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This depends a bit on how you actually want to model this. Time discrete or time continuous ? Differential equations or difference equations? Impulse excitation gives you directly the impulse response ...
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Echoing what already answered: you are approaching this backwards. SNR is a concept that's very fundamental and applicable to way more things that just channel capacity. If you have a signal $y(t)$ ...
Fundamentally, the signal-to-noise-ratio (SNR or $S/N$) is the ratio of signal power to noise power and that power ratio and is usually expressed in $dB$. Shannon and Hartley (likely collaborating at ...