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I have a full synchronization chain in a digital receiver that consists of:

  1. Coarse frequency synchronization: this is a blind balanced quadricorrelator
  2. Frame synchronization: this is data-aided correlation with a unique word
  3. Symbol timing synchronization: this is data-aided early-late gate (ELG)
  4. Fine Frequency Synchronization: this is data-aided Luise & Regiannini (L&R) algorithm
  5. Phase interpolator: which calculates the carrier phase between two pilot fields (we assume each frame has multiple pilot fields separated by $N$ data symbols), and then interpolate over data symbols in between

To obtain accurate frame synchronization, we need to bring down the frequency offset down using the coarse frequency estimation block, and to do accurate timing synchronization we need accurate frame synchronization. Also, fine frequency synchronization depends on both timing synchronization and coarse frequency synchronization.

Do we do the acquisition stage in 3 different non overlapped stages like this:

  1. first let the coarse converge before we start frame synchronization
  2. let the frame synchronization converge before we start timing synchronization
  3. let timing synchronization converges before we do fine frequency synchronization?

In other words, if it took $N_1$ frames for the coarse frequency estimator to converge, and $N_2$ frames for the frame synchronization block to converge, and $N_3$ frames for the fine frequency synchronization to converge, then I would need $N_1+N_2+N_3$ frames for the acquisition stage, or can I make these steps overlap? How this is done in real systems?

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How this is done in real systems?

In many different ways, depending on the needs of the system. This alone is rarely a good question to ask, because other real systems have different needs than your real system. Always ask what the considerations are! There's simply no "one size fits all" solutions.

Do we do the acquisition stage in 3 different non overlapped stages like this:

No. You can have correlator banks to deal with different frequency offset hypotheses, you can use "unique words" that are designed to make it trivial to find fine and coarse frequency estimates without the other, and of course there's nothing stopping you from trying to demodulate the frames before your previous stages have fully synchronized (assuming you have appropriate checksums and error correction, which you probably do!); imperfect synchronization just makes symbol errors more likely, but that doesn't mean that none of the resulting frames can be decoded successfully, and if you get lucky and do get a successful decoding, that can be used to estimate many parameters quite precisely, to feed them back into the synchronization stages.

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