The Fourier transform X(j$$\omega$$) of the signal $$x(t) = {t \over {{{(1 + {t^2})}^2}}}$$ is ____________.

Let x₁(t) = e^$$-$$t u(t) and x₂(t) = u(t) $$-$$ u(t $$-$$ 2), where u( . ) denotes the unit step function. If y(t) denotes the convolution of x₁(t) and x₂(t), then $$\mathop {\lim }\limits_{t \to \infty } y(t)$$ = __________ (rounded off to one decimal place).

The outputs of four systems (S₁, S₂, S₃ and S₄) corresponding to the input signal sin(t), for all time t, are shown in the figure.

Based on the given information, which of the four systems is/are definitely NOT LTI (linear and time-invariant)?

For a vector $$\overline x $$ = [x[0], x[1], ....., x[7]], the 8-point discrete Fourier transform (DFT) is denoted by $$\overline X $$ = DFT($$\overline x $$) = [X[0], X[1], ....., X[7]], where

$$X[k] = \sum\limits_{n = 0}^7 {x[n]\exp \left( { - j{{2\pi } \over 8}nk} \right)} $$.

Here, $$j = \sqrt { - 1} $$. If $$\overline x $$ = [1, 0, 0, 0, 2, 0, 0, 0] and $$\overline y $$ = DFT (DFT($$\overline x $$)), then the value of y[0] is __________ (rounded off to one decimal place).