function FourierSeries

N=50;
Resolution = 1e-6;
TimeDomain = 2*pi;
SaveName = 'try';

GlobalError=zeros(N,1);
for k=0:N
    
    a0 = quad(@(x) f(x).* Cos0(x), 0,TimeDomain, 1e-8) / TimeDomain;
    for i=1:k
        a(i) = quad(@(x) f(x).* CosN(x,i), 0,TimeDomain, 1e-8) / TimeDomain * 2;
        b(i) = quad(@(x) f(x).* SinN(x,i), 0,TimeDomain, 1e-8) / TimeDomain * 2;
    end
    
    
    MeshX = 0:Resolution:2*pi;
    FV = f(MeshX);
    
    FT = a0*Cos0(MeshX);
    for i=1:k
        FT = FT + a(i)*CosN(MeshX,i);
        FT = FT + b(i)*SinN(MeshX,i);
    end
    
    Frequencies = 0:1:k;
    Amplitudes  = zeros(size(Frequencies));
    Amplitudes(1) = a0;
    for i=1:k
        Amplitudes(i+1)=sqrt( a(i)^2 + b(i)^2 );
    end
    
    fig = figure(1);
        set(fig,'Position',[680 560 1120 420]);
    
    subplot(1,2,1); hold on;
        set(gca,'NextPlot','replace');
            pFV = plot(MeshX,FV);
        set(gca,'NextPlot','add');
            pFT = plot(MeshX,FT);
            set(gca,'Box','on','XGrid','on','YGrid','on','XLim',[0 TimeDomain]);
            set(gca,'FontSize',12,'FontName','Times','FontWeight','bold');
        xlabel('t(s)','FontSize',16,'FontName','Times');
        ylabel('f(t)','FontSize',16,'FontName','Times');
            set(pFV,'LineWidth',2,'Color',[0 0 0]);
            set(pFT,'LineWidth',2,'Color',[1 0 0]); hold off;
    subplot(1,2,2);
        spectra = bar(Frequencies,Amplitudes);
            set(gca,'Box','on','XGrid','on','YGrid','on','XLim',[0 10],'YLim',[0 1.4]);
            set(gca,'FontSize',12,'FontName','Times','FontWeight','bold');
        xlabel('\omega_i(rad/s)','FontSize',16,'FontName','Times');
        ylabel('A(\omega)','FontSize',16,'FontName','Times');
            set(spectra,'BarWidth',0.2,'FaceColor',[0 0 0]);
    pause(0.1)
%     saveas(fig, strcat(SaveName,num2str(k)), 'png');
    
    ErrorFunction    = abs( FV - FT );
    GlobalError(k+1) = max(ErrorFunction);
end
        
ListOfN=0:1:N;

figure(3);
err=plot(ListOfN,GlobalError);
    set(gca,'Box','on','XGrid','on','YGrid','on','XLim',[0 N],'YScale','log');
    set(gca,'FontSize',12,'FontName','Times','FontWeight','bold');
        xlabel('N','FontSize',16,'FontName','Times');
        ylabel('E','FontSize',16,'FontName','Times');
    set(err,'LineWidth',2,'Color',[0 0 0]);


% saveas(fig, strcat(SaveName,num2str(N)), 'png');






function y=f(x)

% y = 3./(5-4*cos(x));
% y = abs(-x + pi);
y = sign(x - pi);
% y = 3*cos(x) + 5*sin(5*x);

function y=Cos0(x)
y=1 + 0*x;

function y=CosN(x,N)
y=cos(N*x);

function y=SinN(x,N)
y=sin(N*x);