Closed form not the same as the discrete form

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Tworit Dash
Tworit Dash on 14 Jun 2022
Commented: Tworit Dash on 14 Jun 2022
clear;
lambda = 3e-2;
x = 4 * pi/lambda * linspace(eps, 15, 100000);
T = 5e-3;
t = [0:0.001e-3:T] ; % 0.1:1e-3:0.1+T];
u = 3;
a = 4*pi/lambda * u;
for i = 1:length(x)
Z(i) = sum(-((cos(a.*t) - cos(x(i).*t)).^2 + (sin(a.*t) - sin(x(i).*t)).^2));
end
% Z1 = csc((a+x)/2) .* sin(1/2.*(2.*T + 1).*(a + x)) -1/2 .* csc(a) .* sin(2.*a.*T+a) - 1/2 .* csc(x) .* sin(2.*T.*x+x)
Z1 = csc((a-x)/2) .* sin((T+1/2).*(a-x))/(2*pi) - 2*T - 1;
figure; plot(x*lambda/(4 * pi), (Z));
hold on; plot(x*lambda/(4 * pi), (Z1), '*');
The above code does two things. one is a summation over `t` and stores it in `Z` for all `x` individually with a for loop. The second thing it is doing is trying an analytical function for the summation in `t` and and just use the vector `x` to find the same quantity as `Z`, but analytically. it is stored in `Z1`. Somehow they both are not the same. The expression of the analytical form can be found in https://mathworld.wolfram.com/DeltaFunction.html equation (41) on that page. There is also a plot representing the function there. However, the matlab plot is very different. However, it is interesting to notice that `Z` and `Z1` are the same at the limiting case `a == x`. The plot I get is shown below. It should have a maximum at `a = x`, that is the value it should take at that point is `Z = Z1 = 0` and for the rest of the `x`, it should be less than 0. However, the blue plot satisfies it, but not the red one.

Accepted Answer

Paul
Paul on 14 Jun 2022
Hi Tworit,
I'm not quite following how the code is trying to implement equation 41 from the linked page. Here is one way to create a plot similar to that just above equation (41). Maybe you can modify this as needed? Also, where does the summation equation for Z come from?
n = 10:10:50; % different values of n
x = (-1:.001:1).';
x(x==0) = eps; % avoid divide by zero to keep things simple
Z1 = csc(x/2).*sin((n+1/2).*x)/2/pi;
figure
plot(x,Z1)
To shift the whole thing by 'a' :
a = 1; % for example
n = 10:10:50;
x = a + (-1:.001:1).';
x(x==a) = a+eps; % avoid divide by zero
Z1 = csc((x-a)/2).*sin((n+1/2).*(x-a))/2/pi;
figure
plot(x,Z1)

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