A continuous-time filter $H(s)=(2s+6)/(s^2+6s+8)$ is converted to $$G(z)=\frac{2z^2-0.5032z}{z^2-0.5032z+k}.$$ Its impulse response, sampled at 2 Hz, equals the discrete filter's impulse response at sampling instants. The value of k is ________.
Factor the analog denominator as $(s+2)(s+4)$. With sampling interval T=1/2 s, impulse invariance maps the poles to $e^{-2T}=e^{-1}$ and $e^{-4T}=e^{-2}$.
The digital denominator is $z^2-(e^{-1}+e^{-2})z+e^{-3}$. The sum is 0.5032147, consistent with the stated coefficient. Therefore **$k=e^{-3}=0.049787$**.
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