})}{\left(\sum\limits_{k\in\{1,2\}}w_{%k}(\mathbf{q}_{0},\mathbf{q}_{k})\right)^{2}}}.italic_σ start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT ( bold_q start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , bold_q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , bold_q start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) = square-root start_ARG divide start_ARG ∑ start_POSTSUBSCRIPT italic_k ∈ { 1 , 2 } end_POSTSUBSCRIPT italic_w start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( bold_q start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , bold_q start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_h start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( bold_q start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , bold_q start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) end_ARG start_ARG ( ∑ start_POSTSUBSCRIPT italic_k ∈ { 1 , 2 } end_POSTSUBSCRIPT italic_w start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( bold_q start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , bold_q start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG end_ARG .(15)

II-B Communication Performance

We consider real-time offloading of the radar data to a \acgs, where the master and slave \acpuav employ \acfdma. The instantaneous communication transmit power consumed by \acuav Uksubscript𝑈𝑘U_{k}italic_U start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT is given by 𝐏com,k=(Pcom,k[1],,Pcom,k[N])TN,k{0,1,2}formulae-sequencesubscript𝐏com𝑘superscriptsubscript𝑃com𝑘delimited-[]1subscript𝑃com𝑘delimited-[]𝑁𝑇superscript𝑁𝑘012\mathbf{P}_{\mathrm{com},k}=(P_{\mathrm{com},k}[1],...,P_{\mathrm{com},k}[N])^%{T}\in\mathbb{R}^{N},k\in\{0,1,2\}bold_P start_POSTSUBSCRIPT roman_com , italic_k end_POSTSUBSCRIPT = ( italic_P start_POSTSUBSCRIPT roman_com , italic_k end_POSTSUBSCRIPT [ 1 ] , … , italic_P start_POSTSUBSCRIPT roman_com , italic_k end_POSTSUBSCRIPT [ italic_N ] ) start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT , italic_k ∈ { 0 , 1 , 2 }.We denote the location of the \acgs by 𝐠=(gx,gy,gz)T3𝐠superscriptsubscript𝑔𝑥subscript𝑔𝑦subscript𝑔𝑧𝑇superscript3\mathbf{g}=(g_{x},g_{y},g_{z})^{T}\in\mathbb{R}^{3}bold_g = ( italic_g start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT , italic_g start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT , italic_g start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT and the distance from Uksubscript𝑈𝑘U_{k}italic_U start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT to the \acgs by dk,n(𝐪k)=𝐪k[n]𝐠2,k{0,1,2},n.formulae-sequencesubscript𝑑𝑘𝑛subscript𝐪𝑘subscriptnormsubscript𝐪𝑘delimited-[]𝑛𝐠2for-all𝑘012for-all𝑛d_{k,n}(\mathbf{q}_{k})=||\mathbf{q}_{k}[n]-\mathbf{g}||_{2},\forall k\in\{0,1%,2\},\forall n.italic_d start_POSTSUBSCRIPT italic_k , italic_n end_POSTSUBSCRIPT ( bold_q start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) = | | bold_q start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT [ italic_n ] - bold_g | | start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , ∀ italic_k ∈ { 0 , 1 , 2 } , ∀ italic_n . Thus, adopting the free-space path loss model and \acfdma, theinstantaneous throughput from Uk,k{0,1,2},subscript𝑈𝑘for-all𝑘012U_{k},\forall k\in\{0,1,2\},italic_U start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , ∀ italic_k ∈ { 0 , 1 , 2 } , to the \acgs is given by:

Rk,n(𝐪k,𝐏com,k)=Bc,klog2(1+Pcom,k[n]βc,kdk,n2(𝐪k)),n,subscript𝑅𝑘𝑛subscript𝐪𝑘subscript𝐏com𝑘subscript𝐵𝑐𝑘subscript21subscript𝑃com𝑘delimited-[]𝑛subscript𝛽𝑐𝑘superscriptsubscript𝑑𝑘𝑛2subscript𝐪𝑘for-all𝑛\displaystyle R_{k,n}(\mathbf{q}_{k},\mathbf{P}_{\mathrm{com},k})=B_{c,k}\;%\log_{2}\left(1+\frac{P_{\mathrm{com},k}[n]\;\beta_{c,k}}{d_{k,n}^{2}(\mathbf{%q}_{k})}\right),\forall n,italic_R start_POSTSUBSCRIPT italic_k , italic_n end_POSTSUBSCRIPT ( bold_q start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , bold_P start_POSTSUBSCRIPT roman_com , italic_k end_POSTSUBSCRIPT ) = italic_B start_POSTSUBSCRIPT italic_c , italic_k end_POSTSUBSCRIPT roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( 1 + divide start_ARG italic_P start_POSTSUBSCRIPT roman_com , italic_k end_POSTSUBSCRIPT [ italic_n ] italic_β start_POSTSUBSCRIPT italic_c , italic_k end_POSTSUBSCRIPT end_ARG start_ARG italic_d start_POSTSUBSCRIPT italic_k , italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( bold_q start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) end_ARG ) , ∀ italic_n ,(16)

where Bc,ksubscript𝐵𝑐𝑘B_{c,k}italic_B start_POSTSUBSCRIPT italic_c , italic_k end_POSTSUBSCRIPT is Uksubscript𝑈𝑘U_{k}italic_U start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT’s fixed communication bandwidth and βc,ksubscript𝛽𝑐𝑘\beta_{c,k}italic_β start_POSTSUBSCRIPT italic_c , italic_k end_POSTSUBSCRIPT is the reference channel gain222The reference channel gain is the channel power gain at a reference distance of 1 m. divided by the noise variance.

III Problem Formulation

In this paper, we aim to minimize the height error of the final \acdem σhsubscript𝜎\sigma_{h}italic_σ start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT by jointly optimizing the \acuav formation 𝒬={𝐪k,k{0,1,2}}𝒬subscript𝐪𝑘for-all𝑘012\mathcal{Q}=\{\mathbf{q}_{k},\forall k\in\{0,1,2\}\}caligraphic_Q = { bold_q start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , ∀ italic_k ∈ { 0 , 1 , 2 } } and the instantaneous communication transmit powers 𝒫={𝐏com,k,k{0,1,2}}𝒫subscript𝐏com𝑘for-all𝑘012\mathcal{P}=\{\mathbf{P}_{\mathrm{com},k},\forall k\in\{0,1,2\}\}caligraphic_P = { bold_P start_POSTSUBSCRIPT roman_com , italic_k end_POSTSUBSCRIPT , ∀ italic_k ∈ { 0 , 1 , 2 } }, while satisfying communication and sensing quality-of-service constraints. To this end, we formulate the following optimization problem:

(P):min𝒬,𝒫σh(𝒬):Psubscript𝒬𝒫subscript𝜎𝒬\displaystyle(\mathrm{P}):\min_{\mathcal{Q},\mathcal{P}}\hskip 8.53581pt\sigma%_{h}(\mathcal{Q})( roman_P ) : roman_min start_POSTSUBSCRIPT caligraphic_Q , caligraphic_P end_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT ( caligraphic_Q )
s.t.C1:zminzkzmax,k{0,1,2},\displaystyle\mathrm{C1:}\;z_{\mathrm{min}}\leq z_{k}\leq z_{\mathrm{max}},%\forall k\in\{0,1,2\},C1 : italic_z start_POSTSUBSCRIPT roman_min end_POSTSUBSCRIPT ≤ italic_z start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ≤ italic_z start_POSTSUBSCRIPT roman_max end_POSTSUBSCRIPT , ∀ italic_k ∈ { 0 , 1 , 2 } ,
C2:x0=xtz0tan(θ0),:C2subscript𝑥0subscript𝑥𝑡subscript𝑧0subscript𝜃0\displaystyle\mathrm{C2}:\;x_{0}=x_{t}-z_{0}\tan(\theta_{0}),C2 : italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT - italic_z start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT roman_tan ( italic_θ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) ,
C3:θminθk(𝐪k)θmax,k{1,2},\displaystyle\mathrm{C3}:\;\theta_{\mathrm{min}}\leq\theta_{k}(\mathbf{q}_{k})%\leq\theta_{\mathrm{max}},\forall k\in\{1,2\},C3 : italic_θ start_POSTSUBSCRIPT roman_min end_POSTSUBSCRIPT ≤ italic_θ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( bold_q start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) ≤ italic_θ start_POSTSUBSCRIPT roman_max end_POSTSUBSCRIPT , ∀ italic_k ∈ { 1 , 2 } ,
C4:𝐪i𝐪j2dmin,ij{0,1,2},:C4formulae-sequencesubscriptnormsubscript𝐪𝑖subscript𝐪𝑗2subscript𝑑minfor-all𝑖𝑗012\displaystyle\mathrm{C4}:||\mathbf{q}_{i}-\mathbf{q}_{j}||_{2}\geq d_{\mathrm{%min}},\forall i\neq j\in\{0,1,2\},C4 : | | bold_q start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - bold_q start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT | | start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ≥ italic_d start_POSTSUBSCRIPT roman_min end_POSTSUBSCRIPT , ∀ italic_i ≠ italic_j ∈ { 0 , 1 , 2 } ,
C5:Sk(𝐪k)Smin,k{0,1,2},:C5formulae-sequencesubscript𝑆𝑘subscript𝐪𝑘subscript𝑆minfor-all𝑘012\displaystyle\mathrm{C5}:S_{k}(\mathbf{q}_{k})\geq S_{\rm min},\forall k\in\{0%,1,2\},C5 : italic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( bold_q start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) ≥ italic_S start_POSTSUBSCRIPT roman_min end_POSTSUBSCRIPT , ∀ italic_k ∈ { 0 , 1 , 2 } ,
C6:γSNR,k(𝐪0,𝐪k)γSNRmin,k{1,2},:C6formulae-sequencesubscript𝛾SNR𝑘subscript𝐪0subscript𝐪𝑘superscriptsubscript𝛾SNRminfor-all𝑘12\displaystyle\mathrm{C6}:\gamma_{\mathrm{SNR},k}(\mathbf{q}_{0},\mathbf{q}_{k}%)\geq\gamma_{\rm SNR}^{\mathrm{min}},\forall k\in\{1,2\},C6 : italic_γ start_POSTSUBSCRIPT roman_SNR , italic_k end_POSTSUBSCRIPT ( bold_q start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , bold_q start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) ≥ italic_γ start_POSTSUBSCRIPT roman_SNR end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_min end_POSTSUPERSCRIPT , ∀ italic_k ∈ { 1 , 2 } ,
C7:γRg,k(𝐪k)γRgmin,k{1,2},:C7formulae-sequencesubscript𝛾Rg𝑘subscript𝐪𝑘superscriptsubscript𝛾Rgminfor-all𝑘12\displaystyle\mathrm{C7}:\gamma_{\mathrm{Rg},k}(\mathbf{q}_{k})\geq\gamma_{\rmRg%}^{\mathrm{min}},\forall k\in\{1,2\},C7 : italic_γ start_POSTSUBSCRIPT roman_Rg , italic_k end_POSTSUBSCRIPT ( bold_q start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) ≥ italic_γ start_POSTSUBSCRIPT roman_Rg end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_min end_POSTSUPERSCRIPT , ∀ italic_k ∈ { 1 , 2 } ,
C8:hamb,k(𝐪0,𝐪k)hambmin,k{1,2},:C8formulae-sequencesubscriptamb𝑘subscript𝐪0subscript𝐪𝑘superscriptsubscriptambminfor-all𝑘12\displaystyle\mathrm{C8}:\;h_{\mathrm{amb},k}(\mathbf{q}_{0},\mathbf{q}_{k})%\geq h_{\mathrm{amb}}^{\mathrm{min}},\forall k\in\{1,2\},C8 : italic_h start_POSTSUBSCRIPT roman_amb , italic_k end_POSTSUBSCRIPT ( bold_q start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , bold_q start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) ≥ italic_h start_POSTSUBSCRIPT roman_amb end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_min end_POSTSUPERSCRIPT , ∀ italic_k ∈ { 1 , 2 } ,
C9:0Pcom,k[n]Pcommax,k{0,1,2},n,\displaystyle\mathrm{C9}:0\leq P_{\mathrm{com},k}[n]\leq P_{\mathrm{com}}^{%\mathrm{max}},\forall\;k\in\{0,1,2\},\forall n,C9 : 0 ≤ italic_P start_POSTSUBSCRIPT roman_com , italic_k end_POSTSUBSCRIPT [ italic_n ] ≤ italic_P start_POSTSUBSCRIPT roman_com end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_max end_POSTSUPERSCRIPT , ∀ italic_k ∈ { 0 , 1 , 2 } , ∀ italic_n ,
C10:Rk,n(𝐪k,𝐏com,k)Rmin,k,k{0,1,2},n,:C10formulae-sequencesubscript𝑅𝑘𝑛subscript𝐪𝑘subscript𝐏com𝑘subscript𝑅min𝑘for-all𝑘012for-all𝑛\displaystyle\mathrm{C10}:R_{k,n}(\mathbf{q}_{k},\mathbf{P}_{\mathrm{com},k})%\geq R_{\mathrm{min},k},\forall\;k\in\{0,1,2\},\forall n,C10 : italic_R start_POSTSUBSCRIPT italic_k , italic_n end_POSTSUBSCRIPT ( bold_q start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , bold_P start_POSTSUBSCRIPT roman_com , italic_k end_POSTSUBSCRIPT ) ≥ italic_R start_POSTSUBSCRIPT roman_min , italic_k end_POSTSUBSCRIPT , ∀ italic_k ∈ { 0 , 1 , 2 } , ∀ italic_n ,
C11:n=1NPcom,k[n]Ecommax,k{0,1,2}.:C11formulae-sequencesuperscriptsubscript𝑛1𝑁subscript𝑃com𝑘delimited-[]𝑛superscriptsubscript𝐸commaxfor-all𝑘012\displaystyle\mathrm{C11}:\sum_{n=1}^{N}P_{\mathrm{com},k}[n]\leq E_{\rm com}^%{\rm max},\forall\;k\in\{0,1,2\}.C11 : ∑ start_POSTSUBSCRIPT italic_n = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT italic_P start_POSTSUBSCRIPT roman_com , italic_k end_POSTSUBSCRIPT [ italic_n ] ≤ italic_E start_POSTSUBSCRIPT roman_com end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_max end_POSTSUPERSCRIPT , ∀ italic_k ∈ { 0 , 1 , 2 } .

