1MLE for TDOA/FDOA Location•Overview• Estimating TDOA/FDOA• Estimating Geo-Location2)(tstjetts1)(1ω−Data LinkData Linktjetts2)(2ω−tjetts3)(3ω−MULTIPLE-PLATFORM LOCATIONEmitter to be located3)(tstjetts1)(1ω−Data LinkData Linkτ23= t2–t3= constantTDOATime-Difference-Of-Arrivalτ21 = t2 – t1 = constanttjetts2)(2ω−tjetts3)(3ω−ν21 = ω2 – ω1 = constantν23 = ω2 – ω3 = constantFDOAFrequency-Difference-Of-ArrivalTDOA/FDOA LOCATION4Estimating TDOA/FDOA5SIGNAL MODEL! Will Process Equivalent Lowpass signal, BW = B Hz– Representing RF signal with RF BW = B Hz! Sampled at Fs > B complex samples/sec! Collection Time T sec! At each receiver:BPFADCMakeLPE SignalEqualizecos(ω1t)fXRF(f)X(f)ffXfLPE(f)B/2-B/26TxRxs(t) sr(t) = s(t – τ(t))R(t)Propagation Time: τ(t) = R(t)/c!+++=2)2/()( tavtRtRoUse linear approximation – assumes small change in velocity over observation interval)/]/1([)/][()( cRtcvscvtRtstsoor−−=+−=Time ScalingTime Delay: τdFor Real BP Signals:DOPPLER & DELAY MODEL7Analytic Signals Model)]([)()(~ttjcetEtsφ+ω=Now what? Notice that v << c " (1 – v/c) ≈ 1Say v = –300 m/s (–670 mph) then v/c = –300/3x108= –10-6 " (1 – v/c)=1.000001Now assume E(t) & φ(t) vary slowly enough that )()]/1([)()]/1([ttcvtEtcvEφ≈−φ≈−For the range of vof interestDOPPLER & DELAY MODEL (continued)Analytic Signal of Tx)}]/1([)]/1([{)]/1([)]/1([~)(~ddctcvtcvjddretcvEtcvstsτ−−φ+τ−−ωτ−−=τ−−=Analytic Signal of RxCalled Narrowband Approximation8)()/()}()/({)()()(~dccdcddccctjdtjtcvjjttcvtjdretEeeeetEtsτ−φωω−τω−τ−φ+τω−ω−ωτ−=τ−=ConstantPhaseTermα= –ωcτdDopplerShiftTermωd= ωcv/cCarrierTermTransmitted Signal’sLPE SignalTime-Shifted by τdNarrowband Analytic Signal ModelNarrowband Lowpass Equivalent Signal Model)(ˆ)(ˆdtjjrtseetsdτ−=ω−αThis is the signal that actually gets processed digitallyDOPPLER & DELAY MODEL (continued)9CRLB for TDOAWe already showed that the CRLB for the active sensor case is:But here we need to estimate the delay between two noisy signals rather than between a noisy one and a clean one.The only difference in the result is: replace SNR by an effective SNR given by},min{1111212121SNRSNRSNRSNRSNRSNRSNReff≈++=∑∑−−=−−==12/2/212/2/222][][1NNkNNkrmsnSkSkNB2281)(rmsBSNRNTDOAC××=πwhere Brmsis an effective bandwidth of the signal computed from the DFT values S[k].10CRLB for TDOA (cont.)A more familiar form for this is in terms of the C-T version of the problem: SNRBT B 1 effrmsTDOA×≥22πσdffSdffSf Brms∫∫=2222)()(“seconds”BT = Time-Bandwidth Product (≈ N, number of samples in DT)B = Noise Bandwidth of Receiver (Hz)T = Collection Time (sec)BT is called “Coherent Processing Gain”(Same effect as the DFT Processing Gain on a sinusoid)For a signal with rectangular spectrum of RF width of Bs, then the bound becomes:SNRBT B effsTDOA×≥255.0σS. Stein, “Algorithms for Ambiguity Function Processing,” IEEE Trans. on ASSP, June 198111CRLB for FDOAHere we take advantage of the time-frequency duality if the FT:where Trmsis an effective duration of the signal computed from the signal samples s[k].