Using Hilbert Transform Code

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Joaquin dela Cruz
Joaquin dela Cruz 2020-7-15
编辑： Joaquin dela Cruz 2020-7-17
I am attempting to calculate the hilbert transform of
𝑍(𝜏,𝜔)=1/(1+(𝜏𝜔)^2)
with respect to 𝜔 for various values of 𝜏, using a given code:
function h = hilb1(F, N, b, y)
n = [-N:N-1]'; % Compute the collocation points ...
x = b*tan(pi*(n+1/2)/(2*N));
FF = eval(F); % ... and sample the function.
a = fft(fftshift(FF.*(b-1i*x))); % These three lines compute the
a = exp(-1i*n*pi/(2*N)).*fftshift(a); % expansion coefficients.
a = flipud(1i*(sign(n+1/2).*a))/(2*N);
z = (b+1i*y)./(b-1i*y); % The evaluation of the transform
h = polyval(a,z)./(z.^N.*(b-1i*y)); % reduces to polynomial evaluation
% in the variable z.
with an output in the ranges of: omega_vec = logspace(-4, 4, 81)
A brief explanation of the code can be found here: http://appliedmaths.sun.ac.za/~weideman/research/hilbert.html
However, being new to this area of maths and to Matlab, I do not know where to start.
I understand that I should plug in the above function into F, and used the run function on Matlab to test out other known inputs and values.
However, given the information above I am unsure of what N, b, and y should be? I have experimented with multiple values but most result in some sort of error.
Apologies for what could be an elementary question; I am very much a beginner to both the content and to the code and would greatly appreciate a breakdown of how to approach this task or what the code is even doing.

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