Time constant as a function of

Recall the gain of the simple 2-point averager is

$\displaystyle G(\omega) = \vert 1+ e^{-j\omega T}\vert$

The gain of the weighted 2-point averager is

$\displaystyle G(S; \omega)$	$\displaystyle =$	$\displaystyle \vert(1-S) + Se^{-j\omega T}\vert$
	$\displaystyle =$	$\displaystyle \left\vert(1-S) + S[\cos(\omega T) + j\sin(\omega T))]\right\vert$
	$\displaystyle =$	$\displaystyle \sqrt{\left[(1-S) + S(\cos(\omega T)\right]^2 + S^2\sin^2(\omega T))}$
	$\displaystyle =$	$\displaystyle \sqrt{(1-S)^2 + 2S(1-S)\cos(\omega T) + S^2}.$

The time constant is

$\displaystyle \tau = -\frac{1}{f_0\ln(G(S;\omega))} = -\frac{N + S}{f_s \ln(G(S; \omega))}$