$X(\omega)$ is the Fourier transform of $x(t)$ shown below. The value of $\int\limits_{-\infty}^{\infty}|X(\omega)|^2 d \omega$ (rounded off to two decimal places) is $\_\_\_\_$ .

The transfer function of a stable discrete - time LTI system is $H(z)=\frac{K(z-\alpha)}{(z+0.5)}$ where $K$ and $\alpha$ are real numbers. The value of $\alpha$ (rounded off to one decimal place) with $|\alpha|>1$, for which magnitude response of the system is constant over all frequencies, is $\_\_\_\_$ .

A finite duration discrete-time signal $x[n]$ is obtained by sampling a continuous - time signal $x(t)=\cos (200 \pi t)$ at sampling instants $t=\frac{n}{400}, n=0,1, \ldots ., 7$. The 8-point discrete Fourier transform (DFT) is defined as

$$ X[k]=\sum_{n=0}^7 x[n] e^{-j \pi n k / 4} \text { for } k=0,1, \ldots ., 7 $$

Which one of the following statements is TRUE?

He was not only accused of theft $\_\_\_\_$ of conspiracy.