7.10 MLE ExamplesEx. 1 Range Estimation ProblemRange Estimation D-T Signal ModelRange Estimation Likelihood FunctionRange Estimation ML ConditionRange Estimation MLE ViewpointEx. 2 Sinusoid Parameter Estimation ProblemSinusoid Parameter Estimation ML ConditionSinusoid Parms. Exact MLE ProcedureSinusoid Parms. Approx. MLE ProcedureEx. 3 Bearing Estimation MLE17.10 MLE ExamplesWe’ll now apply the MLE theory to several examples of practical signal processing problems.These are the same examples for which we derived the CRLB in Ch. 3 1. Range Estimation – sonar, radar, robotics, emitter location2. Sinusoidal Parameter Estimation (Amp., Frequency, Phase)– sonar, radar, communication receivers (recall DSB Example), etc.3. Bearing Estimation – sonar, radar, emitter location4. Autoregressive Parameter Estimation– speech processing, econometricsSee BookWe WillCover2Ex. 1 Range Estimation ProblemTransmit Pulse: s(t) nonzero over t∈[0,Ts]Receive Reflection: s(t –τo)Measure Time Delay:τomax,);(0)()()(ostsoTTttwtstxoτττ+=≤≤+−=!"!#$C-T Signal ModeltTss(t)tTs(t –τo)BandlimitedWhite GaussianBPF& Ampx(t)PSD of w(t)f B–BNo/23Range Estimation D-T Signal ModelSample Every ∆ = 1/2B secw[n] = w(n∆)DT White Gaussian NoiseVar σ2= BNof ACF of w(t)τ1/2BB–B1/B 3/2BPSD of w(t)No/2σ2= BNo1,,1,0][][][−=+−= Nnnwnnsnxo…s[n;no]… has M non-zero samples starting at nono ≈τo /∆−≤≤+−+≤≤+−−≤≤=1][1][][10][][NnMnnwMnnnnwnnsnnnwnxooooo4Range Estimation Likelihood FunctionWhite and Gaussian ⇒ Independent ⇒ Product of PDFs3 different PDFs – one for each subinterval23#1222#1221#1022212][exp2])[][(exp2][exp);(πσσσσ=−•−−−•−=∏∏∏−++=−+=−=CnxCnnsnxCnxCnpMnMnnMnnnonnoooooo!!!!&!!!!'(!!!!!!!&!!!!!!!'(!!!&!!!'(xExpand to get an x2[n] term… group it with the other x2[n] term−+−−−•−=∑∑−+=−=!!!!!!!"!!!!!!!#$!!!"!!!#$12022102])[][][2(21exp2][exp);(MnnnoNnNooonnsnnsnxnxCnpσσxmust minimize this or maximize its negative over values of noDoes not depend on no5Range Estimation ML Condition!!!"!!!#$!!!"!!!#$∑∑−+=−+=−+−1210][][][MnnnoMnnnoooonnsnnsnxDoesn’t depend on no! …Summand moves with the limits as nochanges. Because s[n – no] = 0 outside summation range… so can extend it!2So maximize this:∑−=−100][][NnnnsnxSo maximize this:So…. MLE Implementation is based on Cross-correlation: “Correlate” Received signal x[n] with transmitted signal s[n]{},][][][][maxargˆ100∑−=−≤≤−==NnxsxsMNmomnsnxmCmCn6Range Estimation MLE ViewpointmnoCxs[m],][][][10∑−=−=NnxsmnsnxmCDoesn’t depend on no! …Summand moves with the limits as nochanges. Warning: When signals are complex (e.g., ELPS) take find peak of |Cxs[m] |• Think of this as an inner product for each m• Compare data x[n] to all possible delays of signal s[n]! pick no to make them most alike7Ex. 