Estimating Damping Factor (Q) from Noisy Measurements
Our system has the impulse response (why we changed to imaginary instead of real? :) ):
$$h(t)=\mathbb{Im}(Ae(iwt)e(-bt))=Ae^{-bt}sin(wt)$$

With the following structure (ref.) (discretized):
$$H(s)=\frac{Kw}{(s+b)^2+w^2}$$

Under matlab, you only need to use procest:

procest(data,'P2U');


Which models the same structure under the form:
$$H(s)=\frac{K}{(1+2\zeta T_ps+T_p^2s^2)}$$