How to understand haw many matched filters to use?

Question

I have a question that i can not find a concrete answer. Haw many matched filters should be used for 8-PAM/16-QAM/64-QAM/BPSK? Can somebody shortly explain the logic we should follow? Thank you.

Answer 1

Only one matched filter is required (if it can filter complex signals), or two (if not).

A receiver's front end produces two signals, one called in-phase (I) and the other called quadrature (Q). Assuming an AWGN channel: $$ I(t) = \sum_k a_k s(t-kT_p) + n_I(t) $$ and $$ Q(t) = \sum_k b_k s(t-kT_p) + n_Q(t), $$ where $T_p$ is the pulse interval. The task of the receiver is to recover $a_k$ and $b_k$ for all $k$. This can be done by matched-filtering $I(t)$ to obtain $a_k$ and, separately, $Q(t)$ to obtain $b_k$.

However, these two signals can be combined into a single complex signal $R(t) = I(t) + jQ(t)$ which can be filtered by a single matched filter to produce $a_k + jb_k$. In practice, there is little benefit to doing so, since complex arithmetic is in general not more computationally efficient than real arithmetic.

In the specific case of BPSK, $Q(t) = 0$ and only one real filter is required.