Question 24.2: Use Equation (24.3) to find the Fourier integral representat......
Use Equation (24.3) to find the Fourier integral representation of the function defined by
\begin{array}{l}{{f(t)=\frac{1}{2\pi}\int_{-\infty}^{\infty}\!F(\omega)\,\mathrm{e}^{\mathrm{j}\omega t} \mathrm{d}\omega}}\end{array} (24.3)
f(t)={\left\{\begin{array}{l l}{1}&{-1\leqslant t\leqslant 1}\\ {0}&{|t|\gt 1}\end{array}\right.}
Using Equation (24.3) we find
f(t)=\frac{1}{2\pi}\int_{-\infty}^{\infty}F(\omega)\,\mathrm{e}^{\mathrm{j}\omega t}\,\mathrm{d}\omega
where
F(\omega)=\int_{-\infty}^{\infty}f(t)\,\mathrm{e}^{-\mathrm{j}\omega t}\,\mathrm{d}t
=\int_{-1}^{1}1\mathrm{e}^{-\mathrm{j}\omega t}\,\mathrm{d}t since f (t ) is zero outside [−1, 1]
=\left[{\frac{\mathrm{e}^{-\mathrm{j}\omega t}}{-\mathrm{j}\omega}}\right]_{-1}^{1}
=\frac{{\bf e}^{-\mathrm{j}\omega}-{\bf e}^{\mathrm{j}\omega}}{-\mathrm{j}\omega}
=\frac{\mathrm{e}^{\mathrm{j}\omega}-\mathrm{e}^{-\mathrm{j}\omega}}{\mathrm{j}\omega}
Using Euler’s relation (Section 9.6)
\sin\theta={\frac{\mathrm{e}^{\mathrm{j}\theta}-\mathrm{e}^{-\mathrm{j}\theta}}{2\mathrm{j}}}
we find
F(\omega)={\frac{2\sin\omega}{\omega}}
so that
f(t)=\frac{1}{2\pi}\int_{-\infty}^{\infty}\frac{2\sin\omega}{\omega}\,\mathrm{e}^{\mathrm{j}\omega t}\,\mathrm{d}\omega
is the required integral representation. Note that F(\omega)={\frac{2\sin\omega}{\omega}} is the Fourier transform of f (t ). The function \frac{\sin{\omega}}{\omega} occurs frequently and is often referred to as the sinc function (see Section 3.5).