clear
dx = 2*pi/500;
x = -pi:dx:pi;

figure(1)
clf

subplot(2,1,1)
hold on
k = 4; y = sin((k+1/2)*x)./sin(x/2); plot(x,y,'LineWidth',2);
k = 3; y = sin((k+1/2)*x)./sin(x/2); plot(x,y,'LineWidth',2);
k = 2; y = sin((k+1/2)*x)./sin(x/2); plot(x,y,'LineWidth',2);
k = 1; y = sin((k+1/2)*x)./sin(x/2); plot(x,y,'LineWidth',2);
axis([-pi,pi,-2.5,10])
title('Dirichlet kernel: N = 1, 2, 3, 4')

subplot(2,1,2)
hold on
k = 4; y = sin((k+1/2)*x)./sin(x/2); plot(x,y);
k = 8; y = sin((k+1/2)*x)./sin(x/2); plot(x,y);
k = 12; y = sin((k+1/2)*x)./sin(x/2); plot(x,y);
k = 16; y = sin((k+1/2)*x)./sin(x/2); plot(x,y);
axis([-pi,pi,-10,40])
title('Dirichlet kernel: N = 4, 8, 12, 16')

print -dps Dirichlet_kernel.ps

% to see what's happening at a particular point I fix a point
% xx = -pi + 2*pi*60/500 
xx = x(61)
% and look at D_n(xx) as a function of n
for n=1:500, a(n) = sin((n+1/2)*x(60))./sin(x(60)/2); end
figure(2)
clf
plot(a)
xlabel('N','FontSize',16);
ylabel('D_N(xx)','FontSize',16);
title('fix a point and look at D_N at that point as a function of N');