Spent some time on the problem for homework in my Communication Theory Course. The problem is a Fourier Transform of a damped sinusoidal wave. I plan on adding Matlab Graphs in a couple days…

Here, g(t) is the signal in the time domain. G(f) is the transform to the frequency domain. Enjoy!

g(t)=e^{-t}sin(2\pi{f_{c}}t)u(t)

G(f)=\int_{\small-\infty}^{\small\infty}e^{-t}sin(2\pi{f_{c}}t)u(t)e^{-j2\pi{f}t}dt

G(f)=\int_{\small0}^{\small\infty}e^{\small-t}\frac{e^{j2\pi{f_{c}}t}-e^{-j2\pi{f_{c}}t}}{2j}e^{-j2\pi{f}t}dt

G(f)=\frac{1}{2j}\int_{\small0}^{\small\infty}e^{j2\pi{f_{c}}t-t-j2\pi{f}t}-e^{-j2\pi{f_{c}}t-t-j2\pi{f}t}dt

G(f)=\frac{1}{2j}\biggl[\frac{e^{-t(-j2\pi{f_{c}}+1+j2\pi{f})}}{-j2\pi{f_{c}}+1+j2\pi{f}}\biggr]_0^{\infty}-\frac{1}{2j}\biggl[\frac{e^{-t(j2\pi{f_{c}}+1+j2\pi{f})}}{j2\pi{f_{c}}+1+j2\pi{f}}\biggr]_0^{\infty}

G(f)=\frac{1}{2j}\biggl[\frac{1}{j2\pi{f_{c}}-1-j2\pi{f}}\biggr]-\frac{1}{2j}\biggl[\frac{1}{-j2\pi{f_{c}}-1-j2\pi{f}}\biggr]

G(f)=\frac{1}{2j}\biggl[\frac{-j2\pi{f_{c}}-1-j2\pi{f}-j2\pi{f_{c}}+1+j2\pi{f}}{(j2\pi{f_{c}}-1-j2\pi{f})(-j2\pi{f_{c}}-1-j2\pi{f})}\biggr]

G(f)=\frac{1}{2j}\biggl[\frac{j4\pi{f_{c}}}{4\pi^2f_{c}^2+4\pi^2f_{c}f+1+j2\pi{f}-4\pi^2f_{c}f+j2\pi{f}-4\pi^2{f}^2}\biggr]

G(f)=\frac{2\pi{f_{c}}}{1+4\pi^2({f_{c}^2}-f^2)+j4\pi{f}}

Graph Damped sin with Frequency of 30Hz
Damped Sinusoidal Wave

Graph of Frequency Response
Damped Sinusoidal Wave Frequency Response