Let us use the CT LTI system:

$ y(t) = 3x(t) + 7x(t+3) $


The impulse response, h(t), of this system is computed using the following:

$ x(t) = \delta (t) $

$ h(t) = 3\delta (t) + 7\delta (t+3) $

The system function, H(s) is:

$ H(j w) = \int^{\infty}_{- \infty} h(t) e^{- j w t} dt $

$ H(j w) = \int^{\infty}_{- \infty} [3\delta (t) + 7\delta (t+3)]e^{- j w t} $

$ H(j w) = 3 e^{0} + 7 e^{- j 3 t} $

$ H( jw) = 3 + 7 e^{- j 3 t} $


The signal used in question 1:

$ x(t) = 3cos(4\pi t) + e^{j\frac{2\pi}{5}t} $

$ = \frac{3}{2}(e^{j 4 \pi t} + e^{-j 4 \pi t}) + e^{j \frac{2 \pi}{5} t} $

The response, y(t) = H(jw)*x(t)

$ y(t) = (3 + 7 e^{- j 3 t}) * (\frac{3}{2}(e^{j 4 \pi t} + e^{-j 4 \pi t}) + e^{j \frac{2 \pi}{5} t}) $

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