\section*{Problem 5}
Consider a function $f ( t )$ of a real number $t$, where $| f ( t ) |$ and $| f ( t ) | ^ { 2 }$ are integrable. Let $F ( \omega ) = \mathcal { F } [ f ( t ) ]$ denote the Fourier transform of $f ( t )$. It is defined as

$$F ( \omega ) = \mathcal { F } [ f ( t ) ] = \int _ { - \infty } ^ { \infty } f ( t ) \exp ( - i \omega t ) \mathrm { d } t$$

where $\omega$ is a real number and $i$ is the imaginary unit. Then, the following equation is satisfied:

$$\int _ { - \infty } ^ { \infty } | F ( \omega ) | ^ { 2 } \mathrm {~d} \omega = 2 \pi \int _ { - \infty } ^ { \infty } | f ( t ) | ^ { 2 } \mathrm {~d} t$$

Also, let $R _ { f } ( \tau )$ denote the autocorrelation function of $f ( t )$. It is defined as

$$R _ { f } ( \tau ) = \int _ { - \infty } ^ { \infty } f ( t ) f ( t - \tau ) \mathrm { d } t$$

where $\tau$ is a real number.

I. Consider a case where $f ( t )$ is defined as follows:

$$f ( t ) = \begin{cases} \cos ( a t ) & \left( | t | \leq \frac { \pi } { 2 a } \right) \\ 0 & \left( | t | > \frac { \pi } { 2 a } \right) \end{cases}$$

Here, $a$ is a positive real constant. Find the followings:

\begin{enumerate}
  \item $F ( \omega )$,
  \item $R _ { f } ( \tau )$,
  \item $\mathcal { F } \left[ R _ { f } ( \tau ) \right]$.
\end{enumerate}

II. Find the values of the following integrals. Here, you may use the results of I.

\begin{enumerate}
  \item $\int _ { - \infty } ^ { \infty } \frac { \cos ^ { 2 } \frac { \pi x } { 2 } } { \left( x ^ { 2 } - 1 \right) ^ { 2 } } \mathrm {~d} x$,
  \item $\int _ { - \infty } ^ { \infty } \frac { \cos ^ { 4 } \frac { \pi x } { 2 } } { \left( x ^ { 2 } - 1 \right) ^ { 4 } } \mathrm {~d} x$.
\end{enumerate}