Convolution Problem

Question

while working on a signal processing problem i've reached to the following:

So my aproach was:

Am I doing something wrong? Is it valid

Y(f)=[X(f) x H(f)]*W(f)=X(f) x [H(f)*W(f)]

If you could help me move further simplyfing Y(f) Thank you in advance

Answer 1

Just remember that

$H(f) \ast \delta(f-a) = H(f+a)$,

and that

$( H(f)+X(f) ) \ast W(f) = H(f)\ast W(f)+X(f)\ast W(f)$

So

$$\left\{ \left| \cos\left(\frac{\pi f}{50}\right)\right|\times\text{rect}\left( \frac{f}{50}\right)\right\} \ast \left\{ \delta(f-250) + \delta(f+250)\right\} (f)$$ $$ = \left\{ \left| \cos\left(\frac{\pi f}{50}\right)\right|\times\text{rect}\left( \frac{f}{50}\right)\right\} \ast \left\{ \delta(f-250)\right\}(f)$$ $$+\left\{ \left| \cos\left(\frac{\pi f}{50}\right)\right|\times\text{rect}\left( \frac{f}{50}\right)\right\} \ast \left\{ \delta(f+250)\right\}(f)$$

$$=\left| \cos\left(\frac{\pi (f+250)}{50}\right)\right|\times\text{rect}\left( \frac{f+250}{50}\right)+\left| \cos\left(\frac{\pi (f-250)}{50}\right)\right|\times\text{rect}\left( \frac{f-250}{50}\right)$$