∑∑−−=−−==12/2/212/2/222][][1NNnNNnrmsnsnskNT2281)(rmseffTSNRNFDOAC××=πAgain… we use the same effective SNR:},min{1111212121SNRSNRSNRSNRSNRSNRSNReff≈++=12CRLB for FDOA (cont.)A more familiar form for this is in terms of the C-T version of the problem: SNRBT T 1 effrmsFDOA×≥22πσdttsdttst Trms∫∫=2222)()(“Hz”For a signal with constant envelope of duration Ts, then the bound becomes:SNRBT T effsFDOA×≥255.0σS. Stein, “Algorithms for Ambiguity Function Processing,” IEEE Trans. on ASSP, June 198113Interpreting CRLBs for TDOA/FDOAA more familiar form for this is in terms of the C-T version of the problem: SNRBT T 1 effrmsFDOA×≥22πσSNRBT B 1 effrmsTDOA×≥22πσ• BT pulls the signal up out of the noise•Large Brmsimproves TDOA accuracy•Large Trmsimproves FDOA accuracySNR1SNR2T = TsB = BsσTDOAσFDOA3 dB 30 dB 1 ms 1 MHz 17.4 ns 17.4 Hz3 dB 30 dB 100 ms 10 kHz1.7 µs0.17 HzTwo Examples of Accuracy Bounds:14MLE for TDOA/FDOAS. Stein, “Differential Delay/Doppler ML Estimation with Unknown Signals,” IEEE Trans. on SP, August 1993We already showed that the ML Estimate of delay for the active sensor case is the Cross-Correlation of the time signals. By the time-frequency duality the ML estimate for doppler shift should be Cross-Correlation of the FT, which is mathematically equivalent todttstsCT∫+=021)()()(ττdtetstsCTtj∫−=021)()()(ωωdtetstsATtj∫−+=021)()(),(ωττωThe ML estimate of the TDOA/FDOA has been shown to be:Find Peak of |C(τ)|Find Peak of |C(ω)|Find Peak of |A(ω,τ)|15Ambiguity FunctionτωωdτdFindPeakof|A(ω,τ)|ML Estimator for TDOA/FDOA (cont.))(1 dtjjtseedτωα−=DelayτDopplerω“Compare”SignalsFor all Delays & Dopplers)(1ts)(2tsLPE RxSignalsAt Two ReceiversCalled:• Ambiguity Function• Complex Ambiguity Function (CAF) • Cross-Correlation Surface16ML Estimator for TDOA/FDOA (cont.)How well do we expect the Cross-Correlation Processing to perform?Well… it is the ML estimator so it is not necessarily optimum.But… we know that an ML estimate is asymptotically• Unbiased & Efficient (that means it achieves the CRLB)• Gaussian ))(,(~ˆ1θIθθ−NMLThose are some VERY nice properties that we can make use of in our location accuracy analysis!!!17• Consider when τ = τd[]∫ω−ω=τωTtjtjddteetsAd02)(),(like windowed FT of sinusoidwhere window is |s(t)|2ωωd|A(ω,τd)|width ∼ 1/T• Consider when ω = ωd∫τ+τ−=τωTdddttstsA0)()(),(correlation|A(ωd,τ)|ττdwidth ∼ 1/BWProperties of the CAF18TDOA Accuracy depends on:» Effective SNR: SNReff» RMS Widths: Brms= RMS BandwidthTDOA ACCURACY REVISITED dffSdffSf Brms∫∫=2222)()(XCorr Function~1/BrmsTDOANarrow BrmsCasePoor AccuracyWide BrmsCaseGood AccuracyTDOAXCorr FunctionLow Effective SNR Causes Spurious PeaksOn Xcorr FunctionNarrow Xcorr FunctionLess Susceptible to Spurious Peaks19FDOA Accuracy depends on:» Effective SNR: SNReff» RMS Widths: Drms= RMS DurationFDOA ACCURACY REVISITEDdffSdffSf Brms∫∫=2222)()(XCorr Function~1/DrmsFDOANarrow DrmsCasePoor AccuracyWide DrmsCaseGood AccuracyFDOAXCorr FunctionLow Effective SNR Causes Spurious PeaksOn Xcorr FunctionNarrow Xcorr FunctionLess Susceptible to Spurious Peaks20COMPUTING THE AMBIGUITY FUNCTIONDirect