Magnitude spectrum for discrete-time signals I am doing the following question (it is not homework, I am preparing for an exam).


  The time discrete signals $x_1(n)$ and $x_2(n)$ is created by sampling the continuous signal $x_a(t) = \cos (2 \pi 300t) + \cos (2 \pi 600 t)$ with sampling frequency $F_s = 1000\textrm{ Hz}$, $x_1$ with anti aliasing filter and $x_2$ without aliasing filter.
  
  
  a) Calculate $X_a(f)$ for the time continuous signal $x_a(t)$.
  b) Calculate the magnitude spectrum for the time discrete signals $x_1(n)$ and $x_2(n)$ i.e $\lvert X_1(f)\rvert$ and $\lvert X_2(f)\rvert$, with $-1&lt;f&lt;1$. 
  


I have done a), I used the table for Fourier Transform to conclude that $$X_a(f) = \frac{1}{2}\Big(\delta(f+300) + \delta(f-300) + \delta(f+600) + \delta(f-600)\Big).$$

But I can not solve b), I believe that I can use the information/answer from a) that is why I provided it.
This looks like a HW problem, so I can only give you a hint. Recall how the sampling process works - it's a convolution with a spike train i.e. a sequence of Dirac delta functions spaced at $F_s$. So the analog spectrum (that you derived in part (a) of the problem) gets replicated many many times centered around integer multiples of $F_s$. Also be careful about aliasing - if the largest frequency in your analog signal exceeds $F_s/2$ it means that this replication process will leak signal energy into the lower frequencies (draw a picture and convince yourself of this).
Finally, analog frequencies in the interval $[-F_s/2, F_s/2]$ get mapped to digital frequencies in the interval $[-1/2, 1/2]$.