Downsampling/Decimation - When is well done?
Ignoring your original question, and referring to your comment:


  I am observing the FFT of the downsampled signal presents an attenuation. I don't understand why.


The output of FFT (or DFT in general) isn't normalized - it is relative to the signal length (the total amount of samples being transformed). When decimating a signal (say, by a factor of M), you end up with a signal with 1/M length the original signal's length.

So, because DFT/FFT results are relative to signal length, a decimated signal with a 1/M length would have DFT/FFT components "attenuated" by a factor of M as well, even if the original signal was properly downsampled (i.e. bandlimited before being decimated).

To properly compare between the DFT of different signals, you should divide the output of DFT by the length of the transformed signal, to normalize the transform.