# Required information An LTI system has an impulse response

Required information
An LTI system has an impulse response
$$\displaystyle{g{{\left({t}\right)}}}={5}{e}^{{-{3}}}{u}{\left({t}\right)}$$
Find the numerical value of $$\displaystyle{y}{\left({t}\right)}\ {a}{t}\ {t}={0.6}$$.
The numerical value of $$\displaystyle{y}{\left({t}\right)}\ {a}{t}\ {t}={0.6}$$ is ?

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James Etheridge
Step 1
The given impulse response
$$\displaystyle{h}{\left({t}\right)}={5}{e}^{{-{3}{t}}}{u}{\left({t}\right)}$$
Take Inverse Laplace
$$\displaystyle{H}{\left({s}\right)}={\frac{{{5}}}{{{s}+{3}}}}$$
Step 2
$$\displaystyle{y}{\left({t}\right)}={h}{\left({t}\right)}\cdot{u}{\left({t}\right)}$$
Convert in s domain
$$\displaystyle{Y}{\left({s}\right)}={H}{\left({s}\right)}{U}{\left({s}\right)}$$
$$\displaystyle{Y}{\left({s}\right)}={\frac{{{5}}}{{{s}{\left({s}+{3}\right)}}}}$$
Take partial fractions
$$\displaystyle{Y}{\left({s}\right)}={\frac{{\frac{{5}}{{3}}}}{{{s}}}}+{\frac{{-\frac{{5}}{{3}}}}{{{s}+{3}}}}={\frac{{{5}}}{{{3}}}}{\left({\frac{{{1}}}{{{s}}}}-{\frac{{{1}}}{{{s}+{3}}}}\right)}$$
$$\displaystyle{y}{\left({t}\right)}={\frac{{{5}}}{{{3}}}}{\left({1}-{e}^{{-{3}{t}}}\right)}$$
$$\displaystyle{y}{\left({0.6}\right)}={\frac{{{5}}}{{{3}}}}{\left({1}-{e}^{{-{3}\times{0.6}}}={1.4}\right.}$$