HW 05

Find the following Fourier Transforms

• $F[e^{j\omega _{0}t}]$
• $F[\cos {\omega _{0}t}]\,\!$
• $F[\sum _{-\infty }^{\infty }\alpha _{n}e^{j2\pi nt/T}]$
• $F[\sin {\omega _{0}t}]\,\!$

Solutions

$F[e^{j\omega _{0}t}]$	$=\int _{-\infty }^{\infty }e^{j\omega _{0}t}e^{-j\omega t}dt$
	$=\int _{-\infty }^{\infty }e^{j(\omega _{0}-\omega )t}dt$
	$=2\pi \left[{\frac {1}{2\pi }}\int _{-\infty }^{\infty }e^{j(\omega _{0}-\omega )t}dt\right]$
	$=2\pi \delta (\omega _{0}-\omega )\,\!$
$F[\cos {\omega _{0}t}]\,\!$	$=\int _{-\infty }^{\infty }{\frac {e^{j\omega _{0}t}+e^{-j\omega _{0}t}}{2}}e^{-j\omega t}dt$
	$={\frac {1}{2}}\int _{-\infty }^{\infty }\left(e^{j\omega _{0}t}+e^{-j\omega _{0}t}\right)2e^{-j\omega t}dt$
	$={\frac {1}{2}}\int _{-\infty }^{\infty }2e^{j(\omega _{0}-\omega )t}+2e^{-j(\omega _{0}+\omega )t}dt$
	$=\int _{-\infty }^{\infty }e^{j(\omega _{0}-\omega )t}+e^{-j(\omega _{0}+\omega )t}$
	$=\delta (\omega _{0}-\omega )+\delta (\omega _{0}+\omega )\,\!$
$F[\sin {\omega _{0}t}]\,\!$	$=\int _{-\infty }^{\infty }{\frac {e^{j\omega _{0}t}-e^{-j\omega _{0}t}}{2j}}e^{-j\omega t}dt$