So I have a couple of fourier transforms that I have notice that occur many times in my signals course . I tried to use an online calculator to confirm my results but it did not work . I am going to write down how I computed them . The point of this question is to point out mistakes or propose another approach (maybe quicker)
- $v_1(t)=2sin(2\pi f_1t)x(t)$
$ \mapsto v(f)=2x(f)*\frac{1}{2j}{\delta(f-f_1)-\delta(f+f_1)}=\frac{2}{2j}[x(f)* \delta(f-f_1)-x(f)* \delta(f+f_1)]=-j[x(f-f_1) + x(f+f_1)]$
- $v_2(t)=2sin(4 \pi f_1 t)v_1(t)$ ,
so if my previous calculation was right then $v_1(t) \mapsto x(t)$
and $sin(4 \pi f_1 t) \to \frac{1}{|2|}\frac{1}{2j}[\delta(f-f_1) -\delta(f+f_1)]$ from the rule : $g(at) \to \frac{1}{|a|}G(\frac{f}{a})$ .
So we have: $v_2(f)= 2\frac{1}{4j}[\delta(f-f_1) -\delta(f+f_1)]*v_1(f)$
$\to v_2(f)=\frac{1}{2j}[\delta(f-f_1) -\delta(f+f_1)]*v_1(f)=\frac{-j}{2}[v_1(f-f_1) + v_1(f+f_1)]$