Consider the signal $\cos(30t)$ sampled at $w_s=40 rad/s$ using a unit impulse train. The sampled signal is filtered with an ideal low pass filter with unity gain and cutoff frequency $w_c = 40rad/s$. Find the resulting output.
The solution is given as $\frac{20}{\pi}(cos(30t)+cos(10t))$
Attempt:
Taking the Fourier transform of $cos(30t)$ gives $\frac{\pi}{2} (\delta(\omega-30)+\delta(\omega+30))$.
Using the unit impulse train: $X(\omega)_{T_s} = \frac{1}{T_s}\sum X(\omega-k\omega_s)$ but this gives $\frac{1}{T_s}\sum \frac{\pi}{2} (\delta(\omega-70)+\delta(\omega+10))$.
If this is an acceptable approach, it's unclear where the cosine arguments and factor of $\frac{20}{\pi}$ is coming from. Help is appreciated.