Let's suppose that I send the following sequence of bits:
$x = [1, 0]$
I use a 7 length Barker code ($b = [−1,−1,−1,1,1,−1,1]$) in a BPSK modulation scheme, resulting in the following sequence:
$x_B =\\ [−1,−1,−1,\:\:1,\:\:1,−1,\:\:1\\ \:\:\:1,\:\:\:1,\:\:1,-1,-1,\:\:\:1,-1]$
We send it, through a AWGN channel, and after decoding there are some errors in the sequence: $y_B =\\ [\:\:1,−1,\:\:1,\:\:1,\:\:1,−1,\:\:1\\ \:\:\:1,-1,\:\:1,-1,-1,\:\:\:1,-1]$
How would I decode this sequence?