# The impulse response h (t) of a LTI system is given by h (t) = exp (-2t) u (t) where, u (t) denotes the unit step function. The output of this system to the sinusoidal input x(t) = 2 cos(2t) for all t, is _____________

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The impulse response h (t) of a LTI system is given by h (t) = exp (-2t) u (t) where, u (t) denotes the unit step function.  The output of this system to the sinusoidal input x(t) = 2 cos(2t) for all t, is _____________

(a) 0

(b) 2^-0.25 cos(2t-0.125π)

(c) 2^-0.5 cos(2t-0.125π)

(d) 2^-0.5 cos(2t-0.25π)

My question is taken from Region of Convergence topic in chapter Laplace Transform and System Design of Signals and Systems

by (42.1k points)
The correct option is (d) 2^-0.5 cos(2t-0.25π)

To elaborate: X (t) = 2 cos2t

H (ω) = $\frac{1}{2+jω}$

|H (ω)| = $\frac{1}{\sqrt{z^2+ω^2}} = \frac{1}{2\sqrt{2}}$

Hence y (t) = |H (ω)| × 2 cos [2t + *H (ω)]

= $\frac{1}{2\sqrt{2}} × 2 cos⁡(2t – \frac{π}{4})$

= 2^-0.5 cos (2t-0.25π).

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