RF Source Seeking using Frequency Measurements
aa r X i v : . [ ee ss . SP ] M a r RF Source Seeking using Frequency Measurements
Muhammed Faruk Gencel ∗ , Upamanyu Madhow † , Jo˜ao Pedro Hespanha ‡ Department of Electrical and Computer EngineeringUniversity of California Santa BarbaraSanta Barbara, California 93106Email: ∗ [email protected], † [email protected], ‡ [email protected] Abstract —In this paper, we consider a problem motivatedby search-and-rescue applications, where an unmanned aerialvehicle (UAV) seeks to approach the vicinity of a distant quasi-stationary radio frequency (RF) emitter surrounded by localscatterers. The UAV employs only measurements of the Dopplerfrequency of the received RF signal, along with its own bearing,to continuously adapt its trajectory. We propose and evaluate atrajectory planning approach that addresses technical difficultiessuch as the unknown carrier frequency offset between the emitterand the UAV’s receiver, the frequency drifts of the local oscillatorsover time, the direction ambiguity in Doppler, and the noise inthe observations. For the initial trajectory, the UAV estimatesthe direction of the emitter using a circular motion, whichresolves direction ambiguity. The trajectory is then continuouslyadapted using feedback from frequency measurements obtainedby perturbing the bearing around the current trajectory. Weshow that the proposed algorithm converges to the vicinity ofthe emitter, and illustrate its efficacy using simulations.
Index Terms —Source seeking, Maximum Doppler estimation,RF trail following, UAV navigation
I. I
NTRODUCTION
We consider a scenario motivated by search-and-rescue, orother emergency applications, where a UAV seeks to approachan RF source, starting from an initially large distance.TheUAV is equipped with a single omnidirectional antenna, anddoes not rely on GPS or on being able to decode messagesfrom the emitter. The source may be surrounded by localscatterers. In the approach proposed here, the UAV adapts itstrajectory towards the emitter using frequency measurementson the received beacon. In an ideal line of sight (LoS), a singleomni-directional antenna can extract the angle of the arrival θ between the velocity vector of the mobile node and LoS tothe source by measuring the Doppler frequency f d = v cos θc f c ,where v is the velocity and c is the speed of light. Thus, anatural approach is for the UAV to follow the trajectory thatmaximizes the Doppler shift (which corresponds to θ = 0 ).However, translating this intuition into a working approachrequires that we address the following technical challenges:1) The scattering environment around the source causes multi-path fading, resulting in large spatial variations of the receivedsignal power. This can often lead to errors in frequencymeasurements, especially at the low received signal-to-noiseratio (SNR) obtained at large distances.2) The local oscillators at the emitter and UAV are not synchro-nized, and drift over time. Thus, the frequency measurementsmade by the UAV are a sum of the Doppler shift and a slowlydrifting carrier frequency offset. 3) Even in ideal LoS settings, Doppler estimates have directionambiguity: if the trajectory makes an angle θ with the LoS,then the Doppler shift is proportional to cos θ , and cannottherefore enable us to distinguish between + θ and − θ .4) Any trajectory adaptation done by the UAV should befeasible, avoiding sharp direction changes.The main contribution of this paper is to show that wecan indeed overcome the preceding difficulties to obtain ascheme that reaches the vicinity of the emitter, with netdistance traversed being only a small fraction (of the orderof 10%) larger than the initial LoS distance between theUAV and the emitter. We consider a small UAV that flies ataround 100 m altitude, listening to a beacon in commercialfrequency bands (the carrier frequency is set to 2 GHz in ournumerical examples). The initial distance between the UAVand the source is of the order of 5 km. In our proposedapproach, the UAV obtains an initial trajectory estimate byfinding the direction of maximum Doppler when executing acircular motion. Subsequently, it employs feedback control tocontinuously