Doppler TrackingProblem StatementAn AdmissionDoppler Shift ModelTrajectory ModelMeasurement Model and Estimation ProblemEstimation ProblemLinearize the Nonlinear ModelBack-Propagate to Get Predicted FrequenciesConverting to Linear LS Problem FormIterating to the Solution1/11Doppler TrackingPassive Tracking of an Airborne Radar:An Example of Least Squares “State” Estimation2/11Problem StatementAirborne radar to be located follows a trajectory X(t), Y(t), Z(t) with velocities Vx(t), Vy(t), Vz(t)It is transmitting a radar signal whose carrier frequency is fo. Signal is intercepted by a non-moving receiver at known location Xp, Yp, Zp.Problem: Estimate the trajectory X(t), Y(t), Z(t)Solution Here: • Measure received frequency at instants t1, t2, … , tN• Assume a simple model for the aircraft’s motion• Estimate model parameters to give estimates of trajectoryunknown3/11An AdmissionThis problem is somewhat of a “rigged” application…• Unlikely it would be done in practice just like this • Because it will lead to poorly observable parameters • The H matrix is likely to be less than full rank In real practice we would likely need either:• Multiple Doppler sensors or • A single sensor that can measure other things in addition to Doppler (e.g., bearing). We present it this way to maximize the similarity to the example of locating a non-moving radar from a moving platform • Can focus on the main characteristic that arises when the parameter to be estimated is a varying function (i.e. state estimation).4/11Doppler Shift ModelRelative motion between the emitter and receiver… Doppler ShiftFrequency observed at time t is related to the unknown transmitted frequency of foby:()()()()()()−+−+−−+−+−−=222)()()()()()()()()(tZZtYYtXXtZZVtYYtVtXXtVcfftfppppzpypxooWe measure this at time instants t1, t2, … , tN)()()(~iiitvtftf +=And group them into a measurement vector:Frequency Measurement “Noise”[]TNtftftf )(~)(~)(~~21!=fBut what are we trying to estimate from this data vector???5/11Trajectory ModelWe can’t estimate arbitrary trajectory functions… like X(t), Y(t), etc.Need a trajectory model… to reduce the problem to estimating a few parametersHere we will choose the simplest… Constant-Velocity ModelNNzNNyNNxZttVtZYttVtYXttVtX+−×=+−×=+−×=)()()()()()(Final Positions in Observation BlockVelocity ValuesNow, given measurements of frequencies f(t1), f(t2), … , f(tN) ……we wish to estimate the 7-parameter vector:TozyxNNNfVVVZYX ][=x6/11Measurement Model and Estimation ProblemSubstituting the Trajectory Model into the Doppler Modelgives our measurement model:[]()[]()[]()[]()[]()[]()+−×−++−×−++−×−+−×−++−×−++−×−−=222)()()()()()()(),(NNzpNNypNNxpNNzpzNNypyNNxpxooZttVZYttVYXttVXZttVZVYttVYtVXttVXVcfftf xDependence on parameter vector)(),(),(~tvtftf += xx[]vxfxxxxf+==)(),(~),(~),(~)(~21TNtftftf !Noisy Frequency MeasurementNoisy MeasurementVectorNoise-FreeFrequency VectorNoise Vector7/11Estimation ProblemGiven: Noisy Data Vector:Sensor Position: Estimate:Parameter Vector: []TNtftftf ),(~),(~),(~)(~21xxxxf !=pppZYX ,,TozyxNNNfVVVZYX ][=xThis is a nonlinear problem…Although we could use ML to attack this we choose LS here partly because we aren’t given an explicit noise model and partly because LS is “easily” applied here!!!“Nuisance” parameter8/11Linearize the Nonlinear ModelWe have a non-linear measurement model here…so we choose to linearize our model(as before):[]vxxHxfxf +−+≈nnˆ)ˆ()(~where…nxˆis the “current” estimate of the parameter vector)ˆ(nxfis the “predicted” frequencymeasurements computed using the Doppler & Trajectory models with “back-propagation” (see next)[]7654321ˆ||||||),( hhhhhhhxtxHxx=∂∂==nfis the N×7 Jacobian matrix evaluated at the current estimate9/11Back-Propagate to Get Predicted FrequenciesGiven the current parameter estimate:TozyxNNNnnfnVnVnVnZnYnX )](ˆ)(ˆ)(ˆ)(ˆ)(ˆ)(ˆ)(ˆ[ˆ=xBack-Propagate to get the current trajectoryestimate:)(ˆ)()(ˆ)(ˆ)(ˆ)()(ˆ)(ˆ)(ˆ)()(ˆ)(ˆnZttnVtZnYttnVtYnXttnVtXNNznNNynNNxn+−×=+−×=+−×=Use Back-Propagated trajectoryto get predicted frequencies:()()()()()()−+−+−−+−+−−=222)(ˆ)(ˆ)(ˆ)(ˆ)(ˆ)(ˆ)(ˆ)(ˆ)(ˆ)(ˆ)(ˆ)ˆ,(tZZtYYtXXtZZnVtYYnVtXXnVcnfnftfnpnpnpnpznpynpxoonix10/11Converting to Linear LS Problem FormFrom the linearized model and the back-propagated trajectory estimate we get:"vxHxfxxxfxf+∆≈∆−−nnnnˆ)ˆ()(~)ˆ(#$#%&“Update” Vector“Residual” VectorThis is in standard form of Linear LS… so the solution is:())ˆ(ˆ111nTTnxfRHHRHx ∆=∆−−−This LS estimated “update” is then used to get an updated parameter estimate:nnnxxxˆˆˆ1∆+=+R is the covariance matrix of the measurements11/11Iterating to the Solution• n=0: Start with some initial estimate• Loop until stopping criterion satistfied–n ← n+1– Compute Back-Propagated Trajectory– Compute Residual– Compute Jacobian– Compute Update– Check Update for smallness of norm• If Update small enough… stop• Otherwise, update estimate and