frequency - Doppler shift in time domain?

I have some clarifications required in this topic So initially I know that to shift a signal $X(f)$ by $f_0$ we need to take its inverse fourier transform and multiply it with a factor of $\exp(−j2\pi f_0)$.

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  Now as I understand it, in the frequency domain the spectrum is shifted by the value of $f_0$ from its original frequency, correct?

  
  



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  So therefore, is the frequency of my original signal $x(t)$ actually increasing/decreasing?(depending again on the value specified by $f_0$)

  
  

Now coming to my problem at hand, I'm trying to design a receiver which compensates for doppler effect.

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  And as I've read up a doppler shift is nothing but an increase in frequency so am I on the right track in considering a frequency shift or should I actually consider a frequency scaling approach?

  
  


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  Or can someone tell me what exactly does a Doppler shift in frequency have an effect in the time domain?