2 Sinusoid Parameter Estimation ProblemGiven DT signal samples of a sinusoid in noise….Estimate its amplitude, frequency, and phase1,,1,0][)cos(][−=++Ω= NnnwnAnxo…φΩois DT frequency in cycles/sample: 0 < Ωo < πDT White Gaussian NoiseZero Mean & Variance of σ2Multiple parameters… so parameter vector:ToA ][φΩ=θThe likelihood function is:),,())cos(][(21exp);(1022φφσoNnoNAJnAnxCpΩ=+Ω−−=∆−=∑θxFor MLE: Minimize This8Sinusoid Parameter Estimation ML ConditionTo make things easier… Define an equivalent parameter set: [α1α2Ωo]Tα1= Acos(φ) α2= –Asin(φ)Then… J'(α1 ,α2,Ωo) = J(A,Ωo,φ) α = [α1α2]TDefine:c(Ωo) = [1 cos(Ωo) cos(Ωo2) … cos(Ωo(N-1))]Ts(Ωo) = [0 sin(Ωo) sin(Ωo2) … sin(Ωo(N-1))]Tand…H(Ωo) = [c(Ωo) s(Ωo)] an Nx2 matrix9Then: J'(α1 ,α2,Ωo)= [x – H (Ωo) α]T[x – H (Ωo) α]Looks like the linear model case… except for Ωodependence of H (Ωo)Thus, for any fixed Ωovalue, the optimal α estimate is[]xHHHα )()()(ˆ1oTooTΩΩΩ=−Then plug that into J'(α1 ,α2,Ωo):[][][][][][][]!!!!!!!"!!!!!!!#$!!!!!!!!"!!!!!!!!#$ooTooTooTooToTToTooToTooTTToTooJΩ−ΩΩΩΩ−=−ΩΩΩΩ−=ΩΩΩΩ−=Ω−Ω−=Ω−Ω−=Ω′− w.r.t.minimize1)()()()(2121)()()()()()()()(ˆ)()(ˆˆ)(ˆ)(),ˆ,ˆ(1xHHHHxxxxHHHHIxαHxHαxαHxαHxHHHHIαα10Sinusoid Parms. Exact MLE Procedure[]ΩΩΩΩ=Ω−≤Ω≤xHHHHx )()()()(minargˆ10oTooToTooπoΩˆStep 1: Minimize “this term” over Ωoto findStep 2: Use result of Step 1 to get []xHHHα )ˆ()ˆ()ˆ(ˆ1oTooTΩΩΩ=−Done NumericallyStep 3: Convert Step 2 result by solving)ˆsin(ˆˆ)ˆcos(ˆˆ21φαφαAA−==φˆ&ˆfor A11Sinusoid Parms. Approx. MLE ProcedureFirst we look at a specific structure:[]ΩΩΩΩΩΩΩΩΩΩΩΩ=ΩΩΩΩ−−xsxcsscsscccxsxcxHHHHx)()()()()()()()()()()()()()()()(11oToTooTooTooTooTToToToTooToT!!!!!!"!!!!!!#$Then… if Ωo is not near 0 or π, then approximately12002−≈NNand Step 1 becomes{}202100)(minarg)exp(][2minargˆΩ=Ω−=Ω≤Ω≤−=≤Ω≤∑XnjnxNNnoooππand Steps 2 & 3 becomeDTFT of Data x[n])ˆ(ˆ)ˆ(2ˆooXXNAΩ∠=Ω=φ12The processing is implemented as follows:Given the data: x[n], n = 0, 1, 2, … , N-11. Compute the DFT X[m], m = 0, 1, 2, … , M-1 of the data• Zero-pad to length M = 4N to ensure dense grid of frequency points• Use the FFT algorithm for computational efficiency2. Find location of peak• Use quadratic interpolation of |X[m]|3. Find height at peak• Use quadratic interpolation of |X[m]|4. Find angle at peak• Use linear interpolation of ∠X[m]oΩˆ|X(Ω)|Ω∠X(Ω)ΩoΩˆ13Figure 3.8 from textbook:)2cos()(φπ+=tfAtsotEx. 3 Bearing Estimation MLEEmits or reflects signal s(t)Simple model Grab one “snapshot” of all M sensors at a single instant ts:()][~cos][)(][ nwnAnwtsnxssn++Ω=+=φSame as Sinusoidal Estimation!! So… Compute DFT and Find Location of Peak!!If emitted signal is not a sinusoid… then you get a different
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