Objective : Compute $y(t)$ from $Y(j\omega)=H(j\omega)X(j\omega)$ where : $$ x(t)=\left(\frac{\sin(2t)}{\pi t}\right)^{2} $$ and $$ H(j\omega)=\begin{cases}e^{-j\omega}&\text{if $|\omega|<4$}\\0&\text{if otherwise}\end{cases} $$ what I did is find FT of $\left(\frac{\sin(2t)}{\pi t}\right)\left(\frac{\sin(2t)}{\pi t}\right)$ as follow : $$ \mathcal{F}\{s\cdot s\}(t)=\frac{1}{2\pi}[S*S](j\omega) $$
We have : $$S(j\omega)=\mathcal{F}\left\{\frac{\sin(2t)}{\pi t}\right\}=\begin{cases}1&\text{if $|\omega|<2$}\\0&\text{if otherwise}\end{cases}$$ I have computed the modulation : $$ X(j\omega)=\frac{1}{2\pi}[S(j\omega)*S(j\omega)]=\frac{1}{2\pi}\int_{-\infty}^{\infty}S(j\Omega)S(j(\omega-\Omega))\;\text{d}\Omega=\begin{cases}\frac{\omega}{2\pi}&\text{if $0<\omega<2$}\\\frac{2-\omega}{2\pi}&\text{if $2<\omega<4$}\\0&\text{if otherwise}\end{cases} $$
The first issue I have is multiplying $H(j\omega)$ with $X(j\omega)$ since I have two piecewise functions. The second issue is how to thus compute its inverse Fourier transform.
I would hope someone can please help me. Thank you