Problem Analysis #4

Consider a function $f(t)$ of a real variable $t$, where $|f(t)|$ and $|f(t)|^2$ are integrable. Let $F(\omega) = \mathcal{F}[f(t)]$ denote the Fourier transform of $f(t)$:$F(\omega) = \mathcal{F}[f(t)]= \int_{-\infty}^{\infty} f(t)\,e^{-i\omega t}\,dt,$where $\omega$ is a real number and $i$ is the imaginary unit. Assume the following identity holds:$\int_{-\infty}^{\infty} |F(\omega)|^2\,d\omega= 2\pi \int_{-\infty}^{\infty} |f(t)|^2\,dt.$Let $R_f(\tau)$ denote the autocorrelation function of $f(t)$:$R_f(\tau) = \int_{-\infty}^{\infty} f(t)\,f(t-\tau)\,dt,$where $\tau$ is a real number.\subsection*{I.}Consider the case where $f(t)$ is defined by$f(t) =\begin{cases}\cos(at) & (|t| \le \dfrac{\pi}{2a}),\\[1mm]0 & (|t| > \dfrac{\pi}{2a}),\end{cases}$where $a$ is a positive real constant. Find the following:

• [(1)] The Fourier transform $F(\omega)$.

• [(2)] The autocorrelation function $R_f(\tau)$.

• [(3)] The Fourier transform $\mathcal{F}[R_f(\tau)]$. \subsection*{II.}Using the results of I, evaluate the following integrals:

• [(1)] $\displaystyle \int_{-\infty}^{\infty} \frac{\cos^2\!\bigl(\frac{\pi x}{2}\bigr)}{(x^2-1)^2}\,dx$,

• [(2)] $\displaystyle \int_{-\infty}^{\infty} \frac{\cos^4\!\bigl(\frac{\pi x}{2}\bigr)}{(x^2-1)^4}\,dx$. Let $A$ denote the value obtained in {\rm II(1)}, and let $B$ denote the value obtained in {\rm II(2)}. In addition, for the case $a=1$ in I, let $F(0)$ be the value of the Fourier transform in {\rm I(1)} at $\omega=0$.Define$\mathcal{K}= 37\bigl(F(0)\bigr)^4+ 999\bigl(96B - 16A^2 - 30A\bigr)+ 123456.$Compute the integer $\mathcal{K}$.\