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 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.

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.

Is there a reason you don't think that the general formulae are:

$$ G = \sum_{k=0}^{\infty} h[k] $$

and

$$ G = (\sum_{k=0}^{\infty} h^2[k] )^\frac{1}{2} $$

??

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 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.