Constraint C1C1\mathrm{C1}C1 specifies the maximum and minimum allowed flying altitude, denoted by zmaxsubscript𝑧maxz_{\mathrm{max}}italic_z start_POSTSUBSCRIPT roman_max end_POSTSUBSCRIPT and zminsubscript𝑧minz_{\mathrm{min}}italic_z start_POSTSUBSCRIPT roman_min end_POSTSUBSCRIPT, respectively. Constraint C2C2\mathrm{C2}C2 ensures that the beam footprint of the master \acuav is centered around 𝐩t[n]subscript𝐩𝑡delimited-[]𝑛\mathbf{p}_{t}[n]bold_p start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT [ italic_n ]. Constraint C3C3\mathrm{C3}C3 specifies the minimum and maximum slave look angle, denoted by θminsubscript𝜃min\theta_{\rm min}italic_θ start_POSTSUBSCRIPT roman_min end_POSTSUBSCRIPT and θmaxsubscript𝜃max\theta_{\rm max}italic_θ start_POSTSUBSCRIPT roman_max end_POSTSUBSCRIPT, respectively. Constraint C4C4\mathrm{C4}C4 ensures a minimum safety distance dminsubscript𝑑mind_{\mathrm{min}}italic_d start_POSTSUBSCRIPT roman_min end_POSTSUBSCRIPT between any two \acpuav. Constraint C5C5\mathrm{C5}C5 imposes a minimum radar swath width Sminsubscript𝑆minS_{\mathrm{min}}italic_S start_POSTSUBSCRIPT roman_min end_POSTSUBSCRIPT. Constraints C6C6\mathrm{C6}C6 and C7C7\mathrm{C7}C7 ensure minimum thresholds for \acsnr and baseline decorrelation, γSNRminsuperscriptsubscript𝛾SNRmin\gamma_{\rm SNR}^{\mathrm{min}}italic_γ start_POSTSUBSCRIPT roman_SNR end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_min end_POSTSUPERSCRIPT and γRgminsuperscriptsubscript𝛾Rgmin\gamma_{\rm Rg}^{\mathrm{min}}italic_γ start_POSTSUBSCRIPT roman_Rg end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_min end_POSTSUPERSCRIPT, respectively. Constraint C8C8\mathrm{C8}C8 imposes a minimum \achoa, hambminsuperscriptsubscriptambminh_{\mathrm{amb}}^{\mathrm{min}}italic_h start_POSTSUBSCRIPT roman_amb end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_min end_POSTSUPERSCRIPT, required for phase unwrapping [9]. Constraint C9C9\mathrm{C9}C9 imposes a maximum communication transmit power, Pcommaxsuperscriptsubscript𝑃commaxP_{\mathrm{com}}^{\mathrm{max}}italic_P start_POSTSUBSCRIPT roman_com end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_max end_POSTSUPERSCRIPT. Constraint C10C10\mathrm{C10}C10 ensures the minimum required sensing data rate for Uksubscript𝑈𝑘U_{k}italic_U start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT, Rmin,k,k{0,1,2}subscript𝑅min𝑘for-all𝑘012R_{\mathrm{min},k},\forall k\in\{0,1,2\}italic_R start_POSTSUBSCRIPT roman_min , italic_k end_POSTSUBSCRIPT , ∀ italic_k ∈ { 0 , 1 , 2 }. Constraint C11C11\mathrm{C11}C11 limits the total communication energy of Uksubscript𝑈𝑘U_{k}italic_U start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT to Ecommax,k{0,1,2}superscriptsubscript𝐸commaxfor-all𝑘012E_{\mathrm{com}}^{\rm max},\forall k\in\{0,1,2\}italic_E start_POSTSUBSCRIPT roman_com end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_max end_POSTSUPERSCRIPT , ∀ italic_k ∈ { 0 , 1 , 2 }.

Problem (P)P\mathrm{(P)}( roman_P ) is a non-convex optimization problem. The non-convexity is caused by the objective function and constraints C4,C5C4C5\mathrm{C4},\mathrm{C5}C4 , C5, C7C7\mathrm{C7}C7, and C8C8\mathrm{C8}C8. In fact, the height error is simultaneously \achoa- and coherence-dependent, making the objective function challenging. Moreover, the lower bound on an Euclidean distance in C4C4\mathrm{C4}C4 and the trigonometric functions in C5C5\mathrm{C5}C5, C7C7\mathrm{C7}C7, and C8C8\mathrm{C8}C8 make these constraints non-convex and difficult to handle.

IV Solution of the Optimization Problem

Sensing Accuracy Optimization for Communication-assisted Dual-baseline UAV-InSAR ††thanks: This work was supported in part by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) GRK 2680 – Project-ID 437847244. (2)

To balance performance and complexity, we propose a low-complexity sub-optimal solution that minimizes an upper bound on problem (P)P\mathrm{(P)}( roman_P ) based on \acao. We divide problem (P)P\mathrm{(P)}( roman_P ) into 3 sub-problems: (P.0)P.0\mathrm{(P.0)}( roman_P .0 ), (P.1)P.1\mathrm{(P.1)}( roman_P .1 ), and (P.2)P.2\mathrm{(P.2)}( roman_P .2 ). In (P.0)P.0\mathrm{(P.0)}( roman_P .0 ), we optimize the position and communication power of U0subscript𝑈0U_{0}italic_U start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, whereas in (P.1)P.1\mathrm{(P.1)}( roman_P .1 ) and (P.2)P.2\mathrm{(P.2)}( roman_P .2 ), we optimize the positions and communication powers of U1subscript𝑈1U_{1}italic_U start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and U2subscript𝑈2U_{2}italic_U start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT, respectively. Due to symmetry, we focus on (P.0)P.0\mathrm{(P.0)}( roman_P .0 ) and (P.1)P.1\mathrm{(P.1)}( roman_P .1 ), as (P.2)P.2\mathrm{(P.2)}( roman_P .2 ) can be solved similarly to (P.1)P.1\mathrm{(P.1)}( roman_P .1 ), see Figure 2.

IV-A Master \acuav Optimization

In this sub-section, we optimize the position and communication transmit power of the master \acuav, denoted by 𝐪0subscript𝐪0\mathbf{q}_{0}bold_q start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT and 𝐏com,0subscript𝐏com0\mathbf{P}_{\rm com,0}bold_P start_POSTSUBSCRIPT roman_com , 0 end_POSTSUBSCRIPT, respectively, for fixed {𝐪1,𝐪2,𝐏com,1,𝐏com,2}subscript𝐪1subscript𝐪2subscript𝐏com1subscript𝐏com2\{\mathbf{q}_{1},\mathbf{q}_{2},\mathbf{P}_{\rm com,1},\mathbf{P}_{\rm com,2}\}{ bold_q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , bold_q start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , bold_P start_POSTSUBSCRIPT roman_com , 1 end_POSTSUBSCRIPT , bold_P start_POSTSUBSCRIPT roman_com , 2 end_POSTSUBSCRIPT }. The resulting sub-problem, denoted by (P.0)P.0\mathrm{(P.0)}( roman_P .0 ), is still non-convex due to its objective function and C4C4\mathrm{C4}C4. Yet, we leverage \acsca to provide a low-complexity solution for (P.0)P.0(\mathrm{P.0})( roman_P .0 ).
As the master look angle θ0subscript𝜃0\theta_{0}italic_θ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT is fixed, the perpendicular baseline is independent of 𝐪0subscript𝐪0\mathbf{q}_{0}bold_q start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT and is given by [11]:

b,k(𝐪k)=|(xtxk)tan(θ0)zk|tan(θ0)2+1,k{1,2}.b_{\bot,k}(\mathbf{q}_{k})=\frac{\Big{|}(x_{t}-x_{k})-\tan(\theta_{0})z_{k}%\Big{|}}{\sqrt{\tan(\theta_{0})^{2}+1}},\forall k\in\{1,2\}.italic_b start_POSTSUBSCRIPT ⊥ , italic_k end_POSTSUBSCRIPT ( bold_q start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) = divide start_ARG | ( italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT - italic_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) - roman_tan ( italic_θ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) italic_z start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT | end_ARG start_ARG square-root start_ARG roman_tan ( italic_θ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + 1 end_ARG end_ARG , ∀ italic_k ∈ { 1 , 2 } .(17)
Proposition 1.

The height error of the final \acdem, σhsubscript𝜎\sigma_{h}italic_σ start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT, can be upper bounded based on the worst-case coherence as follows:

σh(𝒬)σh¯(𝒬)λ2r02(𝐪0)sin2(θ0)(1𝒜2)8π2𝒜2nL(b,12(𝐪1)+b,22(𝐪2)),subscript𝜎𝒬¯subscript𝜎𝒬absentabsentsuperscript𝜆2subscriptsuperscript𝑟20subscript𝐪0superscript2subscript𝜃01superscript𝒜28superscript𝜋2superscript𝒜2subscript𝑛𝐿subscriptsuperscript𝑏2bottom1subscript𝐪1subscriptsuperscript𝑏2bottom2subscript𝐪2\sigma_{h}(\mathcal{Q})\leq\overline{\sigma_{h}}(\mathcal{Q})\triangleq\frac{}%{}\sqrt{\frac{\lambda^{2}r^{2}_{0}(\mathbf{q}_{0})\sin^{2}(\theta_{0})(1-%\mathcal{A}^{2})}{8\pi^{2}\mathcal{A}^{2}n_{L}\left(b^{2}_{\bot,1}(\mathbf{q}_%{1})+b^{2}_{\bot,2}(\mathbf{q}_{2})\right)}},italic_σ start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT ( caligraphic_Q ) ≤ over¯ start_ARG italic_σ start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT end_ARG ( caligraphic_Q ) ≜ divide start_ARG end_ARG start_ARG end_ARG square-root start_ARG divide start_ARG italic_λ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_r start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( bold_q start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) roman_sin start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_θ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) ( 1 - caligraphic_A start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) end_ARG start_ARG 8 italic_π start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT caligraphic_A start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT ( italic_b start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT ⊥ , 1 end_POSTSUBSCRIPT ( bold_q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) + italic_b start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT ⊥ , 2 end_POSTSUBSCRIPT ( bold_q start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ) end_ARG end_ARG ,(18)

where 𝒜=γRgminγSNRminγother𝒜superscriptsubscript𝛾Rgminsuperscriptsubscript𝛾SNRminsubscript𝛾other\mathcal{A}=\gamma_{\rm Rg}^{\mathrm{min}}\gamma_{\rm SNR}^{\mathrm{min}}%\gamma_{\rm other}caligraphic_A = italic_γ start_POSTSUBSCRIPT roman_Rg end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_min end_POSTSUPERSCRIPT italic_γ start_POSTSUBSCRIPT roman_SNR end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_min end_POSTSUPERSCRIPT italic_γ start_POSTSUBSCRIPT roman_other end_POSTSUBSCRIPT.

