# Sketch the Fourier transforms of the following functions assuming |omega_1|>|omega_2| a) f(t)=[cos(omega_1 t)+i sin(omega_1 t)]-[cos(omega_2 t)+i sin(omega_2 t)] b) f(t)=[cos(omega_0 t)]e^{-frac{t}{T_2}}

Sketch the Fourier transforms of the following functions assuming $|{\omega }_{1}|>|omeg{a}_{2}|$
$a\right)f\left(t\right)=\left[\mathrm{cos}\left({\omega }_{1}t\right)+i\mathrm{sin}\left({\omega }_{1}t\right)\right]-\left[\mathrm{cos}\left({\omega }_{2}t\right)+i\mathrm{sin}\left({\omega }_{2}t\right)\right]$
$b\right)f\left(t\right)=\left[\mathrm{cos}\left({\omega }_{0}t\right)\right]{e}^{-\frac{t}{{T}_{2}}}$
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Step 1
Consider the equation
Step 2
$\left(a\right)f\left(t\right)=\left[\mathrm{cos}\left({\omega }_{1}t\right)+i\mathrm{sin}\left({\omega }_{1}t\right)\right]-\left[\mathrm{cos}\left({\omega }_{2}t\right)+i\mathrm{sin}\left({\omega }_{2}t\right)\right]$ F is a function of a real variable omega ,the function value $F\left(\omega \right)$ is a complex number.
$|F\left(t\right)|$ is called amplitude specturm of f
F=f(T) means F is the Fourier transform of f as for Laplace transform Laplace transform of f
$f\left(t\right)=\left[\mathrm{cos}\left({\omega }_{1}t\right)+i\mathrm{sin}\left({\omega }_{1}t\right)\right]-\left[\mathrm{cos}\left({\omega }_{2}t\right)+i\mathrm{sin}\left({\omega }_{2}t\right)\right]$
$=\mathrm{cos}\left({\omega }_{1}t\right)+i\mathrm{sin}\left({\omega }_{1}t\right)-\mathrm{cos}\left({\omega }_{2}t\right)+i\mathrm{sin}\left({\omega }_{2}t\right)$
(b) $f\left(t\right)=\left[\mathrm{cos}\left({\omega }_{0}t\right)\right]{e}^{-\frac{t}{{T}_{2}}}$
F is a function of real variable t the function value F(t) is a complex number, F=f(t) means F is the Fourier transform of f as for Laplace transform
$f\left(t\right)=\left[\mathrm{cos}\left({\omega }_{0}t\right)\right]{e}^{-\frac{t}{{T}_{2}}}$