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Erin's question via email about a Fourier Transform

Prove It

Well-known member
MHB Math Helper
Jan 26, 2012
1,403
Find the Fourier Transform of $\displaystyle \begin{align*} f \left( t \right) = \begin{cases} 1 - \frac{t}{2} \textrm{ if } 0 \leq t \leq 2 \\ 1 + \frac{t}{2} \textrm{ if } -2 \leq t < 0 \\ 0 \textrm{ elsewhere } \end{cases} \end{align*}$
$\displaystyle \begin{align*} F \left( \omega \right) &= \mathcal{F} \left\{ f \left( t \right) \right\} \\
&= \int_{-\infty}^{\infty}{ f\left( t \right) \mathrm{e}^{-\mathrm{j}\,\omega \, t}\,\mathrm{d}t } \\
&= \int_{-\infty}^{-2}{ 0\,\mathrm{d}t } + \int_{-2}^0{ \left( 1 + \frac{t}{2} \right) \mathrm{e}^{-\mathrm{j}\,\omega\,t}\,\mathrm{d}t } + \int_0^2{ \left( 1 - \frac{t}{2} \right) \mathrm{e}^{-\mathrm{j}\,\omega\,t}\,\mathrm{d}t } + \int_2^{\infty}{0\,\mathrm{d}t} \\
&= 0 + \left[ \left( 1 + \frac{t}{2} \right) \left( \frac{\mathrm{e}^{-\mathrm{j}\,\omega\,t}}{-j\,\omega} \right) \right]_{-2}^0 - \int_{-2}^0{ \frac{\mathrm{e}^{-\mathrm{j}\,\omega\,t}}{-2\,\mathrm{j}\,\omega}\,\mathrm{d}t } + \left[ \left( 1 - \frac{t}{2} \right) \left( \frac{\mathrm{e}^{-\mathrm{j}\,\omega\,t}}{-\mathrm{j}\,\omega} \right) \right]_0^2 - \int_0^2{ \frac{\mathrm{e}^{-\mathrm{j}\,\omega\,t}}{2\,\mathrm{j}\,\omega}\,\mathrm{d}t } + 0 \\
&= \frac{1}{-\mathrm{j}\,\omega} - \left[ \frac{\mathrm{e}^{-\mathrm{j}\,\omega\,t}}{2\,\mathrm{j}^2\,\omega^2} \right] _{-2}^0 - \left( \frac{1}{-\mathrm{j}\,\omega} \right) - \left[ \frac{\mathrm{e}^{-\mathrm{j}\,\omega\,t}}{-2\,\mathrm{j}^2\,\omega^2} \right] _0^2 \\
&= \left[ \frac{\mathrm{e}^{-\mathrm{j}\,\omega\,t}}{2\,\omega^2} \right] _{-2}^0 - \left[ \frac{\mathrm{e}^{-\mathrm{j}\,\omega\,t}}{2\,\omega^2} \right] _0^2 \\ &= \frac{1}{2\,\omega^2} \left\{ \left[ \mathrm{e}^{-\mathrm{j}\,\omega\,t } \right] _{-2}^0 - \left[ \mathrm{e}^{-\mathrm{j}\,\omega\,t} \right] _0^2 \right\} \\
&= \frac{1}{2\,\omega^2} \left[ \left( 1 - \mathrm{e}^{2\,\mathrm{j}\,\omega } \right) - \left( \mathrm{e}^{-2\,\mathrm{j}\,\omega\,t} - 1 \right) \right] \\ &= \frac{1}{2\,\omega^2} \left[ 2 - \left( \mathrm{e}^{2\,\mathrm{j}\,\omega} + \mathrm{e}^{-2\,\mathrm{j}\,\omega} \right) \right] \\ &= \frac{1}{\omega^2} \left[ 1 - \left( \frac{\mathrm{e}^{2\,\mathrm{j}\,\omega} + \mathrm{e}^{-2\,\mathrm{j}\,\omega}}{2} \right) \right] \\ &= \frac{1 - \cos{ \left( 2\,\omega \right) }}{\omega ^2} \end{align*}$
 
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