Proof.

Please refer to Appendix B.∎

Therefore, we relax the complex objective function of (P.0)P.0\mathrm{(P.0)}( roman_P .0 ) by minimizing instead the upper bound on the height error, denoted by σh¯¯subscript𝜎\overline{\sigma_{h}}over¯ start_ARG italic_σ start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT end_ARG and provided in Proposition 1. In iteration i𝑖iitalic_i of the \acsca algorithm, constraint C4C4\mathrm{C4}C4 is tackled using a surrogate function for the Euclidean distance 𝐪0𝐪k2subscriptnormsubscript𝐪0subscript𝐪𝑘2||\mathbf{q}_{0}-\mathbf{q}_{k}||_{2}| | bold_q start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT - bold_q start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT | | start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT around 𝐪0(i)superscriptsubscript𝐪0𝑖\mathbf{q}_{0}^{(i)}bold_q start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_i ) end_POSTSUPERSCRIPT as follows [15]:

C4~:2𝐪0T(𝐪0(i)𝐪k)𝐪0(i)22+𝐪k22dmin,i,k{1,2}.:~C4formulae-sequence2superscriptsubscript𝐪0𝑇superscriptsubscript𝐪0𝑖subscript𝐪𝑘superscriptsubscriptnormsuperscriptsubscript𝐪0𝑖22superscriptsubscriptnormsubscript𝐪𝑘22subscript𝑑minfor-all𝑖for-all𝑘12\mathrm{\widetilde{C4}}:2\mathbf{q}_{0}^{T}(\mathbf{q}_{0}^{(i)}-\mathbf{q}_{k%})-||\mathbf{q}_{0}^{(i)}||_{2}^{2}+||\mathbf{q}_{k}||_{2}^{2}\geq d_{\rm min}%,{\forall i,}\forall k\in\{1,2\}.over~ start_ARG C4 end_ARG : 2 bold_q start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ( bold_q start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_i ) end_POSTSUPERSCRIPT - bold_q start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) - | | bold_q start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_i ) end_POSTSUPERSCRIPT | | start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + | | bold_q start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT | | start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≥ italic_d start_POSTSUBSCRIPT roman_min end_POSTSUBSCRIPT , ∀ italic_i , ∀ italic_k ∈ { 1 , 2 } .(19)

The resulting sub-problem is denoted by (P.0)~~P.0\widetilde{\mathrm{(P.0)}}over~ start_ARG ( roman_P .0 ) end_ARG and is given by:

(P.0)~:min𝐪0,𝐏com,0σh¯(𝒬):~P.0subscriptsubscript𝐪0subscript𝐏com0¯subscript𝜎𝒬\displaystyle\widetilde{\mathrm{(P.0)}}:\min_{\mathbf{q}_{0},\mathbf{P}_{\rmcom%,0}}\hskip 8.53581pt\overline{\sigma_{h}}(\mathcal{Q})over~ start_ARG ( roman_P .0 ) end_ARG : roman_min start_POSTSUBSCRIPT bold_q start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , bold_P start_POSTSUBSCRIPT roman_com , 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT over¯ start_ARG italic_σ start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT end_ARG ( caligraphic_Q )
s.t.C1C3,C4~,C5,C6,C8C11.C1C3~C4C5C6C8C11\displaystyle\mathrm{C1-C3,\widetilde{\mathrm{C4}},C5,C6,C8-C11}.C1 - C3 , over~ start_ARG C4 end_ARG , C5 , C6 , C8 - C11 .

Problem (P.0)~~P.0\widetilde{\mathrm{(P.0)}}over~ start_ARG ( roman_P .0 ) end_ARG is convex and can be solved using the Python CVXPY library [16]. The solution procedure to solve (P.0)P.0\mathrm{(P.0)}( roman_P .0 ) is summarized in Algorithm 1, which converges to a local optimum of the upper bound on problem (P.0)P.0\mathrm{(P.0)}( roman_P .0 ), σh¯¯subscript𝜎\overline{\sigma_{h}}over¯ start_ARG italic_σ start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT end_ARG, in polynomial computational timecomplexity [17]. Algorithm 1 involves N+2𝑁2N+2italic_N + 2 optimization variables, resulting in a computational complexity of 𝒪(M0(N+2)3.5)𝒪subscript𝑀0superscript𝑁23.5\mathcal{O}(M_{0}(N+2)^{3.5})caligraphic_O ( italic_M start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_N + 2 ) start_POSTSUPERSCRIPT 3.5 end_POSTSUPERSCRIPT ), where M0subscript𝑀0M_{0}italic_M start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT is the number of iterations needed for convergence [17].

1:For fixed {𝐪1,𝐪2,𝐏com,1,𝐏com,2}subscript𝐪1subscript𝐪2subscript𝐏com1subscript𝐏com2\{\mathbf{q}_{1},\mathbf{q}_{2},\mathbf{P}_{\rm com,1},\mathbf{P}_{\rm com,2}\}{ bold_q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , bold_q start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , bold_P start_POSTSUBSCRIPT roman_com , 1 end_POSTSUBSCRIPT , bold_P start_POSTSUBSCRIPT roman_com , 2 end_POSTSUBSCRIPT }, set initial point 𝐪0(1)superscriptsubscript𝐪01\mathbf{q}_{0}^{(1)}bold_q start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT, iteration index i=1𝑖1i=1italic_i = 1, and error tolerance 0<ϵ010subscriptitalic-ϵ0much-less-than10<{\epsilon_{0}}\ll 10 < italic_ϵ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ≪ 1.

2:repeat

3:Determine sensing worst-case accuracy σh¯(𝐪0,𝐪1,𝐪2),𝐪0,¯subscript𝜎subscript𝐪0subscript𝐪1subscript𝐪2subscript𝐪0\overline{\sigma_{h}}(\mathbf{q}_{0},\mathbf{q}_{1},\mathbf{q}_{2}),\mathbf{q}%_{0},over¯ start_ARG italic_σ start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT end_ARG ( bold_q start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , bold_q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , bold_q start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) , bold_q start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , and 𝐏com,0subscript𝐏com0\mathbf{P}_{\mathrm{com,0}}bold_P start_POSTSUBSCRIPT roman_com , 0 end_POSTSUBSCRIPT by solving (P.0)~~P.0\mathrm{\widetilde{(P.0)}}over~ start_ARG ( roman_P .0 ) end_ARG around point 𝐪0(i)superscriptsubscript𝐪0𝑖\mathbf{q}_{0}^{(i)}bold_q start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_i ) end_POSTSUPERSCRIPT using

4:CVXPY

5:Set i=i+1𝑖𝑖1i=i+1italic_i = italic_i + 1 and 𝐪0(i)=𝐪0superscriptsubscript𝐪0𝑖subscript𝐪0\mathbf{q}_{0}^{(i)}=\mathbf{q}_{0}bold_q start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_i ) end_POSTSUPERSCRIPT = bold_q start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT

6:until |σh¯(𝐪0(i),𝐪1,𝐪2)σh¯(𝐪0(i1),𝐪1,𝐪2)σh¯(𝐪0(i),𝐪1,𝐪2)|<ϵ0¯subscript𝜎superscriptsubscript𝐪0𝑖subscript𝐪1subscript𝐪2¯subscript𝜎superscriptsubscript𝐪0𝑖1subscript𝐪1subscript𝐪2¯subscript𝜎superscriptsubscript𝐪0𝑖subscript𝐪1subscript𝐪2subscriptitalic-ϵ0\big{|}\frac{\overline{\sigma_{h}}(\mathbf{q}_{0}^{(i)},\mathbf{q}_{1},\mathbf%{q}_{2})-\overline{\sigma_{h}}(\mathbf{q}_{0}^{(i-1)},\mathbf{q}_{1},\mathbf{q%}_{2})}{\overline{\sigma_{h}}(\mathbf{q}_{0}^{(i)},\mathbf{q}_{1},\mathbf{q}_{%2})}\big{|}<{\epsilon_{0}}| divide start_ARG over¯ start_ARG italic_σ start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT end_ARG ( bold_q start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_i ) end_POSTSUPERSCRIPT , bold_q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , bold_q start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) - over¯ start_ARG italic_σ start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT end_ARG ( bold_q start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_i - 1 ) end_POSTSUPERSCRIPT , bold_q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , bold_q start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) end_ARG start_ARG over¯ start_ARG italic_σ start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT end_ARG ( bold_q start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_i ) end_POSTSUPERSCRIPT , bold_q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , bold_q start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) end_ARG | < italic_ϵ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT

7:return solution {𝐪0,𝐏com,0subscript𝐪0subscript𝐏com0\mathbf{q}_{0},\mathbf{P}_{\mathrm{com,0}}bold_q start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , bold_P start_POSTSUBSCRIPT roman_com , 0 end_POSTSUBSCRIPT}

IV-B Slave \acuav Optimization

Next, we optimize the position and communication transmit power of slave \acuav U1subscript𝑈1U_{1}italic_U start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT, denoted by 𝐪1subscript𝐪1\mathbf{q}_{1}bold_q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and 𝐏com,1subscript𝐏com1\mathbf{P}_{\rm com,1}bold_P start_POSTSUBSCRIPT roman_com , 1 end_POSTSUBSCRIPT, respectively, for fixed {𝐪0,𝐪2,𝐏com,0,𝐏com,2}subscript𝐪0subscript𝐪2subscript𝐏com0subscript𝐏com2\{\mathbf{q}_{0},\mathbf{q}_{2},\mathbf{P}_{\rm com,0},\mathbf{P}_{\rm com,2}\}{ bold_q start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , bold_q start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , bold_P start_POSTSUBSCRIPT roman_com , 0 end_POSTSUBSCRIPT , bold_P start_POSTSUBSCRIPT roman_com , 2 end_POSTSUBSCRIPT }. The resulting problem, denoted by (P.1)P.1\mathrm{(P.1)}( roman_P .1 ),is non-convex due to the objective function and constraints C4,C5,C7,C4C5C7\mathrm{C4},\mathrm{C5},\mathrm{C7},C4 , C5 , C7 , and C8C8\mathrm{C8}C8. To tackle this sub-problem, we employ again \acsca. First, we adopt the upper bound σh¯¯subscript𝜎\overline{\sigma_{h}}over¯ start_ARG italic_σ start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT end_ARG provided by Proposition 1. Furthermore, it can be shown that minimizing σh¯¯subscript𝜎\overline{\sigma_{h}}over¯ start_ARG italic_σ start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT end_ARG for fixed 𝐪0subscript𝐪0\mathbf{q}_{0}bold_q start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT and 𝐪2subscript𝐪2\mathbf{q}_{2}bold_q start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT is equivalent to maximizing the perpendicular baseline b,1(𝐪1)subscript𝑏bottom1subscript𝐪1b_{\bot,1}(\mathbf{q}_{1})italic_b start_POSTSUBSCRIPT ⊥ , 1 end_POSTSUBSCRIPT ( bold_q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ). Moreover, in each \acsca iteration j𝑗jitalic_j, non-convex constraint C4C4\mathrm{C4}C4 is replaced with convex constraint C4~~~~C4\widetilde{\widetilde{\mathrm{C4}}}over~ start_ARG over~ start_ARG C4 end_ARG end_ARG based on a surrogate function similar to (19). Constraint C5C5\mathrm{C5}C5 is convexified based on a Taylor approximation around point 𝐪1(j)superscriptsubscript𝐪1𝑗\mathbf{q}_{1}^{(j)}bold_q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_j ) end_POSTSUPERSCRIPT as follows:

C5~:r12(𝐪1(j))+(r12𝐪1(𝐪1(j)))T(𝐪1𝐪1(j))Sminz1Θ3dB,j.:~C5superscriptsubscript𝑟12superscriptsubscript𝐪1𝑗superscriptsuperscriptsubscript𝑟12subscript𝐪1superscriptsubscript𝐪1𝑗𝑇subscript𝐪1superscriptsubscript𝐪1𝑗subscript𝑆minsubscript𝑧1subscriptΘ3dBfor-all𝑗\mathrm{\widetilde{C5}}:r_{1}^{2}(\mathbf{q}_{1}^{(j)})+\left(\frac{\partial r%_{1}^{2}}{\partial\mathbf{q}_{1}}(\mathbf{q}_{1}^{(j)})\right)^{T}(\mathbf{q}_%{1}-\mathbf{q}_{1}^{(j)})\geq\frac{S_{\rm min}z_{1}}{\Theta_{\rm 3dB}},{%\forall j}.over~ start_ARG C5 end_ARG : italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( bold_q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_j ) end_POSTSUPERSCRIPT ) + ( divide start_ARG ∂ italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG ∂ bold_q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG ( bold_q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_j ) end_POSTSUPERSCRIPT ) ) start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ( bold_q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - bold_q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_j ) end_POSTSUPERSCRIPT ) ≥ divide start_ARG italic_S start_POSTSUBSCRIPT roman_min end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG start_ARG roman_Θ start_POSTSUBSCRIPT 3 roman_d roman_B end_POSTSUBSCRIPT end_ARG , ∀ italic_j .(20)

The resulting problem is given by:

(P.1)~:max𝐪1,𝐏com,1b,1(𝐪1):~P.1subscriptsubscript𝐪1subscript𝐏com1subscript𝑏bottom1subscript𝐪1\displaystyle\mathrm{\widetilde{(P.1)}}:\max_{\mathbf{q}_{1},\mathbf{P}_{\rmcom%,1}}\hskip 8.53581ptb_{\bot,1}(\mathbf{q}_{1})over~ start_ARG ( roman_P .1 ) end_ARG : roman_max start_POSTSUBSCRIPT bold_q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , bold_P start_POSTSUBSCRIPT roman_com , 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT ⊥ , 1 end_POSTSUBSCRIPT ( bold_q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT )
s.t.C1,C3,C4~~,C5~,C6C11.C1C3~~C4~C5C6C11\displaystyle\mathrm{C1,C3,\widetilde{\widetilde{C4}},\widetilde{C5},{C6}-C11}.C1 , C3 , over~ start_ARG over~ start_ARG C4 end_ARG end_ARG , over~ start_ARG C5 end_ARG , C6 - C11 .