adapt its trajectory, using the change in measuredfrequency offset as it executes designed piecewise lineardeviations in bearing from the nominal trajectory.The proposed approach performs significantly better thanRF source following using received signal strength (RSS)measurements [1]. While RSS measurements are simpler tomake and do not require coherent processing at the receiver,the sensitivity of RSS change as a function of range to theemitter is small even in ideal settings, since it is proportionalto the inverse square of the range. In addition, local scatterersaround the emitter lead to slow, and deep, spatial variationsin RSS due to fading. The approach in [1] employs theobservation that the rate of change of RSS due to fading isminimum in the LoS direction, along with a random walkinspired by bacterial chemotaxis. For a setting similar to ours,the RSS-based scheme requires the UAV to traverse a distancethat is about three times larger than the shortest path betweenthe initial UAV location and the emitter. Furthermore, thetrajectories employed in [1], both for initialization and for therandom walk, are non-smooth and difficult to execute.RSS measurements are more effective if supplemented with directional information. A rotating UAV was employed in[2], with the angle of arrival to the emitter estimated as thedirection of maximum RSS. This approach is not applicable tofixed wing UAVs with omnidirectional antennas as consideredhere.hile we consider the problem of approaching the emitter,there is a significant literature on localizing the emitter usinga mobile platform [3], [4] or multiple mobile platforms [5],[6]. Particle filter based algorithms for tracking the posteriordistribution of the emitter are investigated in [7], [8]. Inaddition to having a different design goal from ours, it isworth noting that these approaches require that the mobileplatform always knows its own absolute location, say usingGPS. Our problem formulation requires the UAV to trackchanges in its own bearing, but does not require that it know itsabsolute location, and hence is applicable even in GPS-deniedenvironments. Fig. 1. 2D system model, with scatterers in a disk around the source
II. S
YSTEM M ODEL
The problem of drawing the UAV to the emitter locationis three-dimensional, but we restrict the problem to twodimensions for simplicity. As shown in Figure 1, the sourceis located at the the origin of the 2D plane, and is surroundedby L local scatterers. The UAV coordinates at any giventime are denoted by p = ( x, y ) with velocity components v x = v cos( φ ) and v y = v sin( φ ) . The distance between themobile receiver and the source is d . The scatterers are insidean annulus with outer radius R and inner radius R in , and theangles { α i } are uniformly distributed between − π and π foreach scatterer.We consider a narrowband flat fading channel at carrierfrequency f c , and assume that the receiver will compute itsestimates based on known pilot signals transmitted by thesource. The complex baseband channel seen by the mobilenode can be expressed as the sum of LoS and scattered components and can be written as h ( t ) = s Kσ h K + 1 e j (2 πf d,max cos( α − φ ) t +2 πf o ( t ) t + ψ ) + s σ h K + 1 L X i =1 e j (2 πf d,max cos( α si − φ ) t +2 πf o ( t ) t + ψ i ) + n ( t ) (1)where K is the ratio of power between the direct path andthe scattered paths, f d,max = vf c c is the maximum Dopplerfrequency, f o ( t ) is the carrier frequency offset drifting overtime, ψ and ψ i are the phase of LoS and scattered signalcomponents respectively, σ h is the received signal power and n ( t ) represents the additive noise at the receiver with thevariance σ n .The received signal strength σ h is governed by the distancebetween source and mobile receiver d , along with spatialvariations due to multipath fading. The effect of Dopplerfrequency on the received signal profile is negligible at theselow speeds since f c ≫ f d,max . We model the received signalstrength by modeling the electric field at the mobile node ata point P in polar coordinates ( d, α ) as [1]: EF ( d, α ) = e − jβd d + L X i =1 Γ i e − jβ ( d i + r i ) d i + r i (2)where β = πλ and λ is the wavelength, d i + r i