I am working on an application for which a matched filter seems like the right concept. In the derivation for the matched filter which I went through (here), they define the SNR as the ratio of signal power to noise power as the basis for the derivation of the expression for the matched filter. I think that for my application, the "amplitude" SNR would be a better target for optimization. However, it looks like I reach a mathematical dead-end that is avoided by the use of the power SNR. Below is my work which roughly follows the derivation from wiki:
Is there a manipulation I am missing that could get me past this obstacle or does someone know if this result has been solved for already? I have not found it anywhere.