Yet, the expressions for the perpendicular baseline in (17) and for the baseline decorrelation still present obstacles for solving (P.1)~~P.1\mathrm{\widetilde{(P.1)}}over~ start_ARG ( roman_P .1 ) end_ARG. Thus, we divide the search space of problem (P.1)~~P.1\mathrm{\widetilde{(P.1)}}over~ start_ARG ( roman_P .1 ) end_ARG into two disjoint sets, denoted by ak={𝐪k;θ0θk(𝐪k)}superscriptsubscript𝑎𝑘subscript𝐪𝑘subscript𝜃0subscript𝜃𝑘subscript𝐪𝑘\mathcal{I}_{a}^{k}=\{\mathbf{q}_{k};\theta_{0}\geq\theta_{k}(\mathbf{q}_{k})\}caligraphic_I start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT = { bold_q start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ; italic_θ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ≥ italic_θ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( bold_q start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) } and bk={𝐪k;θ0<θk(𝐪k)},k{1,2}formulae-sequencesubscriptsuperscript𝑘𝑏subscript𝐪𝑘subscript𝜃0subscript𝜃𝑘subscript𝐪𝑘𝑘12\mathcal{I}^{k}_{b}=\{\mathbf{q}_{k};\theta_{0}<\theta_{k}(\mathbf{q}_{k})\},k%\in\{1,2\}caligraphic_I start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT = { bold_q start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ; italic_θ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT < italic_θ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( bold_q start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) } , italic_k ∈ { 1 , 2 }. The solution that maximizes σh¯¯subscript𝜎\overline{\sigma_{h}}over¯ start_ARG italic_σ start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT end_ARG over a1subscriptsuperscript1𝑎\mathcal{I}^{1}_{a}caligraphic_I start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT and b1subscriptsuperscript1𝑏\mathcal{I}^{1}_{b}caligraphic_I start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT is selected, see Figure 2.

Proposition 2.

Constraint C7C7\mathrm{C7}C7 is equivalent to the following convex constraints:

{C7a:zkαa(xtxk)0,if𝐪kakC7b:(xtxk)zkαb0,if𝐪kbk,k{1,2},cases:C7aformulae-sequencesubscript𝑧𝑘subscript𝛼𝑎subscript𝑥𝑡subscript𝑥𝑘0ifsubscript𝐪𝑘superscriptsubscript𝑎𝑘otherwise:C7bformulae-sequencesubscript𝑥𝑡subscript𝑥𝑘subscript𝑧𝑘subscript𝛼𝑏0ifsubscript𝐪𝑘superscriptsubscript𝑏𝑘otherwisefor-all𝑘12\displaystyle\quad\begin{cases}\mathrm{C7a}:z_{k}\alpha_{a}-(x_{t}-x_{k})\leq 0%,\text{ if }\mathbf{q}_{k}\in\mathcal{I}_{a}^{k}\\\mathrm{C7b}:(x_{t}-x_{k})-z_{k}\alpha_{b}\leq 0,\text{ if }\mathbf{q}_{k}\in%\mathcal{I}_{b}^{k}\end{cases},\forall k\in\{1,2\},{ start_ROW start_CELL C7a : italic_z start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_α start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT - ( italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT - italic_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) ≤ 0 , if bold_q start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∈ caligraphic_I start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT end_CELL start_CELL end_CELL end_ROW start_ROW start_CELL C7b : ( italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT - italic_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) - italic_z start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_α start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ≤ 0 , if bold_q start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∈ caligraphic_I start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT end_CELL start_CELL end_CELL end_ROW , ∀ italic_k ∈ { 1 , 2 } ,(21)

where αa=tan(arcsin(2h(γRgmin)h(γRgmin)sin(θ0)))subscript𝛼𝑎2superscriptsubscript𝛾Rgminsuperscriptsubscript𝛾Rgminsubscript𝜃0\alpha_{a}=\tan\left(\arcsin\left(\frac{2-h(\gamma_{\rm Rg}^{\rm min})}{h(%\gamma_{\rm Rg}^{\rm min})}\sin(\theta_{0})\right)\right)italic_α start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT = roman_tan ( roman_arcsin ( divide start_ARG 2 - italic_h ( italic_γ start_POSTSUBSCRIPT roman_Rg end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_min end_POSTSUPERSCRIPT ) end_ARG start_ARG italic_h ( italic_γ start_POSTSUBSCRIPT roman_Rg end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_min end_POSTSUPERSCRIPT ) end_ARG roman_sin ( italic_θ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) ) ), αb=tan(arcsin(h(γRgmin)2h(γRgmin)sin(θ0)))subscript𝛼𝑏superscriptsubscript𝛾Rgmin2superscriptsubscript𝛾Rgminsubscript𝜃0\alpha_{b}=\tan\left(\arcsin\left(\frac{h(\gamma_{\rm Rg}^{\rm min})}{2-h(%\gamma_{\rm Rg}^{\rm min})}\sin(\theta_{0})\right)\right)italic_α start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT = roman_tan ( roman_arcsin ( divide start_ARG italic_h ( italic_γ start_POSTSUBSCRIPT roman_Rg end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_min end_POSTSUPERSCRIPT ) end_ARG start_ARG 2 - italic_h ( italic_γ start_POSTSUBSCRIPT roman_Rg end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_min end_POSTSUPERSCRIPT ) end_ARG roman_sin ( italic_θ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) ) ), and function h(x)=xBp2BpBp2xBp𝑥𝑥subscript𝐵𝑝2subscript𝐵𝑝subscript𝐵𝑝2𝑥subscript𝐵𝑝h(x)=\frac{xB_{p}-2-B_{p}}{B_{p}-2-xB_{p}}italic_h ( italic_x ) = divide start_ARG italic_x italic_B start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT - 2 - italic_B start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT end_ARG start_ARG italic_B start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT - 2 - italic_x italic_B start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT end_ARG.

Proof.

Please refer to Appendix C.∎

Constraint C7C7\mathrm{C7}C7 is transformed based on Proposition 2. Then, for 𝐪1a1subscript𝐪1subscriptsuperscript1𝑎\mathbf{q}_{1}\in\mathcal{I}^{1}_{a}bold_q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∈ caligraphic_I start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT, constraint C7C7\mathrm{C7}C7 is replaced by C7aC7a\mathrm{C7a}C7a and the resulting problem is denoted by (P.1.a)~~formulae-sequenceP.1a\mathrm{\widetilde{(P.1.a)}}over~ start_ARG ( roman_P .1 . roman_a ) end_ARG. Simlarly, sub-problem (P.1.b)~~formulae-sequenceP.1b\mathrm{\widetilde{(P.1.b)}}over~ start_ARG ( roman_P .1 . roman_b ) end_ARG denotes sub-problem (P.1)~~P.1\mathrm{\widetilde{(P.1)}}over~ start_ARG ( roman_P .1 ) end_ARG for 𝐪1b1subscript𝐪1subscriptsuperscript1𝑏\mathbf{q}_{1}\in\mathcal{I}^{1}_{b}bold_q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∈ caligraphic_I start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT, where constraint C7bC7b\mathrm{C7b}C7b replaces C7C7\mathrm{C7}C7.

The proposed \acsca algorithm to solve (P.1)P.1\mathrm{(P.1)}( roman_P .1 ) is omitted due to space limitation, but is similar to Algorithm 1, where the convex approximations (P.1.a~)~formulae-sequenceP.1a\mathrm{(\widetilde{P.1.a})}( over~ start_ARG roman_P .1 . roman_a end_ARG ) and (P.1.b~)~formulae-sequenceP.1b\mathrm{(\widetilde{P.1.b})}( over~ start_ARG roman_P .1 . roman_b end_ARG ) are solved in parallel using CVXPY [16], with precision ϵ1=ϵ0subscriptitalic-ϵ1subscriptitalic-ϵ0\epsilon_{1}=\epsilon_{0}italic_ϵ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_ϵ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT. The algorithm converges to a local optimum of the upper bound on sub-problem (P.1)P.1\mathrm{(P.1)}( roman_P .1 ) entailing computational complexity 𝒪(2M1(N+2)3.5)𝒪2subscript𝑀1superscript𝑁23.5\mathcal{O}(2M_{1}(N+2)^{3.5})caligraphic_O ( 2 italic_M start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_N + 2 ) start_POSTSUPERSCRIPT 3.5 end_POSTSUPERSCRIPT ), where M1subscript𝑀1M_{1}italic_M start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT is the required number of iterations [17].

1:Set initial formation 𝒬(1)={𝐪0(1),𝐪1(1),𝐪2(1)}superscript𝒬1superscriptsubscript𝐪01superscriptsubscript𝐪11superscriptsubscript𝐪21\mathcal{Q}^{(1)}=\{\mathbf{q}_{0}^{(1)},\mathbf{q}_{1}^{(1)},\mathbf{q}_{2}^{%(1)}\}caligraphic_Q start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT = { bold_q start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT , bold_q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT , bold_q start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT }, initial communication transmit powers 𝒫(1)={𝐏com,0(1),𝐏com,1(1),𝐏com,2(1)}superscript𝒫1subscriptsuperscript𝐏1com0subscriptsuperscript𝐏1com1subscriptsuperscript𝐏1com2\mathcal{P}^{(1)}=\{\mathbf{P}^{(1)}_{\mathrm{com,0}},\mathbf{P}^{(1)}_{%\mathrm{com,1}},\mathbf{P}^{(1)}_{\mathrm{com,2}}\}caligraphic_P start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT = { bold_P start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_com , 0 end_POSTSUBSCRIPT , bold_P start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_com , 1 end_POSTSUBSCRIPT , bold_P start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_com , 2 end_POSTSUBSCRIPT }, iteration index m=1𝑚1m=1italic_m = 1, and error tolerance 0<ϵ210subscriptitalic-ϵ2much-less-than10<{\epsilon_{2}}\ll 10 < italic_ϵ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ≪ 1.

