Sensing Performance of Cooperative Joint Sensing-Communication UAV Network

We consider a cooperative JSC UAV network that conducts downward-looking detection and detection data communication simultaneously. As illustrated in Fig. 1, the network consists of an FCUAV and $M$ SUs, where all UAVs hover within a constrained two-dimensional (2D) plane at a fixed altitude $h$. Each UAV has an antenna array¹¹1All antenna elements are isotropic. that consists of a communication array (ComA) and a sensing array (SenA) to generate a ComB and a SenB, respectively. SenB is pointed downward to detect the targets on the ground (or the sea), and ComB is pointed to UAVs to transmit sensing data. Due to the significant difference between the sensing direction and the communication direction, ComB and SenB can be easily formed orthogonally and operate simultaneously to achieve beam sharing scheme. FCUAV integrates all the sensing data from SUs with ComB and detects targets simultaneously with SenB. SUs detect the targets with SenB while simultaneously transmitting the sensing data to FCUAV with ComB.

The antenna array of each UAV has available radio power $P$, i.e., the sum of SenB power and ComB power is $P$. The SenB power is given by

where $\beta_{R}$ is defined as the sensing power ratio (SPR). The ComB power is thus $P_{c}={1-\beta_{R}}P$. OFDM waveform is adopted as the communication and sensing waveform to achieve joint communication and sensing [4]. Time division multiple access (TDMA) is adopted by the JSC UAV network to exploit the same frequency band to transmit sensing data. To ensure that FCUAV integrates the intact sensing data of each SU, the communication capacity between each SU and FCUAV has to be larger than the generating rate of sensing data. Therefore, the distance between FCUAV and each SU must be smaller than MCR, denoted by $x_{Q}$. Each UAV also has a guard radius ${R_{g}}$ for safety, within which no other UAVs can hover. Thus, SUs distribute within a concentric circle area centered at FCUAV, which is defined as the maximum cooperation area (MCA) whose inner and outer radii are ${R_{g}}$ and ${x_{Q}}$, respectively.

To satisfy the constraints on the false alarm rate and the detection probability for effective sensing [20], each UAV has to decrease the loss of sensing signal power and reduce interference to other UAVs in order to maintain the minimum SINR of sensing [21]. Thus, there is the maximum sensing range for each UAV, which is denoted by $R_{max}$.

As shown in Fig. 1, the region on the ground that the JSC UAV network has to detect is defined as the detected plane. The region in the detected plane that can be effectively detected²²2i.e., the false alarm rate is below the maximum and detection probability is higher than the minimum. by a UAV is defined as the sensing zone (SZ) of the UAV. The maximum radius of SZ is determined by $R_{max}$ and $h$. We propose the average union area of SZs of all UAVs to be the sensing performance metric of the JSC UAV network. The outage capacity of the links between SUs and FCUAV is defined as the communication performance metric of the JSC UAV network.

The beam sharing scheme is used to realize communication and sensing simultaneously. Let ($\varphi$,$\theta$) be a certain direction in the three-dimensional (3D) space, where $\theta$ denotes the elevation angle and $\varphi$ denotes the azimuth angle. ComA of each UAV generates ComB with elevation beamwidth³³3In this paper, the term “beamwidth” refers to the -3dB half-power beamwidth. $\Delta\theta_{c}$ and azimuth beamwidth $\Delta\phi_{c}$. SenA of each UAV generates SenB with elevation beamwidth $\Delta\theta_{r}$ and azimuth beamwidth $\Delta\phi_{r}$. Then the average directional gain of a 3D beam can be approximated by [11] [22]

where $\Delta\theta$ and $\Delta\phi$ are the azimuth beamwidth and elevation beamwidth, respectively.

The antenna array designed to achieve beam sharing consists of two parts, as illustrated in Fig. 2. The upper linear subarray is ComA that has $M_{com}$ antenna elements with inter-distance half-wavelength. The lower subarray is SenA composed of two decoupled circular subarrays, where one is transmitting array (TxA) and the other is receiving array (RxA). TxA and RxA are used to generate transmitting SenB and receiving SenB, respectively. The electronic shielding plate is placed between two subarrays of SenA and the pulse response between two subarrays is also accurately measured to eliminate self-interference between TxA and RxA of SenA in real-time manner [23].

