Frequency-Based Location (i.e. Doppler Location)Single Platform Emitter LocationAOA(DF) FOA Interferometery TOASBI LBIEmitter Location is Two Estimation Problems in One:1) Estimate Signal Parameter(s) that Depend on Emitter’s Location:a) Time-of-Arrival (TOA) of Pulsesb) Phase Interferometery: Phase is measured between two different signals received at nearby antennas•SBI– Short Baseline Interferometery (antennas are close enough together that phase is measured without ambiguity)•LBI– Long Baseline Interferometery (antennas are far enough apart that phase is measured with ambiguity; ambiguity resolved either using processing or so-called self-resolved)c) Frequency-of-Arrival(FOA) or Dopplerd) Angle-of-Arrival (AOA)2) Use Signal Parameters Measured at Several Instants to Estimate LocationFrequency-Based Location (i.e. Doppler Location)The Problem• Emitter assumed non-moving and at position (X,Y,Z)– Transmitting a radar signal at unknown carrier frequency is fo• Signal is intercepted by a receiver on a single aircraft– A/C dynamics are considered to be perfectly known as a function of time• Nav Data: Position Xp(t), Yp(t), Zp(t) and Velocity Vx(t), Vy(t), Vz(t)• Relative motion between the Tx and Rx causes Doppler shift– Received carrier frequency differs from transmitted carrier frequency– Thus, the carrier frequency of the received signal will change with time • For a given set of nav data, how the frequency changes dependS on the transmitter’s carrier frequency foand the emitter’s position (X,Y,Z)– Parameter Vector: x = [X Y Z fo]T– fo is a “nuisance” parameter• Received frequency is a function of time as well as parameter vector x()()()()()())1()()()()()()()()()(),(222−+−+−−+−+−−=ZtZYtYXtXZtZtVYtYtVXtXtVcfftfppppzpypxoox• Make noisy frequency measurements at t1, …, tN:• Problem: Given noisy frequency measurements and the nav data, estimate x• What PDF model do we use for our data????)(),(),(~iiitvtftf += xxIn the TDOA/FDOA case… we had an ML estimator for TDOA/FDOA so we could claim that the measurements were asymptotically Gaussian. Because we then had a well-specified PDF for the TDOA/FDOA we could hope to use ML for the location processing. However, here we have no ML estimatorfor the instantaneous frequency so claiming that the inst. freq. estimates are Gaussian is a bit of a stretch.So we could:1. Outright ASSUME Gaussian and then use ML approach2. Resort to LS… which does not even require a PDF viewpoint!Both paths get us to the exact same place:Find the estimate that minimizes∑=−=NieieietftfJ12)]ˆ,(),(~[)ˆ( xxxIf we Assume Gaussian… we could choose: • Newton-Raphson MLE approach: leads to double derivatives of the measurement model f (ti,xe).If we Resort to LS… we could choose either:• Newton-Raphson approach, which in this case is identical to N-R under the Gaussian assumption• Gauss-Newtonapproach, which needs only first derivatives of the measurement model f (ti,xe).We’ll resort to LS and use Gauss-NewtonTimeMeasured FrequencyFrequency ComputedUsing Measured Navand Poor Assumed Loc.Frequency ComputedUsing Measured Navand Good Assumed Loc.∑=−=NiiitftfJ12)]ˆ,(),(~[ xxLS Approach: Find the estimate such that the corresponding computed frequency measurements are “close” to the actual measurements:– Minimize xˆ)ˆ,( xitfThe SolutionMeasurement model in (1) is nonlinear in x ! no closed form solution– Newton-Raphson: Linearize the derivative of the cost function– Gauss-Newton: Linearize the measurement modelThus: ! (A Linear Model)where…Get LS solution for update and then update current estimate:[]vxxHxfxf +−+≈nnˆ)ˆ()(vxHxf +∆≈∆ )ˆ(n~[]4321ˆ|||),( hhhhxtxHxx=∂∂==nf())ˆ(ˆ111nTTxfRHHRHx ∆=∆−−−xxxˆˆˆ1∆+=+ nnUnder the condition that the frequency measurement errors are Gaussian, then the CRLB for the problem can be shown to be ()11}var{−−≥ HRHxTCan use this to investigate performance under geometries of interest…even when the measurement errors aren’t truly GaussianThe AlgorithmInitialization: • Use the average of the measured frequencies as an initial transmitter frequency estimate. • To get an initial estimate of the emitter’s X,Y,Z components there are several possibilities:– Perform a grid search– Use some information from another sensor (e.g., if other on-board sensors can give a rough angle use that together with a typical range)– Pick several typical initial locations (e.g., one in each quadrant with some typical range)• Let the initial estimate be]ˆˆˆˆ[ˆ0,0000 ofZYX=xIteration: For n = 0, 1, 2, …1. Compute the vector of predicted frequencies at times {t1, t2, …, tN} using the current nthestimate and the nav info:()()()()()()−+−+−−+−+−−=222,,ˆ)(ˆ)(ˆ)(ˆ)()(ˆ)()(ˆ)()(ˆˆ)ˆ,(ˆnjpnjpnjpnjpjznjpjynjpjxnononjZtZYtYXtXZtZtVYtYtVXtXtVcfftf x[]TnNnnntftftf )ˆ,(ˆ)ˆ,(ˆ)ˆ,(ˆ)ˆ(21xxxxf !=2. Compute the residual vector by subtracting the predicted frequency vector from the measured frequency vector: )ˆ(ˆ)(~)ˆ(nnxfxfxf −=∆3. Compute Jacobian matrix H using the nav info and the current estimate:[]4321ˆ|||),( hhhhxtxHxx=∂∂==nf,)(ˆ)(ˆ)(ˆ)(ˆˆ)()(ˆˆ)()(ˆˆ)()(ˆ222jnjnjnjnnjpjnnjpjnnjpjntZtYtXtRZtZtZYtYtYXtXtX∆+∆+∆=−=∆−=∆−=∆Define:[]∆+∆+∆∆+−−=∂∂==3ˆ1ˆ)(ˆ)()(ˆ)()(ˆ)()(ˆˆ)(ˆ),()(jjnjzjnjyjnjxjnjjxojRtZtVtYtVtXtVtXRtVcftfXjnxxxh[]∆+∆+∆∆+−−=∂∂==3ˆ2ˆ)(ˆ)()(ˆ)()(ˆ)()(ˆˆ)(ˆ),()(jjnjzjnjyjnjxjnjjyojRtZtVtYtVtXtVtYRtVcftfXjnxxxh[]∆+∆+∆∆+−−=∂∂==3ˆ3ˆ)(ˆ)()(ˆ)()(ˆ)()(ˆˆ)(ˆ),()(jjnjzjnjyjnjxjnjjzojRtZtVtYtVtXtVtZRtVcftfZjnxxxh1),()(ˆ4≈∂∂==njotffjxxxh4. Compute the estimate update:())ˆ(ˆ111nTTnxfCHHCHx ∆=∆−−−C is the covariance of the frequency measurements; usually assumed to be diagonal with measurement variances on the diagonalIn practice you would implement this inverse using Singular Value Decomposition (SVD) due to numerical issues of H being near to singular (MATLAB will give you a warning when this is a problem)See pp. 676-677 of the book “Numerical Recipes …”5. Update the estimate usingnnnxxxˆˆˆ1∆+=+6. Check for convergence of solution: look to see if update is small in some specified sense.If