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

$F\left[e^{j\omega _{0}t}\right]$	$=\int _{-\infty }^{\infty }e^{j\omega _{0}t}e^{-j\omega t}dt$
	$=\int _{-\infty }^{\infty }e^{j(\omega _{0}-\omega )t}dt$
	$=2\pi \left[{\frac {1}{2\pi }}\int _{-\infty }^{\infty }e^{j(\omega _{0}-\omega )t}dt\right]$
	$=2\pi \delta (\omega _{0}-\omega )\,\!$
$F\left[\cos {\omega _{0}t}\right]\,\!$	$=\int _{-\infty }^{\infty }{\frac {e^{j\omega _{0}t}+e^{-j\omega _{0}t}}{2}}e^{-j\omega t}dt$
	$={\frac {1}{2}}\int _{-\infty }^{\infty }\left(e^{j\omega _{0}t}+e^{-j\omega _{0}t}\right)e^{-j\omega t}dt$
	$={\frac {1}{2}}\int _{-\infty }^{\infty }\left[e^{j(\omega _{0}-\omega )t}+e^{-j(\omega _{0}+\omega )t}\right]dt$
	$=\pi \left[{\frac {1}{2\pi }}\int _{-\infty }^{\infty }\left(e^{j(\omega _{0}-\omega )t}+e^{-j(\omega _{0}+\omega )t}\right)\,dt\right]$
	$=\pi \delta (\omega _{0}-\omega )+\pi \delta (\omega _{0}+\omega )\,\!$
$F\left[\sin {\omega _{0}t}\right]\,\!$	$=\int _{-\infty }^{\infty }{\frac {e^{j\omega _{0}t}-e^{-j\omega _{0}t}}{2j}}e^{-j\omega t}dt$
	$={\frac {1}{2j}}\int _{-\infty }^{\infty }\left(e^{j\omega _{0}t}-e^{-j\omega _{0}t}\right)e^{-j\omega t}dt$
	$={\frac {1}{2j}}\int _{-\infty }^{\infty }\left(e^{j(\omega _{0}-\omega )t}-e^{-j(\omega _{0}+\omega )t}\right)dt$
	$={\frac {\pi }{j}}\left[{\frac {1}{2\pi }}\int _{-\infty }^{\infty }\left(e^{j(\omega _{0}-\omega )t}-e^{-j(\omega _{0}+\omega )t}\right)\,dt\right]$
	$=-j\pi \delta (\omega _{0}-\omega )+j\pi \delta (\omega _{0}+\omega )\,\!$
$F\left[\sum _{-\infty }^{\infty }\alpha _{n}e^{j2\pi nt/T}\right]$	$=\int _{-\infty }^{\infty }\left(\sum _{-\infty }^{\infty }\alpha _{n}e^{j2\pi nt/T}\right)e^{-j\omega t}dt$
	$=\sum _{-\infty }^{\infty }\alpha _{n}\left(\int _{-\infty }^{\infty }e^{j2\pi nt/T}e^{-j2\pi ft}dt\right)$
	$=\sum _{-\infty }^{\infty }\alpha _{n}\left(\int _{-\infty }^{\infty }e^{j2\pi t({\frac {n}{T}}-f)}dt\right)$
	$=\sum _{-\infty }^{\infty }\alpha _{n}\delta \left({\frac {n}{T}}-f\right)$