# signals and systems,matched filter concept

The system with impulse response hi(t) is known as the matched filter for the signal Xi(t) because the impulse response is tuned to xi(t) in order to produce the maximum output signal. My intuition says for y(4) to be maximum,x(t)=h(-t+4). But I can not come up with a more rational mathematical proof.Can anyone help me with this

• Please do your own homework, bypassing that doesn’t really help you. But if there is something you don’t understand after you have done all the prerequisite investment of your own time we’re happy to help with ours. Jun 9 at 13:27
• @DanBoschen I have uploaded my approach to the question sir,can you please tell me if my approach is correct. Or can you suggest me how to approach the problem in a better way Jun 9 at 13:54
• It looks like Hilmar has helped you! Thanks for the update Jun 9 at 14:57
• Take a look at this answer. Jun 9 at 17:02

$$y(4) = h(1)\cdot x(4) + h(2)\cdot x(3) + h(3)\cdot x(2) + h(4)\cdot x(1)$$
Given that both $$|h| \le 1$$ and $$|x| \le 1$$ it should be pretty obvious what's the largest that $$y(4)$$ can be and what you need to do to each individual term to get the max result.