is the totaldistance of the path that goes from the source to receiverthrough i th scatterer and Γ i is the reflection coefficient forthe i th scatterer [9].Frequency estimation accuracy in a flat fading channel,assuming all paths have the same frequency offset, is propor-tional to the received SNR [10]. This assumption is a goodapproximation for our model when d ≫ R . We use low-complexity single tone frequency estimation [11], selecting themaximum frequency over the DFT grid, and then interpolatingusing a quadratic fit. As the UAV gets closer to the source,each path sees a different frequency offset due to the differencein the reflection angles, and the frequency estimate degrades.Estimation accuracy in this region could potentially be furtherimproved with frequency estimates derived from second orderstatistics [12], or by employing super-resolution techniques[13]. However, at shorter ranges, more sophisticated frequencyestimation should be coupled with more detailed anisotropicreflection models, hence we leave this as an interesting topicfor future work (e.g., on how to track a moving emitter inurban canyons).III. A LGORITHM FOR S OURCE S EEKING
In this section, we describe and justify the strategy forplanning the UAV trajectory by using frequency measurementsand show that the proposed algorithm will converge to thetrue source direction with no prior information on the sourcelocation. The purpose is to draw UAV to the vicinity of theemitter as quickly as possible. We set d v ≪ d as the requireddistance between UAV and the source at which we declarehe tracking process successful. We assume constant speedthrough the trajectory of the UAV and use a feasible trajectoryfor the motion of UAV. The goal is to minimize flight time.We assume prior knowledge of the emitted signal carrierfrequency f c , but not of the carrier frequency offset f o , whichalso drifts over time. The pilot beacon for a given frequencymeasurement contains N symbols, with symbol period T s ,so that the measurement interval for frequency estimation is T = N T s . The pilot beacons are repeated with the period of T slot .For the n th received beacon, frequency measurements areobtained by applying FFT to the N complex baseband samplesof (1), and the peak frequency ˆ ω i i ∈ , · · · , N F F T is refinedby using a quadratic interpolation with adjacent samples: ˜ ω n = ˆ ω i + ˆ ω i − − ˆ ω i +1 ω i − + ˆ ω i +1 − ω i ) 2 πT s N F F T (3)Thus, we obtain a noisy estimate of the sum of carrierfrequency offset, Doppler frequency and the frequency drift.We model this, together with the bearing angle measured bythe UAV sensors, as ˜ ω n = ω n + n ω,n ˜ φ n = φ n + n φ,n . (4)where n ω,n and n φ,n are frequency measurement and bear-ing measurement noises, modeled as zero mean independentGaussian random variables with variances σ ω,n and σ φ,n ,respectively. The bearing measurement error variance σ φ,n is assumed to be constant throughout the flight. However,the frequency measurement error variance σ ω,n can vary: itincreases as RSS drops during fades, and as Doppler spreadincreases as the UAV approaches the emitter.While we do not model the UAV dynamics, we will restrictthe algorithm described in the next section to use trimmingtrajectories, which have the desirable property that the trackingerror dynamics and kinematics is time invariant and for whichthere are well-developed trajectory tracking controllers [14]–[16]. The orientation tracking error about the desired trimmingtrajectory is included in error parameter n φ,n . A. Trajectory adaptation
The source tracking algorithm can be divided into twostages, discussed in more detail below. In the first stage, theUAV gets a rough estimate of the direction of the source bydoing a circular motion. This stage is optional, and can beremoved at the expense of some inefficiency in flight time.The second stage involves piecewise linear trajectories withperturbations of bearing, with change in Doppler providingfeedback signal for continuous trajectory corrections.
Stage 1 - Circular motion for initial trajectory estimate:
