Unmanned vehicles (UVs) have attracted growing attention in the past decade, generating new opportunities in a variety of emerging Internet of Things (IoT) applications, including precision farming [1], aerial imaging [2], smart manufacturing [3], maintenance in harsh locations [4], first response, and assisted living [5, 6]. However, for UVs to play a key role in all these applications, it is necessary to precisely localize their positions in a variety of operational settings without compromising their battery life. In this regard, the global positioning system (GPS) [7] has been a key resource in the last decades for navigation and localization in outdoor settings. Nevertheless, GPS is often unavailable in indoor and underground environments, and its localization accuracy may exceed the size of today's UVs by orders of magnitude. Consequently, growing interest has been devoted to alternative methodologies for the localization of UVs in GPS-denied environments. In this regard, ranging techniques based on Light-Detection-and-Ranging (LIDAR) [8], ultrasound detection [9] and frequency-modulated-continuous-wave (FMCW) radars [10] have been extensively investigated, offering enhanced localization capabilities. Nonetheless, these techniques are power-hungry, making them unsuitable for low-power applications. Also, they require complex designs for the interrogation nodes, in addition to sophisticated pattern recognition algorithms [11].
Driven by the on-going Radio-Frequency-Identification (RFID) [12] revolution, increased attention has been also paid lately to ranging techniques based on radiofrequency (RF) passive tags. In particular, the adoption of RF passive tags has been recently proposed for localizing UVs in GPS-denied environments through a low-cost monitoring system not requiring any power from the targeted UVs [13–16]. A directional wireless transceiver can interrogate an RF passive tag onboard of a UV with a continuous-wave (CW) signal and leverage the received signal strength indicator (RSSI) [17] or the phase [18] of the backscattering signal to retrieve the distance between the tag and the transceiver. However, the RSSI of passive tags is inevitably distorted by multipath interference affecting the backscattered signal, which can be strong in indoor or underground settings as shown in Fig. 1(a) [19]. Such interference causes ranging errors that can become severe, especially when electrically small passive tags are used to fit into compact UVs. Relying on the phase difference between the received backscattered signal and the interrogation signal also leads to a low-ranging accuracy due to multipath, as well as to cycle-ambiguity [20]. Cycle-ambiguity arises because multiple distances between the tag and the transceiver can yield to the same phase difference between interrogation and backscattered signals. [21]. Furthermore, since conventional passive tags operate in the linear regime, their backscattered signals have the same frequency as the interrogation signals. As a result, significant ranging inaccuracies can also be caused by electromagnetic clutter and by self-interference at the reader [22] as in Fig. 1(b-c). In this context, a new category of tags operating in the nonlinear regime and known as harmonic-tags (HTs) has been recently proposed for ranging applications [13, 21, 23, 24]. Unlike linear passive tags, HTs can employ the nonlinearities of varactors or diodes to generate backscattered signals at twice the frequency of the interrogation signal as in Fig. 1(d-e). This feature provides HTs’ readers with an immunity to both electromagnetic clutter and self-interference, beyond the limits of conventional linear passive tags [23]. Yet, the accuracy that a reader can ultimately achieve when remotely monitoring its distance from an HT in an indoor or underground setting remains inevitably limited by the multipath interference [24] affecting the HT’s backscattered signal (see Fig. 1(a)) [19]. In fact, the only way for a reader to accurately assess its distance from an HT is to rely on power-hungry wideband transmitters and on intense signal processing [21]. Therefore, a new class of passive tags is needed to overcome the limits of existing counterparts used for ranging, enabling an accuracy insensitive to multipath interference affecting its backscattered signal, as well as to readers’ self-interference and clutter.
