AdvancedSignalandImageProcessingProfessor:A.Mohammad–DjafariExercisenumber1:SignalandImagerepresentationandLinearTransformsPart1:SignalandImagerepresentation1.Considerthefollowingsignals:(a)f(t)=asin(ωt)(b)f(t)=acos(ωt)K(c)f(t)=k=1[aksin(ωkt)+bksin(ωkt)]PK(d)f(t)=k=1akexp{−j(ωkt)}(e)f(t)=aPexp−t2K12(f)f(t)=k=1akexp−2(t−mk)/vk(g)f(t)=aPsin(ωt)/(ωt)(h)f(t)=1,if|t|<a,0elsewhereForeachofthesesignals,firstcomputetheirFourierTransformF(ω),thenwriteaMatlabprogramtoplotthesesignalsandtheircorresponding|F(ω)|.2.Considerthefollowingsignals:(a)f(t)=aexp{−t/τ},t>0(b)f(t)=0,t≤0,1Foreachofthesesignals,computetheirLaplaceTransformg(s).3.Considerthefollowingimages:(a)f(x,y)=asin(ωxx)+bcos(ωyx)(b)f(x,y)=aexp−(x2+y2)K122(c)f(x,y)=k=1akexp−2[(x−mxk)/vxk+(y−myk)/vyk](d)f(x,y)=1P,if|x|<a&|y|<b,0elsewhere(e)f(x,y)=1,ifx2+y2<a,0elsewhereForeachoftheseimages,firstcomputetheirFourierTransformF(u,v),thenwriteaMatlabprogramtoplotthesesignalsandtheircorresponding|F(u,v)|.4.Considerthefollowingimages:(a)f(x,y)=δ(x−a)δ(y−b)(b)f(x,y)=aexp−(x2+y2)K122(c)f(x,y)=k=1akexp−2[(x−mxk)/vxk+(y−myk)/vyk](d)f(x,y)=1P,x2+y2<a,0elsewhereForeachoftheseimages,firstcomputetheirRadonTransformg(r,φ),thenwriteaMatlabprogramtoplottheseimagesandtheircorrespondingRadonTransformg(r,φ).1Part2:Linearsystems:ConvolutionLetconsiderthefollowingsystem:——R—————f(t)Cg(t)————————G(ω)1.WritetheexpressionofthetransferfunctionH(ω)=F(ω)2.Writetheexpressionoftheimpulseresponseh(t)3.Writetheexpressionoftherelationlinkingtheoutputg(t)totheinputf(t)andtheimpulseresponseh(t)4.WritetheexpressionoftherelationlinkingtheFouriertransformsG(ω),F(ω)andH(ω)5.WritetheexpressionoftherelationlinkingtheLaplacetransformsG(s),F(s)andH(s)6.Givetheexpressionoftheoutputwhentheinputisf(t)=δ(t)7.Givetheexpressionoftheoutputwhentheinputisastepfunctionf(t)=u(t)=0∀t<0,1∀t≥08.Givetheexpressionoftheoutputwhentheinputisf(t)=asin(ω0t)9.Givetheexpressionoftheoutputwhentheinputisf(t)