# SNR of filter output

I have a filter where the input is:

$$x(t) = s(t) + w(t)$$ which is signal plus white Gaussian noise and the output is $$y(t)$$. If the impulse response of the filter is $$h(t)$$, then:

$$y(t) = s(t)*h(t) + w(t)*h(t)$$

I have given the following equation to calculate the SNR:

$$SNR = \frac{\int_{-\infty}^\infty|s(t)*h(t)|^2 dt}{\int_{-\infty}^\infty|w(t)*h(t)|^2 dt}$$

And in frequency domain, following the Parseval's theorem:

$$SNR = \frac{\int_{-\infty}^\infty|S(f)H(f)|^2df}{\int_{-\infty}^\infty|W(f)H(t)|^2 df}$$

I wonder how this represents the signal power and noise power at numerator and denominator, respectively? Cause it is more like calculating energy than power.

Using

$$\int\limits_{-\infty}^\infty|s(t)*h(t)|^2 \,\mathrm dt$$

requires that this integral exists – and that requires, unless an infinite amount of the energy of $$s$$ is in $$\ker((*h))$$ (i.e. gets mapped to 0) that $$s$$ is an energy signal, not only a power signal (as you're probably used to dealing with). (An energy signal is exactly a signal that is magnitude-integrable.)

For that, defining SNR as quotient of energies makes sense. Also, your $$S(f), H(f)$$ are then energy spectral densities, instead of power spectral densities.

Problem: WGN is not an energy signal, and it's by definition not in the kernel of any deterministic filter. Therefore,

$$\int\limits_{-\infty}^\infty|w(t)*h(t)|^2 \,\mathrm dt$$

simply doesn't exist and the formula doesn't "work out", as it has a diverging denominator.

I can only guess where the formula came from, something like from the derivation of (average) power of a signal $$\xi$$ being $$\lim\limits_{T\to\infty}\frac1T\int_{T/2}^{T/2} |\xi(t)|^2 \,\mathrm dt$$, and then eager cancelation of the $$\frac 1T$$ from denominator and numerator?