Frequency combs have been extensively studied in the last 20 years as they provide robust and equally spaced spectral comb lines that can be used as frequency references. Frequency combs can be generated in nonlinear optical systems through mode-locking [25], Kerr nonlinearities [26, 27] and electro-optic modulation [28, 29]. In microresonator-based frequency combs, which are amenable for integration, the comb spacing is determined by the free spectral range of resonators, and its range is typically within GHz to THz levels when operating at optical frequencies [30]. Such frequency synthesizers have opened several opportunities in the field of metrology, holding the promise for extremely precise timing and sensing applications, as well as for optical ranging [31, 32]. Particularly, dual-comb systems can provide unprecedented ranging resolutions at fast rates. However, they typically rely on power-hungry free-space lasers incident on targets, and they are susceptible to large path losses, scattering losses and beam dispersion [33, 34]. Frequency combs in the microwave range have also been recently demonstrated through strong light-matter interactions [35] in optical microresonators, as well as in microelectromechanical systems (MEMS) utilizing nonlinear three-wave mixing [36–39], nonlinear friction forces [40, 41] and strong nonlinear electromechanical couplings [42]. Refs [43, 44] demonstrated comb generation in Josephson-Junction circuits for quantum engineering applications. However, to our knowledge, there have been no attempts to generate and employ frequency combs in passive RF systems for ranging applications. To this end, simultaneously leveraging the strong nonlinearities of solid-state components and the large quality factor (> 1,000) of microacoustic piezoelectric resonators [45–48] offers an ideal platform for low-power comb generation in the RF range.
In this Article we demonstrate a nonlinear RF passive tag engineered to realize comb-based ranging with accuracy intrinsically immune to both self and multipath interference affecting its backscattered signal. Such a passive tag, referred to as a quasi-Harmonic Tag (qHT), receives an interrogation signal with frequency \({\omega _p}\) and power \({P_r}\), and it responds by passively generating a frequency comb through its nonlinear behavior, as shown in Fig. 1(f). The comb is symmetrically distributed around half the frequency of the interrogation signal (i.e. \({\omega _p}/2\)), with a comb line spacing \(\left( {\Delta f} \right)\) that is a function of \({P_r}\) as in Fig. 1(g). This differs from frequency comb generation in optical systems, where the comb-line spacing remains independent of the input power level. The inverse proportionality between received power \(\left( {{P_r}} \right)\) and its distance from the interrogating node (d), described by the Friis transmission relation plotted in Fig. 1(h) [49], can be exploited by qHTs to retrieve d. This can be done by simply measuring the comb line spacing, as sketched in Fig. 1(i). In this regard, any measured \(\Delta f\) value univocally maps onto one specific\({P_r}\) level, and consequently to a specific value of d. Therefore, readers with a directional transmitter can remotely assess their distance from a qHT by simply extracting \(\Delta f\), and the accuracy of such extraction is not degraded by multipath. In fact, undesired scattering caused by multipath can distort both the RSSI and the phase of the current RF passive tags’ backscattered signal, but they have no effect on \(\Delta f\) in qHTs. Moreover, qHTs’ ability to utilize separate bands for receiving the interrogation signal and transmitting their backscattering signal confers immunity to electromagnetic clutter. Also, it enhances the resilience of qHTs’ readers to their own self-interference by allowing them to filter all the harmonics of the transmitted interrogation signals that are generated due to nonlinearities in the power amplifier of their transmitter. In turn, a minimum level of received power, designated as \({P_{th}}\), is required to activate the generation of a frequency comb in qHTs. In this regard, the value of \({P_{th}}\) ultimately sets the maximum distance from which qHTs can be remotely interrogated.
A typical application scenario leveraging the \(\Delta f -\)\({P_r}\) dependence characterizing the operation of qHTs is sketched in Fig. 1(j), where we envision a qHT mounted onboard of a drone that is sequentially interrogated by different beacons located at predetermined positions. The qHT generates unique frequency comb patterns corresponding to its distance \(d\) from each interrogating beacon. Hence, by extracting\(\Delta f\)from each one of the beacon’s received signals, and by using trilateration, we can identify the azimuthal position of the qHT within the common area covered by the beacons.
