The output $y[n]$ of a discrete - time system for an input $x[n]$ is

$$ y[n]=\max\limits_{-\infty \leq k \leq n}|x[k]| $$

The unit impulse response of the system is

Which one of the following pole-zero plots corresponds to the transfer function of an LTI system characterized by the input-output difference equation given below?

$$ y[n]=\sum_{k=0}^3(-1)^k x[n-k] $$

$X(\omega)$ 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 $\_\_\_\_$ .