Two subarrays of SenA are used to solve the problem of minimum detection range by achieving simultaneous transmitting and receiving of radar sensing signal[21]. If SenA adopts only one subarray, then it has to arrange time slots for transmitting and receiving the radar signal. In this case, the radar echo can come back to SenA before the slot for receiving, and then the missing of the target will happen. Only when the distance between UAV and the target is larger than the minimum value can the target be possibly detected, which is inapplicable to the situation where the target is close to the JSC units. By contrast, if we adopt double subarrays to form SenA, then the radar sensing signal can be transmitted and received simultaneously. Thus, the problem of minimum detection range is solved.

As illustrated in Fig. 3, SenA has antennas arranged in concentric circles. There are $N_{A}$ circles in each SenA, and we define each circle as a layer. From center to periphery, SenA has the 0th layer to the ($N_{A}-$1)th layer. Except for the 0th layer with only one antenna element that is the phase reference antenna (PhRefA), there are ${{2}^{b}}$ antenna elements in each layer, where $b$ is a positive integer.

Fig. 4 shows the adjacent layers of SenA. The distance between the antennas that locate at the same polar angle of adjacent layers is $d$. Antennas in each layer locate at the evenly split angles, and are referred to as the $0$th to the ($2^{b}-1$)th antenna, anti-clockwise.

OFDM signal is adopted in the JSC UAV network to exploit its advantage of the frequency diversity and obtain high processing gain [19]. The baseband OFDM JSC signal is modeled as [19]

where ${M_{s}}$ is the number of OFDM symbols in one frame, ${N_{c}}$ is the number of subcarriers, $d_{Tx}\left({m{N_{c}}+q}\right)$ is the transmit symbol on the $q$th subcarrier of the $m$th OFDM block, $B$ is the bandwidth of the JSC signal, ${f_{q}}=\frac{qB}{{{N_{c}}}}$ is the baseband frequency of the $q$th subcarrier, and $T$ is the duration time of each OFDM symbol that contains guard interval and the duration of OFDM block.

The phase difference between the transmitted and the received OFDM symbols is used to estimate the Doppler frequency shift and the time delay between the target and the UAV.

The Doppler shift results from the radial relative velocity between the UAV and the target, and can be presented as

where ${c_{0}}$ is the speed of light, ${f_{c}}$ is the carrier frequency of the OFDM signal, and ${v_{rel}}$ is the radial relative velocity between UAV and the target. The time delay is expressed as

where $R_{r}$ is the distance between the target and the UAV. The complex ratio of the received to the transmitted OFDM signal is expressed as [19]

where $(\cdot)_{m,q}$ is the entry of a matrix at the $m$th row and the $q$th column, ${{d_{Rx}}\left({m{N_{c}}+q}\right)}$ is the received OFDM symbol corresponding to ${{d_{Tx}}\left({m{N_{c}}+q}\right)}$, and $({\bf{H}})_{m,q}$ is the complex fading factor for the $q$th subcarrier of the $m$th OFDM symbol. The matrix form of (13) is given by

where $m=0,1,...,M_{s}-1$, $q=0,1,...,N_{c}-1$, ${{\bf k}_{R}}\left(q\right)=\exp\left({-j2\pi{f_{q}}\tau_{s}}\right)$, and ${{{{\bf k}_{D}}}}\left(m\right)=\exp\left({j2\pi mT{f_{d,s}}}\right)$.

By applying discrete Fourier transform (DFT) to each column of ${{\bf{D}}_{div}}$ and inverse discrete Fourier transform (IDFT) to each row of ${{\bf{D}}_{div}}$, ${f_{d,s}}$ and ${R_{r}}$ can be obtained respectively.

