The signal x(n) is upsampled by 2 and then filtered. The signal and the filter have the following spectrum:
After upsampling the spectrum of X would look the same except the x axis would be divided by 2. So the limit of the spectrum would be at pi/4. Since the filter has amplitude 2 would the amplitude of X be increased from 1 to 2 or will the amplitude of X remain unchanged?
Finally, for the multiplication of the cosine it would be equivalent to multiply with 2 deltas located at -pi/2 and pi/2. Would the multiplication with this deltas cause a kind of mirroring effect on the spectrum?