The UAV picks a random point at a distance R c and follows acircular trajectory, as shown in Figure 2, saving the frequencymeasurements ˜ ω n with corresponding bearing measurements ˜ φ n . The largest frequency measurement corresponds to themaximum Doppler f d,max and bearing angle that correspondsto the desired direction is approximately π + α in an ideal setting. The smallest frequency measurements corresponds tothe − f d,max at the direction of α .The SNR is low when UAV is very distant, and multipathfading may occasionally result in large outliers in the fre-quency measurements ˜ ω n . We apply outlier rejection to thefrequency measurements as ˇ ω n = ( ˜ ω n , | ˜ ω n − ˜ ω n − | < π vf c c ˜ ω n − , otherwise (5)and apply a moving average filter with length T slot . Then,the initial direction for UAV is determined by finding thedirection of maximum frequency estimate as follows i = argmax n ˇ ω n θ = ˜ φ i . (6) Fig. 2. Initial circular motion of the UAV
Stage 2 - Continuous updates:
In this stage, the UAVderives information for feedback control of its trajectory indiscrete time steps spanning M T slot for each step. If theestimated direction towards the emitter is θ k from the previousdirection, the UAV moves in the direction θ k + δ k for a timeinterval with length M T slot , yielding frequency measurements { ˇ ω m , m = 1 , ..., M } , and then in the direction θ k − δ k for the same duration, yielding measurements { ˇ ω m , m = M + 1 , ..., M } . The difference between these two sets offrequency measurements is used to update θ k , as follows: θ k +1 = θ k + 1 M (cid:0) M X m =1 ˇ ω m − M X m = M +1 ˇ ω m (cid:1) δ k πf d,max . (7)Taking the difference in this fashion allows significant reduc-tion of the effect of carrier frequency offset and drift, whichvary slowly relative to the iteration step duration M T slot . Ad-ditional robustness against measurement noise can be obtainedby increasing the perturbation δ k , at the cost of increased traveldistance. . Analysis We now analyze the convergence of the algorithm in stage2. Straightforward trigonometry shows that the update step inequation (7) with a constant δ k = δ can be written as θ k +1 = θ k + (cos( θ k − θ ∗ k + δ ) − cos( θ k − θ ∗ k − δ )) δ = θ k − θ k − θ ∗ k ) sin ( δ ) δ. (8)Let the error term ˜ θ k = θ k − θ ∗ k . For d ≫ || p k +1 − p k || (rangemuch larger than the distance between consecutive iterations),we have θ ∗ k ≈ θ ∗ k +1 , which yields ˜ θ k +1 = ˜ θ k − θ k ) sin ( δ ) δ (9)Intuitive insight is obtained for small ˜ θ k and δ by using theapproximation sin x ≈ x : ˜ θ k +1 ≈ ˜ θ k (1 − δ ) , correspondingto exponential decrease in estimation error.For a rigorous proof of convergence, we pick ˜ θ k as aLyapunov function. From (9), we obtain that the change inone time step is given by ˜ θ k +1 − ˜ θ k = − α sin ˜ θ k (˜ θ k − α sin ˜ θ k ) where α = δ sin δ . Note that sin x ( x − α sin x ) > for < | x | < π and < α < . Thus, ˜ θ k is a strictly decreasingfunction for α < , which provides a wide range of choicesfor δ . Fig. 3. Trajectory updates using direction perturbations ± δ k IV. S
IMULATION R ESULTS
The simulation parameters are given in Table I. We apply N F F T = 4096 point FFT to N = 1000 data chunks inevery T slot = 50 ms for frequency estimation. The averagereceived SNR at the initial distance of 5 km is set to 0 dB.Figure 4 shows an example UAV trajectory. Figure 5 showsthe estimated frequency in the presence of multipath, CFOand frequency drift for that particular trajectory. Figure 5 alsoshows the received signal power profile through the trajectoryand the spatial variations at the received power. We observethat the frequency estimation error increases as the UAV getscloser to the source due to increased Doppler spread.Figure 6 shows the histogram of the total distance traveledwith Monte Carlo simulations of 1000 runs for the samescenario. The average distance traveled is 5.5 km, which is ParametersParameter Symbol Valued 5000 mR 200 m R in
100 m f c σ n -70 dB T slot
50 ms T
10 ms T s
10 us N F F T δ o d v mM TABLE IS
IMULATION PARAMETERS -500 0 500 1000 1500 2000 2500 3000 3500 x (m) -500050010001500200025003000350040004500 y ( m ) System model
SourceMobile node initialScatterersMobile node path
Fig. 4. Example trajectory (0 dB average initial SNR, initial distance 5 km).