Let ${{\bf{D}}_{m,IDFT}}$ denote the IDFT of the $m$th row of ${{\bf{D}}_{div}}$, and ${\bf{D}}_{q,DFT}$ denote the DFT of the $q$th column of ${{\bf{D}}_{div}}$. The index corresponding to the peak of ${{\bf{D}}_{m,IDFT}}$ depends on $f_{d,s}$ and can be obtained by one-dimensional exhaustive search as follows [4]:

Similarly, the index corresponding to the peak of ${\bf{D}}_{q,DFT}$ depends on $R_{r}$ and can be obtained by one-dimensional exhaustive search as follows [4]:

Based on (15) and (16), we can obtain $f_{d,s}$ and $R_{r}$, respectively. The accuracy of the estimation depends on the duration time of each OFDM symbol and the bandwidth of signal [4]. This radar echo processing technique can generate a processing gain as [4]

In order to model the mutual sensing interference, the power spectrum of the JSC OFDM signal needs to be considered first, which is the sum of the power spectra of $N_{c}$ subcarriers [24]. According to [24], the power spectrum of the $q$th subcarrier is

where $T_{s}=\frac{1}{B}$ is the sampling cycle of the OFDM signal. The aggregate power spectrum of all subcarriers is noise-like [4]. For reasonable simplification, we assume that the transmitting power of the OFDM sensing signal concentrates in the baseband spectrum $[0,B]$. UAVs receive the sensing interference power imposed by all other UAVs’ scattered sensing signals.

FCUAV is chosen as the reference to analyze the average sensing interference that a UAV receives from other UAVs. As illustrated in Fig. 5, denote ${R_{i}}$ as the distance between SU_i and FCUAV,⁴⁴4$R_{i}$ is an independently and identically distributed (i.i.d.) variable. $Pos_{i}$ as the intersection point between the SenB direction of SU_i and SZ of SU_i, ${R_{i,1}}$ as the distance between SU_i and $Pos_{i}$, ${R_{i,2}}$ as the distance between $Pos_{i}$ and FCUAV, and $r_{i}$ as the distance between $Pos_{i}$ and the projection point of SU_i on SZ. Assuming that the interfering sensing signals are random due to the random scattering, the sensing interfering power imposed by SU_i on FCUAV is formulated as [11, 25]

where ${g_{ts}}$ and ${g_{rs}}$ are the gains of transmitting and receiving sensing beams respectively, $\overline{\sigma}$ is the mean of radar cross section of target, $\lambda$ is the wavelength of the carrier of JSC transceiver, and $R_{i,1}$ and $R_{i,2}$ are expressed as follows:

The maximal point of $I_{sen,i}$ with regard to $r_{i}$ is $r_{i}=\frac{{{R_{i}}}}{2}$, which is easily obtained by the first-order and the second-order derivative of $I_{sen,i}$ versus $r_{i}$. Thus, we set ${R_{i,1}}={R_{i,2}}={\left[{{{\left({\frac{{{R_{i}}}}{2}}\right)}^{2}}+{h^{2}}}\right]^{\frac{1}{2}}}$ to obtain the upper bound of the average sensing interference imposed on FCUAV by SU_i.

The expectation of sensing interference imposed on FCUAV by $M$ SUs can be upper-bounded by

The second order derivative of ${{\left({{R_{i}}^{2}+4{h^{2}}}\right)}^{-2}}$ with regard to $R_{i}$ is $24{R_{i}}^{2}({R_{i}}^{2}+4h^{2})^{-4}$ for $R_{i},h>0$, which is larger than 0. Thus, ${I_{sen}}$ is convex of $R_{i}$ for $R_{i}>0$. By using the Jensen inequality, we have

where ${E\left\{{{{\left({{R_{i}}^{2}+4{h^{2}}}\right)}^{{}^{-2}}}}\right\}}$ can be obtained as

The proof of (23) is given in Theorem 1 in appendix. Then we have

As stated before, the power spectrum of OFDM signal is noise-like. Thus, the interference-and-noise imposed on sensing receiver of each UAV follows the Gaussian distribution, i.e., ${A_{in}}\sim N\left(0,N_{s}+{I_{sen}}\right)$⁵⁵5$N\left(\mu,\sigma^{2}\right)$ is the Gaussian distribution with $\mu$ as mean and $\sigma^{2}$ as variance., where $N_{s}$ is the thermal noise power.