2:repeat

3: Set m=m+1𝑚𝑚1m=m+1italic_m = italic_m + 1

4: Determine σh¯(𝐪0,𝐪1(m1),𝐪2(m1))¯subscript𝜎subscript𝐪0superscriptsubscript𝐪1𝑚1superscriptsubscript𝐪2𝑚1\overline{\sigma_{h}}(\mathbf{q}_{0},\mathbf{q}_{1}^{(m-1)},\mathbf{q}_{2}^{(m%-1)})over¯ start_ARG italic_σ start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT end_ARG ( bold_q start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , bold_q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_m - 1 ) end_POSTSUPERSCRIPT , bold_q start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_m - 1 ) end_POSTSUPERSCRIPT ) and set 𝐪0(m)=𝐪0superscriptsubscript𝐪0𝑚subscript𝐪0\mathbf{q}_{0}^{(m)}=\mathbf{q}_{0}bold_q start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_m ) end_POSTSUPERSCRIPT = bold_q start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT and 𝐏com,0(m)=𝐏com,0subscriptsuperscript𝐏𝑚com0subscript𝐏com0\mathbf{P}^{(m)}_{\mathrm{com,0}}=\mathbf{P}_{\mathrm{com,0}}bold_P start_POSTSUPERSCRIPT ( italic_m ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_com , 0 end_POSTSUBSCRIPT = bold_P start_POSTSUBSCRIPT roman_com , 0 end_POSTSUBSCRIPT after solving (P.0)P.0\rm(P.0)( roman_P .0 ) for

5: fixed {𝐪1(m1),𝐪2(m1),𝐏com,1(m1),𝐏com,2(m1)}superscriptsubscript𝐪1𝑚1superscriptsubscript𝐪2𝑚1subscriptsuperscript𝐏𝑚1com1subscriptsuperscript𝐏𝑚1com2\{\mathbf{q}_{1}^{(m-1)},\mathbf{q}_{2}^{(m-1)},\mathbf{P}^{(m-1)}_{\mathrm{%com,1}},\mathbf{P}^{(m-1)}_{\mathrm{com,2}}\}{ bold_q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_m - 1 ) end_POSTSUPERSCRIPT , bold_q start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_m - 1 ) end_POSTSUPERSCRIPT , bold_P start_POSTSUPERSCRIPT ( italic_m - 1 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_com , 1 end_POSTSUBSCRIPT , bold_P start_POSTSUPERSCRIPT ( italic_m - 1 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_com , 2 end_POSTSUBSCRIPT } using Algorithm 1

6: Determine σh¯(𝐪0(m1),𝐪1,𝐪2(m1))¯subscript𝜎superscriptsubscript𝐪0𝑚1subscript𝐪1superscriptsubscript𝐪2𝑚1\overline{\sigma_{h}}(\mathbf{q}_{0}^{(m-1)},\mathbf{q}_{1},\mathbf{q}_{2}^{(m%-1)})over¯ start_ARG italic_σ start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT end_ARG ( bold_q start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_m - 1 ) end_POSTSUPERSCRIPT , bold_q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , bold_q start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_m - 1 ) end_POSTSUPERSCRIPT ) and set 𝐪1(m)=𝐪1superscriptsubscript𝐪1𝑚subscript𝐪1\mathbf{q}_{1}^{(m)}=\mathbf{q}_{1}bold_q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_m ) end_POSTSUPERSCRIPT = bold_q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and 𝐏com,1(m)=𝐏com,1subscriptsuperscript𝐏𝑚com1subscript𝐏com1\mathbf{P}^{(m)}_{\mathrm{com,1}}=\mathbf{P}_{\mathrm{com,1}}bold_P start_POSTSUPERSCRIPT ( italic_m ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_com , 1 end_POSTSUBSCRIPT = bold_P start_POSTSUBSCRIPT roman_com , 1 end_POSTSUBSCRIPT after solving (P.1)P.1\rm(P.1)( roman_P .1 )

7: for fixed {𝐪0(m1),𝐪2(m1),𝐏com,0(m1),𝐏com,2(m1)}superscriptsubscript𝐪0𝑚1superscriptsubscript𝐪2𝑚1subscriptsuperscript𝐏𝑚1com0subscriptsuperscript𝐏𝑚1com2\{\mathbf{q}_{0}^{(m-1)},\mathbf{q}_{2}^{(m-1)},\mathbf{P}^{(m-1)}_{\mathrm{%com,0}},\mathbf{P}^{(m-1)}_{\mathrm{com,2}}\}{ bold_q start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_m - 1 ) end_POSTSUPERSCRIPT , bold_q start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_m - 1 ) end_POSTSUPERSCRIPT , bold_P start_POSTSUPERSCRIPT ( italic_m - 1 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_com , 0 end_POSTSUBSCRIPT , bold_P start_POSTSUPERSCRIPT ( italic_m - 1 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_com , 2 end_POSTSUBSCRIPT } using \acsca

8: Determine σh¯(𝐪0(m1),𝐪1(m1),𝐪2)¯subscript𝜎superscriptsubscript𝐪0𝑚1superscriptsubscript𝐪1𝑚1subscript𝐪2\overline{\sigma_{h}}(\mathbf{q}_{0}^{(m-1)},\mathbf{q}_{1}^{(m-1)},\mathbf{q}%_{2})over¯ start_ARG italic_σ start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT end_ARG ( bold_q start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_m - 1 ) end_POSTSUPERSCRIPT , bold_q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_m - 1 ) end_POSTSUPERSCRIPT , bold_q start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) and set 𝐪2(m)=𝐪2superscriptsubscript𝐪2𝑚subscript𝐪2\mathbf{q}_{2}^{(m)}=\mathbf{q}_{2}bold_q start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_m ) end_POSTSUPERSCRIPT = bold_q start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT and 𝐏com,2(m)=𝐏com,2subscriptsuperscript𝐏𝑚com2subscript𝐏com2\mathbf{P}^{(m)}_{\mathrm{com,2}}=\mathbf{P}_{\mathrm{com,2}}bold_P start_POSTSUPERSCRIPT ( italic_m ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_com , 2 end_POSTSUBSCRIPT = bold_P start_POSTSUBSCRIPT roman_com , 2 end_POSTSUBSCRIPT after solving (P.2)P.2\rm(P.2)( roman_P .2 )

9: for fixed {𝐪0(m1),𝐪1(m1),𝐏com,0(m1),𝐏com,1(m1)}superscriptsubscript𝐪0𝑚1superscriptsubscript𝐪1𝑚1subscriptsuperscript𝐏𝑚1com0subscriptsuperscript𝐏𝑚1com1\{\mathbf{q}_{0}^{(m-1)},\mathbf{q}_{1}^{(m-1)},\mathbf{P}^{(m-1)}_{\mathrm{%com,0}},\mathbf{P}^{(m-1)}_{\mathrm{com,1}}\}{ bold_q start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_m - 1 ) end_POSTSUPERSCRIPT , bold_q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_m - 1 ) end_POSTSUPERSCRIPT , bold_P start_POSTSUPERSCRIPT ( italic_m - 1 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_com , 0 end_POSTSUBSCRIPT , bold_P start_POSTSUPERSCRIPT ( italic_m - 1 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_com , 1 end_POSTSUBSCRIPT } using \acsca

10:until |σh¯(𝐪0(m),𝐪1(m),𝐪2(m))σh¯(𝐪0(m1),𝐪1(m1),𝐪2(m1))σh¯(𝐪0(m),𝐪1(m),𝐪2(m))|ϵ2¯subscript𝜎superscriptsubscript𝐪0𝑚superscriptsubscript𝐪1𝑚superscriptsubscript𝐪2𝑚¯subscript𝜎superscriptsubscript𝐪0𝑚1superscriptsubscript𝐪1𝑚1superscriptsubscript𝐪2𝑚1¯subscript𝜎superscriptsubscript𝐪0𝑚superscriptsubscript𝐪1𝑚superscriptsubscript𝐪2𝑚subscriptitalic-ϵ2\big{|}\frac{\overline{\sigma_{h}}(\mathbf{q}_{0}^{(m)},\mathbf{q}_{1}^{(m)},%\mathbf{q}_{2}^{(m)})-\overline{\sigma_{h}}(\mathbf{q}_{0}^{(m-1)},\mathbf{q}_%{1}^{(m-1)},\mathbf{q}_{2}^{(m-1)})}{\overline{\sigma_{h}}(\mathbf{q}_{0}^{(m)%},\mathbf{q}_{1}^{(m)},\mathbf{q}_{2}^{(m)})}\big{|}\leq{\epsilon_{2}}| divide start_ARG over¯ start_ARG italic_σ start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT end_ARG ( bold_q start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_m ) end_POSTSUPERSCRIPT , bold_q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_m ) end_POSTSUPERSCRIPT , bold_q start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_m ) end_POSTSUPERSCRIPT ) - over¯ start_ARG italic_σ start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT end_ARG ( bold_q start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_m - 1 ) end_POSTSUPERSCRIPT , bold_q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_m - 1 ) end_POSTSUPERSCRIPT , bold_q start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_m - 1 ) end_POSTSUPERSCRIPT ) end_ARG start_ARG over¯ start_ARG italic_σ start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT end_ARG ( bold_q start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_m ) end_POSTSUPERSCRIPT , bold_q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_m ) end_POSTSUPERSCRIPT , bold_q start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_m ) end_POSTSUPERSCRIPT ) end_ARG | ≤ italic_ϵ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT

11:return solution {𝒬,𝒫}={𝐪0(m),𝐪1(m),𝐪2(m),𝐏com,0(m),𝐏com,1(m),𝐏com,2(m)}𝒬𝒫superscriptsubscript𝐪0𝑚superscriptsubscript𝐪1𝑚superscriptsubscript𝐪2𝑚subscriptsuperscript𝐏𝑚com0subscriptsuperscript𝐏𝑚com1subscriptsuperscript𝐏𝑚com2\{\mathcal{Q},\mathcal{P}\}=\{\mathbf{q}_{0}^{(m)},\mathbf{q}_{1}^{(m)},%\mathbf{q}_{2}^{(m)},\mathbf{P}^{(m)}_{\mathrm{com,0}},\mathbf{P}^{(m)}_{%\mathrm{com,1}},\mathbf{P}^{(m)}_{\mathrm{com},2}\}{ caligraphic_Q , caligraphic_P } = { bold_q start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_m ) end_POSTSUPERSCRIPT , bold_q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_m ) end_POSTSUPERSCRIPT , bold_q start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_m ) end_POSTSUPERSCRIPT , bold_P start_POSTSUPERSCRIPT ( italic_m ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_com , 0 end_POSTSUBSCRIPT , bold_P start_POSTSUPERSCRIPT ( italic_m ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_com , 1 end_POSTSUBSCRIPT , bold_P start_POSTSUPERSCRIPT ( italic_m ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_com , 2 end_POSTSUBSCRIPT }

IV-C Solution to Problem (P)P\mathrm{(P)}( roman_P )

To solve problem (P)P\mathrm{(P)}( roman_P ), we use \acao by solving sub-problems (P.0)P.0\mathrm{(P.0)}( roman_P .0 ), (P.1)P.1\mathrm{(P.1)}( roman_P .1 ), and (P.2)P.2\mathrm{(P.2)}( roman_P .2 ) iteratively, see Figure 2. In Algorithm 2, we summarize all steps used to solve problem (P)P\mathrm{(P)}( roman_P ). Algorithm 2 converges to a local optimum of the worst-case height error, σh¯¯subscript𝜎\overline{\sigma_{h}}over¯ start_ARG italic_σ start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT end_ARG, with time complexity 𝒪(M2(2M1+M0)(N+2)3.5)𝒪subscript𝑀22subscript𝑀1subscript𝑀0superscript𝑁23.5\mathcal{O}(M_{2}(2M_{1}+M_{0})(N+2)^{3.5})caligraphic_O ( italic_M start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( 2 italic_M start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_M start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) ( italic_N + 2 ) start_POSTSUPERSCRIPT 3.5 end_POSTSUPERSCRIPT ), where M2subscript𝑀2M_{2}italic_M start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT is the required number of iterations [17].