Model: Frequency comb generation in qHTs
As mentioned previously, a fundamental characteristic of qHTs is the generation of frequency combs with line spacing that varies according to the received power level. This feature allows readers to accurately determine their distance from a designated qHT. Consequently, it is important to ascertain the key parameters that control the comb generation in a qHT, as these parameters play a pivotal role in the design of these tags. With no loss of generality, a qHT can be described as a parametric oscillator (PO) [50], coupled to a high quality factor \(\left( Q \right)\) piezoelectric microacoustic resonator with resonance frequency matching or closely matching the PO’s output frequency. We can describe this system with a toy model formed by two electrical modes and a mechanical mode, as shown in Fig. 2(a). The two electrical modes, namely the signal mode \({a_1}\) and the pump mode \({a_2}\), are parametrically coupled via a nonlinearity, while the mechanical mode b is linearly coupled to the signal mode with electromechanical coupling rate G [51, 52]. The pump mode \({a_2}\) is driven by a microwave pump signal at \({\omega _p}\), which indirectly excites the signal mode \({a_1}\) through parametric down-conversion [Fig. 2(b-left)]. This interaction is made possible by a second-order nonlinearity \(\left( {{\chi ^{\left( 2 \right)}}} \right)\), leading to \({\omega _2}=2{\omega _1}\), where \({\omega _1}\) and \({\omega _2}\) denote the natural resonant angular frequencies of \({a_1}\) and \({a_2}\), respectively [53]. Since the loss rate of \({a_2}\) is larger than the one of \({a_1}\), we adiabatically eliminate the pump mode \({a_2}\) from the equations of motion, reducing them to a single equation for the nonlinear interaction (see detailed analysis in Supplementary Section I). The mechanical mode, with resonant frequency close to the frequency of \({a_1}\)\(\left( {{\omega _b} \approx {\omega _1}} \right)\), is linearly coupled to \({a_1}\) as shown in Fig. 2(b-right). Hence, we can rewrite the equations of motion as shown in Fig. 2c:
$$\begin{gathered} {{\dot {a}}_1}= - \left( {\frac{{{\kappa _1}}}{2}+i{\Delta _a}} \right){a_1} - iGb - {r_1}{\left| {{a_1}} \right|^2}{a_1} - i{r_2}{a_1}^{*}, \hfill \\ \dot {b}= - \left( {\frac{\Gamma }{2}+i{\Delta _b}} \right)b - iG{a_1}, \hfill \\ \end{gathered}$$
1
where \({r_1}\) is the gain saturation coefficient and \({r_2}\) is the small-signal gain due to parametric amplification. Additionally, the detuning coefficients are given by\(\,\,{\Delta _a}={\omega _1} - {{{\omega _p}} \mathord{\left/ {\vphantom {{{\omega _p}} 2}} \right. \kern-0pt} 2}\) and \({\Delta _b}={\omega _b} - {{{\omega _p}} \mathord{\left/ {\vphantom {{{\omega _p}} 2}} \right. \kern-0pt} 2}\), while \({\kappa _1}\) and \(\Gamma\) are the loss rates of \({a_1}\) and b; hence, the mechanical quality factor is \(Q \equiv {{{\omega _b}} \mathord{\left/ {\vphantom {{{\omega _b}} \Gamma }} \right. \kern-0pt} \Gamma }\). We can simplify Eqs. (1), combining them into a single equation by integrating the coupling term:
$${\dot {a}_1}= - \left( {\frac{\kappa }{2}+i{\Delta _a}} \right){a_1} - {r_1}{\left| a \right|^2}{a_1} - i{r_2}{a_1}^{*}+F\left( t \right),$$
2
where \(F\left( t \right) \equiv - {G^2}\int\limits_{0}^{t} {d\tau } f\left( \tau \right){a_1}\left( {t - \tau } \right)\) represents the external stimuli applied to \({a_1}\), and \(f\left( \tau \right) \equiv {e^{ - \left( {\frac{\Gamma }{2}+i{\Delta _b}} \right)\tau }}\) is the linear response function of the mechanical mode [43]. The stimuli\(F\left( t \right)\) plays a crucial role in the generation of frequency combs, with comb generation contingent upon the strength of the coupling rate G as discussed below.
