Matched filters

Working on my communication class right now, and I'm having a bit of a problem with the matched filters.

As I see it the matched filters "eliminate" some of the noise. We project the received signal on our orthonormal base and everything that can't be generated by it is considered to be noise.

Now let X(t) be our only signal we send, time-limited to [0;T]. The matched filter at the receiver's end would have an impulse response of h(t) = X(T-t). After our received signal R(t) passes through this filter we sample at ts=T and we get our Y(t).

After all that we can make our final decision based on Y(t) and the expected X(t), to see if the transmitter sent X(t) or zero.

Now my question is why do we use a shifted time-reversed function, instead of just X(t)? Someone told me that it's because having h(t)=X(t) is too costly in the real world on the hardware level. But I'm not really sure it's this. Or maybe I'm completely wrong and and with h(t)=X(t) it would work or be less effective.

• The short answer is because you want to implement a correlation, which is equivalent to convolution (i.e., filtering) with a time reverse of the function you want to correlate with. Since the time reverse of the transmitted pulse is non-causal, you shift it to the right to make it causal, hence $X(T-t)$ instead of $X(-t)$. – Matt L. Jun 19 '15 at 16:02
• Also note that, in most (if not all) practical systems, $X(t)$ is symmetric, so the matched filter's response is indeed $X(t)$ (since $X(t)=X(T-t)$). – MBaz Jun 19 '15 at 16:09