V Simulation Results

ParameterValue ParameterValue ParameterValueN𝑁Nitalic_N80808080 Pcommaxsuperscriptsubscript𝑃commaxP_{\mathrm{com}}^{\mathrm{max}}italic_P start_POSTSUBSCRIPT roman_com end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_max end_POSTSUPERSCRIPT9 dB λ𝜆\lambdaitalic_λ0.12 mδtsubscript𝛿𝑡\delta_{t}italic_δ start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT1 s Ecommaxsuperscriptsubscript𝐸commaxE_{\mathrm{com}}^{\rm max}italic_E start_POSTSUBSCRIPT roman_com end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_max end_POSTSUPERSCRIPT600 J τp×PRFsubscript𝜏𝑝PRF\tau_{p}\times\mathrm{PRF}italic_τ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT × roman_PRF10-4zminsubscript𝑧minz_{\mathrm{min}}italic_z start_POSTSUBSCRIPT roman_min end_POSTSUBSCRIPT1 m Rmin,0subscript𝑅min0R_{\mathrm{min},0}italic_R start_POSTSUBSCRIPT roman_min , 0 end_POSTSUBSCRIPT10 Mbit/s θdsubscript𝜃𝑑\theta_{d}italic_θ start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT45°zmaxsubscript𝑧maxz_{\mathrm{max}}italic_z start_POSTSUBSCRIPT roman_max end_POSTSUBSCRIPT100 m Rmin,1subscript𝑅min1R_{\mathrm{min},1}italic_R start_POSTSUBSCRIPT roman_min , 1 end_POSTSUBSCRIPT17 Mbit/s Tsyssubscript𝑇sysT_{\mathrm{sys}}italic_T start_POSTSUBSCRIPT roman_sys end_POSTSUBSCRIPT400 Khambminsuperscriptsubscriptambminh_{\mathrm{amb}}^{\mathrm{min}}italic_h start_POSTSUBSCRIPT roman_amb end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_min end_POSTSUPERSCRIPT1.2 m Rmin,2subscript𝑅min2R_{\mathrm{min},2}italic_R start_POSTSUBSCRIPT roman_min , 2 end_POSTSUBSCRIPT1 Mbit/s BRgsubscript𝐵RgB_{\mathrm{Rg}}italic_B start_POSTSUBSCRIPT roman_Rg end_POSTSUBSCRIPT3 GHzhambmaxsuperscriptsubscriptambmaxh_{\mathrm{amb}}^{\mathrm{max}}italic_h start_POSTSUBSCRIPT roman_amb end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_max end_POSTSUPERSCRIPT2.2 m Bc,k,ksubscript𝐵𝑐𝑘for-all𝑘B_{c,k},\forall kitalic_B start_POSTSUBSCRIPT italic_c , italic_k end_POSTSUBSCRIPT , ∀ italic_k1 GHz F𝐹Fitalic_F5 dBxtsubscript𝑥𝑡x_{t}italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT20 m γ𝛾\gammaitalic_γ18.75 dB L𝐿Litalic_L6 dBgxsubscript𝑔𝑥g_{x}italic_g start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT70 m γRgminsuperscriptsubscript𝛾Rgmin\gamma_{\mathrm{Rg}}^{\mathrm{min}}italic_γ start_POSTSUBSCRIPT roman_Rg end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_min end_POSTSUPERSCRIPT0.8 f0subscript𝑓0f_{0}italic_f start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT2.5 GHzgysubscript𝑔𝑦g_{y}italic_g start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT150 m γSNRminsuperscriptsubscript𝛾SNRmin\gamma_{\mathrm{SNR}}^{\mathrm{min}}italic_γ start_POSTSUBSCRIPT roman_SNR end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_min end_POSTSUPERSCRIPT0.80.80.80.8 ϵ0=ϵ1=ϵ2subscriptitalic-ϵ0subscriptitalic-ϵ1subscriptitalic-ϵ2\epsilon_{0}=\epsilon_{1}=\epsilon_{2}italic_ϵ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = italic_ϵ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_ϵ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT10-2gzsubscript𝑔𝑧g_{z}italic_g start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT25 m γothersubscript𝛾other\gamma_{\rm other}italic_γ start_POSTSUBSCRIPT roman_other end_POSTSUBSCRIPT0.6 θ3dBsubscript𝜃3dB\theta_{3\mathrm{dB}}italic_θ start_POSTSUBSCRIPT 3 roman_d roman_B end_POSTSUBSCRIPT30°dminsubscript𝑑mind_{\mathrm{min}}italic_d start_POSTSUBSCRIPT roman_min end_POSTSUBSCRIPT2 m σ0subscript𝜎0\sigma_{0}italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT-3 dB θminsubscript𝜃min\theta_{\rm min}italic_θ start_POSTSUBSCRIPT roman_min end_POSTSUBSCRIPT37.24°Sminsubscript𝑆minS_{\rm min}italic_S start_POSTSUBSCRIPT roman_min end_POSTSUBSCRIPT55 m Ptsubscript𝑃𝑡P_{t}italic_P start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT23232323 dBm θmaxsubscript𝜃max\theta_{\rm max}italic_θ start_POSTSUBSCRIPT roman_max end_POSTSUBSCRIPT48.7°vysubscript𝑣𝑦v_{y}italic_v start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT4.3 m/s Gr=Gtsubscript𝐺𝑟subscript𝐺𝑡G_{r}=G_{t}italic_G start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT = italic_G start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT5 dBi nLsubscript𝑛𝐿n_{L}italic_n start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT4

This section presents simulation results for Algorithm 2, using parameters from Table I, unless stated otherwise. To evaluate performance, we adopt the next benchmark schemes:

  • Benchmark scheme 1: Here, a single-baseline \acuav-\acinsar system consisting only of U0subscript𝑈0U_{0}italic_U start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT and U1subscript𝑈1U_{1}italic_U start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT is considered [11]. The upper bound on the height error σh1subscript𝜎subscript1\sigma_{h_{1}}italic_σ start_POSTSUBSCRIPT italic_h start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT is minimized based on a two-step \acao algorithm.

  • Benchmark scheme 2: In this scheme, we fix the position of the master \acuav at 𝐪0=𝐪0fixedsubscript𝐪0superscriptsubscript𝐪0fixed\mathbf{q}_{0}=\mathbf{q}_{0}^{\rm fixed}bold_q start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = bold_q start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_fixed end_POSTSUPERSCRIPT, which is feasible for (P)P\mathrm{(P)}( roman_P ), and optimize the remaining variables.

  • Benchmark scheme 3: Here, we apply a static and constant communication power allocation (i.e., Pcom,k[n]=EcommaxN,n,ksubscript𝑃com𝑘delimited-[]𝑛superscriptsubscript𝐸commax𝑁for-all𝑛for-all𝑘P_{\mathrm{com},k}[n]=\frac{E_{\rm com}^{\rm max}}{N},\forall n,\forall kitalic_P start_POSTSUBSCRIPT roman_com , italic_k end_POSTSUBSCRIPT [ italic_n ] = divide start_ARG italic_E start_POSTSUBSCRIPT roman_com end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_max end_POSTSUPERSCRIPT end_ARG start_ARG italic_N end_ARG , ∀ italic_n , ∀ italic_k), and optimize the remaining variables.

Sensing Accuracy Optimization for Communication-assisted Dual-baseline UAV-InSAR ††thanks: This work was supported in part by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) GRK 2680 – Project-ID 437847244. (3)

In Figure 3, we present the height error of the final \acdem σhsubscript𝜎\sigma_{h}italic_σ start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT and its upper bound σh¯¯subscript𝜎\overline{\sigma_{h}}over¯ start_ARG italic_σ start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT end_ARG versus the minimum \achoa, hambminsuperscriptsubscriptambminh_{\rm amb}^{\rm min}italic_h start_POSTSUBSCRIPT roman_amb end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_min end_POSTSUPERSCRIPT. The figure shows that the sensing accuracy degrades with stricter requirements on the minimum \achoa, which is due to the relation between the \achoa and the height error, see (13). We note that the tightness of the upper bound on the height error in (18) increases with stricter sensing requirements, i.e., if γSNRminγRgminγother1superscriptsubscript𝛾SNRminsuperscriptsubscript𝛾Rgminsubscript𝛾other1\gamma_{\rm SNR}^{\rm min}\gamma_{\rm Rg}^{\rm min}\gamma_{\rm other}\to 1italic_γ start_POSTSUBSCRIPT roman_SNR end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_min end_POSTSUPERSCRIPT italic_γ start_POSTSUBSCRIPT roman_Rg end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_min end_POSTSUPERSCRIPT italic_γ start_POSTSUBSCRIPT roman_other end_POSTSUBSCRIPT → 1, then σhσh¯subscript𝜎¯subscript𝜎\sigma_{h}\to\overline{\sigma_{h}}italic_σ start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT → over¯ start_ARG italic_σ start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT end_ARG. Nevertheless, Figure 3reveals that even for system parameters, for which the upper bound is not tight, it is a useful metric for optimization. In fact, the proposed scheme consistently achieves a gain of at least 39.3% compared to benchmark scheme 1. This gain is due to averaging the height error, which improves the sensing accuracy and highlights the importance of using multiple \acpuav for acquisition. Additionally, optimizing the \acuav formation and the communication power allows the proposed solution to outperform both benchmark schemes 2 and 3, with a gain that varies with the \achoa. For instance, for hambmin=1.2superscriptsubscriptambmin1.2h_{\rm amb}^{\rm min}=1.2italic_h start_POSTSUBSCRIPT roman_amb end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_min end_POSTSUPERSCRIPT = 1.2 m, respective gains of 20.5% and 12.5% are observed.

Sensing Accuracy Optimization for Communication-assisted Dual-baseline UAV-InSAR ††thanks: This work was supported in part by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) GRK 2680 – Project-ID 437847244. (4)

Figure 4 depicts the final height error and its upper bound versus the minimum data rate of U1subscript𝑈1U_{1}italic_U start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT. The figure shows that higher data rate requirements lead to worse accuracy. This can be explained by U1subscript𝑈1U_{1}italic_U start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT’s extended range when lower data rates are sufficient. In contrast, increasing the data rate requirement limits the U1subscript𝑈1U_{1}italic_U start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT-\acgs distance, which affects the perpendicular baseline and increases the height estimation error. Furthermore, Figure 4 highlights the need to properly allocate the communication power to ensure real-time data offloading to the \acgs. A static power allocation, as in benchmark scheme 3, negatively affects the height error and eventually leads to infeasibility of problem (P)P\mathrm{(P)}( roman_P ) starting from Rmin,1=17.1subscript𝑅min117.1R_{\rm min,1}=17.1italic_R start_POSTSUBSCRIPT roman_min , 1 end_POSTSUBSCRIPT = 17.1 Mbit/s, indicated by the red colored region in Figure 4. Figure 4 confirms that the proposed scheme outperforms all benchmark schemes.

VI Conclusion

In this work, we studied a dual-baseline \acuav-based \acinsar system using a swarm of three drones to generate two independent \acpdem of a target area. The final \acdem is obtained with a weighted averaging technique, improving sensing precision. We proposed a low-complexity algorithm that minimizes an upper bound on the height estimation error of the final \acdem by jointly optimizing the \acuav formation and communication power allocation, while meeting sensing and communication constraints. Simulation results showed that the proposed scheme significantly improves sensing accuracy compared to single-baseline systems and other benchmark schemes.

Appendix A Bi-static \acsnr Approximation

In this appendix, we provide the detailed steps for deriving the bi-static \acsnr approximation in (9). To this end, we start with the bi-static radar \acsnr expression for distributed targets, denoted by SNRrsubscriptSNR𝑟\mathrm{SNR}_{r}roman_SNR start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT and given by [13]:

SNRr=σ0AresPtGtGrλ2(4π)3RTx2RRx2kbBnoiseTsysFL,subscriptSNR𝑟subscript𝜎0subscript𝐴ressubscript𝑃𝑡subscript𝐺𝑡subscript𝐺𝑟superscript𝜆2superscript4𝜋3subscriptsuperscript𝑅2Txsubscriptsuperscript𝑅2Rxsubscript𝑘𝑏subscript𝐵noisesubscript𝑇sys𝐹𝐿\mathrm{SNR}_{r}=\frac{\sigma_{0}A_{\rm res}P_{t}G_{t}G_{r}\lambda^{2}}{(4\pi)%^{3}R^{2}_{\rm Tx}R^{2}_{\rm Rx}k_{b}B_{\rm noise}T_{\rm sys}FL},roman_SNR start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT = divide start_ARG italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT roman_res end_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT italic_G start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT italic_G start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT italic_λ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG ( 4 italic_π ) start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT italic_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_Tx end_POSTSUBSCRIPT italic_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_Rx end_POSTSUBSCRIPT italic_k start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT italic_B start_POSTSUBSCRIPT roman_noise end_POSTSUBSCRIPT italic_T start_POSTSUBSCRIPT roman_sys end_POSTSUBSCRIPT italic_F italic_L end_ARG ,(22)

where RTxsubscript𝑅TxR_{\rm Tx}italic_R start_POSTSUBSCRIPT roman_Tx end_POSTSUBSCRIPT and RRxsubscript𝑅RxR_{\rm Rx}italic_R start_POSTSUBSCRIPT roman_Rx end_POSTSUBSCRIPT represent the radar transmit and receive slant ranges, respectively, Aressubscript𝐴resA_{\rm res}italic_A start_POSTSUBSCRIPT roman_res end_POSTSUBSCRIPT is the size of one resolution cell, i.e., the size of the smallest area the \acsar system can distinguish, Bnoisesubscript𝐵noiseB_{\rm noise}italic_B start_POSTSUBSCRIPT roman_noise end_POSTSUBSCRIPT is the effective noise bandwidth, and L𝐿Litalic_L denotes the total radar losses. After coherent integration of the radar signal to form the \acsar image, the radar \acsnr is improved by a factor Nprocsubscript𝑁procN_{\rm proc}italic_N start_POSTSUBSCRIPT roman_proc end_POSTSUBSCRIPT as follows:

