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PPM Variant - Analytic symbol error probability

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WUID

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Hi all,
I'm trying to calculate symbol error probability of variant of pulse position modulation considering only AWGN channel. The decoder is maximum likelihood so the implementation of the detector is by using vector correlators.
$$
\arg \max_{x} <y,x>.
$$
Attached is the waveforms vector representation of the transceiver & receiver. I didn't find any general term for the inner product that will lead me to solve the probability problem and from there to find the analytic solution. Any help will be appreciated.

oe9Tc.jpg
 

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