# Matched Filter Impulse Response

At the receiver end of a digital communication system, the matched filter is designed with impulse response matched to the effective signal that is given as : $$x_{(z,1)}^{*}(T_p-t)-x_{(z,0)}^{*}(T_p-t)$$

where $$x_{(z,1)}(t)$$ corresponds to a 1 and $$x_{(z,0)}(t)$$ corresponds to a zero.

But every where I check in the notes or the textbook I see that h(t) is gets conveniently modified from $$x_{(z,1)}^{*}(T_p-t)-x_{(z,0)}^{*}(T_p-t)$$

to

$$x_{(z,1)}^{*}(t)-x_{(z,0)}^{*}(t)$$

and the square of this is calculated as the euclidean distance.

I do not understand the rationale behind this.

• Ah, I don't think I'd call what you have there a matched filter, but pulse-shape decision or something (I'm assuming $x_{(z,0)}$ and $x_{(z,1)}$ are pretty much orthogonal – otherwise writing it like this has little benefit over comparing the sampled output to constellation points) – Marcus Müller Jan 11 at 9:13
• point is – if my terminology would be different than yours, chances are I'm just misunderstanding you (a shame!). Could you explain what these $x$ actually are, and what do you do with its output? – Marcus Müller Jan 11 at 9:19
• $x_{(z,0)}$ and $x_{(z,1)}$ are the transmitted signals representing zero and one.... this is a single bit demodulator implementation... – Orpheus Jan 11 at 15:43
• but these two have the same wave form, right? – Marcus Müller Jan 11 at 16:33
• Hint: It is not the square of the difference signal but rather the energy that is a measure of the )squared) Euclldean distance, and the energy does.not depend on whether the signal is running backwards or forwards. – Dilip Sarwate Jan 14 at 23:01