Background: I have a variety of BPSK which uses a raised cosine transmitted pulse shape:
$$ h(t) = {1 \over 2}\, (1+\cos(\pi t))\, \Pi(t/2) $$
After matched filtering in the receiver, the end result has ISI. Decimated to 1 sample per symbol, the pulse shape works out to [1/6, 1, 1/6].
So, I might filter and decimate this signal to 1 sample per symbol, then apply an equalizing filter to mitigate that ISI.
My question: modulo the potential consequences to computational complexity, is this any different from combining the matched filtering and equalization into a single filter?
I know filtering by two successive FIR filters is equivalent to one filter with the impulse response of the two convolved together. But with the decimation step between, I'm less sure.