The impulse response ht of a linear time-invariant continuous time system is described by ht - exp- t u t - exp- t u-t- where ut denotes the unit step function- and - and - are real constants - This system is stable if

H(t) = e^α t u(t) + e^β t u( t) For the system to be stable,∫_ ∞^∞ h(t)dt lt; ∞ For the above condition, h(t) should be as shown below. Therefore α lt; 0 ...

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Stability Property of LTI System

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Question

The impulse response h(t) of a linear time-invariant continuous time system is described by $h (t) = e x p (α t) u (t) + e x p (β t) u (- t),$ where u(t) denotes the unit step function, and $α a n d β$ are real constants . This system is stable if

α is negative and β is negative

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α is positive and β is negative

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α is positive and β is positive

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α is negative and β is positive

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Solution

The correct option is D
$α is negative and β is positive$
$h (t) = e^{α t} u (t) + e^{β t} u (- t)$
For the system to be stable, $\infty \int - \infty h (t) d t < \infty$
For the above condition, h(t) should be as shown below.

$Therefore α < 0 and β > 0$

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The impulse response h(t) of a linear time-invariant continuous time system is described by h(t)=exp(αt)u(t)+exp(βt)u(−t), where u(t) denotes the unit step function, and α and β are real constants . This system is stable if

The correct option is D α is negative and β is positiveh(t)=eαt u(t)+eβt u(−t) For the system to be stable,∞∫−∞ h(t)dt<∞ For the above condition, h(t) should be as shown below. Therefore α < 0 and β > 0

The impulse response h(t) of a linear time-invariant continuous time system is described by $h (t) = e x p (α t) u (t) + e x p (β t) u (- t),$ where u(t) denotes the unit step function, and $α a n d β$ are real constants . This system is stable if

The correct option is D
$α is negative and β is positive$
$h (t) = e^{α t} u (t) + e^{β t} u (- t)$
For the system to be stable, $\infty \int - \infty h (t) d t < \infty$
For the above condition, h(t) should be as shown below.

$Therefore α < 0 and β > 0$