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The impulse response of a system is \(h\left( t \right)\; = \;tu\left( t \right)\). For an input \(u\left( {t - 1} \right)\)the output is
1. \(\frac{{{t^2}u\left( t \right)}}{2}\)
2. \(\frac{{t\left( {t - 1} \right)}}{2}\;u\left( {t - 1} \right)\)
3. \(\frac{{{{\left( {t - 1} \right)}^2}}}{2}u\left( {t - 1} \right)\)
4. \(\frac{{{t^2} - 1}}{2}\;u\left( {t - 1} \right)\)

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Correct Answer - Option 3 : \(\frac{{{{\left( {t - 1} \right)}^2}}}{2}u\left( {t - 1} \right)\)

\(y\left( t \right) = \;u\left( {t - 1} \right)*tu\left( t \right)\)

Taking Laplace Transform,

\(Y\left( s \right) = \frac{{{e^{ - s}}}}{s} \times \frac{1}{{{s^2}}}\)

\(Y\left( s \right) = \frac{{{e^{ - s}}}}{{{s^3}}}\)

\(Y\left( s \right)\mathop \to \limits^{ILT} \frac{{{{\left( {t - 1} \right)}^2}}}{2}u\left( {t - 1} \right)\)

[As \(\frac{1}{{{s^2}}}\mathop \to \limits^{ILT} t\;u\left( t \right)\)&  shifting in time leads to phase in frequency]

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