SNR=SNRrNproc=SNRrNpulseNa,SNRsubscriptSNR𝑟subscript𝑁procsubscriptSNR𝑟subscript𝑁pulsesubscript𝑁𝑎\mathrm{SNR}=\mathrm{SNR}_{r}N_{\rm proc}=\mathrm{SNR}_{r}N_{\rm pulse}N_{a},roman_SNR = roman_SNR start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT italic_N start_POSTSUBSCRIPT roman_proc end_POSTSUBSCRIPT = roman_SNR start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT italic_N start_POSTSUBSCRIPT roman_pulse end_POSTSUBSCRIPT italic_N start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ,(23)

where Npulsesubscript𝑁pulseN_{\rm pulse}italic_N start_POSTSUBSCRIPT roman_pulse end_POSTSUBSCRIPT is the number of samples per pulse and Nasubscript𝑁𝑎N_{a}italic_N start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT is the number of samples within the synthetic aperture length, which is denoted by Lasubscript𝐿𝑎L_{a}italic_L start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT. Note that Npulse=τpBRgτpBnoisesubscript𝑁pulsesubscript𝜏𝑝subscript𝐵Rgsubscript𝜏𝑝subscript𝐵noiseN_{\rm pulse}=\tau_{p}B_{\rm Rg}\approx\tau_{p}B_{\rm noise}italic_N start_POSTSUBSCRIPT roman_pulse end_POSTSUBSCRIPT = italic_τ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT italic_B start_POSTSUBSCRIPT roman_Rg end_POSTSUBSCRIPT ≈ italic_τ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT italic_B start_POSTSUBSCRIPT roman_noise end_POSTSUBSCRIPT and Na=LaPRFvysubscript𝑁𝑎subscript𝐿𝑎PRFsubscript𝑣𝑦N_{a}=\frac{L_{a}\mathrm{PRF}}{v_{y}}italic_N start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT = divide start_ARG italic_L start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT roman_PRF end_ARG start_ARG italic_v start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT end_ARG. Thus, the image \acsnr can be written as:

SNR=σ0AresPtGtGrλ2τpPRFLa(4π)3RTx2RRx2vykbTsysFL.SNRsubscript𝜎0subscript𝐴ressubscript𝑃𝑡subscript𝐺𝑡subscript𝐺𝑟superscript𝜆2subscript𝜏𝑝PRFsubscript𝐿𝑎superscript4𝜋3subscriptsuperscript𝑅2Txsubscriptsuperscript𝑅2Rxsubscript𝑣𝑦subscript𝑘𝑏subscript𝑇sys𝐹𝐿\mathrm{SNR}=\frac{\sigma_{0}A_{\rm res}P_{t}G_{t}G_{r}\lambda^{2}\tau_{p}%\mathrm{PRF}L_{a}}{(4\pi)^{3}R^{2}_{\rm Tx}R^{2}_{\rm Rx}v_{y}k_{b}T_{\rm sys}%FL}.roman_SNR = divide start_ARG italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT roman_res end_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT italic_G start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT italic_G start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT italic_λ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_τ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT roman_PRF italic_L start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT end_ARG start_ARG ( 4 italic_π ) start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT italic_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_Tx end_POSTSUBSCRIPT italic_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_Rx end_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT italic_k start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT italic_T start_POSTSUBSCRIPT roman_sys end_POSTSUBSCRIPT italic_F italic_L end_ARG .(24)

The \acsnr expression depends on the resolution cell size Aressubscript𝐴resA_{\rm res}italic_A start_POSTSUBSCRIPT roman_res end_POSTSUBSCRIPT, which is related to the range and azimuth resolutions, denoted by δrsubscript𝛿𝑟\delta_{r}italic_δ start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT and δasubscript𝛿𝑎\delta_{a}italic_δ start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT, respectively. For a mono-static system with slant range RTxsubscript𝑅TxR_{\rm Tx}italic_R start_POSTSUBSCRIPT roman_Tx end_POSTSUBSCRIPT and incidence angle θ𝜃\thetaitalic_θ, Ares=δrδa=λRTxc4BRgLasin(θ)subscript𝐴ressubscript𝛿𝑟subscript𝛿𝑎𝜆subscript𝑅Tx𝑐4subscript𝐵Rgsubscript𝐿𝑎𝜃A_{\rm res}=\delta_{r}\delta_{a}=\frac{\lambda R_{\rm Tx}c}{4B_{\rm Rg}L_{a}%\sin(\theta)}italic_A start_POSTSUBSCRIPT roman_res end_POSTSUBSCRIPT = italic_δ start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT italic_δ start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT = divide start_ARG italic_λ italic_R start_POSTSUBSCRIPT roman_Tx end_POSTSUBSCRIPT italic_c end_ARG start_ARG 4 italic_B start_POSTSUBSCRIPT roman_Rg end_POSTSUBSCRIPT italic_L start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT roman_sin ( italic_θ ) end_ARG leads to the expression in (8) [10]. However, calculating Aressubscript𝐴resA_{\rm res}italic_A start_POSTSUBSCRIPT roman_res end_POSTSUBSCRIPT for the bi-static case is more complex. According to [18], in this case, the ground range resolution is given by:

δr=cBRg(sin(θTx)+sin(θRx)),subscript𝛿𝑟𝑐subscript𝐵Rgsubscript𝜃Txsubscript𝜃Rx\delta_{r}=\frac{c}{B_{\rm Rg}\left(\sin(\theta_{\rm Tx})+\sin(\theta_{\rm Rx}%)\right)},italic_δ start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT = divide start_ARG italic_c end_ARG start_ARG italic_B start_POSTSUBSCRIPT roman_Rg end_POSTSUBSCRIPT ( roman_sin ( italic_θ start_POSTSUBSCRIPT roman_Tx end_POSTSUBSCRIPT ) + roman_sin ( italic_θ start_POSTSUBSCRIPT roman_Rx end_POSTSUBSCRIPT ) ) end_ARG ,(25)

where θTxsubscript𝜃Tx\theta_{\rm Tx}italic_θ start_POSTSUBSCRIPT roman_Tx end_POSTSUBSCRIPT and θRxsubscript𝜃Rx\theta_{\rm Rx}italic_θ start_POSTSUBSCRIPT roman_Rx end_POSTSUBSCRIPT denote the transmit and receive incidence angles, respectively. Furthermore, the azimuth resolution is given by:

δa=lRRxRTx+RRx.subscript𝛿𝑎𝑙subscript𝑅Rxsubscript𝑅Txsubscript𝑅Rx\delta_{a}=l\frac{R_{\rm Rx}}{R_{\rm Tx}+R_{\rm Rx}}.italic_δ start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT = italic_l divide start_ARG italic_R start_POSTSUBSCRIPT roman_Rx end_POSTSUBSCRIPT end_ARG start_ARG italic_R start_POSTSUBSCRIPT roman_Tx end_POSTSUBSCRIPT + italic_R start_POSTSUBSCRIPT roman_Rx end_POSTSUBSCRIPT end_ARG .(26)

In the case of across-track interferometry applications, since a lower bound is imposed on the baseline decorrelation by constraint C7C7\mathrm{C7}C7, the bi-static angle Δθ=|θTxθRx|Δ𝜃subscript𝜃Txsubscript𝜃Rx\Delta\theta=|\theta_{\rm Tx}-\theta_{\rm Rx}|roman_Δ italic_θ = | italic_θ start_POSTSUBSCRIPT roman_Tx end_POSTSUBSCRIPT - italic_θ start_POSTSUBSCRIPT roman_Rx end_POSTSUBSCRIPT | is relatively small (i.e., on the order of few degrees). Thus, the range resolution can be approximated by δrc2BRgsin(θTx)subscript𝛿𝑟𝑐2subscript𝐵Rgsubscript𝜃Tx\delta_{r}\approx\frac{c}{2B_{\rm Rg}\sin(\theta_{\rm Tx})}italic_δ start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ≈ divide start_ARG italic_c end_ARG start_ARG 2 italic_B start_POSTSUBSCRIPT roman_Rg end_POSTSUBSCRIPT roman_sin ( italic_θ start_POSTSUBSCRIPT roman_Tx end_POSTSUBSCRIPT ) end_ARG and σ0(θRx)σ0(θTxfixed angle)=σ0subscript𝜎0subscript𝜃Rxsubscript𝜎0superscriptsubscript𝜃Txfixed anglesubscript𝜎0\sigma_{0}(\theta_{\rm Rx})\approx\sigma_{0}(\overbrace{\theta_{\rm Tx}}^{%\text{fixed angle}})=\sigma_{0}italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_θ start_POSTSUBSCRIPT roman_Rx end_POSTSUBSCRIPT ) ≈ italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( over⏞ start_ARG italic_θ start_POSTSUBSCRIPT roman_Tx end_POSTSUBSCRIPT end_ARG start_POSTSUPERSCRIPT fixed angle end_POSTSUPERSCRIPT ) = italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT. Based on the bi-static effective range Reff=RRx+RTx2subscript𝑅effsubscript𝑅Rxsubscript𝑅Tx2R_{\rm eff}=\frac{R_{\rm Rx}+R_{\rm Tx}}{2}italic_R start_POSTSUBSCRIPT roman_eff end_POSTSUBSCRIPT = divide start_ARG italic_R start_POSTSUBSCRIPT roman_Rx end_POSTSUBSCRIPT + italic_R start_POSTSUBSCRIPT roman_Tx end_POSTSUBSCRIPT end_ARG start_ARG 2 end_ARG, we approximate the azimuth resolution as:

δaλReffRRxLa(RTx+RRx),subscript𝛿𝑎𝜆subscript𝑅effsubscript𝑅Rxsubscript𝐿𝑎subscript𝑅Txsubscript𝑅Rx\delta_{a}\approx\frac{\lambda R_{\rm eff}R_{\rm Rx}}{L_{a}(R_{\rm Tx}+R_{\rmRx%})},italic_δ start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ≈ divide start_ARG italic_λ italic_R start_POSTSUBSCRIPT roman_eff end_POSTSUBSCRIPT italic_R start_POSTSUBSCRIPT roman_Rx end_POSTSUBSCRIPT end_ARG start_ARG italic_L start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ( italic_R start_POSTSUBSCRIPT roman_Tx end_POSTSUBSCRIPT + italic_R start_POSTSUBSCRIPT roman_Rx end_POSTSUBSCRIPT ) end_ARG ,(27)

which leads to the following approximation of the resolution cell size:

Ares=δrδaλRRxc4LaBRgsin(θTx).subscript𝐴ressubscript𝛿𝑟subscript𝛿𝑎𝜆subscript𝑅Rx𝑐4subscript𝐿𝑎subscript𝐵Rgsubscript𝜃TxA_{\rm res}=\delta_{r}\delta_{a}\approx\frac{\lambda R_{\rm Rx}c}{4L_{a}B_{\rmRg%}\sin(\theta_{\rm Tx})}.italic_A start_POSTSUBSCRIPT roman_res end_POSTSUBSCRIPT = italic_δ start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT italic_δ start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ≈ divide start_ARG italic_λ italic_R start_POSTSUBSCRIPT roman_Rx end_POSTSUBSCRIPT italic_c end_ARG start_ARG 4 italic_L start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT italic_B start_POSTSUBSCRIPT roman_Rg end_POSTSUBSCRIPT roman_sin ( italic_θ start_POSTSUBSCRIPT roman_Tx end_POSTSUBSCRIPT ) end_ARG .(28)

Finally, the approximated bi-static \acsnr expression is obtained by inserting (28) in (24).

