About understanding a Matched Filter
I just want to comment on the confusion between $p(t)$ and $h(t)$. What happens is that we transmit a signal $s(t)$ where

$$s(t)=\sum_ks_kg(t-kT_s)$$

where $s_k$ are the data symbols, $g(t)$ is the transmit pulse shape, and $T_s$ is the symbol time. 

Transmitting this over a wireless channel with a channel impulse response $h(t)$, the channel acts acts as a filter, and thus the channel output is the convolution between the transmitted signal and the channel, i.e.,

$$y(t) = s(t)\star h(t) = \sum_ks_k\underbrace{\left[g(t-kT_s)\star h(t)\right]}_{p(t-kT_s)}$$

So, the convolution of the transmit pulse shape and the channel is considered in the matched filter design.

By the way, the purpose of the matched filter is not to "remove the noise", but rather to maximize the signal-to-noise ratio (SNR) by maximizing the received signal power at the output of the matched filter.