The useful sensing signal power is

where ${\gamma_{s}}$ denotes SINR of sensing, and $G_{p}$ is presented in (17). We can obtain the following detection problem:

where hypothesis $H_{1}$ is that there is a target in the direction of sensing beam, hypothesis $H_{0}$ is the opposite, and $y$ is the output signal of radar after processing ${M_{s}}$ OFDM symbols. The decision rule of whether there is a target in the sensing direction can be expressed as [20]

where $\eta^{\prime}$ is the decision threshold. When $y>\eta^{\prime}$, the sensor accepts the hypothesis $H_{1}$; otherwise, the sensor accepts the hypothesis $H_{0}$.

The false alarm rate and the detection probability are the metrics that have to be considered for detection [20], and are presented by

and

respectively, where $Q\left(\cdot\right)$ is the monotonically decreasing Q-function [20]. According to (28), we have

where $Q^{-1}\left(\cdot\right)$ is the inverse function of Q-fucntion. By substituting (25) and (30) into (29), we can obtain $P_{D}$ as

For effective detection, $P_{F}$ and $P_{D}$ must be constrained as [20]

where $\alpha_{f}$ is the maximum false alarm rate, and ${\alpha_{D}}$ is the minimum detection probability. Based on (31) and (32), the minimum SINR to meet the constraints can be expressed as

According to [21], the maximum sensing range of UAV is

where $\beta_{R}$ is SPR, $P$ is the total available radio power, ${g_{ts}}$ and ${g_{rs}}$ are the transmitting and the receiving antenna gains of SenB respectively, ${f}$ is the carrier frequency of transceiver, $\overline{\sigma}$ is the mean radar cross section, $L_{s}$ is the aggregated power loss at the propagation medium, $I_{sen}$ is given in (24), $k$ is the Boltzmann’s constant, $F_{n}$ is the noise figure of the receiver, $B$ is the bandwidth of JSC transceiver, and ${T_{0}}=$ 290 K (in absolute temperature) is the standard temperature.

The time that SU requires to transmit the sensing data to FCUAV is defined as the ratio of the amount of sensing data to the communication throughput, which is

where $V_{data}$ is the generating rate of the sensing data, $\tau_{\det}$ is the detection time, and ${{T_{c}}\left(R_{i}\right)}$ is the throughput of the communication link between FCUAV and SU_i. With the increase of $R_{i}$, ${{T_{c}}\left(R_{i}\right)}$ decreases because the received ComB power decreases.

We adopt TDMA for SUs to transmit the sensing data to FCUAV. Each SU must transmit the sensing data in the assigned time slots of length ${\tau_{c\max}}$. SUs carry out downward-looking detection constantly and transmit its sensing data in the assigned slots.

Fig. 6 illustrates the time slots allocated to SUs for transmitting sensing data to FCUAV. Let ${\tau_{det\_i}}$ and ${\tau_{cmax\_i}}$ denote the sensing slots and communication slots for SU_i, respectively. All UAVs are provided with the same sensing time ${\tau_{det}}$ and transmission time ${\tau_{cmax}}$. In order to ensure the consecutive detection of each UAV, we have

In this case, after SU_i completes the sensing tasks in $\tau_{det\_i}$, the next slot for SU_i to transmit sensing data comes again.

Based on (35), (36) and ${\tau_{c}}\leq{\tau_{cmax}}$, we obtain ${{{{T_{c}}\left(R_{i}\right)}}\geq M\times{V_{data}}}$. Thus, the distance between FCUAV and each SU is no larger than MCR, i.e.,

where $T_{c}^{-1}(\cdot)$ is the inverse function of ${T_{c}}\left(R_{i}\right)$ and will be presented in the next section.

