Given a captured noisy BPSK signal, how do I formulate the maximum-likelihood problem so as to estimate the matched filter? The model of interest would be baseband BPSK with AWGN:

$$c(t) = \sum_{\ell=-\infty}^\infty a_\ell s(t - \ell T) + N(\mu,\sigma)$$

Where $a_\ell \in \{-1,1\}$ and $s(t)$ is the pulse function that I would like to estimate from samples of $c(t)$.

There are a lot of maximum-likelihood estimators for the different signal parameters (delay, phase, and the symbols themselves), but I've not found ML treatment of the pulse shape itself.


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