The N-point DFT X of a sequence x[n] 0 ≤ n ≤ N − 1 is given by
$$X\left[ k \right] = {1 \over {\sqrt N }}\,\,\sum\limits_{n = 0}^{N - 1} x \,[n\,]e{\,^{ - j{{2\pi } \over N}nk}}$$, 0$$ \le k \le N - 1$$
Denote this relation as X = DFT(x). For N= 4 which one of the following sequences satisfies DFT (DFT(x) ) = ___________.

The sequence x $$\left[ n \right]$$ = $${0.5^n}$$ u[n], where u$$\left[ n \right]$$ is the unit step sequence, is convolved with itself to obtain y $$\left[ n \right]$$ . Then $$\sum\limits_{n = \infty }^{ + \infty } y \left[ n \right]$$ is ____________.

A real - values signal x(t) limited to the frequency band $$\left| f \right| \le {W \over 2}$$ is passed through a linear time invariant system whose frequency response is
$$H(f) = \left\{ {\matrix{ {{e^{ - j4\pi f}},} & {\left| f \right| \le \,{W \over 2}} \cr {0,} & {\left| f \right| > \,{W \over 2}} \cr } } \right.$$

The output of the system is

A Fourier transform pair is given by $${\left( {{2 \over 3}} \right)^n}$$ u $$\left[ {n + 3} \right]\,\mathop \Leftrightarrow \limits^{FT} \,{{A{e^{ - j6\pi f}}} \over {1 - \left( {{2 \over 3}} \right){e^{ - j2\pi f}}}}$$ ,
where u(n) donotes the unit step sequence. The values of A is_______________.