Required information An LTI system has an impulse response

philosphy111of 2021-11-21 Answered
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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Expert Answer

James Etheridge
Answered 2021-11-22 Author has 916 answers
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
\(\displaystyle{Y}{\left({s}\right)}={\frac{{\frac{{5}}{{3}}}}{{{s}}}}+{\frac{{-\frac{{5}}{{3}}}}{{{s}+{3}}}}={\frac{{{5}}}{{{3}}}}{\left({\frac{{{1}}}{{{s}}}}-{\frac{{{1}}}{{{s}+{3}}}}\right)}\)
\(\displaystyle{y}{\left({t}\right)}={\frac{{{5}}}{{{3}}}}{\left({1}-{e}^{{-{3}{t}}}\right)}\)
\(\displaystyle{y}{\left({0.6}\right)}={\frac{{{5}}}{{{3}}}}{\left({1}-{e}^{{-{3}\times{0.6}}}={1.4}\right.}\)
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