Appendix B Proof of Proposition 1

It can be shown that σΦksubscript𝜎subscriptΦ𝑘\sigma_{\Phi_{k}}italic_σ start_POSTSUBSCRIPT roman_Φ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT is monotonically decreasing \acwrt the total coherence γksubscript𝛾𝑘\gamma_{k}italic_γ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT. Therefore, given the worst-case coherence value 𝒜=γRgminγSNRminγother𝒜superscriptsubscript𝛾Rgminsuperscriptsubscript𝛾SNRminsubscript𝛾other\mathcal{A}=\gamma_{\rm Rg}^{\mathrm{min}}\gamma_{\rm SNR}^{\mathrm{min}}%\gamma_{\rm other}caligraphic_A = italic_γ start_POSTSUBSCRIPT roman_Rg end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_min end_POSTSUPERSCRIPT italic_γ start_POSTSUBSCRIPT roman_SNR end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_min end_POSTSUPERSCRIPT italic_γ start_POSTSUBSCRIPT roman_other end_POSTSUBSCRIPT, the following inequality holds:

σΦk(𝐪0,𝐪k)1𝒜1𝒜22nL,k{1,2}.formulae-sequencesubscript𝜎subscriptΦ𝑘subscript𝐪0subscript𝐪𝑘1𝒜1superscript𝒜22subscript𝑛𝐿for-all𝑘12\sigma_{\Phi_{k}}(\mathbf{q}_{0},\mathbf{q}_{k})\leq\frac{1}{\mathcal{A}}\sqrt%{\frac{1-\mathcal{A}^{2}}{2n_{L}}},\forall k\in\{1,2\}.italic_σ start_POSTSUBSCRIPT roman_Φ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( bold_q start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , bold_q start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) ≤ divide start_ARG 1 end_ARG start_ARG caligraphic_A end_ARG square-root start_ARG divide start_ARG 1 - caligraphic_A start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG 2 italic_n start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT end_ARG end_ARG , ∀ italic_k ∈ { 1 , 2 } .(29)

Based on (13), (17), and (29), we construct an upper bound on the height error σhksubscript𝜎subscript𝑘\sigma_{h_{k}}italic_σ start_POSTSUBSCRIPT italic_h start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT of interferometric pair (U0,Uk)subscript𝑈0subscript𝑈𝑘(U_{0},U_{k})( italic_U start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_U start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) as follows:

σhk(𝐪0,𝐪k)λr0(𝐪0)sin(θ0)2πb,k(𝐪k)𝒜1𝒜22nL,k{1,2}.formulae-sequencesubscript𝜎subscript𝑘subscript𝐪0subscript𝐪𝑘𝜆subscript𝑟0subscript𝐪0subscript𝜃02𝜋subscript𝑏bottom𝑘subscript𝐪𝑘𝒜1superscript𝒜22subscript𝑛𝐿for-all𝑘12\sigma_{h_{k}}(\mathbf{q}_{0},\mathbf{q}_{k})\leq\frac{\lambda r_{0}(\mathbf{q%}_{0})\sin(\theta_{0})}{2\pi b_{\bot,k}(\mathbf{q}_{k})\mathcal{A}}\sqrt{\frac%{1-\mathcal{A}^{2}}{2n_{L}}},\forall k\in\{1,2\}.italic_σ start_POSTSUBSCRIPT italic_h start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( bold_q start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , bold_q start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) ≤ divide start_ARG italic_λ italic_r start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( bold_q start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) roman_sin ( italic_θ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) end_ARG start_ARG 2 italic_π italic_b start_POSTSUBSCRIPT ⊥ , italic_k end_POSTSUBSCRIPT ( bold_q start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) caligraphic_A end_ARG square-root start_ARG divide start_ARG 1 - caligraphic_A start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG 2 italic_n start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT end_ARG end_ARG , ∀ italic_k ∈ { 1 , 2 } .(30)

The proof is concluded by inserting the upper bound (30) into the expression for the height error of the final \acdem, given in (15).

Appendix C Proof of Proposition 2

Let function f(x)=1Bp[2+Bp1+x2Bp1+1x]𝑓𝑥1subscript𝐵𝑝delimited-[]2subscript𝐵𝑝1𝑥2subscript𝐵𝑝11𝑥f(x)=\frac{1}{B_{p}}\left[\frac{2+B_{p}}{1+x}-\frac{2-B_{p}}{1+\frac{1}{x}}\right]italic_f ( italic_x ) = divide start_ARG 1 end_ARG start_ARG italic_B start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT end_ARG [ divide start_ARG 2 + italic_B start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT end_ARG start_ARG 1 + italic_x end_ARG - divide start_ARG 2 - italic_B start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT end_ARG start_ARG 1 + divide start_ARG 1 end_ARG start_ARG italic_x end_ARG end_ARG ] such that f(𝒳(𝐪k))𝑓𝒳subscript𝐪𝑘f(\mathcal{X}(\mathbf{q}_{k}))italic_f ( caligraphic_X ( bold_q start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) ) represents the baseline decorrelation. Note that 𝐪k,0𝒳(𝐪k)2for-allsubscript𝐪𝑘0𝒳subscript𝐪𝑘2\forall\mathbf{q}_{k},0\leq\mathcal{X}(\mathbf{q}_{k})\leq 2∀ bold_q start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , 0 ≤ caligraphic_X ( bold_q start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) ≤ 2, therefore, we focus on function f𝑓fitalic_f in the domain [0,2]02[0,2][ 0 , 2 ]. It can be shown that function f𝑓fitalic_f is a decreasing and invertible function such that:

f1(x)=h(x)=Bpx2BpBp2xBp.superscript𝑓1𝑥𝑥subscript𝐵𝑝𝑥2subscript𝐵𝑝subscript𝐵𝑝2𝑥subscript𝐵𝑝f^{-1}(x)=h(x)=\frac{B_{p}x-2-B_{p}}{B_{p}-2-xB_{p}}.italic_f start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_x ) = italic_h ( italic_x ) = divide start_ARG italic_B start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT italic_x - 2 - italic_B start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT end_ARG start_ARG italic_B start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT - 2 - italic_x italic_B start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT end_ARG .(31)

Therefore, constraint C7C7\mathrm{C7}C7 is equivalent to the following constraint:

C7:f(𝒳(𝐪k))γRgmin𝒳(𝐪k)h(γRgmin).:C7𝑓𝒳subscript𝐪𝑘superscriptsubscript𝛾Rgminiff𝒳subscript𝐪𝑘superscriptsubscript𝛾Rgmin\mathrm{C7}:f(\mathcal{X}(\mathbf{q}_{k}))\geq\gamma_{\rm Rg}^{\rm min}\iff%\mathcal{X}(\mathbf{q}_{k})\leq h(\gamma_{\rm Rg}^{\rm min}).C7 : italic_f ( caligraphic_X ( bold_q start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) ) ≥ italic_γ start_POSTSUBSCRIPT roman_Rg end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_min end_POSTSUPERSCRIPT ⇔ caligraphic_X ( bold_q start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) ≤ italic_h ( italic_γ start_POSTSUBSCRIPT roman_Rg end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_min end_POSTSUPERSCRIPT ) .(32)

Based on (11) and (32), constraint C7C7\mathrm{C7}C7 is equivalent to the following constraints:

{C7a:sin(θk(𝐪k))2h(γRgmin)h(γRgmin)sin(θ0),if𝐪kak,C7b:sin(θk(𝐪k))h(γRgmin)2h(γRgmin)sin(θ0),if𝐪kbk.cases:C7aformulae-sequencesubscript𝜃𝑘subscript𝐪𝑘2superscriptsubscript𝛾Rgminsuperscriptsubscript𝛾Rgminsubscript𝜃0ifsubscript𝐪𝑘superscriptsubscript𝑎𝑘otherwise:C7bformulae-sequencesubscript𝜃𝑘subscript𝐪𝑘superscriptsubscript𝛾Rgmin2superscriptsubscript𝛾Rgminsubscript𝜃0ifsubscript𝐪𝑘superscriptsubscript𝑏𝑘otherwise\begin{dcases}\mathrm{C7a}:\sin(\theta_{k}(\mathbf{q}_{k}))\geq\frac{2-h(%\gamma_{\rm Rg}^{\rm min})}{h(\gamma_{\rm Rg}^{\rm min})}\sin(\theta_{0}),%\text{if }\mathbf{q}_{k}\in\mathcal{I}_{a}^{k},\\\mathrm{C7b}:\sin(\theta_{k}(\mathbf{q}_{k}))\leq\frac{h(\gamma_{\rm Rg}^{\rmmin%})}{2-h(\gamma_{\rm Rg}^{\rm min})}\sin(\theta_{0}),\text{if }\mathbf{q}_{k}%\in\mathcal{I}_{b}^{k}.\end{dcases}{ start_ROW start_CELL C7a : roman_sin ( italic_θ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( bold_q start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) ) ≥ divide start_ARG 2 - italic_h ( italic_γ start_POSTSUBSCRIPT roman_Rg end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_min end_POSTSUPERSCRIPT ) end_ARG start_ARG italic_h ( italic_γ start_POSTSUBSCRIPT roman_Rg end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_min end_POSTSUPERSCRIPT ) end_ARG roman_sin ( italic_θ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) , if bold_q start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∈ caligraphic_I start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT , end_CELL start_CELL end_CELL end_ROW start_ROW start_CELL C7b : roman_sin ( italic_θ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( bold_q start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) ) ≤ divide start_ARG italic_h ( italic_γ start_POSTSUBSCRIPT roman_Rg end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_min end_POSTSUPERSCRIPT ) end_ARG start_ARG 2 - italic_h ( italic_γ start_POSTSUBSCRIPT roman_Rg end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_min end_POSTSUPERSCRIPT ) end_ARG roman_sin ( italic_θ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) , if bold_q start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∈ caligraphic_I start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT . end_CELL start_CELL end_CELL end_ROW(33)

Note that constraints C7aC7a\mathrm{C7a}C7a and C7bC7b\mathrm{C7b}C7b are in general non-convex, however, knowing that 0θk(𝐪k)π2,𝐪k,kformulae-sequence0subscript𝜃𝑘subscript𝐪𝑘𝜋2for-allsubscript𝐪𝑘for-all𝑘0\leq\theta_{k}(\mathbf{q}_{k})\leq\frac{\pi}{2},\forall\mathbf{q}_{k},\forallk0 ≤ italic_θ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( bold_q start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) ≤ divide start_ARG italic_π end_ARG start_ARG 2 end_ARG , ∀ bold_q start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , ∀ italic_k, then we can apply increasing functions arcsin\arcsinroman_arcsin and tan\tanroman_tan to (33), which results in the following equivalent convex constraints:

{C7a:xtxkzktan(arcsin((2h(γRgmin))sin(θ0)h(γRgmin))),if𝐪kak,k,C7b:xtxkzktan(arcsin(h(γRgmin)sin(θ0)2h(γRgmin))),if𝐪kbk,k.cases:C7aformulae-sequencesubscript𝑥𝑡subscript𝑥𝑘subscript𝑧𝑘2superscriptsubscript𝛾Rgminsubscript𝜃0superscriptsubscript𝛾Rgminifsubscript𝐪𝑘superscriptsubscript𝑎𝑘for-all𝑘otherwise:C7bformulae-sequencesubscript𝑥𝑡subscript𝑥𝑘subscript𝑧𝑘superscriptsubscript𝛾Rgminsubscript𝜃02superscriptsubscript𝛾Rgminifsubscript𝐪𝑘superscriptsubscript𝑏𝑘for-all𝑘otherwise\begin{dcases}\mathrm{C7a}:\frac{x_{t}-x_{k}}{z_{k}}\geq\tan\left(\arcsin\left%(\frac{\left(2-h(\gamma_{\rm Rg}^{\rm min})\right)\sin(\theta_{0})}{h(\gamma_{%\rm Rg}^{\rm min})}\right)\right),\text{if }\mathbf{q}_{k}\in\mathcal{I}_{a}^{%k},{\forall k,}\\\mathrm{C7b}:\frac{x_{t}-x_{k}}{z_{k}}\leq\tan\left(\arcsin\left(\frac{h(%\gamma_{\rm Rg}^{\rm min})\sin(\theta_{0})}{2-h(\gamma_{\rm Rg}^{\rm min})}%\right)\right),\text{if }\mathbf{q}_{k}\in\mathcal{I}_{b}^{k},{\forall k.}\end%{dcases}{ start_ROW start_CELL C7a : divide start_ARG italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT - italic_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_ARG start_ARG italic_z start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_ARG ≥ roman_tan ( roman_arcsin ( divide start_ARG ( 2 - italic_h ( italic_γ start_POSTSUBSCRIPT roman_Rg end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_min end_POSTSUPERSCRIPT ) ) roman_sin ( italic_θ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) end_ARG start_ARG italic_h ( italic_γ start_POSTSUBSCRIPT roman_Rg end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_min end_POSTSUPERSCRIPT ) end_ARG ) ) , if bold_q start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∈ caligraphic_I start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT , ∀ italic_k , end_CELL start_CELL end_CELL end_ROW start_ROW start_CELL C7b : divide start_ARG italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT - italic_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_ARG start_ARG italic_z start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_ARG ≤ roman_tan ( roman_arcsin ( divide start_ARG italic_h ( italic_γ start_POSTSUBSCRIPT roman_Rg end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_min end_POSTSUPERSCRIPT ) roman_sin ( italic_θ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) end_ARG start_ARG 2 - italic_h ( italic_γ start_POSTSUBSCRIPT roman_Rg end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_min end_POSTSUPERSCRIPT ) end_ARG ) ) , if bold_q start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∈ caligraphic_I start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT , ∀ italic_k . end_CELL start_CELL end_CELL end_ROW(34)

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Sensing Accuracy Optimization for Communication-assisted Dual-baseline UAV-InSAR 

††thanks:  This work was supported in part by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) GRK 2680 – Project-ID 437847244. (2024)
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