What are the formulas for signal and noise power gain of digital filters (FIR and IIR)?
For a FIR, I've seen in Harris' windowing paper that the DC gain is the sum of the filter weights.
$$ G = \sum_{i=0}^{N-1} w_i $$
For a FIR, I've seen that the noise gain is the square root of the sum of the square of the weights. $$ G = (\sum_{k=0}^{N-1} w_i^2 )^\frac{1}{2} $$
What about an IIR?
Is there a reason you don't think that the general formulae are:
$$ G = \sum_{k=0}^{\infty} h[k] $$
and
$$ G = (\sum_{k=0}^{\infty} h^2[k] )^\frac{1}{2} $$
These will work, provided the system is linear and time-invariant (and BIBO stable).
To work with a rational transfer function: $$ H(z) = \frac{\displaystyle\sum_{m=0}^{M} b_m z^{-m}}{\displaystyle\sum_{n=0}^{N} a_n z^{-n}} $$ then the DC gain (at $z = e^{j0}$) will be: $$ G_{DC} = \left. H(z) \right|_{z=1 = e^{j0}} = \frac{\displaystyle\sum_{m=0}^{M} b_m}{\displaystyle\sum_{n=0}^{N} a_n } $$
I'll have to think a bit more about the second gain in your question.
