HW 05

Find the following Fourier Transforms

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

Solutions

$F[e^{j{\omega }_{0}t}]$	$=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{e}^{j{\omega }_{0}t}{e}^{-j\omega t}dt$
	$=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{e}^{j({\omega }_0-\omega )t}dt$
	$=2\pi \left[\frac{1}{2\pi }\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{e}^{j({\omega }_0-\omega )t}dt\right]$
	$=2\pi \delta ({\omega }_0-\omega )$
$F[\cos {\omega }_{0}t]$	$=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\frac{e^{j{\omega }_{0}t}+e^{-j{\omega }_{0}t}}{2}{e}^{-j\omega t}dt$
	$=\frac{1}{2}\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}(e^{j{\omega }_{0}t}+e^{-j{\omega }_{0}t})e^{-j\omega t}dt$
	$=\frac{1}{2}\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}[e^{j({\omega }_0-\omega )t}+e^{-j({\omega }_0+\omega )t}]dt$
	$=\pi \left[\frac{1}{2\pi }\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}(e^{j({\omega }_0-\omega )t}+e^{-j({\omega }_0+\omega )t})dt\right]$
	$=\pi \delta ({\omega }_0-\omega )+\pi \delta ({\omega }_0+\omega )$
$F[\sin {\omega }_{0}t]$	$=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\frac{e^{j{\omega }_{0}t}-e^{-j{\omega }_{0}t}}{2j}{e}^{-j\omega t}dt$
	$=\frac{1}{2j}\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}(e^{j{\omega }_{0}t}-e^{-j{\omega }_{0}t})e^{-j\omega t}dt$
	$=\frac{1}{2j}\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}(e^{j({\omega }_0-\omega )t}-e^{-j({\omega }_0+\omega )t})dt$
	$=\frac{\pi }{j}\left[\frac{1}{2\pi }\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}(e^{j({\omega }_0-\omega )t}-e^{-j({\omega }_0+\omega )t})dt\right]$
	$=-j\pi \delta ({\omega }_0-\omega )+j\pi \delta ({\omega }_0+\omega )$
$F[\sum _{-\mathrm{\infty }}^{\mathrm{\infty }}{\alpha }_n{e}^{j2\pi nt/T}]$	$=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\left(\sum _{-\mathrm{\infty }}^{\mathrm{\infty }}{\alpha }_n{e}^{j2\pi nt/T}\right)e^{-j\omega t}dt$
	$=\sum _{-\mathrm{\infty }}^{\mathrm{\infty }}{\alpha }_n\left(\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{e}^{j2\pi nt/T}{e}^{-j2\pi ft}dt\right)$
	$=\sum _{-\mathrm{\infty }}^{\mathrm{\infty }}{\alpha }_n\left(\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{e}^{j2\pi t(\frac{n}{T}-f)}dt\right)$
	$=\sum _{-\mathrm{\infty }}^{\mathrm{\infty }}{\alpha }_n\delta (\frac{n}{T}-f)$