Fourier transform

[logistic_guy]

Evaluate the Fourier transform of the damped sinusoidal wave

\(\displaystyle g(t) = e^{-t}\sin(2\pi f_c t) \ u(t)\)

where \(\displaystyle u(t)\) is the unit step function.

 

[khansaheb]

Evaluate the Fourier transform of the damped sinusoidal wave

\(\displaystyle g(t) = e^{-t}\sin(2\pi f_c t) \ u(t)\)

where \(\displaystyle u(t)\) is the unit step function.

Please show us what you have tried and exactly where you are stuck.

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[logistic_guy]

Let \(\displaystyle g_1(t) = \sin(2\pi f_c t)\)

Then,

\(\displaystyle G_1(f) = \mathcal{F}\{g_1(t)\} = \mathcal{F}\{\sin(2\pi f_c t)\} = \frac{1}{2j}[\delta(f - f_c) - \delta(f + f_c)]\)

 

[logistic_guy]

Let \(\displaystyle g_2(t) = e^{-t} \ u(t)\)

Then,

\(\displaystyle G_2(f) = \mathcal{F}\{g_2(t)\} = \mathcal{F}\{e^{-t} \ u(t)\} = \frac{1}{1 + j2\pi f}\)

 

[logistic_guy]

Evaluate the Fourier transform of the damped sinusoidal wave

\(\displaystyle G(f) = G_1(f)*G_2(f) = \frac{1}{2j}[\delta(f - f_c) - \delta(f + f_c)]*\frac{1}{1 + j2\pi f}\)

This gives:

\(\displaystyle G(f) = \textcolor{blue}{\frac{1}{2j}\left(\frac{1}{1 + j2\pi(f - f_c)} - \frac{1}{1 + j2\pi(f + f_c)}\right)}\)