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So we're doing a project where we've decided on QAM-n modulation. I've read up a little bit on synchronization and found Barker sequences, which supposedly has nice autocorrelation properties which makes them good for synchronization.

So one option is to use BPSK modulation only in the synchronization phase and then move onto QAM for the data. But to me, it's very unintuitive that we couldn't construct even NICER synchronization sequences (wrt autocorrelation peakedness etc) if we utilize more of the signal space. Maybe something like simultaneous Barker BPSK for both I and Q?

Is there a nice explanation for why it's generally best to use BPSK for the synchronization?

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Longer preambles make it easier for the demodulator to lock onto the signal well. This is due to getting more preamble signal energy and because longer preambles make it easier to detect stuff like carrier offset.

Barker sequences are nice for all the reasons you state, but they are also very limited. The longest Barker sequence is 13 bits and there is only one of them. The longest Barker sequence length that has two sequences is four, so you'd be limiting yourself to a four bit long preamble, which is not a very useful preamble.

You may say to yourself, "I could just use the same 13-bit Barker sequence for both the I and Q." If you do that, though, you have made a BPSK constellation, only you've rotated it by $\frac{\pi}{4}$.

If you want to use a QPSK preamble, then I would use pseudo-random sequences from linear feedback shift registers (LFSR's). You can find the polynomials for maximal length LFSR's here.

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