- #1

Captain1024

- 45

- 2

## Homework Statement

Evaluate the Fourier Transform of the damped sinusoidal wave [itex]g(t)=e^{-t}sin(2\pi f_ct)u(t)[/itex] where u(t) is the unit step function.

## Homework Equations

[itex]\omega =2\pi f[/itex]

[itex]G(f)=\int ^{\infty}_{-\infty} g(t)e^{-j2\pi ft}dt[/itex]

[itex]sin(\omega _ct)=\frac{e^{j\omega _ct}-e^{-j\omega _ct}}{2j}[/itex]

## The Attempt at a Solution

[itex]G(f)=\int ^{\infty}_{-\infty}e^{-t}sin(2\pi f_ct)u(t)e^{-j2\pi ft}dt[/itex]

[itex]sin(2\pi f_ct)=\frac{e^{j2\pi f_ct}-e^{-j2\pi f_ct}}{2j}[/itex]

[itex]G(f)=\frac{1}{2j}\int ^{\infty}_{0}e^{-t}(e^{j2\pi f_ct}-e^{-j2\pi f_ct})e^{-f2\pi ft}dt[/itex]

[itex]G(f)=\frac{1}{2j}\int ^{\infty}_{0}e^{-j2\pi ft-t}(e^{j2\pi f_ct}-e^{-j2\pi f_ct})dt[/itex]

[itex]G(f)=\frac{1}{2j}\int ^{\infty}_{0}e^{-j2\pi ft-t+j2\pi f_ct}-e^{-j2\pi ft-t-j2\pi f_ct}dt[/itex]

[itex]G(f)=\frac{1}{2j}\int ^{\infty}_{0}e^{j2\pi t(f_c-f)-t}-e^{-j2\pi t(f+f_c)-t}dt[/itex]

I am not feeling confident on my algebra and I also feel like I should be able to simplify this more before I integrate.