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AdvancedSignalandImageProcessingProfessor:A.Mohammad–DjafariExercisenumber2:ModelingandparameterestimationPart1:ParametricmodelingandestimationCaseofsinusoids:Weobservedasignalf(t)whichweconsidertobeperiodicandmodelitbyf(t)=asin(2πt/T+φ).Wehavesamplesofthissignalevery1hour∆=1handhaveobserveditduring4days(4×24=96hours).1.AssumeweknowT=24h.Proposemethodstoestimateaandφ.TestyourmethodusingMatlabprogramming[a,phi]=estimate1Sinusa(f,T);2.Now,proposemethodstoestimateT,aandφ.TestyourmethodusingMatlabprogramming[T,a,phi]=estimate1sinus(f);3.Generalisethesetwoprogramsforthecasewhere∆isanyvalue[a,phi]=estimate1Sinusa(f,T,Delta);and[T,a,phi]=estimate1sinus(f,Delta);NowconsiderthemoregeneralmodelKf(t)=X[akcos(kπt/T)+bksin(kπt/T)]k=11.AssumeweknowT=24handweknowK.Proposemethodstoestimateakandbk.TestyourmethodusingMatlabprogramming[a,b]=estimateKSinusa(f,K,T);2.AssumeKisgiven,proposemethodstoestimateT,akandbk.TestyourmethodusingMatlabprogramming[T,a,b]=estimateKsinus(f,K);3.Now,proposeamethodtoestimateKtoo.CaseofGaussianshapesignals:Weobservedasignalf(t)whichwemodelitbyf(t)=aN(m,v).Wehavesamplesofthissignalevery1hour∆=1handhaveobserveditduring4days(4×24=96hours).1.Assumeweknowm=48,v=24.Proposemethodstoestimatea.TestyourmethodusingMatlabprogramminga=estimate1Gaussa(f,m,v);2.Now,proposemethodstoestimatem,vanda.TestyourmethodusingMatlabprogramming[m,v,a]=estimate1Gauss(f);NowconsiderthemoregeneralmodelKf(t)=XakN(mk,vk)k=11.AssumeweknowK,mkandvk.Proposemethodstoestimateak.TestyourmethodusingMatlabprogramminga=estimateKGaussa(f,m,v);12.AssumeKisgiven,proposemethodstoestimatemkvkandak.TestyourmethodusingMatlabprogramming[m,v,a]=estimateKGauss(f,K);3.Now,proposeamethodtoestimateKtoo.Part2ProbabilisticParametricmodelingandestimationCaseofMAmodels:Considerasignalf(t)whichismodeledbythefollowingMAmodelKf(t)=Xbkǫ(t−k∆)k=1whereǫ(t)∼N(0,v),∀tandδ=1.1.AssumeweknowKandv.Proposemethodstoestimatebk.TestyourmethodusingMatlabp
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