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I was reading Chapter 2 from "Telecommunication Breakdown" by Johnson and Sethares and came across this paragraph:

However, sometimes it is more cost effective to perform certain tasks in analog circuitry. For example, if the transmitter modulates to a very high frequency, then it may cost too much to sample fast enough. Currently, it is common practice to perform some frequency translation and some out-of-band signal reduction in the analog portion of the receiver. Sometimes the analog portion may translate the received signal all the way back to baseband. Other times, the analog portion translates to some intermediate frequency, and then the digital portion finishes the translation.

And so I was wondering, if I wanted to do carrier recovery digitally, where in the receiver chain would I perform such task? I guess I'd need to translate input RF signal to IF and then perform carrier recovery digitally but in order to do so, I need to know the frequency and phase of the RF signal to translate it to IF, don't I? And I can only do that by carrier sync/recovery.

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You don't need to know the phase at all and you only need to know the frequency approximately. It is the job of your digital carrier recovery scheme to estimate and remove the phase and residual frequency offset.

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