I have a signal of period $T_0=8$, let's say $x(t)$, and it has the following Fourier coefficients:
$$ a_k=\frac{1}{4} \mathrm{sinc}^2\left(\frac{3k}{8}\right) e^{ik\frac{\pi}{2}} $$ for $1\le|k|\le6 $ and $a_k=0$ for the rest.
I also have a system which has a frequency response : $$H(j\omega)=\frac{j\omega}{j\omega+\frac{3\pi}{2}}$$
Using MATLAB, what's the best way apply this filter to this signal? I got the signal $x(t)$ from the coefficients and now I don't know how to proceed from here. I'll leave my code here:
T=8;
w0=2*pi/T;
t=0:0.001:2*T;
exp_jwt=exp(1j*w0*t);
k=-6:6;
ak=0.25*(sinc(3*k/8)).^2.*exp(1j*k*pi/2);
ak(k==0)=0;
x=zeros(size(t));
for (i=1:length(k))
x=x+ak(i).*(exp_jwt).^(k(i));
end