V. C
ONCLUSIONS
We have shown that a UAV with a single omnidirectionalantenna can approach an RF source using only frequency andbearing measurements, in a manner that is robust to multipathfading (via rejection of outliers in frequency measurements)and carrier frequency offset and drift (via averaging anddifferencing frequency measurements over relatively shortintervals). Our analysis shows exponential convergence to thecorrect approach angle towards the source. While the receiveroperations required are more sophisticated than required forextracting RSS, the performance is far superior to that ofa previously proposed RSS-based scheme. There are severalinteresting directions for future work, including improvedalgorithms for determining the maximum Doppler, especially
100 200 300 400 500 600 time (s) -300-200-1000100200 f r equen cy ( H z ) Frequency measurements over time frequency estimatefrequency drift time (s) -140-120-100-80-60-40-20 r e c e i v ed po w e r ( d B ) Received signal power over time
Fig. 5. Frequency measurements and RSS for the route in Figure 4 distance traveled (m)
Mean path length = 5504.60 meters
Fig. 6. Histogram of total distance traveled to get to within 200m of thesource (mean ∼ . km) in the presence of Doppler spread; detailed modeling ofthe propagation environments at shorter range, in order tounderstand the impact on both the frequency measurementsand the trajectory updates (e.g., if the LoS is blocked, theUAV may follow a strong reflected path until it sees a LoSpath again); and more detailed accounting of UAV dynamics.A CKNOWLEDGMENTS
This research was supported by the National Science Foun-dation under grant CCF-1302114.R
EFERENCES[1] A. Wadhwa, U. Madhow, J. Hespanha, and B. M. Sadler, “Followingan RF trail to its source,” in
Communication, Control, and Computing(Allerton), 2011 49th Annual Allerton Conference on . IEEE, 2011, pp.580–587.[2] S. Venkateswaran, J. T. Isaacs, K. Fregene, R. Ratmansky, B. M. Sadler,J. P. Hespanha, and U. Madhow, “RF source-seeking by a micro aerialvehicle using rotation-based angle of arrival estimates,” in
AmericanControl Conference (ACC), 2013 . IEEE, 2013, pp. 2581–2587. distance traveled (m)
Mean path length = 5985.57 meters
Fig. 7. Histogram of total distance traveled to get to within 200m of thesource without stage 1 (mean ∼ km)[3] K. Becker, “Passive localization of frequency-agile radars from angleand frequency measurements,” IEEE Transactions on Aerospace andElectronic Systems , vol. 35, no. 4, pp. 1129–1144, 1999.[4] N. H. Nguyen and K. Do˘ganc¸ay, “Single-platform passive emitter local-ization with bearing and doppler-shift measurements using pseudolinearestimation techniques,”
Signal Processing , vol. 125, pp. 336–348, 2016.[5] A. Amar and A. J. Weiss, “Localization of narrowband radio emittersbased on doppler frequency shifts,”
IEEE Transactions on Signal Pro-cessing , vol. 56, no. 11, pp. 5500–5508, 2008.[6] A. Tahat, G. Kaddoum, S. Yousefi, S. Valaee, and F. Gagnon, “A lookat the recent wireless positioning techniques with a focus on algorithmsfor moving receivers,”
IEEE Access , vol. 4, pp. 6652–6680, 2016.[7] H. Witzgall, B. Pinney, and M. Tinston, “Doppler geolocation withdrifting carrier,” in
MILITARY COMMUNICATIONS CONFERENCE,2011-MILCOM 2011 . IEEE, 2011, pp. 193–198.[8] H. Witzgall, J. Covington, and A. Pierce, “Single aircraft passive dopplerlocation of radios,” in
Aerospace Conference, 2015 IEEE . IEEE, 2015,pp. 1–8.[9] A. Aragon-Zavala,
Antennas and propagation for wireless communica-tion systems . John Wiley & Sons, 2008.[10] V. M. Baronkin, Y. V. Zakharov, and T. C. Tozer, “Cramer-rao lowerbound for frequency estimation in multipath rayleigh fading chan-nels,” in
Acoustics, Speech, and Signal Processing, 2001. Proceed-ings.(ICASSP’01). 2001 IEEE International Conference on , vol. 4.IEEE, 2001, pp. 2557–2560.[11] D. R. Brown III, Y. Liao, and N. Fox, “Low-complexity real-time single-tone phase and frequency estimation,”
IEEE Military Communication ,2010.[12] M. Souden, S. Affes, J. Benesty, and R. Bahroun, “Robust doppler spreadestimation in the presence of a residual carrier frequency offset,”
IEEETransactions on Signal processing , vol. 57, no. 10, pp. 4148–4153, 2009.[13] B. Mamandipoor, D. Ramasamy, and U. Madhow, “Newtonized orthog-onal matching pursuit: Frequency estimation over the continuum.”
IEEETrans. Signal Processing , vol. 64, no. 19, pp. 5066–5081, 2016.[14] I. Kaminer, A. Pascoal, E. Hallberg, and C. Silvestre, “Trajectorytracking for autonomous vehicles: An integrated approach to guidanceand control,”
Journal of Guidance, Control, and Dynamics , vol. 21,no. 1, pp. 29–38, 1998.[15] D. R. Nelson, D. B. Barber, T. W. McLain, and R. W. Beard, “Vectorfield path following for miniature air vehicles,”
IEEE Transactions onRobotics , vol. 23, no. 3, pp. 519–529, 2007.[16] P. Sujit, S. Saripalli, and J. B. Sousa, “An evaluation of UAV pathfollowing algorithms,” in