What is the decay rate (time constant) of two One-Pole LPF's cascaded in series?
If both filters have the same time constant, i.e., if their individual impulse responses are
$$h(t)=e^{-\alpha t}u(t),\qquad \alpha&gt;0\tag{1}$$
then the impulse response of the cascade is
$$h_{tot}(t)=te^{-\alpha t}u(t)\tag{2}$$
If both filter have different time constants with individual impulse responses given by
$$h_i(t)=e^{-\alpha_it}u(t),\qquad \alpha_i&gt;0\tag{3}$$
the total impulse response is given by
$$h_{tot}(t)=\frac{e^{-\alpha_1t}-e^{-\alpha_2t}}{\alpha_2-\alpha_1}u(t)\tag{4}$$
I don't think it makes much sense to ask for an &quot;effective&quot; time constant of the cascaded systems. How should one define such a time constant? Of course, for large $t$ the impulse response $(2)$ decays as $e^{-\alpha t}$, and the impulse response $(4)$ decays as $e^{-\min(\alpha_1,\alpha_2)t}$, but that doesn't really say much about the systems' behavior.