In the weak electromechanical coupling rate regime \(\left( {G<{\Gamma \mathord{\left/ {\vphantom {\Gamma 2}} \right. \kern-0pt} 2}} \right)\), the last term in Eq. (2) simplifies to \(F\left( t \right) \approx - \frac{{{G^2}}}{{{\Gamma \mathord{\left/ {\vphantom {\Gamma 2}} \right. \kern-0pt} 2}+i{\Delta _b}}}{a_1}\left( t \right)\). Thus, Eq. (2) is characterized by an effective loss \(\bar {\kappa }=\kappa +\frac{{{G^2}}}{{{{\left( {{\Gamma \mathord{\left/ {\vphantom {\Gamma 2}} \right. \kern-0pt} 2}} \right)}^2}+{\Delta _b}^{2}}}\Gamma\) and an effective detuning \({\bar {\Delta }_a}={\Delta _a} - \frac{{{G^2}}}{{{{\left( {{\Gamma \mathord{\left/ {\vphantom {\Gamma 2}} \right. \kern-0pt} 2}} \right)}^2}+{\Delta _b}^{2}}}{\Delta _b}\). Linear stability theory can be utilized to further analyze Eq. (2) using the perturbative series \({a_1}\left( t \right) \to \alpha +\delta {a_1}\left( t \right)\), in order to derive the pole of the system (see Supplementary Section II). For large pump power levels, i.e., \({r_1}^{2}{\left| \alpha \right|^4}+{r_2}>{\overline {\Delta } _a}^{2}\), the real part of the pole is \(\operatorname{Re} \left[ s \right]= - \frac{{\overline {\kappa } }}{2} - 2{r_1}{\left| \alpha \right|^2} \pm \sqrt {{{\left( {{r_1}^{2}{{\left| \alpha \right|}^4}+{r_2}} \right)}^2} - {{\overline {\Delta } }_a}^{2}}\) while the imaginary part is zero. Consequently, a single-frequency oscillation is observed at \({{{\omega _p}} \mathord{\left/ {\vphantom {{{\omega _p}} 2}} \right. \kern-0pt} 2}\), as shown in Fig. 2(d-left). The synchronization of the signal mode \({a_1}\) at half frequency of the pump results from the typical response of degenerate parametric systems, which do not exhibit multiple oscillations at frequencies uncorrelated from their pump frequency [50]. Thus, frequency combs cannot be generated by our system for G < Γ/2.
On the other hand, when the system is operated in the strong coupling regime \(\left( {G>{\Gamma \mathord{\left/ {\vphantom {\Gamma 2}} \right. \kern-0pt} 2}} \right)\), the investigation of its stability requires applying the Laplace transform to Eq. (2) after its linearization. This allows to verify that frequency combs are generated at \(\operatorname{Re} \left[ s \right]=0\), with a maximum comb spacing \(\Delta {f_{\hbox{max} }}=2\sqrt {{G^2} - {{\left( {{\Gamma \mathord{\left/ {\vphantom {\Gamma 2}} \right. \kern-0pt} 2}} \right)}^2}}\)that is obtained when the pump reaches the threshold for comb generation (i.e., when \({P_r}={P_{th}}\)). Such combs cannot be observed in the weak coupling regime. Figure 2(d) illustrates the numerically evaluated spectrum of \({a_1}\) at \(\operatorname{Re} \left[ s \right]=0\) where the comb starts emerging, and the analytically calculated maximum spacing is remarkably close to the corresponding numerically extracted value. By increasing the small signal gain \({r_2}\), the pole can be further pushed above the real axis, causing the comb spacing to decrease and progressively driving the system into a synchronized period-doubling regime. In other words, the synchronization is frustrated for a wide range of \({r_2}\) under the strong coupling regime. In this range, frequency combs are observed and the comb-line spacing is a monotonic function of \({r_2}\), which is the key feature to produce ranging measurements without ambiguities. More details on our analytical model are provided in Supplementary Section II.