Let there be a triangular pulse $f(t)$ in time domain (as shown in figure). The triangular pulse can be expressed as a convolution of a function $g(t)$ i.e. $f(t)=(g*g)(t)$. Find $g(t)$.
my attempt:
i used property of similar triangle $$\frac{f(t)}{a-|t|}=\frac{1}{a}$$ $$f(t)=1-\frac{|t|}{a}$$ $$\therefore f(t)=\begin{cases}{ 1-\frac{|t|}{a}\ \text{if}\ |t|\le a \\ 0 \ \text{if}\ |t|>a} \end{cases}$$
$$f(t)=(g*g)(t)$$ i take fourier transform $$F(f(t))=F(g(t))F(g(t))$$$$F(f(t))=(F(g(t)))^2$$ I found $F(f(t))=a sinc^2(a\omega/2)$ so i get $$F(g(t))=\sqrt{F(f(t))}=\sqrt{a} sinc (a\omega /2)$$ $$g(t)=F^{-1}(\sqrt{a} sinc (a\omega /2))$$
I got stuck here and I am not sure if everything correct upto this point. Can someone please help me to get $g(t)$.
thank you