Fourier transform

logistic_guy

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Evaluate the Fourier transform of the damped sinusoidal wave

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

where u(t)\displaystyle u(t) is the unit step function.
 
Evaluate the Fourier transform of the damped sinusoidal wave

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

where u(t)\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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Let g1(t)=sin(2πfct)\displaystyle g_1(t) = \sin(2\pi f_c t)

Then,

G1(f)=F{g1(t)}=F{sin(2πfct)}=12j[δ(ffc)δ(f+fc)]\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)]
 
Let g2(t)=et u(t)\displaystyle g_2(t) = e^{-t} \ u(t)

Then,

G2(f)=F{g2(t)}=F{et u(t)}=11+j2πf\displaystyle G_2(f) = \mathcal{F}\{g_2(t)\} = \mathcal{F}\{e^{-t} \ u(t)\} = \frac{1}{1 + j2\pi f}
 
Evaluate the Fourier transform of the damped sinusoidal wave
G(f)=G1(f)G2(f)=12j[δ(ffc)δ(f+fc)]11+j2πf\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:

G(f)=12j(11+j2π(ffc)11+j2π(f+fc))\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)}
 
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