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I'm currently creating a project in Matlab where I'm simulating a communication, based on the SSB modulation, between a transmitter and a receiver.

I've added a Barker sequence in the trasmitted message in order to Synchronize the receiver. The receiver starts to demodulate the message from the istant when the cross-correlation beetween the Barker sequence and the received message is maximum.

The project works with no problems, but I have a question...which kind of autocorrelation function should have a sequence used for this scope? I've found that a good sequence should have an autocorrelation function similar to a pulse...is it correct? If it is...why?

Thanks for the attention!

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I've found that a good sequence should have an autocorrelation function similar to a pulse...is it correct? If it is...why?

You want to pinpoint one specific delay – the right time synchronization "point". From that, it directly follows that the ACF should be as "dirac-y" as possible to minimize ambiguities and maximize energy in the one right time.

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The properties of the autocorrelation of sequences that are useful for this are ones that have a maximum magnitude at $\tau=0$ and minimum for any other value of $\tau$ for the reasons already given by Marcus. The Barker sequences are one set of sequences that provide this, but there are not that many and they are relatively short. Other sequences are maximum-length pseudo-random sequences generated from linear feedback shift registers configured with feedback taps associated with irreducible polynomials.

Below is a graphic demonstrating this with a 15 chip LFSR sequence. Correlation is a sum of products, and here we use the math given by Galois Fields (GF{2}). Without the need to get into that, simply convert the zero symbol to -1 and the one symbol to +1 and use the basic rules of arithmetic:

LFSR Correlation properties

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