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Many modern digital communication systems repeat a preamble at the start of a frame/burst which is used by the receiver for timing and frequency synchronization. One example is 802.11a. Why are the preambles repeated versus using a non-repeating synchronization sequence? I can see how the repetition aids in a larger carrier frequency offset estimation, but it also adds ambiguity in the detector at the output of the cross correlation.

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In case of IEEE802.11a, that's Schmidl&Cox synchronization. I go into a few effects of having that in this answer; and as I say there,

If in doubt, read the original Schmidl&Cox Paper Robust Frequency and Timing Synchronization for OFDM. It's actually a good read, and a good paper with respect to demonstration of state of the art, proposed algorithm, theory, and simulation. I highly recommend it.

But let's address a few things:

but it also adds ambiguity in the detector at the output of the cross correlation.

Not really; you correlate with the whole first training symbol, not just with half of it. Ideally, you correlate with the both. No downside to it, these are known at the receiver. Sure, you get side lobes at half a symbol shift, but that will be robustly lower than the main cross-correlation peak.

I can see how the repetition aids in a larger carrier frequency offset estimation

Ah, but it's actually used to implement a fine frequency offset correction! Do a fixed lag correlation of your RX signal: take a vector half a symbol worth of time samples, and another one of the same length taken from directly behind that. Dot product: you get a correlation coefficient for a lag of half a symbol. That coefficient

  1. will be high if your signal, at the point where you do that correlation, contains a repeated sequence of half a symbol length
  2. will have a phase that's proportional to frequency, because every sample in the second half has the same value, but a phase shift of $f_\text{offset}\cdot \,l_\text{symbol}/2\mod 2\pi$, due to an unknown frequency offset. That actually gives you a very robust frequency estimate, because you're using so many samples for a correlation – but it comes with ambiguities depending on the symbol duration $l_\text{symbol}$. So, it's a good fractional, and not a useful coarse frequency estimate.

The second symbol now isn't repetitive, so that indeed makes your timing metric more robust, and more precise.

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  • $\begingroup$ I agree that with the example I provided, 802.11a, you would correlate with the entire preamble to resolve any correlation ambiguity. But what about if the preamble only contained N repeats of the same sequence? The output of the correlator would have ambiguous spikes for each repetition. $\endgroup$ Nov 30, 2022 at 18:57
  • $\begingroup$ For the fixed lag correlation, do you mean to say autocorrelation (i.e., received signal dot product with itself)? Or do you mean with the preamble sequence? $\endgroup$ Nov 30, 2022 at 19:07
  • $\begingroup$ with itself, as I described! $\endgroup$ Nov 30, 2022 at 19:08
  • $\begingroup$ but you don't do that repeated halves all over again. The whole point of doing it once is using Schmidl&Cox. No system that does not exploit the repetition would do the repetition. $\endgroup$ Nov 30, 2022 at 19:10
  • $\begingroup$ The dot product with half a symbol period delay to resolve frequency error is a great idea! $\endgroup$ Nov 30, 2022 at 19:23

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