Clear[rep,X,Theta,No,Y,PlotX,PlotN,PlotY];rep=40000;For[i=1;X={Random[NormalDistribution[0,1]]},i≤rep,i++,X=Join[X,{Exp[-1/1000]*X[[i]]+Random[NormalDistribution[0,Sqrt[1-Exp[-1/2000]]]]}]];For[i=1;No={Random[NormalDistribution[0,1]]},i≤rep,i++,No=Join[No,{Exp[-1/1000]*No[[i]]+Random[NormalDistribution[0,Sqrt[1-Exp[-1/2000]]]]}]];Theta=Random[UniformDistribution[{-Pi,Pi}]];No=Table[Sqrt[2]*No[[i]]*Cos[Theta-15+10*(i-1)/1000],{i,1,rep+1}];PlotX=ListPlot[X,Ticks->{{{1,"-15"},{5001,"-10"},{10001,"-5"},{15001,"0"},{20001,"5"},{25001,"10"},{30001,"15"},{35001,"20"},{40001,"25"}},Automatic},AxesLabel->{"t","X(t)"},AxesOrigin->{15001,0}];PlotN=ListPlot[No,Ticks->{{{1,"-15"},{5001,"-10"},{10001,"-5"},{15001,"0"},{20001,"5"},{25001,"10"},{30001,"15"},{35001,"20"},{40001,"25"}},Automatic},AxesLabel->{"t","N(t)"},AxesOrigin->{15001,0}];Y=X+No;PlotY=ListPlot[Y,Ticks->{{{1,"-15"},{5001,"-10"},{10001,"-5"},{15001,"0"},{20001,"5"},{25001,"10"},{30001,"15"},{35001,"20"},{40001,"25"}},Automatic},AxesLabel->{"t","Y(t)"},AxesOrigin->{15001,0}];

Show[PlotX]

Show[PlotN]

Show[PlotY]

Clear[Sxx,Snn];Sxx[omega_]:=2/(1+omega^2);Snn[omega_]:=(Sxx[omega-10]+Sxx[omega+10])/2;Plot1=Plot[Sxx[omega],{omega,-25,25},PlotRange->{-0.01,2.01}];Plot2=Plot[Snn[omega],{omega,-25,25},PlotRange->{-0.01,2.01}];

Show[{Plot1,Plot2}]

FullSimplify[FourierTransform[Sxx[omega]/(Sxx[omega]+Snn[omega]),omega,t]/Sqrt[2*Pi]]

Clear[h2];h2[t_]:=Re[(Sqrt[(2/π)]) ((1/(4 (Sqrt[366])))5 ((((29-15 I (Sqrt[183])) (E^(t Root[5151+48 ((#1)^2)+((#1)^4)&,3])) HeavisideTheta[-t])/(Sqrt[((24+5 I (Sqrt[183]))/π)]))+(((29-15 I (Sqrt[183])) (E^(t Root[5151+48 ((#1)^2)+((#1)^4)&,2])) HeavisideTheta[t])/(Sqrt[((24+5 I (Sqrt[183]))/π)]))+(1/(Sqrt[((24-5 I (Sqrt[183]))/π)]))(29+15 I (Sqrt[183])) (E^(t Root[5151+48 ((#1)^2)+((#1)^4)&,1])) ((E^(t Root[82416+192 ((#1)^2)+((#1)^4)&,4])) HeavisideTheta[-t]+HeavisideTheta[t])))];

Plot[h2[t],{t,-2.01,2.01},PlotRange->{-2.01,1.51}]

Clear[hh2];hh2=Table[N[h2[-2+i/1000]],{i,0,4000}];ListPlot[hh2,Ticks->{{{1,"-2"},{1001,"-1"},{2001,"0"},{3001,"1"},{4001,"2"}},Automatic},AxesLabel->{"t","h(t)-delta(t)/2"},PlotRange->{-2.01,1.51},AxesOrigin->{2001,0}]

Clear[Z];Z=Table[Sum[hh2[[i]]*Y[[j+2000-i]]/1000,{i,1,2000}]+Sum[hh2[[i]]*Y[[j+2000-i]]/1000,{i,2002,4000}]+Y[[j]]/2,{j,15001,25001}];

ListPlot[Z,Ticks->{{{1,"0"},{5001,"5"},{10001,"10"}},Automatic},AxesLabel->{"t","Z(t)"},AxesOrigin->{1,0}]

ListPlot[Table[Y[[j]],{j,15001,25001}],Ticks->{{{1,"0"},{5001,"5"},{10001,"10"}},Automatic},AxesLabel->{"t","Y(t)"},AxesOrigin->{1,0}]

ListPlot[Table[X[[j]],{j,15001,25001}],Ticks->{{{1,"0"},{5001,"5"},{10001,"10"}},Automatic},AxesLabel->{"t","X(t)"},AxesOrigin->{1,0}]

ListPlot[(Table[Y[[j]],{j,15001,25001}]-Table[X[[j]],{j,15001,25001}])^2,Ticks->{{{1,"0"},{5001,"5"},{10001,"10"}},Automatic},AxesLabel->{"t","(Y(t)-X(t))^2"},AxesOrigin->{1,0},PlotRange->{-0.01,3.11}]

ListPlot[(Z-Table[X[[j]],{j,15001,25001}])^2,Ticks->{{{1,"0"},{5001,"5"},{10001,"10"}},Automatic},AxesLabel->{"t","(Z(t)-X(t))^2"},AxesOrigin->{1,0},PlotRange->{-0.01,3.11}]