Trying to compute Hilbert transform numerically

[hunt_mat]

I know the result:
$$\widehat{\mathscr{H}(f)}(k)=-i\sgn (k)\hat{f}(k)$$

I want to use this to compute the Hilbert transform. I have written code for Fourier transform,inverse Fourier transform and that the Hilbert transform. My code is the following:

function y=ft(x,f,k)
n=length(k); %See now long the wave vector is
y=zeros(1,n); %Output is the same length
for i=1:n
    v_1=exp(-sqrt(-1)*k(i)*x); %Compute exp(-ikx)
    v_2=f.*v_1; %Compute integrand of Fourier transform
    y(i)=-trapz(v_2,x); %Compute transform
end

function y=ift(k,f_hat,x) %Inverse Fourier transform
n=length(x); %Compute the length of the x
y=zeros(1,n); %Solution has same length
a=1/(2*pi); %scaling factor
for i=1:n
v_1=exp(sqrt(-1)*k*x(i)); %Compute exp(ikx)
v_2=f_hat.*v_1; %Compute integrand of inverse Fourier transform
y(i)=-a*trapz(v_2,k); %Compute transform
end

function y=hilbert(x,f_x,z)

%This takes numerical input (x,f(x)) and evaluates the Hilbert transform at
%x=z;
k=-4:0.001:4; %Choose the range of the wave number
f_x_hat=ft(x,f_x,k); %Compute Fourier transform
dum=-sqrt(-1)*sign(k).*f_x_hat; %Multiply by -isgn(k)
y=ift(k,dum,z); %Take inverse Fourier transform.

My solution has LOTS of oscillations to it and I have no idea what is going on.

Any suggestions?

 

[gtoguy97]

It looks like you might be missing a step in your code. When computing the Hilbert transform, you should first multiply the Fourier transform by the factor -i sgn(k) and then take the inverse Fourier transform of the resulting expression. This step is missing in your code and it could be causing the oscillations you are seeing.