FFTs and the Power Spectrum are useful for measuring the frequency content of stationary or transient signals. FFTs produce the average frequency content of a signal over the entire time that the signal was acquired. For this reason, you should use FFTs for stationary signal analysis or in cases where you need only the average energy at each frequency line.
Do you remember that the discrete Fourier transform gives the periodic function and the carrier frequency is not always in the first period? Best regards.