We define an SU and FCUAV that are communicating sensing data as an S-C pair. As shown in Fig. 7, the radius of SZ of each UAV is formulated as

where $u\left(\cdot\right)$ is the step function⁶⁶6i.e., when there is $x\geq 0$, $u\left(x\right)=1$; otherwise, $u\left(x\right)=0$.. The overlapped area between SZs of the S-C pair is

Then, the union area of SZs of the S-C pair is

The expectation of ${S_{u1}}\left({{R_{i}}}\right)$, for ${R_{g}}\leq R_{i}<x_{Q}$, is defined as the sensing performance metric of the S-C pair, which is formulated as [1]

where ${f_{R_{i}}}\left(r\right)=\frac{{2r}}{{{x_{Q}}^{2}-{R_{g}}^{2}}}$ for ${R_{g}}<r<{x_{Q}}$, is the probability density function (PDF) of $R_{i}$. Moreover, $F_{1}\left({{x_{Q}}}\right)$ and $F_{2}\left({{x_{Q}}}\right)$ are given in (42) and (43), respectively.

There are $M$ S-C pairs located in MCA. ACSA of the JSC UAV network is defined as the average union area of SZs of $M$ S-C pairs. Because it is extremely intractable to obtain an explicit expression of the average union area of a certain number of randomly distributed circle areas. For tractability, we merely subtract the overlapped sensing area between each SU and FCUAV to obtain the upper bound of ACSA (UB-ACSA) as the sensing performance metric of the JSC UAV network. UB-ACSA is obtained as [1]

where ${S\left({{R_{d}}}\right)}=\pi R_{d}^{2}$ is the area of SZ of each UAV, and $\boldsymbol{R}=$ $(R_{1},R_{2},...,R_{M})$ is the multiple random variables composed of the distances between SUs and FCUAV. Each element of $\boldsymbol{R}$ is an i.i.d. variable.

UB-ACSA can be achieved when all SUs do not have overlapped SZs, which requires FCUAV to coordinate the trajectory of each SU. The larger UB-ACSA is, the better the sensing performance of the JSC UAV network is. The JSC UAV network can detect the zone with area of UB-ACSA in much shorter time than individual UAV can do. The overlapped SZs of SUs can be detected by different SUs from different directions, which can reduce the probability of false detection in the overlapped SZs.

The power of communication signal received by FCUAV is

where $P_{c}$ is the power of ComB, $\alpha$ is the path loss exponent, ${g_{c}}$ is the directional gain of the communication link, $h_{c}$ is the small scale fading factor that follows Rician distribution with Rician factor $K_{h}$, and $x_{0}$ is the distance between the transmitter and FCUAV. We have ${g_{c}}{\rm{=}}{g_{tc}}\times{g_{rc}}$, where ${g_{tc}}$ and ${g_{rc}}$ are the directional gains of transmitting ComB and receiving ComB, respectively. The PDF of $h_{c}$ is expressed as follows [26]:

where ${I_{0}}\left(x\right)=\sum_{n=0}^{\infty}{\frac{\left(x/2\right)^{2n}}{n!\Gamma(n+1)}}$ is the 0-th order modified Bessel function of the first kind, ${K_{h}}={{{v^{2}}}\mathord{\left/{\vphantom{{{v^{2}}}{{\sigma_{K}}^{2}}}}\right.\kern-1.2pt}{{\sigma_{K}}^{2}}}$, $\overline{\Omega}{\rm{=}}2{{{\sigma_{K}}^{2}}}{\rm{+}}{v^{2}}$ is the normalized power, ${v^{2}}$ denotes the strong line-of-sight (LOS) power, and ${{\sigma_{K}}^{2}}$ represents the multipath reflected power [27, 28, 29].

The successful transmission probability (STP) is defined as the probability that the received SNR (or SINR) is larger than a threshold $\gamma$ that is necessary for successful transmission [26],[30], i.e.,

where $N_{com}$ is AWGN of communication receiver. Since $h_{c}$ follows Rician distribution with Rician factor $K_{h}$, STP is formulated as [27]

where $Q\left({{a_{1}},{a_{2}}}\right)$ is the first order Marcum Q-function. The outage probability is the complement of $\rho_{c}^{s}$, i.e.,

The outage capacity is written as [30]

where $B$ is the available bandwidth of the JSC transceiver, $\varepsilon$ is the maximum of $\rho_{c}^{out}$, and ${\gamma_{th}}$ is the SNR threshold that makes the outage probability equal to $\varepsilon$[30]. If $\gamma>\gamma_{th}$, then the outage probability will be larger than $\varepsilon$. Combining (48), (49) and (50), we have